Finite Group Databasebeta

← #33093835

Subgroups — #33093835

Order 18688.

730 conjugacy classes of subgroups.

The subgroup lattice has 638 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22619717C211yesmaximal, characteristic
2#22618870C211yesmaximal, characteristic, complemented
3#22618358C211yesmaximal, characteristic
4#22610873C211yesmaximal, characteristic, complemented
5#22610705C211yesmaximal, characteristic, complemented
6#22610530C211yesmaximal, characteristic, complemented
7#22610209C211yesmaximal, characteristic
8#22610205C211yesmaximal, characteristic
9#22609950C211yesmaximal, characteristic
10#22609814C211yesmaximal, characteristic, complemented
11#22609812C211yesmaximal, characteristic, complemented
12#22609804C211yesmaximal, characteristic, complemented
13#22609772C211yesmaximal, characteristic
14#22609767C211yesmaximal, characteristic
15C73×#1922C211yesmaximal, characteristic, complemented
16C2×#14556599C2211yescharacteristic
17C2×#14556597C2211yescharacteristic
18#17989140C2211yescharacteristic
19#17989055C2211yescharacteristic
20#17988897C411yescharacteristic, complemented
21#17988893C411yescharacteristic, complemented
22#17988890C411yescharacteristic, complemented
23#17988886C411yescharacteristic, complemented
24#17988881C411yescharacteristic, complemented
25#17988876C411yescharacteristic, complemented
26#17988871C411yescharacteristic, complemented
27#17988866C411yescharacteristic, complemented
28#17988716—21no—
29#17988716—21no—
30#17988714—21no—
31#17988714—21no—
32C2×#14556527C2211yescharacteristic
33C2×#14556526C2211yescharacteristic
34#17988623C2211yescharacteristic
35#17988613C2211yescharacteristic
36#17988610C2211yescharacteristic
37C2×#14556520C2211yescharacteristic
38C2×#14556519C2211yescharacteristic
39C2×#14556517C2211yescharacteristic
40#17988584C2211yescharacteristic
41#17988581C2211yescharacteristic
42#17988545C2211yescharacteristic
43#17988544—21no—
44#17988543C2211yescharacteristic
45#17988543C2211yescharacteristic
46#17988541—21no—
47M16×Dic73C2211yescharacteristic
48C4×#11924011—21no—
49C2×#14556502C2211yescharacteristic
50C2×#14556501C2211yescharacteristic
51#97×D73—21no—
52C2×C8×Dic73C2211yescharacteristic
53C4×#11924007C2211yescharacteristic
54#17988351C2211yescharacteristic
55#17988348C2211yescharacteristic
56#17988347C2211yescharacteristic
57#17988346C2211yescharacteristic
58#17988346C2211yescharacteristic
59#17988341C2211yescharacteristic
60#427×C73C2211yescharacteristic
61#424×C73C2211yescharacteristic
62#405×C73C2211yescharacteristic
63#341×C73C2211yescharacteristic
64#339×C73C2211yescharacteristic
65#339×C73C2211yescharacteristic
66#336×C73C2211yescharacteristic
67C2×#11924028C2311yescharacteristic
68#14556599—22no—
69#14556599Q814yescomplemented
70#14556597—22no—
71#14556597—22no—
72C2×#11924012C2311yescharacteristic
73C2×#11924012C2311yescharacteristic
74C2×#11924011—21no—
75C2×#11924011—21no—
76C22×#10596831C2311yescharacteristic
77C22×#10596831C2311yescharacteristic
78C2×C8×D73—21no—
79C2×C8×D73—21no—
80#14556590C2×C411yescharacteristic
81#14556590C2×C411yescharacteristic
82#14556582C2×C411yescharacteristic
83#14556582C2×C411yescharacteristic
84#14556568—21no—
85#14556568—21no—
86C4×Dic146—21no—
87C4×Dic146—21no—
88#14556556C2×C411yescharacteristic
89#14556556C2×C411yescharacteristic
90#14556553C2×C411yescharacteristic
91#14556553C2×C411yescharacteristic
92#14556551—21no—
93#14556551—21no—
94#14556550C2311yescharacteristic
95#14556548—21no—
96C2×#11924007—21no—
97C2×C4×Dic73C2311yescharacteristic
98C2×C4×Dic73C2311yescharacteristic
99C42×D73—21no—
100#14556527—22no—
101#14556526—22no—
102#14556524C2×C411yescharacteristic
103#14556524C2×C411yescharacteristic
104#14556522C2×C411yescharacteristic
105#14556522C2×C411yescharacteristic
106#14556520—21no—
107#14556520—21no—
108#14556520—21no—
109#14556520—21no—
110#14556520C2×C411yescharacteristic
111#14556520C2×C411yescharacteristic
112#14556520C2×C411yescharacteristic
113#14556520C2×C411yescharacteristic
114#14556519—22no—
115#14556517—22no—
116#14556516C2×C411yescharacteristic
117#14556516C2×C411yescharacteristic
118#14556514C2×C411yescharacteristic
119#14556514C2×C411yescharacteristic
120#14556513C2×C411yescharacteristic
121#14556513C2×C411yescharacteristic
122#14556511C2×C411yescharacteristic
123#14556511C2×C411yescharacteristic
124#14556502—21no—
125#14556502—21no—
126#14556502—22no—
127#14556501—21no—
128#14556501—21no—
129#14556501D411yescharacteristic
130#14556501D411yescharacteristic
131#14556501D411yescharacteristic
132#14556501D411yescharacteristic
133C4×#10596831—21no—
134C4×#10596831—21no—
135C8×Dic73—21no—
136C8×Dic73—22no—
137C8×Dic73—21no—
138#14556498—21no—
139C2×M16×C73C2311yescharacteristic
140C2×M16×C73C2311yescharacteristic
141C22×C8×C73C2311yescharacteristic
142C22×C8×C73C2311yescharacteristic
143#117×C73C2311yescharacteristic
144#117×C73C2311yescharacteristic
145C2×C42×C73C2311yescharacteristic
146#107×C73C2×C411yescharacteristic
147#107×C73C2×C411yescharacteristic
148#106×C73C2×C411yescharacteristic
149#106×C73C2×C411yescharacteristic
150C4⋊C8×C73—21no—
151C4⋊C8×C73—21no—
152#97×C73—21no—
153C4×C8×C73—21no—
154#11924028—21no—
155#11924028—21no—
156#11924028C2×Q812yes—
157#11924028#4112yes—
158C2×Dic146—21no—
159C2×Dic146—21no—
160C2×Dic146C22×C411yescharacteristic
161C2×D292C22×C411yescharacteristic
162C2×C22⋊D73—21no—
163C22×Dic73—21no—
164C22×Dic73#4111yescharacteristic
165C22×Dic73#4111yescharacteristic
166C2×C4×D73—21no—
167C2×C4×D73—21no—
168C2×C4×D73—21no—
169#11924012—21no—
170#11924012—21no—
171#11924012—21no—
172#11924012#4112yes—
173#11924011—24no—
174#11924011—24no—
175C2×#10596831—21no—
176C2×#10596831—21no—
177C2×#10596831—21no—
178C2×#10596831—21no—
179C2×#10596831C22×C411yescharacteristic
180C2×#10596831C22×C411yescharacteristic
181C2×#10596831C2×D411yescharacteristic
182C2×#10596831C2×D411yescharacteristic
183C2×#10596831C2×Q812yes—
184C2×#10596831#4112yes—
185C8×D73—24no—
186C8×D73—24no—
187C4⋊Dic73—21no—
188C4⋊Dic73—21no—
189C4⋊Dic73C22×C411yescharacteristic
190C4⋊Dic73C22×C411yescharacteristic
191#11924007—21no—
192#11924007—21no—
193#11924007—21no—
194#11924007—21no—
195#11924007—21no—
196#11924007—21no—
197#11924007—42no—
198C22⋊Dic73—21no—
199#11924005—21no—
200#11924005—21no—
201C4×Dic73—21no—
202C4×Dic73—21no—
203C4×Dic73—21no—
204C4×Dic73—21no—
205C4×Dic73—21no—
206C4×Dic73C22×C411yescharacteristic
207C4×Dic73C22×C411yescharacteristic
208C4×Dic73C2×D411yescharacteristic
209C4×Dic73C2×D411yescharacteristic
210C22×C4×C73—21no—
211C22×C4×C73C2411yescharacteristic
212M16×C73—21no—
213M16×C73—21no—
214M16×C73—21no—
215M16×C73C2×Q812yes—
216C2×C8×C73—21no—
217C2×C8×C73—21no—
218C2×C8×C73—21no—
219C2×C8×C73—21no—
220C2×C8×C73—21no—
221C2×C8×C73—21no—
222C2×C8×C73C22×C411yescharacteristic
223C2×C8×C73C22×C411yescharacteristic
224C2×C8×C73C2×D411yescharacteristic
225C2×C8×C73C2×D411yescharacteristic
226C4⋊C4×C73C22×C411yescharacteristic
227C4⋊C4×C73C22×C411yescharacteristic
228C4⋊C4×C73C22×C411yescharacteristic
229C4⋊C4×C73C22×C411yescharacteristic
230C22⋊C4×C73—21no—
231C22⋊C4×C73—21no—
232C42×C73—21no—
233C42×C73—21no—
234C42×C73—21no—
235C42×C73—21no—
236C23×C73C2×#4111yescharacteristic
237C22×D73—21no—
238C2×C4×C73—21no—
239C2×C4×C73—21no—
240C2×C4×C73—21no—
241C2×C4×C73—21no—
242C2×C4×C73—41no—
243C2×C4×C73—41no—
244C2×C4×C73—41no—
245C2×C4×C73C23×C411yescharacteristic
246C2×C4×C73C22×D411yescharacteristic
247C2×C4×C73C22×Q811yescharacteristic
248C2×C4×C73C2×#4111yescharacteristic
249C2×C4×C73C2×#4111yescharacteristic
250C2×C4×C73C2×#4111yescharacteristic
251C22⋊D73—22no—
252C22⋊D73—22no—
253C2×Dic73—21no—
254C2×Dic73—22no—
255C2×Dic73—21no—
256C2×Dic73—21no—
257C2×Dic73—41no—
258C2×Dic73—41no—
259C2×Dic73—41no—
260C2×Dic73—41no—
261C2×Dic73C4×D411yescharacteristic
262C2×Dic73C4×D411yescharacteristic
263C2×Dic73C4×D411yescharacteristic
264C2×Dic73C4×D411yescharacteristic
265D292—21no—
266D292C4×Q812yes—
267C4×D73—22no—
268C4×D73—22no—
269C4×D73—81no—
270C4×D73—81no—
271Dic146—21no—
272Dic146—81no—
273Dic146—81no—
274Dic146C4×Q812yes—
275C8×C73—21no—
276C8×C73—21no—
277C8×C73—22no—
278C8×C73—21no—
279C8×C73—21no—
280C8×C73—41no—
281#10596831—21no—
282#10596831—21no—
283#10596831—22no—
284#10596831—41no—
285#10596831C22⋊Q812yes—
286#10596831C22⋊Q812yes—
287C22×C73—21no—
288C22×C73—21no—
289C22×C73C2×C4×D411yescharacteristic
290C22×C73#52911yescharacteristic
291C22×C73#53311yescharacteristic
292C2×D73—22no—
293C2×D73—41no—
294C2×D73—41no—
295C4×C73—41no—
296C4×C73—41no—
297C4×C73—41no—
298C4×C73—41no—
299C4×C73—81no—
300C4×C73C2×C4×Q811yescharacteristic
301C4×C73C4×#4111yescharacteristic
302C4×C73C2×C22⋊Q811yescharacteristic
303C4×C73#52511yescharacteristic
304Dic73—21no—
305Dic73—21no—
306Dic73—21no—
307Dic73—21no—
308Dic73—22no—
309Dic73—42no—
310Dic73—41no—
311Dic73—41no—
312#21629—731nomaximal, Sylow
313C2×C73—21no—
314C2×C73—21no—
315C2×C73C4×C22⋊Q811yescharacteristic
316C2×C73#295611yescharacteristic
317C2×C73#295711yescharacteristic
318D73—42no—
319#3076—731no—
320#2961—731no—
321#2875—731no—
322#1986—731no—
323#1950—731no—
324#1922—731no—
325#1922D7311yescharacteristic, complemented
326#1866—731no—
327#1866—731no—
328#1806—731no—
329#1773—731no—
330#1772—731no—
331#1771—731no—
332#1765—731no—
333#1764—731no—
334C73#2162911yescharacteristic, complemented, Sylow
335C2×#131—731no—
336C2×#131—731no—
337#533—731no—
338C4×#41—731no—
339#479—731no—
340#479—731no—
341#478—731no—
342#478—731no—
343#477—731no—
344#476—731no—
345#475—731no—
346#474—731no—
347#446—1461no—
348#446—1461no—
349#446—1461no—
350#446—1461no—
351C2×#108—731no—
352C2×#108—731no—
353#427—731no—
354#427C2×D7311yescharacteristic
355#424—731no—
356#424C2×D7311yescharacteristic
357#423—731no—
358C2×C4⋊C8—731no—
359C2×#104—731no—
360C2×#104—731no—
361#416—731no—
362#416—731no—
363#405—731no—
364#405—731no—
365#405—731no—
366#405—1461no—
367#405—1461no—
368#405C2×D7311yescharacteristic
369C4×M16—731no—
370C4×M16—1461no—
371C2×#97—731no—
372C2×#97—731no—
373C2×#97—1461no—
374C2×C4×C8—731no—
375C4×C4⋊C4—731no—
376#341—731no—
377#341C2×D7311yescharacteristic
378#339—731no—
379#339—731no—
380#339—731no—
381#339—731no—
382#339C2×D7311yescharacteristic
383#339C2×D7311yescharacteristic
384#336—731no—
385#336C2×D7311yescharacteristic
386C2×#41—731no—
387#131—734no—
388#131—1462no—
389#131—1462no—
390#131—1462no—
391C2×M16—731no—
392C2×M16—731no—
393C2×M16—1461no—
394C2×M16—1461no—
395C2×M16C22×D7311yescharacteristic
396C2×M16C22×D7311yescharacteristic
397C22×C8—731no—
398C22×C8—731no—
399C22×C8—1461no—
400C22×C8—1461no—
401C22×C8C22×D7311yescharacteristic
402C22×C8C22×D7311yescharacteristic
403C4⋊Q8—731no—
404C4⋊Q8—731no—
405#125—731no—
406#125—731no—
407C22⋊Q8—1461no—
408C22⋊Q8—1461no—
409C4×Q8—731no—
410C4×Q8—731no—
411C4×Q8—1461no—
412C4×Q8—1461no—
413C4×D4—731no—
414C4×D4—731no—
415C4×D4—1461no—
416C4×D4—1461no—
417#117—731no—
418#117—1461no—
419#117C22×D7311yescharacteristic
420#117C22×D7311yescharacteristic
421C2×C4⋊C4—1461no—
422C2×C42—731no—
423C2×C42—731no—
424C2×C42—1461no—
425C2×C42C22×D7311yescharacteristic
426#108—1462no—
427#108—1462no—
428#107—731no—
429#107—731no—
430#107C4×D7311yescharacteristic
431#107C4×D7311yescharacteristic
432#106—731no—
433#106—731no—
434#106C4×D7311yescharacteristic
435#106C4×D7311yescharacteristic
436C4⋊C8—21no—
437C4⋊C8—21no—
438C4⋊C8—731no—
439C4⋊C8—731no—
440C4⋊C8—731no—
441C4⋊C8—731no—
442C4⋊C8—1461no—
443C4⋊C8—1461no—
444C4⋊C8—1461no—
445C4⋊C8—1461no—
446#104—1462no—
447#104—1462no—
448#103—731no—
449#103—731no—
450#103—731no—
451#103—731no—
452#102—731no—
453#102—731no—
454#102—731no—
455#102—731no—
456#97—21no—
457#97—731no—
458#97—731no—
459#97—731no—
460#97—731no—
461#97—1462no—
462#97—1461no—
463#97—1461no—
464#97—1461no—
465#97—1461no—
466C4×C8—21no—
467C4×C8—1461no—
468C4×C8—1461no—
469C4×C8—1461no—
470C4×C8—1462no—
471C4×C8—1461no—
472#95—1461no—
473#41—732no—
474#41—732no—
475#41—1461no—
476#41—1461no—
477C2×Q8—731no—
478C2×Q8—1461no—
479C2×Q8—1461no—
480C2×D4—731no—
481C2×D4—1461no—
482C22×C4—21no—
483C22×C4—731no—
484C22×C4—731no—
485C22×C4—1461no—
486C22×C4—1461no—
487C22×C4—1461no—
488C22×C4—1461no—
489C22×C4C23×D7311yescharacteristic
490M16—21no—
491M16—21no—
492M16—21no—
493M16—732no—
494M16—1464no—
495M16—1464no—
496M16—1461no—
497M16—1461no—
498M16—1461no—
499M16Q8×D7312yes—
500C2×C8—21no—
501C2×C8—21no—
502C2×C8—21no—
503C2×C8—21no—
504C2×C8—21no—
505C2×C8—21no—
506C2×C8—731no—
507C2×C8—731no—
508C2×C8—732no—
509C2×C8—732no—
510C2×C8—731no—
511C2×C8—731no—
512C2×C8—1464no—
513C2×C8—1464no—
514C2×C8—1461no—
515C2×C8—1461no—
516C2×C8—1461no—
517C2×C8—1461no—
518C2×C8C2×C4×D7311yescharacteristic
519C2×C8C2×C4×D7311yescharacteristic
520C2×C8D4×D7311yescharacteristic
521C2×C8D4×D7311yescharacteristic
522C4⋊C4—2922no—
523C4⋊C4—731no—
524C4⋊C4—731no—
525C4⋊C4—1461no—
526C4⋊C4—1461no—
527C4⋊C4—1461no—
528C4⋊C4—1461no—
529C4⋊C4—1461no—
530C4⋊C4—1461no—
531C4⋊C4—1461no—
532C4⋊C4—1461no—
533C4⋊C4C2×C4×D7311yescharacteristic
534C4⋊C4C2×C4×D7311yescharacteristic
535C4⋊C4C2×C4×D7311yescharacteristic
536C4⋊C4C2×C4×D7311yescharacteristic
537C22⋊C4—21no—
538C22⋊C4—21no—
539C22⋊C4—1461no—
540C22⋊C4—1461no—
541C22⋊C4—1461no—
542C42—21no—
543C42—21no—
544C42—21no—
545C42—21no—
546C42—731no—
547C42—731no—
548C42—731no—
549C42—731no—
550C42—1461no—
551C42—1461no—
552C42—1461no—
553C42—1461no—
554C42—1461no—
555C23—1461no—
556C23#41×D7311yescharacteristic
557Q8—5841no—
558Q8—5841no—
559Q8—732no—
560Q8—1461no—
561D4—732no—
562D4—1462no—
563D4—1462no—
564D4—1461no—
565C2×C4—21no—
566C2×C4—21no—
567C2×C4—21no—
568C2×C4—21no—
569C2×C4—41no—
570C2×C4—41no—
571C2×C4—41no—
572C2×C4—2921no—
573C2×C4—2921no—
574C2×C4—2921no—
575C2×C4—2921no—
576C2×C4—5841no—
577C2×C4—5841no—
578C2×C4—731no—
579C2×C4—731no—
580C2×C4—731no—
581C2×C4—731no—
582C2×C4—1462no—
583C2×C4—1461no—
584C2×C4—1461no—
585C2×C4—1461no—
586C2×C4—1462no—
587C2×C4—1462no—
588C2×C4C22×C4×D7311yescharacteristic
589C2×C4C2×D4×D7311yescharacteristic
590C2×C4C2×Q8×D7311yescharacteristic
591C2×C4#41×D7311yescharacteristic
592C2×C4#41×D7311yescharacteristic
593C2×C4#41×D7311yescharacteristic
594C8—21no—
595C8—21no—
596C8—22no—
597C8—21no—
598C8—21no—
599C8—41no—
600C8—2921no—
601C8—732no—
602C8—732no—
603C8—1461no—
604C8—1461no—
605C8—1462no—
606C22—21no—
607C22—21no—
608C22—2921no—
609C22—2921no—
610C22—1462no—
611C22C4×D4×D7311yescharacteristic
612C22#1798912011yescharacteristic
613C22#1798914311yescharacteristic
614C4—41no—
615C4—41no—
616C4—41no—
617C4—41no—
618C4—81no—
619C4—2922no—
620C4—2921no—
621C4—2921no—
622C4—1461no—
623C4—1461no—
624C4—1461no—
625C4—1461no—
626C4—1462no—
627C4C4×Q8×D7311yescharacteristic
628C4#1798905511yescharacteristic
629C4C22⋊Q8×D7311yescharacteristic
630C4#1798909511yescharacteristic
631C2—21no—
632C2—21no—
633C2—2922no—
634C2#2261246511yescharacteristic
635C2#2261883911yescharacteristic
636C2#2261884311yescharacteristic
subgroupG/H
centerC2×C4C2×D4×D73
commutatorC2×C4×C73C23×C4
FrattiniC22×C4C23×D73
FittingC73×#1922C2
socleC22×C73C2×C4×D4
prime radicalC2×#11924028C23
cosocleC22×C4×C73C24
hypercenter#1922D73
hypocenterC73#21629
LCSG ▷ C2×C4×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C2×C4 ◁ C22×C8 ◁ #1922

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
2. #22618870C2semidirect
4. #22610873C2semidirect
5. #22610705C2semidirect
6. #22610530C2semidirect
10. #22609814C2semidirect
11. #22609812C2semidirect
12. #22609804C2semidirect
15. C73×#1922C2semidirect
20. #17988897C4semidirect
21. #17988893C4semidirect
22. #17988890C4semidirect
23. #17988886C4semidirect
24. #17988881C4semidirect
25. #17988876C4semidirect
26. #17988871C4semidirect
27. #17988866C4semidirect
69. #14556599Q8semidirect
325. #1922D73semidirect
334. C73#21629semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
#226197172
#226188702
#226183582
#226108732
#226107052
#226105302
#226102092
#226102052
#226099502
#226098142
#226098122
#226098042
#226097722
#226097672
C73×#19222
#2162973