Finite Group Databasebeta

← #33144848

Subgroups — #33144848

Order 18688.

818 conjugacy classes of subgroups.

The subgroup lattice has 620 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#17988995C211yesmaximal, characteristic, complemented
2C2×#17988630C211yesmaximal, characteristic
3#22613638C211yesmaximal, characteristic
4#22613634C211yesmaximal, characteristic
5#22611411C211yesmaximal, characteristic, complemented
6#22611405C211yesmaximal, characteristic, complemented
7#22611268C211yesmaximal, characteristic, complemented
8#22611262C211yesmaximal, characteristic, complemented
9#22610668C211yesmaximal, characteristic, complemented
10#22610663C211yesmaximal, characteristic, complemented
11#22610126C211yesmaximal, characteristic
12#22610086C211yesmaximal, characteristic, complemented
13#22609962C211yesmaximal, characteristic
14#22609717C211yesmaximal, characteristic
15C73×#1854C211yesmaximal, characteristic
16C22×#11924012C2211yescharacteristic
17C2×#14556590C2211yescharacteristic
18C2×#14556582C2211yescharacteristic
19C2×#14556568—21no—
20C22×C4⋊Dic73C2211yescharacteristic
21#17988995—24no—
22#17988630—24no—
23C2×#14556524C2211yescharacteristic
24C2×#14556524C2211yescharacteristic
25C2×#14556524C2211yescharacteristic
26C2×#14556523C2211yescharacteristic
27C2×#14556523C2211yescharacteristic
28C2×#14556522C2211yescharacteristic
29C2×#14556522C2211yescharacteristic
30C2×#14556522C2211yescharacteristic
31C2×#14556501C2211yescharacteristic
32C2×C8×Dic73C2211yescharacteristic
33#17988523—21no—
34#17988521—21no—
35#17988510—21no—
36#17988508—21no—
37#17988486C2211yescharacteristic
38#17988486C2211yescharacteristic
39#17988485—21no—
40#17988483C2211yescharacteristic
41#17988483C2211yescharacteristic
42#17988471—21no—
43#17988459—21no—
44#17988456C2211yescharacteristic
45#17988456C2211yescharacteristic
46#17988447C2211yescharacteristic
47C4×C22⋊Dic73C2211yescharacteristic
48#17988350C2211yescharacteristic
49#17988350C2211yescharacteristic
50#17988349C2211yescharacteristic
51#17988349C2211yescharacteristic
52#17988341C2211yescharacteristic
53#17988341C2211yescharacteristic
54C2×#117×C73C2211yescharacteristic
55C2×#107×C73C2211yescharacteristic
56C2×#106×C73C2211yescharacteristic
57C2×C22⋊C8×C73C2211yescharacteristic
58#380×C73C2211yescharacteristic
59#340×C73C2211yescharacteristic
60#340×C73C2211yescharacteristic
61C22×Dic146—21no—
62C23×Dic73—21no—
63C2×#11924012—22no—
64C2×#11924012—22no—
65C2×#11924012Q814yescomplemented
66C22×#10596831C2311yescharacteristic
67C22×#10596831C2311yescharacteristic
68#14556590—22no—
69#14556590—41no—
70#14556582—22no—
71#14556582—41no—
72#14556568—44no—
73C2×C4⋊Dic73—22no—
74C2×C4⋊Dic73—21no—
75C2×C4⋊Dic73—21no—
76C2×C4⋊Dic73—21no—
77C2×C4⋊Dic73—21no—
78C2×C4⋊Dic73—41no—
79C2×C4⋊Dic73C2311yescharacteristic
80C2×C4⋊Dic73C2311yescharacteristic
81C2×C4⋊Dic73C2311yescharacteristic
82C2×C4⋊Dic73C2311yescharacteristic
83C2×#11924007—21no—
84C2×#11924007—21no—
85C2×#11924007—21no—
86C2×#11924007—21no—
87C2×#11924007—21no—
88C2×#11924007—21no—
89C2×#11924007—21no—
90C2×#11924007—21no—
91C2×C22⋊Dic73—21no—
92C2×C22⋊Dic73D411yescharacteristic
93C2×C22⋊Dic73D411yescharacteristic
94C2×C4×Dic73C2311yescharacteristic
95C2×C4×Dic73C2311yescharacteristic
96#14556524—22no—
97#14556524—22no—
98#14556524D414yes—
99#14556523—22no—
100#14556523—41no—
101#14556522—22no—
102#14556522—22no—
103#14556522—22no—
104#14556501—22no—
105C8×Dic73—22no—
106#14556498—21no—
107#14556498—21no—
108#14556498—21no—
109#14556498—21no—
110#14556498—21no—
111#14556498D411yescharacteristic
112#14556498D411yescharacteristic
113C23×C4×C73C2311yescharacteristic
114C22×C8×C73C2311yescharacteristic
115C22×C8×C73C2311yescharacteristic
116#117×C73—22no—
117#117×C73—41no—
118C2×C4⋊C4×C73C2311yescharacteristic
119C2×C4⋊C4×C73C2311yescharacteristic
120C2×C4⋊C4×C73C2311yescharacteristic
121C2×C4⋊C4×C73C2311yescharacteristic
122C2×C22⋊C4×C73—21no—
123C2×C22⋊C4×C73—21no—
124C2×C42×C73—21no—
125#107×C73—22no—
126#106×C73—41no—
127C22⋊C8×C73Q814yescomplemented
128#95×C73—21no—
129C2×Dic146—42no—
130C2×Dic146—42no—
131C2×Dic146—81no—
132C22×Dic73—21no—
133C22×Dic73—21no—
134C22×Dic73—21no—
135C22×Dic73—21no—
136C22×Dic73—42no—
137C22×Dic73—42no—
138C22×Dic73—81no—
139C22×Dic73C2×D411yescharacteristic
140C22×Dic73C2×D411yescharacteristic
141C22×Dic73C2×D411yescharacteristic
142C22×Dic73C2×D411yescharacteristic
143#11924012—28no—
144C2×#10596831—21no—
145C2×#10596831—21no—
146C2×#10596831C2×D412yes—
147C2×#10596831C2×Q812yes—
148C2×#10596831C2×Q812yes—
149C2×#10596831#4112yes—
150C4⋊Dic73—21no—
151C4⋊Dic73—21no—
152C4⋊Dic73—21no—
153C4⋊Dic73—21no—
154C4⋊Dic73—21no—
155C4⋊Dic73—21no—
156C4⋊Dic73—21no—
157C4⋊Dic73—42no—
158C4⋊Dic73—42no—
159C4⋊Dic73—42no—
160C4⋊Dic73—42no—
161C4⋊Dic73C2×D412yes—
162#11924007—42no—
163#11924007—42no—
164#11924007—42no—
165#11924007—42no—
166#11924007—42no—
167#11924007—81no—
168#11924007—81no—
169#11924007—81no—
170C22⋊Dic73—22no—
171C22⋊Dic73—22no—
172C22⋊Dic73—81no—
173C4×Dic73—21no—
174C4×Dic73—21no—
175C4×Dic73—21no—
176C4×Dic73—21no—
177C24×C73C2×D411yescharacteristic
178C22×C4×C73—21no—
179C22×C4×C73—21no—
180C22×C4×C73—21no—
181C22×C4×C73—21no—
182C22×C4×C73—21no—
183C22×C4×C73—21no—
184C22×C4×C73—21no—
185C22×C4×C73—21no—
186C22×C4×C73C2×D411yescharacteristic
187C22×C4×C73C2×Q812yes—
188C22×C4×C73C2×Q812yes—
189C22×C4×C73C2411yescharacteristic
190C2×C8×C73—21no—
191C2×C8×C73—21no—
192C2×C8×C73—21no—
193C2×C8×C73—21no—
194C2×C8×C73C2×Q812yes—
195C2×C8×C73C2×Q812yes—
196C4⋊C4×C73—21no—
197C4⋊C4×C73—21no—
198C4⋊C4×C73—21no—
199C4⋊C4×C73—21no—
200C4⋊C4×C73—21no—
201C4⋊C4×C73—21no—
202C4⋊C4×C73—21no—
203C4⋊C4×C73C2×D412yes—
204C22⋊C4×C73—42no—
205C22⋊C4×C73—42no—
206C42×C73—42no—
207C23×C73—21no—
208C23×C73—21no—
209C23×C73—21no—
210C23×C73—21no—
211C23×C73—21no—
212C23×C73—21no—
213C23×C73C4⋊Q812yes—
214C23×C73C22×D411yescharacteristic
215C2×C4×C73—21no—
216C2×C4×C73—21no—
217C2×C4×C73—22no—
218C2×C4×C73—22no—
219C2×C4×C73—22no—
220C2×C4×C73—22no—
221C2×C4×C73—22no—
222C2×C4×C73—41no—
223C2×C4×C73—41no—
224C2×C4×C73—41no—
225C2×C4×C73—41no—
226C2×C4×C73—41no—
227C2×C4×C73—41no—
228C2×C4×C73—41no—
229C2×C4×C73—41no—
230C2×C4×C73—41no—
231C2×C4×C73—41no—
232C2×C4×C73—41no—
233C2×C4×C73C4⋊Q812yes—
234C2×C4×C73C22×D411yescharacteristic
235C2×C4×C73C22×Q811yescharacteristic
236C2×C4×C73C22×Q811yescharacteristic
237C2×C4×C73C2×#4111yescharacteristic
238C2×C4×C73H(5,𝔽2)11yescharacteristic
239C2×C4×C732−1+411yescharacteristic
240C2×Dic73—21no—
241C2×Dic73—21no—
242C2×Dic73—21no—
243C2×Dic73—21no—
244C2×Dic73—21no—
245C2×Dic73—21no—
246C2×Dic73—22no—
247C2×Dic73—21no—
248C2×Dic73—21no—
249C2×Dic73—21no—
250C2×Dic73—21no—
251C2×Dic73—41no—
252C2×Dic73—41no—
253C2×Dic73—41no—
254C2×Dic73—41no—
255C2×Dic73—41no—
256C2×Dic73—41no—
257C2×Dic73—41no—
258C2×Dic73—41no—
259C2×Dic73—41no—
260C2×Dic73—41no—
261C2×Dic73—41no—
262C2×Dic73—84no—
263C2×Dic73AGL(1,ℤ/8)12yes—
264C2×Dic73#13712yes—
265Dic146—84no—
266C8×C73—22no—
267C8×C73—22no—
268#10596831—22no—
269#10596831C22⋊Q814yescomplemented
270C22×C73—22no—
271C22×C73—22no—
272C22×C73—22no—
273C22×C73—41no—
274C22×C73—41no—
275C22×C73—41no—
276C22×C73—41no—
277C22×C73C2×C4⋊Q811yescharacteristic
278C22×C73D4211yescharacteristic
279C22×C73#54711yescharacteristic
280C22×C73C2×AGL(1,ℤ/8)11yescharacteristic
281C22×C73C2×#13711yescharacteristic
282C22×C73#57711yescharacteristic
283C22×C73#57711yescharacteristic
284C4×C73—22no—
285C4×C73—42no—
286C4×C73—81no—
287C4×C73—81no—
288C4×C73—81no—
289C4×C73C2×C22⋊Q811yescharacteristic
290C4×C73#54411yescharacteristic
291C4×C73D4×Q811yescharacteristic
292C4×C73#55411yescharacteristic
293Dic73—22no—
294Dic73—22no—
295Dic73—41no—
296Dic73—42no—
297Dic73—41no—
298Dic73—81no—
299Dic73—81no—
300Dic73—81no—
301#26078—731nomaximal, Sylow
302C2×C73—42no—
303C2×C73#244011yescharacteristic
304C2×C73#316311yescharacteristic
305C2×C73#316511yescharacteristic
306C2×C73#327211yescharacteristic
307C2×C73#327811yescharacteristic
308C2×C73#334411yescharacteristic
309C2×C73#334511yescharacteristic
310C2×#501—731no—
311C2×#428—731no—
312#2441—731no—
313#2440—731no—
314#2084—731no—
315#2083—731no—
316#2059—731no—
317#2058—731no—
318#1943—731no—
319#1942—731no—
320#1854D7311yescharacteristic
321#1853—731no—
322#1844—731no—
323#1809—731no—
324#1753—731no—
325C73#2607811yescharacteristic, complemented, Sylow
326C22×M16—731no—
327C2×C4⋊Q8—731no—
328C2×#125—731no—
329C2×C22⋊Q8—1461no—
330C2×#117C2×D7311yescharacteristic
331C22×C4⋊C4—731no—
332#501—1464no—
333#428—1464no—
334C2×#107—731no—
335C2×#107—731no—
336C2×#107—731no—
337C2×#107C2×D7311yescharacteristic
338C2×#106—731no—
339C2×#106—731no—
340C2×#106—731no—
341C2×#106—731no—
342C2×#106—731no—
343C2×#106C2×D7311yescharacteristic
344C2×C22⋊C8C2×D7311yescharacteristic
345C2×#97—731no—
346C2×C4×C8—731no—
347#399—1461no—
348#398—1461no—
349#396—1461no—
350#395—1461no—
351#389—731no—
352#389—731no—
353#389—731no—
354#389—731no—
355#389—1461no—
356#387—1461no—
357#385—1461no—
358#384—731no—
359#384—731no—
360#380—731no—
361#380C2×D7311yescharacteristic
362C4×C22⋊C4—731no—
363#340—731no—
364#340—731no—
365#340—731no—
366#340—731no—
367#340C2×D7311yescharacteristic
368#340C2×D7311yescharacteristic
369#336—731no—
370#336—731no—
371C22×Q8—1461no—
372C23×C4—1461no—
373C23×C4C22×D7311yescharacteristic
374C2×M16—734no—
375C2×M16—1462no—
376C2×M16—1462no—
377C22×C8—731no—
378C22×C8—731no—
379C22×C8C22×D7311yescharacteristic
380C22×C8C22×D7311yescharacteristic
381C4⋊Q8—2921no—
382C4⋊Q8—1462no—
383#125—2921no—
384#125—1462no—
385C22⋊Q8—2924no—
386#117—22no—
387#117—41no—
388C2×C4⋊C4—2921no—
389C2×C4⋊C4—731no—
390C2×C4⋊C4—731no—
391C2×C4⋊C4—731no—
392C2×C4⋊C4—731no—
393C2×C4⋊C4—1461no—
394C2×C4⋊C4—1461no—
395C2×C4⋊C4—1461no—
396C2×C4⋊C4—1461no—
397C2×C4⋊C4—1461no—
398C2×C4⋊C4—1461no—
399C2×C4⋊C4—1461no—
400C2×C4⋊C4—1461no—
401C2×C4⋊C4—1461no—
402C2×C4⋊C4—1462no—
403C2×C4⋊C4—1461no—
404C2×C4⋊C4—1461no—
405C2×C4⋊C4—1461no—
406C2×C4⋊C4C22×D7311yescharacteristic
407C2×C4⋊C4C22×D7311yescharacteristic
408C2×C4⋊C4C22×D7311yescharacteristic
409C2×C4⋊C4C22×D7311yescharacteristic
410C2×C22⋊C4—21no—
411C2×C22⋊C4—21no—
412C2×C22⋊C4—731no—
413C2×C22⋊C4—731no—
414C2×C22⋊C4—1461no—
415C2×C42—21no—
416C2×C42—731no—
417C2×C42—731no—
418#107—22no—
419#107—734no—
420#107—1462no—
421#107—1462no—
422#106—41no—
423#106—2921no—
424#106—1462no—
425#106—1462no—
426#106—1462no—
427#106—1462no—
428C22⋊C8Dic14614yescomplemented
429#97—1462no—
430C4×C8—1462no—
431#95—21no—
432#95—731no—
433#95—731no—
434#95—1461no—
435#95—1461no—
436#95—1461no—
437#95—1461no—
438#95—1461no—
439C24D4×D7311yescharacteristic
440C2×Q8—2922no—
441C2×Q8—2922no—
442C2×Q8—5841no—
443C22×C4—21no—
444C22×C4—21no—
445C22×C4—21no—
446C22×C4—21no—
447C22×C4—21no—
448C22×C4—21no—
449C22×C4—21no—
450C22×C4—21no—
451C22×C4—2922no—
452C22×C4—2922no—
453C22×C4—5841no—
454C22×C4—731no—
455C22×C4—731no—
456C22×C4—731no—
457C22×C4—731no—
458C22×C4—1461no—
459C22×C4—1461no—
460C22×C4—1461no—
461C22×C4—1461no—
462C22×C4D4×D7311yescharacteristic
463C22×C4Q8×D7312yes—
464C22×C4C2×Dic14612yes—
465C22×C4C23×D7311yescharacteristic
466M16—1468no—
467C2×C8—21no—
468C2×C8—21no—
469C2×C8—21no—
470C2×C8—21no—
471C2×C8—732no—
472C2×C8—732no—
473C2×C8—732no—
474C2×C8—732no—
475C2×C8—1461no—
476C2×C8—1461no—
477C2×C8C2×Dic14612yes—
478C2×C8C2×Dic14612yes—
479C4⋊C4—21no—
480C4⋊C4—21no—
481C4⋊C4—21no—
482C4⋊C4—21no—
483C4⋊C4—21no—
484C4⋊C4—21no—
485C4⋊C4—21no—
486C4⋊C4—2922no—
487C4⋊C4—2922no—
488C4⋊C4—2922no—
489C4⋊C4—2922no—
490C4⋊C4—2922no—
491C4⋊C4—2922no—
492C4⋊C4—2922no—
493C4⋊C4—2922no—
494C4⋊C4—2922no—
495C4⋊C4—5841no—
496C4⋊C4—5841no—
497C4⋊C4—5841no—
498C4⋊C4—732no—
499C4⋊C4—1461no—
500C4⋊C4—1461no—
501C4⋊C4—1461no—
502C4⋊C4—1461no—
503C4⋊C4—1461no—
504C4⋊C4—1461no—
505C4⋊C4—1461no—
506C4⋊C4D4×D7312yes—
507C22⋊C4—42no—
508C22⋊C4—42no—
509C22⋊C4—5841no—
510C22⋊C4—1462no—
511C22⋊C4—1462no—
512C42—42no—
513C42—1461no—
514C42—1461no—
515C42—1461no—
516C42—1461no—
517C23—21no—
518C23—21no—
519C23—21no—
520C23—21no—
521C23—21no—
522C23—21no—
523C23#1455659012yes—
524C23C2×D4×D7311yescharacteristic
525Q8—5844no—
526C2×C4—21no—
527C2×C4—21no—
528C2×C4—22no—
529C2×C4—22no—
530C2×C4—22no—
531C2×C4—22no—
532C2×C4—22no—
533C2×C4—41no—
534C2×C4—41no—
535C2×C4—41no—
536C2×C4—41no—
537C2×C4—41no—
538C2×C4—41no—
539C2×C4—41no—
540C2×C4—41no—
541C2×C4—41no—
542C2×C4—41no—
543C2×C4—41no—
544C2×C4—2921no—
545C2×C4—2921no—
546C2×C4—2921no—
547C2×C4—2921no—
548C2×C4—2921no—
549C2×C4—2921no—
550C2×C4—2921no—
551C2×C4—2921no—
552C2×C4—2921no—
553C2×C4—2921no—
554C2×C4—2921no—
555C2×C4—5844no—
556C2×C4—732no—
557C2×C4—732no—
558C2×C4—1461no—
559C2×C4—1461no—
560C2×C4—1461no—
561C2×C4—1461no—
562C2×C4—1461no—
563C2×C4—1462no—
564C2×C4—1461no—
565C2×C4—1461no—
566C2×C4—1461no—
567C2×C4—1461no—
568C2×C4—1461no—
569C2×C4#1455659012yes—
570C2×C4C2×D4×D7311yescharacteristic
571C2×C4C2×Q8×D7311yescharacteristic
572C2×C4C22×Dic14611yescharacteristic
573C2×C4#41×D7311yescharacteristic
574C2×C4#1455663611yescharacteristic
575C2×C4#1455663811yescharacteristic
576C8—22no—
577C8—22no—
578C8—734no—
579C8—1462no—
580C22—22no—
581C22—22no—
582C22—22no—
583C22—41no—
584C22—41no—
585C22—41no—
586C22—41no—
587C22C2×#1455659011yescharacteristic
588C22#1798921911yescharacteristic
589C22#1798923511yescharacteristic
590C22AGL(1,ℤ/8)×D7311yescharacteristic
591C22#137×D7311yescharacteristic
592C22#1798939411yescharacteristic
593C22#1798939411yescharacteristic
594C4—22no—
595C4—42no—
596C4—81no—
597C4—81no—
598C4—81no—
599C4—2922no—
600C4—2921no—
601C4—2921no—
602C4—5841no—
603C4—5841no—
604C4—5841no—
605C4—1462no—
606C4—1462no—
607C4C22⋊Q8×D7311yescharacteristic
608C4#1798921611yescharacteristic
609C4D4×Dic14611yescharacteristic
610C4#1798929611yescharacteristic
611C2—42no—
612C2#2261363211yescharacteristic
613C2#2262046311yescharacteristic
614C2#2262047511yescharacteristic
615C2#2262152411yescharacteristic
616C2#2262159011yescharacteristic
617C2#2262237911yescharacteristic
618C2#2262240411yescharacteristic
subgroupG/H
centerC23C2×D4×D73
commutator, cosocleC22×C4×C73C24
FrattiniC22×C4C23×D73
FittingC73×#1854C2
socleC23×C73C22×D4
prime radicalC24×C73C2×D4
hypercenter#1854D73
hypocenterC73#26078
LCSG ▷ C22×C4×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C23 ◁ C23×C4 ◁ #1854

Complements

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

NH ≅ G/Ntype
1. C2×#17988995C2semidirect
5. #22611411C2semidirect
6. #22611405C2semidirect
7. #22611268C2semidirect
8. #22611262C2semidirect
9. #22610668C2semidirect
10. #22610663C2semidirect
12. #22610086C2semidirect
65. C2×#11924012Q8semidirect
127. C22⋊C8×C73Q8semidirect
269. #10596831C22⋊Q8semidirect
325. C73#26078semidirect
428. C22⋊C8Dic146semidirect

Maximal subgroups

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

subgroupindex
C2×#179889952
C2×#179886302
#226136382
#226136342
#226114112
#226114052
#226112682
#226112622
#226106682
#226106632
#226101262
#226100862
#226099622
#226097172
C73×#18542
#2607873