Finite Group Databasebeta

← #33149355

Subgroups — #33149355

Order 18688.

858 conjugacy classes of subgroups.

The subgroup lattice has 642 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#17988877C211yesmaximal, characteristic
2C2×#17988687C211yesmaximal, characteristic
3#22613770C211yesmaximal, characteristic, complemented
4#22611482C211yesmaximal, characteristic, complemented
5#22611413C211yesmaximal, characteristic, complemented
6#22611323C211yesmaximal, characteristic, complemented
7#22611259C211yesmaximal, characteristic, complemented
8#22610666C211yesmaximal, characteristic, complemented
9#22610497C211yesmaximal, characteristic, complemented
10#22610265C211yesmaximal, characteristic
11#22610265C211yesmaximal, characteristic
12#22610198C211yesmaximal, characteristic, complemented
13#22610178C211yesmaximal, characteristic
14#22610178C211yesmaximal, characteristic
15C73×#2349C211yesmaximal, characteristic
16C22×#11924016C2211yescharacteristic
17C2×#14556591C2211yescharacteristic
18C2×C4×Dic146C2211yescharacteristic
19#17988877—24no—
20#17988687—24no—
21C2×#14556524C2211yescharacteristic
22C2×#14556524C2211yescharacteristic
23C2×#14556524C2211yescharacteristic
24C2×#14556522C2211yescharacteristic
25C2×#14556522C2211yescharacteristic
26C2×C4⋊#10596831C2211yescharacteristic
27C2×#14556514C2211yescharacteristic
28C2×#14556514C2211yescharacteristic
29C2×#14556514C2211yescharacteristic
30C2×#14556512C2211yescharacteristic
31C2×#14556512C2211yescharacteristic
32C2×#14556512C2211yescharacteristic
33C2×#14556502C2211yescharacteristic
34#17988521—21no—
35#17988507—21no—
36#17988493C2211yescharacteristic
37#17988493C2211yescharacteristic
38#17988491—21no—
39#17988485C2211yescharacteristic
40#17988485C2211yescharacteristic
41#17988473—21no—
42#17988349C2211yescharacteristic
43#17988349C2211yescharacteristic
44#17988349C2211yescharacteristic
45#17988349C2211yescharacteristic
46#17988342C2211yescharacteristic
47#17988342C2211yescharacteristic
48#17988342C2211yescharacteristic
49#17988342C2211yescharacteristic
50C2×C4×D4×C73C2211yescharacteristic
51C2×C4×D4×C73C2211yescharacteristic
52C2×C4×D4×C73C2211yescharacteristic
53#389×C73C2211yescharacteristic
54#389×C73C2211yescharacteristic
55#385×C73—21no—
56#385×C73—21no—
57#382×C73C2211yescharacteristic
58#380×C73—21no—
59C4×C4⋊C4×C73C2211yescharacteristic
60C2×#95×C73—21no—
61C22×Dic146—21no—
62C22×Dic146C2311yescharacteristic
63C2×#11924016—22no—
64C2×#11924016—22no—
65C2×#11924016Q814yescomplemented
66C22×#10596831C2311yescharacteristic
67C22×#10596831C2311yescharacteristic
68C22×#10596831C2311yescharacteristic
69C22×#10596831C2311yescharacteristic
70#14556591—22no—
71#14556591—41no—
72C4×Dic146—22no—
73C4×Dic146—41no—
74C2×C4⋊Dic73—21no—
75C2×C4⋊Dic73C2311yescharacteristic
76C2×C4⋊Dic73C2311yescharacteristic
77C2×C4⋊Dic73C2311yescharacteristic
78C2×#11924007—21no—
79C2×#11924007—21no—
80C2×#11924007—21no—
81C2×C4×Dic73—21no—
82#14556524—22no—
83#14556524—22no—
84#14556524—41no—
85#14556522—22no—
86#14556522—22no—
87C4⋊#10596831—22no—
88#14556514—22no—
89#14556514—22no—
90#14556514—41no—
91#14556512—22no—
92#14556512—22no—
93#14556512D414yes—
94#14556502—22no—
95#14556498—21no—
96#14556498—21no—
97#14556498—21no—
98#14556498—21no—
99#14556498—21no—
100#14556498—21no—
101C22×D4×C73C2311yescharacteristic
102C23×C4×C73—21no—
103C23×C4×C73—21no—
104C23×C4×C73—21no—
105C4×D4×C73—22no—
106C4×D4×C73—22no—
107C4×D4×C73—22no—
108C4×D4×C73—22no—
109C4×D4×C73—41no—
110C4×D4×C73—41no—
111C2×C4⋊C4×C73—21no—
112C2×C4⋊C4×C73—21no—
113C2×C4⋊C4×C73D411yescharacteristic
114C2×C4⋊C4×C73D411yescharacteristic
115C2×C4⋊C4×C73C2311yescharacteristic
116C2×C4⋊C4×C73C2311yescharacteristic
117C2×C4⋊C4×C73C2311yescharacteristic
118C2×C22⋊C4×C73—21no—
119C2×C22⋊C4×C73—21no—
120C2×C22⋊C4×C73—21no—
121C2×C42×C73C2311yescharacteristic
122C2×C42×C73C2311yescharacteristic
123C2×C42×C73C2311yescharacteristic
124#95×C73—21no—
125#95×C73—21no—
126#95×C73—21no—
127#95×C73—21no—
128#95×C73D411yescharacteristic
129#95×C73D411yescharacteristic
130C2×Dic146—21no—
131C2×Dic146—21no—
132C2×Dic146—21no—
133C2×Dic146—21no—
134C2×Dic146—21no—
135C2×Dic146—42no—
136C2×Dic146—42no—
137C2×Dic146—81no—
138C2×Dic146C2×Q812yes—
139C22×Dic73—21no—
140C22×Dic73—21no—
141C22×Dic73—21no—
142C22×Dic73—21no—
143#11924016—28no—
144C2×#10596831—21no—
145C2×#10596831—21no—
146C2×#10596831—21no—
147C2×#10596831—21no—
148C2×#10596831—21no—
149C2×#10596831—21no—
150C2×#10596831—21no—
151C2×#10596831—21no—
152C2×#10596831—21no—
153C2×#10596831—21no—
154C2×#10596831C2×D412yes—
155C2×#10596831C2×Q812yes—
156C4⋊Dic73—21no—
157C4⋊Dic73—21no—
158C4⋊Dic73—21no—
159C4⋊Dic73—21no—
160C4⋊Dic73—21no—
161C4⋊Dic73—42no—
162C4⋊Dic73C2×D412yes—
163#11924007—42no—
164#11924007—81no—
165#11924007—81no—
166C4×Dic73—42no—
167C24×C73—21no—
168C2×D4×C73—21no—
169C2×D4×C73—21no—
170C2×D4×C73—21no—
171C2×D4×C73C2×D412yes—
172C2×D4×C73C2×Q812yes—
173C2×D4×C73#4112yes—
174C22×C4×C73—21no—
175C22×C4×C73—24no—
176C22×C4×C73—21no—
177C22×C4×C73—21no—
178C22×C4×C73—42no—
179C22×C4×C73—42no—
180C22×C4×C73—42no—
181C22×C4×C73—42no—
182C22×C4×C73—81no—
183C22×C4×C73—81no—
184C22×C4×C73—81no—
185C22×C4×C73—81no—
186C22×C4×C73C2×D411yescharacteristic
187C22×C4×C73C2×D411yescharacteristic
188C22×C4×C73C2×D411yescharacteristic
189C22×C4×C73C2×D411yescharacteristic
190C22×C4×C73C2×D411yescharacteristic
191C22×C4×C73C2×D411yescharacteristic
192C22×C4×C73C2411yescharacteristic
193C4⋊C4×C73—22no—
194C4⋊C4×C73—21no—
195C4⋊C4×C73—21no—
196C4⋊C4×C73—21no—
197C4⋊C4×C73—21no—
198C4⋊C4×C73—21no—
199C4⋊C4×C73—41no—
200C4⋊C4×C73—81no—
201C4⋊C4×C73—81no—
202C4⋊C4×C73#4112yes—
203C22⋊C4×C73—24no—
204C22⋊C4×C73—42no—
205C22⋊C4×C73—42no—
206C42×C73—21no—
207C42×C73—21no—
208C42×C73—21no—
209C42×C73—21no—
210C42×C73—21no—
211C42×C73—21no—
212C23×C73—22no—
213C23×C73—22no—
214C23×C73—22no—
215C23×C73—41no—
216C23×C73—41no—
217C23×C73—41no—
218C23×C73—41no—
219C23×C73C22×D411yescharacteristic
220D4×C73—22no—
221D4×C73—22no—
222D4×C73—22no—
223D4×C73C22⋊Q814yes—
224C2×C4×C73—21no—
225C2×C4×C73—21no—
226C2×C4×C73—21no—
227C2×C4×C73—21no—
228C2×C4×C73—22no—
229C2×C4×C73—21no—
230C2×C4×C73—21no—
231C2×C4×C73—21no—
232C2×C4×C73—21no—
233C2×C4×C73—21no—
234C2×C4×C73—21no—
235C2×C4×C73—21no—
236C2×C4×C73—21no—
237C2×C4×C73—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×C73—41no—
246C2×C4×C73—41no—
247C2×C4×C73—41no—
248C2×C4×C73—41no—
249C2×C4×C73—41no—
250C2×C4×C73—84no—
251C2×C4×C73—84no—
252C2×C4×C73—84no—
253C2×C4×C73#13712yes—
254C2×C4×C73C22×D411yescharacteristic
255C2×C4×C73C22×Q811yescharacteristic
256C2×C4×C73C2×#4111yescharacteristic
257C2×C4×C73C2×#4111yescharacteristic
258C2×C4×C73H(5,𝔽2)11yescharacteristic
259C2×C4×C732−1+411yescharacteristic
260C2×Dic73—22no—
261C2×Dic73—22no—
262C2×Dic73—41no—
263C2×Dic73—41no—
264C2×Dic73—41no—
265C2×Dic73—41no—
266C2×Dic73—41no—
267C2×Dic73—41no—
268C2×Dic73—41no—
269C2×Dic73—41no—
270C2×Dic73—41no—
271C2×Dic73—41no—
272Dic146—22no—
273Dic146—22no—
274Dic146—41no—
275Dic146—41no—
276Dic146—84no—
277#10596831—22no—
278#10596831—22no—
279#10596831—41no—
280#10596831—41no—
281C22×C73—24no—
282C22×C73—42no—
283C22×C73—42no—
284C22×C73—42no—
285C22×C73—42no—
286C22×C73—42no—
287C22×C73—42no—
288C22×C73#52511yescharacteristic
289C22×C73D4211yescharacteristic
290C22×C73D4×Q811yescharacteristic
291C22×C73C2×#13711yescharacteristic
292C22×C73#57511yescharacteristic
293C22×C73#57611yescharacteristic
294C22×C73#57711yescharacteristic
295C4×C73—22no—
296C4×C73—41no—
297C4×C73—41no—
298C4×C73—42no—
299C4×C73—41no—
300C4×C73—41no—
301C4×C73—41no—
302C4×C73—81no—
303C4×C73—81no—
304C4×C73C2×C22⋊Q811yescharacteristic
305C4×C73#53911yescharacteristic
306C4×C73#54711yescharacteristic
307C4×C73#55411yescharacteristic
308Dic73—42no—
309Dic73—42no—
310Dic73—81no—
311Dic73—81no—
312#26396—731nomaximal, Sylow
313C2×C73—44no—
314C2×C73#244911yescharacteristic
315C2×C73#311611yescharacteristic
316C2×C73#311811yescharacteristic
317C2×C73#326611yescharacteristic
318C2×C73#327311yescharacteristic
319C2×C73#338511yescharacteristic
320C2×C73#338811yescharacteristic
321C2×#476—731no—
322C2×#440—731no—
323#2451—731no—
324#2349D7311yescharacteristic
325#2097—731no—
326#2085—731no—
327#2068—731no—
328#2057—731no—
329#1943—731no—
330#1917—731no—
331#1877—731no—
332#1877—731no—
333#1865—731no—
334#1862—731no—
335#1862—731no—
336C73#2639611yescharacteristic, complemented, Sylow
337C22×SD16—731no—
338C2×C4⋊Q8—731no—
339C2×C4×Q8—731no—
340C2×C4×D4C2×D7311yescharacteristic
341C2×C4×D4C2×D7311yescharacteristic
342C2×C4×D4C2×D7311yescharacteristic
343#476—1464no—
344#440—1464no—
345C2×#107—731no—
346C2×#107—731no—
347C2×#107—731no—
348C2×#106—731no—
349C2×#106—731no—
350C2×C4⋊C8—731no—
351C2×#103—731no—
352C2×#103—731no—
353C2×#103—731no—
354C2×#102—731no—
355C2×#102—731no—
356C2×#102—731no—
357C2×#97—731no—
358#398—1461no—
359#395—1461no—
360#391—731no—
361#391—731no—
362#391—1461no—
363#389—731no—
364#389—731no—
365#389C2×D7311yescharacteristic
366#389C2×D7311yescharacteristic
367#387—1461no—
368#385—21no—
369#385—21no—
370#382C2×D7311yescharacteristic
371#380—21no—
372C4×C4⋊C4C2×D7311yescharacteristic
373C2×#95—21no—
374#340—731no—
375#340—731no—
376#340—731no—
377#340—731no—
378#336—731no—
379#336—731no—
380#336—731no—
381#336—731no—
382C22×Q8—731no—
383C22×Q8—1461no—
384C22×D4C22×D7311yescharacteristic
385C23×C4—21no—
386C23×C4—21no—
387C23×C4—21no—
388C2×SD16—734no—
389C2×SD16—1462no—
390C2×SD16—1462no—
391C22×C8—731no—
392C22×C8—731no—
393C22×C8—731no—
394C22×C8—731no—
395C4⋊Q8—2921no—
396C4⋊Q8—1462no—
397C4×Q8—2921no—
398C4×Q8—1462no—
399C4×D4—22no—
400C4×D4—22no—
401C4×D4—22no—
402C4×D4—22no—
403C4×D4—41no—
404C4×D4—41no—
405C2×C4⋊C4—21no—
406C2×C4⋊C4—21no—
407C2×C4⋊C4—731no—
408C2×C4⋊C4—731no—
409C2×C4⋊C4—731no—
410C2×C4⋊C4—1461no—
411C2×C4⋊C4—1461no—
412C2×C4⋊C4—1461no—
413C2×C4⋊C4—1461no—
414C2×C4⋊C4C22⋊D7311yescharacteristic
415C2×C4⋊C4C22⋊D7311yescharacteristic
416C2×C4⋊C4C22×D7311yescharacteristic
417C2×C4⋊C4C22×D7311yescharacteristic
418C2×C4⋊C4C22×D7311yescharacteristic
419C2×C22⋊C4—21no—
420C2×C22⋊C4—21no—
421C2×C22⋊C4—21no—
422C2×C42—1461no—
423C2×C42C22×D7311yescharacteristic
424C2×C42C22×D7311yescharacteristic
425C2×C42C22×D7311yescharacteristic
426#107—2921no—
427#107—1462no—
428#107—1462no—
429#106—1462no—
430#106—1462no—
431C4⋊C8—1462no—
432#103—2921no—
433#103—1462no—
434#103—1462no—
435#102—734no—
436#102—1462no—
437#102—1462no—
438#97—1462no—
439#95—21no—
440#95—21no—
441#95—21no—
442#95—21no—
443#95—1461no—
444#95—1461no—
445#95—1461no—
446#95—1461no—
447#95—1461no—
448#95—1461no—
449#95C22⋊D7311yescharacteristic
450#95C22⋊D7311yescharacteristic
451C24—21no—
452C2×Q8—2922no—
453C2×Q8—2922no—
454C2×Q8—5841no—
455C2×Q8—732no—
456C2×Q8—1461no—
457C2×Q8—1461no—
458C2×Q8—1461no—
459C2×Q8—1461no—
460C2×Q8—1461no—
461C2×D4—21no—
462C2×D4—21no—
463C2×D4—21no—
464C2×D4D4×D7312yes—
465C2×D4Q8×D7312yes—
466C2×D4#1192402812yes—
467C22×C4—21no—
468C22×C4—24no—
469C22×C4—21no—
470C22×C4—21no—
471C22×C4—42no—
472C22×C4—42no—
473C22×C4—42no—
474C22×C4—42no—
475C22×C4—81no—
476C22×C4—81no—
477C22×C4—81no—
478C22×C4—81no—
479C22×C4—1461no—
480C22×C4—1461no—
481C22×C4—1461no—
482C22×C4—1461no—
483C22×C4C2×C22⋊D7311yescharacteristic
484C22×C4C2×C22⋊D7311yescharacteristic
485C22×C4C2×C22⋊D7311yescharacteristic
486C22×C4C2×C22⋊D7311yescharacteristic
487C22×C4C2×C22⋊D7311yescharacteristic
488C22×C4C2×C22⋊D7311yescharacteristic
489C22×C4C23×D7311yescharacteristic
490SD16—1468no—
491C2×C8—732no—
492C2×C8—732no—
493C2×C8—1461no—
494C2×C8—1461no—
495C2×C8—1461no—
496C2×C8—1461no—
497C2×C8—1461no—
498C2×C8—1461no—
499C2×C8—1461no—
500C2×C8—1461no—
501C2×C8—1461no—
502C2×C8—1461no—
503C4⋊C4—22no—
504C4⋊C4—21no—
505C4⋊C4—21no—
506C4⋊C4—21no—
507C4⋊C4—21no—
508C4⋊C4—21no—
509C4⋊C4—41no—
510C4⋊C4—81no—
511C4⋊C4—81no—
512C4⋊C4—2922no—
513C4⋊C4—2922no—
514C4⋊C4—5841no—
515C4⋊C4—5841no—
516C4⋊C4—732no—
517C4⋊C4—1461no—
518C4⋊C4—1461no—
519C4⋊C4—1461no—
520C4⋊C4—1461no—
521C4⋊C4—1461no—
522C4⋊C4#1192402812yes—
523C22⋊C4—24no—
524C22⋊C4—42no—
525C22⋊C4—42no—
526C42—21no—
527C42—21no—
528C42—21no—
529C42—21no—
530C42—21no—
531C42—21no—
532C42—2922no—
533C23—22no—
534C23—22no—
535C23—22no—
536C23—41no—
537C23—41no—
538C23—41no—
539C23—41no—
540C23C22×C22⋊D7311yescharacteristic
541Q8—2921no—
542Q8—2921no—
543Q8—5844no—
544Q8—1462no—
545Q8—1462no—
546D4—22no—
547D4—22no—
548D4—22no—
549D4#1455656914yes—
550C2×C4—21no—
551C2×C4—21no—
552C2×C4—21no—
553C2×C4—21no—
554C2×C4—21no—
555C2×C4—21no—
556C2×C4—21no—
557C2×C4—21no—
558C2×C4—21no—
559C2×C4—21no—
560C2×C4—21no—
561C2×C4—21no—
562C2×C4—21no—
563C2×C4—21no—
564C2×C4—21no—
565C2×C4—21no—
566C2×C4—21no—
567C2×C4—22no—
568C2×C4—41no—
569C2×C4—41no—
570C2×C4—41no—
571C2×C4—41no—
572C2×C4—41no—
573C2×C4—41no—
574C2×C4—41no—
575C2×C4—41no—
576C2×C4—84no—
577C2×C4—84no—
578C2×C4—84no—
579C2×C4—2921no—
580C2×C4—2921no—
581C2×C4—2921no—
582C2×C4—2921no—
583C2×C4—2921no—
584C2×C4—2921no—
585C2×C4—2921no—
586C2×C4—2921no—
587C2×C4—2921no—
588C2×C4—2921no—
589C2×C4—1462no—
590C2×C4—1462no—
591C2×C4#1455662112yes—
592C2×C4C2×D4×D7311yescharacteristic
593C2×C4C2×Q8×D7311yescharacteristic
594C2×C4C2×#1192402811yescharacteristic
595C2×C4C2×#1192402811yescharacteristic
596C2×C4#1455663611yescharacteristic
597C2×C4#1455663911yescharacteristic
598C8—2921no—
599C8—2921no—
600C8—1462no—
601C8—1462no—
602C22—24no—
603C22—42no—
604C22—42no—
605C22—42no—
606C22—42no—
607C22—42no—
608C22—42no—
609C22#1798909811yescharacteristic
610C22D4×C22⋊D7311yescharacteristic
611C22Q8×C22⋊D7311yescharacteristic
612C22C2×#1455662111yescharacteristic
613C22#1798938511yescharacteristic
614C22#1798939011yescharacteristic
615C22#1798939811yescharacteristic
616C4—22no—
617C4—41no—
618C4—41no—
619C4—41no—
620C4—41no—
621C4—42no—
622C4—41no—
623C4—81no—
624C4—81no—
625C4—2922no—
626C4—2922no—
627C4—5841no—
628C4—5841no—
629C4C2×#1455656911yescharacteristic
630C4#1798918411yescharacteristic
631C4#1798923811yescharacteristic
632C4#1798929211yescharacteristic
633C2—44no—
634C2#2261374011yescharacteristic
635C2#2262009311yescharacteristic
636C2#2262011511yescharacteristic
637C2#2262143711yescharacteristic
638C2#2262153811yescharacteristic
639C2#2262284111yescharacteristic
640C2#2262287611yescharacteristic
subgroupG/H
centerC23C22×C22⋊D73
commutator, cosocleC22×C4×C73C24
FrattiniC22×C4C23×D73
FittingC73×#2349C2
socleC23×C73C22×D4
prime radicalC22×D4×C73C23
hypercenter#2349D73
hypocenterC73#26396
LCSG ▷ C22×C4×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C23 ◁ C4×C4⋊C4 ◁ #2349

Complements

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

NH ≅ G/Ntype
3. #22613770C2semidirect
4. #22611482C2semidirect
5. #22611413C2semidirect
6. #22611323C2semidirect
7. #22611259C2semidirect
8. #22610666C2semidirect
9. #22610497C2semidirect
12. #22610198C2semidirect
65. C2×#11924016Q8semidirect
336. C73#26396semidirect

Maximal subgroups

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

subgroupindex
C2×#179888772
C2×#179886872
#226137702
#226114822
#226114132
#226113232
#226112592
#226106662
#226104972
#226102652
#226101982
#226101782
C73×#23492
#2639673