Finite Group Databasebeta

← #10695499

Subgroups — #10695499

Order 768.

792 conjugacy classes of subgroups; only the 85 normal subgroups are shown.

normal subgroups only

This group has 792 conjugacy classes of subgroups — too many to draw. Only its 85 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#3006C211yesmaximal, characteristic
2#84094C211yesmaximal, characteristic, complemented
3#82631C211yesmaximal, characteristic, complemented
4#72138C211yesmaximal, characteristic, complemented
5#71686C211yesmaximal, characteristic, complemented
6#69980C211yesmaximal, characteristic, complemented
7#67685C211yesmaximal, characteristic, complemented
8#9931—31nomaximal, Sylow
9C22×C3×#41C2211yescharacteristic
10C2×C3×AGL(1,ℤ/8)—21no—
11C2×C3×#121C2211yescharacteristic
12C2×#924—21no—
13C2×#924C2211yescharacteristic
14C3×#449—22no—
15C3×#448—21no—
16C3×#447—21no—
17C3×#417—21no—
18C3×#407C2211yescharacteristic
19C2×#837—22no—
20#5230—21no—
21#5222—21no—
22#5222—22no—
23C2×#834—22no—
24#5218—21no—
25#5212—21no—
26#5212C2211yescharacteristic
27C4×#217C2211yescharacteristic
28#5207C2211yescharacteristic
29#5042—22no—
30#5040—21no—
31#5033—21no—
32#4970—21no—
33#4467—22no—
34#3193—31no—
35#3006D311yescharacteristic
36#2616—31no—
37#1964—31no—
38#1789—31no—
39#1508—31no—
40#1305—31no—
41C2×C3×#41—21no—
42C2×C3×#41—21no—
43C2×C3×#41—21no—
44C2×C3×#41—21no—
45C2×C3×#41—44no—
46C22×C3×Q8D411yescharacteristic
47C22×C3×D4—21no—
48C22×C3×D4—21no—
49C22×C3×D4D411yescharacteristic
50C22×C3×D4D411yescharacteristic
51C23×C3×C4—21no—
52C23×C3×C4C2311yescharacteristic
53C22×Dic6D411yescharacteristic
54C3×AGL(1,ℤ/8)—44no—
55C2×C3×SD16—22no—
56C2×C3×D8—22no—
57C2×C3×M16—21no—
58C3×#121—21no—
59C3×#121—21no—
60C3×#121—21no—
61C3×#121—21no—
62C2×C3×C4⋊C4D411yescharacteristic
63C2×C3×C22⋊C4—21no—
64C2×#217—21no—
65C2×#217—22no—
66C2×#217—21no—
67C2×#217—22no—
68C2×#217D412yes—
69C2×#214—22no—
70C2×C4⋊Dic3—21no—
71C2×C4⋊Dic3D411yescharacteristic
72#924—21no—
73#924—21no—
74#924—21no—
75#924—21no—
76#924—44no—
77C2×Dic3⋊C4—22no—
78C2×Dic3⋊C4—21no—
79C2×C4×Dic3—21no—
80C2×#208D412yes—
81C3×#103—22no—
82C3×#102—22no—
83C3×C22⋊C8—21no—
84C3×C22⋊C8D412yes—
85#837—44no—
86#834—44no—
87#832—22no—
88C3⋊#95—22no—
89C3⋊#95—21no—
90C3⋊#95—21no—
91#830—21no—
92#830—21no—
93#810—22no—
94#807—22no—
95C22×#41C2×D311yescharacteristic
96C2×AGL(1,ℤ/8)—21no—
97C2×C22⋊Q8—31no—
98C2×C22⋊Q8—61no—
99C2×#121C2×D311yescharacteristic
100#484—61no—
101#480—61no—
102#468—62no—
103#450—61no—
104#449—22no—
105#448—21no—
106#448—61no—
107#447—21no—
108C2×#104—62no—
109#417—21no—
110C2×#99—62no—
111#407—31no—
112#407C2×D311yescharacteristic
113#395—61no—
114#394—62no—
115#394—61no—
116#380—31no—
117#380—61no—
118C4×C22⋊C4—31no—
119#328—62no—
120C24×C3—21no—
121C24×C3—21no—
122C24×C3C2×D411yescharacteristic
123C3×#41—24no—
124C3×#41—24no—
125C3×#41—42no—
126C3×#41—42no—
127C3×#41—42no—
128C3×#41—42no—
129C3×#41—42no—
130C3×#41—42no—
131C2×C3×Q8—21no—
132C2×C3×Q8—21no—
133C2×C3×Q8—42no—
134C2×C3×D4—21no—
135C2×C3×D4—21no—
136C2×C3×D4—21no—
137C2×C3×D4—21no—
138C2×C3×D4—42no—
139C2×C3×D4—42no—
140C2×C3×D4—42no—
141C2×C3×D4—42no—
142C2×C3×D4—41no—
143C2×C3×D4—41no—
144C2×C3×D4—42no—
145C2×C3×D4—42no—
146C2×C3×D4—41no—
147C2×C3×D4—41no—
148C22×C3×C4—24no—
149C22×C3×C4—21no—
150C22×C3×C4—21no—
151C22×C3×C4—42no—
152C22×C3×C4—42no—
153C22×C3×C4—41no—
154C22×C3×C4—41no—
155C22×C3×C4C2×D411yescharacteristic
156C22×C3×C4C2×D411yescharacteristic
157C22×C3×C4C2×D411yescharacteristic
158C22×C3×C4C2×D411yescharacteristic
159C22×C3×C4C2×D411yescharacteristic
160C22×C3×C4#4111yescharacteristic
161C22×Dic3—22no—
162C22×Dic3—21no—
163C22×Dic3—21no—
164C22×Dic3—41no—
165C2×Dic6—21no—
166C2×Dic6—21no—
167C2×Dic6—42no—
168C3×SD16—44no—
169C3×D8—44no—
170C3×M16—42no—
171C2×C3×C8—22no—
172C3×C4⋊C4—21no—
173C3×C4⋊C4—21no—
174C3×C22⋊C4—41no—
175C3×C22⋊C4—41no—
176#217—22no—
177#217—44no—
178#217—41no—
179#217—41no—
180#217—41no—
181#217—42no—
182#217—82no—
183#214—82no—
184C4⋊Dic3—21no—
185C4⋊Dic3—21no—
186C4⋊Dic3—42no—
187Dic3⋊C4—44no—
188Dic3⋊C4—41no—
189Dic3⋊C4—41no—
190C4×Dic3—42no—
191#208—42no—
192C2×C3⋊C8—22no—
193C2×#41—21no—
194C2×#41—21no—
195C2×#41—21no—
196C2×#41—21no—
197C2×#41—44no—
198C22×Q8—31no—
199C22×Q8#6711yescharacteristic
200C22×D4—21no—
201C22×D4—21no—
202C22×D4#6711yescharacteristic
203C22×D4#6711yescharacteristic
204C23×C4—21no—
205C23×C4C22×D311yescharacteristic
206AGL(1,ℤ/8)—44no—
207C2×SD16—22no—
208C2×SD16—62no—
209C2×D8—22no—
210C2×M16—21no—
211C2×M16—32no—
212C22⋊Q8—61no—
213C22⋊Q8—61no—
214C22⋊Q8—61no—
215C22⋊Q8—61no—
216C22⋊Q8—124no—
217#121—21no—
218#121—21no—
219#121—21no—
220#121—21no—
221C2×C4⋊C4—31no—
222C2×C4⋊C4—62no—
223C2×C4⋊C4—61no—
224C2×C4⋊C4—61no—
225C2×C4⋊C4#6711yescharacteristic
226C2×C22⋊C4—21no—
227C2×C22⋊C4—32no—
228C2×C22⋊C4—61no—
229C2×C22⋊C4—62no—
230C2×C22⋊C4—61no—
231C2×C22⋊C4—62no—
232C2×C42—61no—
233#107—62no—
234#104—124no—
235#103—22no—
236#103—62no—
237#102—22no—
238#102—62no—
239#99—124no—
240C22⋊C8—21no—
241C22⋊C8—61no—
242C22⋊C8—61no—
243C22⋊C8#6712yes—
244#95—62no—
245#95—61no—
246#95—61no—
247C23×C3—21no—
248C23×C3—21no—
249C23×C3—22no—
250C23×C3—41no—
251C23×C3—41no—
252C23×C3—41no—
253C23×C3—41no—
254C23×C3—41no—
255C23×C3—42no—
256C23×C3—41no—
257C23×C3—41no—
258C23×C3—41no—
259C23×C3—41no—
260C23×C3—81no—
261C23×C3—81no—
262C23×C3#12011yescharacteristic
263C23×C3#12111yescharacteristic
264C23×C3#12711yescharacteristic
265C3×Q8—22no—
266C3×Q8—22no—
267C3×Q8—41no—
268C3×Q8—41no—
269C3×D4—22no—
270C3×D4—22no—
271C3×D4—41no—
272C3×D4—41no—
273C3×D4—42no—
274C3×D4—44no—
275C3×D4—44no—
276C3×D4—42no—
277C3×D4—82no—
278C3×D4—81no—
279C3×D4—81no—
280C2×C3×C4—22no—
281C2×C3×C4—22no—
282C2×C3×C4—22no—
283C2×C3×C4—22no—
284C2×C3×C4—41no—
285C2×C3×C4—41no—
286C2×C3×C4—41no—
287C2×C3×C4—44no—
288C2×C3×C4—41no—
289C2×C3×C4—41no—
290C2×C3×C4—44no—
291C2×C3×C4—42no—
292C2×C3×C4—41no—
293C2×C3×C4#12011yescharacteristic
294C2×C3×C4#12011yescharacteristic
295C2×C3×C4#12111yescharacteristic
296C2×C3×C4#12111yescharacteristic
297C2×Dic3—42no—
298C2×Dic3—42no—
299C2×Dic3—42no—
300C2×Dic3—42no—
301C2×Dic3—41no—
302C2×Dic3—41no—
303C2×Dic3—42no—
304C2×Dic3—41no—
305C2×Dic3—41no—
306C2×Dic3—81no—
307C2×Dic3—81no—
308Dic6—42no—
309Dic6—81no—
310C3×C8—42no—
311C3⋊C8—42no—
312C24—21no—
313C24—21no—
314C24D3×D411yescharacteristic
315#41—24no—
316#41—24no—
317#41—42no—
318#41—42no—
319#41—42no—
320#41—42no—
321#41—42no—
322#41—42no—
323C2×Q8—21no—
324C2×Q8—21no—
325C2×Q8—42no—
326C2×Q8—61no—
327C2×Q8—61no—
328C2×Q8—122no—
329C2×D4—21no—
330C2×D4—21no—
331C2×D4—21no—
332C2×D4—21no—
333C2×D4—42no—
334C2×D4—42no—
335C2×D4—42no—
336C2×D4—42no—
337C2×D4—41no—
338C2×D4—41no—
339C2×D4—42no—
340C2×D4—42no—
341C2×D4—41no—
342C2×D4—41no—
343C22×C4—24no—
344C22×C4—21no—
345C22×C4—21no—
346C22×C4—42no—
347C22×C4—42no—
348C22×C4—41no—
349C22×C4—41no—
350C22×C4—62no—
351C22×C4—61no—
352C22×C4—61no—
353C22×C4—121no—
354C22×C4D3×D411yescharacteristic
355C22×C4D3×D411yescharacteristic
356C22×C4#23711yescharacteristic
357C22×C4C2×#6711yescharacteristic
358C22×C4C2×#6711yescharacteristic
359C22×C4C2×#6711yescharacteristic
360SD16—44no—
361SD16—242no—
362D8—44no—
363M16—42no—
364M16—122no—
365C2×C8—22no—
366C2×C8—62no—
367C4⋊C4—21no—
368C4⋊C4—21no—
369C4⋊C4—61no—
370C4⋊C4—61no—
371C4⋊C4—124no—
372C4⋊C4—122no—
373C4⋊C4—121no—
374C4⋊C4—121no—
375C22⋊C4—41no—
376C22⋊C4—41no—
377C22⋊C4—62no—
378C22⋊C4—124no—
379C22⋊C4—121no—
380C22⋊C4—121no—
381C22⋊C4—121no—
382C22⋊C4—122no—
383C22⋊C4—242no—
384C42—122no—
385C22×C3—21no—
386C22×C3—21no—
387C22×C3—21no—
388C22×C3—21no—
389C22×C3—21no—
390C22×C3—21no—
391C22×C3—21no—
392C22×C3—21no—
393C22×C3—42no—
394C22×C3—42no—
395C22×C3—42no—
396C22×C3—81no—
397C22×C3—81no—
398C22×C3—81no—
399C22×C3—81no—
400C22×C3—81no—
401C22×C3—81no—
402C22×C3—81no—
403C22×C3—81no—
404C22×C3—81no—
405C22×C3—162no—
406C22×C3#39211yescharacteristic
407C22×C3#45311yescharacteristic
408C22×C3#45411yescharacteristic
409C3×C4—21no—
410C3×C4—21no—
411C3×C4—21no—
412C3×C4—42no—
413C3×C4—81no—
414C3×C4B(4,𝔽2)12yes—
415Dic3—82no—
416Dic3—82no—
417Dic3—81no—
418Dic3—81no—
419C23—21no—
420C23—21no—
421C23—22no—
422C23—41no—
423C23—41no—
424C23—41no—
425C23—41no—
426C23—41no—
427C23—42no—
428C23—41no—
429C23—41no—
430C23—41no—
431C23—41no—
432C23—81no—
433C23—81no—
434C23#93711yescharacteristic
435C23#93911yescharacteristic
436C23#94011yescharacteristic
437Q8—22no—
438Q8—22no—
439Q8—41no—
440Q8—41no—
441Q8—122no—
442Q8—241no—
443D4—22no—
444D4—22no—
445D4—41no—
446D4—41no—
447D4—42no—
448D4—44no—
449D4—44no—
450D4—42no—
451D4—82no—
452D4—81no—
453D4—81no—
454C2×C4—22no—
455C2×C4—22no—
456C2×C4—22no—
457C2×C4—22no—
458C2×C4—41no—
459C2×C4—41no—
460C2×C4—41no—
461C2×C4—44no—
462C2×C4—41no—
463C2×C4—41no—
464C2×C4—44no—
465C2×C4—42no—
466C2×C4—41no—
467C2×C4—122no—
468C2×C4—122no—
469C2×C4—122no—
470C2×C4—122no—
471C2×C4—122no—
472C2×C4—121no—
473C2×C4—121no—
474C2×C4—121no—
475C2×C4—121no—
476C2×C4—241no—
477C2×C4—241no—
478C2×C4#93711yescharacteristic
479C2×C4#93811yescharacteristic
480C2×C4#93911yescharacteristic
481C2×C4#95311yescharacteristic
482C8—42no—
483C8—122no—
484C2×C3—21no—
485C2×C3—21no—
486C2×C3—41no—
487C2×C3—41no—
488C2×C3—82no—
489C2×C3—161no—
490C2×C3#200611yescharacteristic
491C2×C3#200911yescharacteristic
492C2×C3#202111yescharacteristic
493C22—21no—
494C22—21no—
495C22—21no—
496C22—21no—
497C22—21no—
498C22—21no—
499C22—21no—
500C22—21no—
501C22—42no—
502C22—42no—
503C22—42no—
504C22—81no—
505C22—81no—
506C22—81no—
507C22—81no—
508C22—81no—
509C22—81no—
510C22—81no—
511C22—81no—
512C22—81no—
513C22—162no—
514C22#519811yescharacteristic
515C22#520111yescharacteristic
516C22#522311yescharacteristic
517C4—21no—
518C4—21no—
519C4—21no—
520C4—42no—
521C4—81no—
522C4—242no—
523C4—242no—
524C4—241no—
525C4—241no—
526C4#503212yes—
527C3#993111yescharacteristic, complemented, Sylow
528C2—21no—
529C2—21no—
530C2—41no—
531C2—41no—
532C2—82no—
533C2—161no—
534C2#7167111yescharacteristic
535C2#7209811yescharacteristic
536C2#7212011yescharacteristic
subgroupG/H
centerC22#5223
commutator, cosocleC23×C3×C4C23
FrattiniC23×C4C22×D3
Fitting, prime radicalC3×#3006C2
socleC22×C3#392
hypercenter#3006D3
hypocenterC3#9931
LCSG ▷ C23×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23×C4 ◁ #3006

Complements

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

NH ≅ G/Ntype
2. #84094C2semidirect
3. #82631C2semidirect
4. #72138C2semidirect
5. #71686C2semidirect
6. #69980C2semidirect
7. #67685C2semidirect
527. C3#9931semidirect

Maximal subgroups

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

subgroupindex
C3×#30062
#840942
#826312
#721382
#716862
#699802
#676852
#99313