Finite Group Databasebeta

← #10693224

Subgroups — #10693224

Order 768.

472 conjugacy classes of subgroups; only the 93 normal subgroups are shown.

normal subgroups only

This group has 472 conjugacy classes of subgroups — too many to draw. Only its 93 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
1#82139C211yesmaximal, characteristic
2#78604C211yesmaximal, characteristic
3C3×#1823C211yesmaximal, characteristic, complemented
4#71562C211yesmaximal, characteristic, complemented
5#69508C211yesmaximal, characteristic
6#67679C211yesmaximal, characteristic, complemented
7#67626C211yesmaximal, characteristic, complemented
8#9522—31nomaximal, Sylow
9C22×C3×C4⋊C4C2211yescharacteristic
10#5791C2211yescharacteristic
11C2×C3×C22⋊C8C2211yescharacteristic
12C3×#380C2211yescharacteristic
13#5207C2211yescharacteristic
14#5207C2211yescharacteristic
15#5207C2211yescharacteristic
16#5111—21no—
17C8×#67—21no—
18#4971—21no—
19#4970C412yescomplemented
20#4968C412yescomplemented
21#4967—21no—
22#4966—21no—
23#4964—21no—
24#4729—21no—
25#4728—21no—
26#4727—21no—
27#4720—21no—
28C3×#340—21no—
29C3×#340—21no—
30#4530—21no—
31#4530—21no—
32#4452—21no—
33#4451—21no—
34#3092—31no—
35#2976—31no—
36#1823D311yescharacteristic, complemented
37#1821—31no—
38#1513—31no—
39#1305—31no—
40#1293—31no—
41C23×C3×C4—21no—
42C23×C3×C4C2311yescharacteristic
43C2×#235D411yescharacteristic
44C22×C3×C8D411yescharacteristic
45C22×C3×C8D411yescharacteristic
46C2×C3×C4⋊C4—21no—
47C2×C3×C4⋊C4—21no—
48C2×C3×C4⋊C4—21no—
49C2×C3×C4⋊C4—21no—
50C2×C3×C4⋊C4—21no—
51C2×C3×C4⋊C4—21no—
52C2×C3×C4⋊C4C2×C412yes—
53C2×C3×C22⋊C4—21no—
54#953—21no—
55#930C2×C411yescharacteristic
56#930D411yescharacteristic
57#929—21no—
58C4×#67—21no—
59C4×#67—21no—
60#926D411yescharacteristic
61#924C2×C411yescharacteristic
62#924D411yescharacteristic
63C2×#208—21no—
64C2×#208—21no—
65C2×#208D411yescharacteristic
66C2×#208D411yescharacteristic
67C2×ΓL(1,𝔽25)—21no—
68C2×D3×C8—21no—
69C3×C22⋊C8—21no—
70C3×C22⋊C8—21no—
71C3×#95—21no—
72#830—21no—
73#830—21no—
74#830—21no—
75#830—21no—
76#830—21no—
77#830C2×C412yes—
78(C2×D3)⋊C8—21no—
79(C2×D3)⋊C8—21no—
80#815—21no—
81Dic3⋊C8—21no—
82Dic3⋊C8—21no—
83C8×Dic3—21no—
84#810—22no—
85#809—22no—
86#808—22no—
87#807—22no—
88#525—31no—
89C22×C4⋊C4C2×D311yescharacteristic
90#484—32no—
91#483—61no—
92#481—32no—
93#480—61no—
94#436—61no—
95#435—61no—
96#435—61no—
97C8×D4—61no—
98#433—61no—
99#428—61no—
100#417—61no—
101#408—61no—
102#407—31no—
103#407—31no—
104#407—31no—
105C2×C22⋊C8C2×D311yescharacteristic
106#380C2×D311yescharacteristic
107#343—61no—
108#343—61no—
109#340—21no—
110#340—21no—
111#328—61no—
112#327—61no—
113C24×C3#4111yescharacteristic
114C22×C3×C4—22no—
115C22×C3×C4—41no—
116C22×C3×C4—41no—
117C22×C3×C4—41no—
118C22×C3×C4—41no—
119C22×C3×C4—41no—
120C22×C3×C4—81no—
121C22×C3×C4—81no—
122C22×C3×C4C22×C411yescharacteristic
123C22×C3×C4C2×D411yescharacteristic
124C22×C3×C4C2×D411yescharacteristic
125C22×C3×C4C2×D411yescharacteristic
126C22×C3×C4C2×D411yescharacteristic
127C22×C3×C4#4111yescharacteristic
128C2×#67—21no—
129C2×#67—21no—
130C2×#67—21no—
131#235—41no—
132#235—41no—
133C2×D12C22⋊C411yescharacteristic
134C2×C4×D3—21no—
135C2×Dic6C22⋊C411yescharacteristic
136C2×C3×C8—21no—
137C2×C3×C8—21no—
138C2×C3×C8—41no—
139C2×C3×C8—41no—
140C3×C4⋊C4—22no—
141C3×C4⋊C4—22no—
142C3×C4⋊C4—41no—
143C3×C4⋊C4—41no—
144C3×C4⋊C4—41no—
145C3×C4⋊C4—41no—
146C3×C22⋊C4—81no—
147#217—21no—
148#217—21no—
149#217—21no—
150(C2×D3)⋊C4—21no—
151(C2×D3)⋊C4—21no—
152C4⋊Dic3C22⋊C411yescharacteristic
153C4⋊Dic3C22⋊C411yescharacteristic
154Dic3⋊C4—21no—
155Dic3⋊C4—21no—
156C4×Dic3—21no—
157#208—41no—
158#208—41no—
159#208—41no—
160#208—41no—
161#208—41no—
162#208—41no—
163C2×C3⋊C8—22no—
164C2×C3⋊C8—21no—
165C2×C3⋊C8—21no—
166C2×C3⋊C8—41no—
167ΓL(1,𝔽25)—81no—
168D3×C8—81no—
169C2×#41—31no—
170C23×C4—21no—
171C23×C4C22×D311yescharacteristic
172C2×M16—31no—
173C2×M16—31no—
174C2×M16—61no—
175C2×M16—61no—
176C2×M16—61no—
177C22×C8—61no—
178C22×C8#6711yescharacteristic
179C22×C8#6711yescharacteristic
180#123—61no—
181C22⋊Q8—31no—
182C22⋊Q8—31no—
183#121—31no—
184#121—31no—
185#120—61no—
186C4×D4—61no—
187C4×D4—61no—
188#117—31no—
189C2×C4⋊C4—21no—
190C2×C4⋊C4—21no—
191C2×C4⋊C4—21no—
192C2×C4⋊C4—21no—
193C2×C4⋊C4—21no—
194C2×C4⋊C4—21no—
195C2×C4⋊C4C4×D312yes—
196C2×C22⋊C4—21no—
197#107—62no—
198#106—62no—
199C4⋊C8—61no—
200C4⋊C8—61no—
201#103—62no—
202#102—62no—
203C22⋊C8—21no—
204C22⋊C8—21no—
205C22⋊C8—32no—
206C22⋊C8—61no—
207C22⋊C8—61no—
208C22⋊C8—61no—
209C22⋊C8—61no—
210C22⋊C8—61no—
211C22⋊C8—61no—
212C22⋊C8—61no—
213#97—61no—
214C4×C8—61no—
215#95—21no—
216C23×C3—21no—
217C23×C3—21no—
218C23×C3—41no—
219C23×C3—41no—
220C23×C3C4×D411yescharacteristic
221C23×C3#12111yescharacteristic
222C23×C3#12311yescharacteristic
223C22×D3—21no—
224C2×C3×C4—21no—
225C2×C3×C4—21no—
226C2×C3×C4—21no—
227C2×C3×C4—21no—
228C2×C3×C4—21no—
229C2×C3×C4—21no—
230C2×C3×C4—41no—
231C2×C3×C4—41no—
232C2×C3×C4—41no—
233C2×C3×C4—42no—
234C2×C3×C4—82no—
235C2×C3×C4—81no—
236C2×C3×C4—81no—
237C2×C3×C4—82no—
238C2×C3×C4—81no—
239C2×C3×C4—81no—
240C2×C3×C4—81no—
241C2×C3×C4C2×C22⋊C411yescharacteristic
242C2×C3×C4C4×D411yescharacteristic
243C2×C3×C4#12011yescharacteristic
244C2×C3×C4#12111yescharacteristic
245#67—41no—
246#67—41no—
247#67—81no—
248#67—81no—
249C2×Dic3—21no—
250C2×Dic3—21no—
251C2×Dic3—21no—
252D12—41no—
253C4×D3—41no—
254C4×D3—41no—
255Dic6—41no—
256C3×C8—41no—
257C3×C8—41no—
258C3⋊C8—42no—
259C3⋊C8—41no—
260C3⋊C8—41no—
261C3⋊C8—81no—
262C24#23711yescharacteristic
263#41—121no—
264#41—121no—
265C2×Q8—31no—
266C2×D4—31no—
267C2×D4—61no—
268C2×D4—61no—
269C2×D4—61no—
270C22×C4—22no—
271C22×C4—41no—
272C22×C4—41no—
273C22×C4—41no—
274C22×C4—41no—
275C22×C4—41no—
276C22×C4—61no—
277C22×C4—81no—
278C22×C4—81no—
279C22×C4C2×C4×D311yescharacteristic
280C22×C4#23511yescharacteristic
281C22×C4D3×D411yescharacteristic
282C22×C4D3×D411yescharacteristic
283C22×C4D3×D411yescharacteristic
284C22×C4C2×#6711yescharacteristic
285M16—121no—
286M16—121no—
287M16—121no—
288M16—121no—
289M16—121no—
290M16—121no—
291M16—241no—
292C2×C8—21no—
293C2×C8—21no—
294C2×C8—41no—
295C2×C8—41no—
296C2×C8—62no—
297C2×C8—61no—
298C2×C8—61no—
299C2×C8—121no—
300C2×C8—241no—
301C4⋊C4—22no—
302C4⋊C4—22no—
303C4⋊C4—31no—
304C4⋊C4—31no—
305C4⋊C4—41no—
306C4⋊C4—41no—
307C4⋊C4—41no—
308C4⋊C4—41no—
309C4⋊C4—61no—
310C4⋊C4—61no—
311C22⋊C4—61no—
312C22⋊C4—61no—
313C22⋊C4—61no—
314C22⋊C4—61no—
315C22⋊C4—61no—
316C22⋊C4—81no—
317C42—61no—
318C22×C3—21no—
319C22×C3—21no—
320C22×C3—21no—
321C22×C3—21no—
322C22×C3—21no—
323C22×C3—21no—
324C22×C3—21no—
325C22×C3—21no—
326C22×C3—42no—
327C22×C3—42no—
328C22×C3#38611yescharacteristic
329C22×C3#47111yescharacteristic
330C22×C3#47211yescharacteristic
331C2×D3—41no—
332C2×D3—41no—
333C2×D3—41no—
334C3×C4—21no—
335C3×C4—21no—
336C3×C4—21no—
337C3×C4—82no—
338C3×C4—161no—
339C3×C4#45811yescharacteristic
340C3×C4#45811yescharacteristic
341Dic3—41no—
342Dic3—41no—
343Dic3—41no—
344C23—21no—
345C23—21no—
346C23—41no—
347C23—41no—
348C23—61no—
349C23#67⋊C411yescharacteristic
350C23#88311yescharacteristic
351C23#93811yescharacteristic
352Q8—121no—
353D4—121no—
354D4—121no—
355D4—121no—
356D4—241no—
357D4—241no—
358C2×C4—21no—
359C2×C4—21no—
360C2×C4—21no—
361C2×C4—21no—
362C2×C4—21no—
363C2×C4—21no—
364C2×C4—41no—
365C2×C4—41no—
366C2×C4—41no—
367C2×C4—42no—
368C2×C4—61no—
369C2×C4—61no—
370C2×C4—61no—
371C2×C4—82no—
372C2×C4—81no—
373C2×C4—81no—
374C2×C4—82no—
375C2×C4—81no—
376C2×C4—81no—
377C2×C4—81no—
378C2×C4—121no—
379C2×C4—121no—
380C2×C4D3×C22⋊C411yescharacteristic
381C2×C4#88411yescharacteristic
382C2×C4C4×#6711yescharacteristic
383C2×C4#93711yescharacteristic
384C8—41no—
385C8—41no—
386C8—122no—
387C8—121no—
388C8—121no—
389C8—241no—
390C2×C3—21no—
391C2×C3—21no—
392C2×C3—41no—
393C2×C3—41no—
394C2×C3#189011yescharacteristic
395C2×C3#190211yescharacteristic
396C2×C3#190711yescharacteristic
397D3—81no—
398C22—21no—
399C22—21no—
400C22—21no—
401C22—21no—
402C22—21no—
403C22—21no—
404C22—21no—
405C22—21no—
406C22—42no—
407C22—42no—
408C22—121no—
409C22—121no—
410C22—121no—
411C22#495611yescharacteristic
412C22#516011yescharacteristic
413C22#517711yescharacteristic
414C4—21no—
415C4—21no—
416C4—21no—
417C4—82no—
418C4—121no—
419C4—121no—
420C4—121no—
421C4—161no—
422C4#505311yescharacteristic
423C4#505311yescharacteristic
424C3#952211yescharacteristic, complemented, Sylow
425C2—21no—
426C2—21no—
427C2—41no—
428C2—41no—
429C2—241no—
430C2#6953811yescharacteristic
431C2#7021811yescharacteristic
432C2#7034911yescharacteristic
subgroupG/H
centerC22#4956
commutatorC22×C3×C4C22×C4
FrattiniC23×C4C22×D3
FittingC3×#1823C2
socleC22×C3#386
prime radical#5791C22
cosocleC23×C3×C4C23
hypercenter#1823D3
hypocenterC3#9522
LCSG ▷ C22×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1823

Complements

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

NH ≅ G/Ntype
3. C3×#1823C2semidirect
4. #71562C2semidirect
6. #67679C2semidirect
7. #67626C2semidirect
19. #4970C4semidirect
20. #4968C4semidirect
36. #1823D3semidirect
424. C3#9522semidirect

Maximal subgroups

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

subgroupindex
#821392
#786042
C3×#18232
#715622
#695082
#676792
#676262
#95223