Finite Group Databasebeta

← #10693253

Subgroups — #10693253

Order 768.

482 conjugacy classes of subgroups; only the 95 normal subgroups are shown.

normal subgroups only

This group has 482 conjugacy classes of subgroups — too many to draw. Only its 95 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#82746C211yesmaximal, characteristic
2#82727C211yesmaximal, characteristic
3C3×#1511C211yesmaximal, characteristic
4#69975C211yesmaximal, characteristic
5#69534C211yesmaximal, characteristic
6#67686C211yesmaximal, characteristic
7#67628C211yesmaximal, characteristic
8#9526—31nomaximal, Sylow
9C2×C3×C22⋊Q8C2211yescharacteristic
10C2×#926C2211yescharacteristic
11C3×#407C2211yescharacteristic
12C3×#381C2211yescharacteristic
13#5229—21no—
14#5225—21no—
15#5211C2211yescharacteristic
16C4×#217C2211yescharacteristic
17#5207C2211yescharacteristic
18#5045—22no—
19#5044C412yescomplemented
20#5043C412yescomplemented
21#4980—21no—
22#4980—21no—
23Dic3×C4⋊C4—21no—
24#4964—21no—
25#4951—21no—
26#4951—21no—
27C3×#328—22no—
28#4555—21no—
29#4555—21no—
30#4527—21no—
31#4527—21no—
32#3095—31no—
33#2332—31no—
34#1826—31no—
35#1820—31no—
36#1511D311yescharacteristic
37#1306—31no—
38#1284—31no—
39C22×C3×Q8D411yescharacteristic
40C23×C3×C4C2311yescharacteristic
41C2×C3×M16—21no—
42C3×C22⋊Q8—21no—
43C3×C22⋊Q8—21no—
44C3×C22⋊Q8C2×C412yes—
45C3×C22⋊Q8D412yes—
46C2×C3×C4⋊C4—21no—
47C2×C3×C4⋊C4D411yescharacteristic
48C2×C3×C22⋊C4—21no—
49C2×C3×C22⋊C4—21no—
50C2×#217—21no—
51C2×#217—21no—
52C2×#217D411yescharacteristic
53C2×#217D411yescharacteristic
54#926—21no—
55#926—21no—
56#926—21no—
57#926—21no—
58C2×C4⋊Dic3—21no—
59C2×C4⋊Dic3—21no—
60C2×C4⋊Dic3C2×C411yescharacteristic
61C2×C4⋊Dic3C2×C411yescharacteristic
62C2×C4×Dic3—21no—
63C2×C4×Dic3—21no—
64C2×C4×Dic3—21no—
65C2×C4×Dic3D411yescharacteristic
66C2×C4×Dic3D411yescharacteristic
67C2×#208—21no—
68C3×C22⋊C8—21no—
69C3×C22⋊C8—21no—
70C3×#95—21no—
71C3×#95—21no—
72#835—22no—
73#835—22no—
74C3⋊#95—21no—
75C3⋊#95—21no—
76C3⋊#95—21no—
77C3⋊#95—21no—
78C3⋊#95—21no—
79C3⋊#95—21no—
80C3⋊#95—21no—
81#830—21no—
82#830C2×C412yes—
83#808—22no—
84#807—22no—
85C2×C22⋊Q8C2×D311yescharacteristic
86C2×#117—31no—
87#485—62no—
88#484—32no—
89#483—32no—
90#428—61no—
91#419—61no—
92#407—31no—
93#407C2×D311yescharacteristic
94#391—61no—
95#388—61no—
96#388—61no—
97#387—61no—
98#387—61no—
99#381—31no—
100#381C2×D311yescharacteristic
101C4×C4⋊C4—61no—
102C4×C22⋊C4—31no—
103#342—61no—
104#342—61no—
105#339—61no—
106#339—61no—
107#328—22no—
108C24×C3C2×D411yescharacteristic
109C2×C3×Q8—21no—
110C2×C3×Q8—41no—
111C2×C3×Q8—41no—
112C2×C3×Q8C22⋊C412yes—
113C22×C3×C4—21no—
114C22×C3×C4—21no—
115C22×C3×C4—41no—
116C22×C3×C4—41no—
117C22×C3×C4—41no—
118C22×C3×C4C22×C411yescharacteristic
119C22×C3×C4C2×D411yescharacteristic
120C22×C3×C4C2×D411yescharacteristic
121C22×C3×C4C2×D411yescharacteristic
122C22×C3×C4#4111yescharacteristic
123C22×C3×C4#4111yescharacteristic
124C22×Dic3—21no—
125C22×Dic3—21no—
126C22×Dic3—21no—
127C22×Dic3—21no—
128C22×Dic3—21no—
129C22×Dic3—21no—
130C3×M16—41no—
131C3×M16—41no—
132C2×C3×C8—41no—
133C3×C4⋊C4—22no—
134C3×C4⋊C4—22no—
135C3×C4⋊C4—21no—
136C3×C4⋊C4C22⋊C412yes—
137C3×C22⋊C4—22no—
138C3×C22⋊C4—22no—
139C3×C22⋊C4—41no—
140C3×C22⋊C4—41no—
141#217—42no—
142#217—41no—
143#217—41no—
144#217—41no—
145#217—41no—
146C4⋊Dic3—21no—
147C4⋊Dic3—21no—
148C4⋊Dic3—21no—
149C4⋊Dic3—21no—
150C4⋊Dic3—42no—
151C4⋊Dic3—81no—
152C4×Dic3—42no—
153C4×Dic3—41no—
154C4×Dic3—41no—
155C4×Dic3—41no—
156C4×Dic3—41no—
157C4×Dic3—81no—
158#208—41no—
159#208—41no—
160C2×C3⋊C8—22no—
161C22×Q8D1211yescharacteristic
162C23×C4C22×D311yescharacteristic
163C2×M16—21no—
164C2×M16—61no—
165C22⋊Q8—21no—
166C22⋊Q8—21no—
167C22⋊Q8C4×D312yes—
168C22⋊Q8#6712yes—
169#117—61no—
170#117—61no—
171#117—61no—
172#117—61no—
173C2×C4⋊C4—21no—
174C2×C4⋊C4—31no—
175C2×C4⋊C4—31no—
176C2×C4⋊C4—61no—
177C2×C4⋊C4—61no—
178C2×C4⋊C4D1211yescharacteristic
179C2×C22⋊C4—21no—
180C2×C22⋊C4—21no—
181C2×C22⋊C4—31no—
182C2×C22⋊C4—31no—
183C2×C22⋊C4—61no—
184C2×C22⋊C4—61no—
185C2×C42—31no—
186C2×C42—31no—
187C2×C42—61no—
188C2×C42—61no—
189C2×C42—61no—
190#107—62no—
191#106—62no—
192#103—62no—
193#103—62no—
194C22⋊C8—21no—
195C22⋊C8—21no—
196C22⋊C8—32no—
197C22⋊C8—61no—
198#95—21no—
199#95—21no—
200#95—61no—
201#95—61no—
202#95—61no—
203#95—61no—
204#95—61no—
205#95—61no—
206#95—61no—
207C23×C3—21no—
208C23×C3—21no—
209C23×C3—41no—
210C23×C3—41no—
211C23×C3C4×D411yescharacteristic
212C23×C3#12011yescharacteristic
213C23×C3#12111yescharacteristic
214C3×Q8—42no—
215C3×Q8—81no—
216C2×C3×C4—21no—
217C2×C3×C4—21no—
218C2×C3×C4—21no—
219C2×C3×C4—21no—
220C2×C3×C4—22no—
221C2×C3×C4—22no—
222C2×C3×C4—41no—
223C2×C3×C4—41no—
224C2×C3×C4—41no—
225C2×C3×C4—41no—
226C2×C3×C4—41no—
227C2×C3×C4—41no—
228C2×C3×C4—41no—
229C2×C3×C4—41no—
230C2×C3×C4—81no—
231C2×C3×C4—81no—
232C2×C3×C4—81no—
233C2×C3×C4C2×C22⋊C411yescharacteristic
234C2×C3×C4C4×D411yescharacteristic
235C2×C3×C4#12111yescharacteristic
236C2×C3×C4#12311yescharacteristic
237C2×Dic3—22no—
238C2×Dic3—41no—
239C2×Dic3—41no—
240C2×Dic3—41no—
241C2×Dic3—41no—
242C2×Dic3—41no—
243C2×Dic3—41no—
244C2×Dic3—41no—
245C2×Dic3—41no—
246C2×Dic3—41no—
247C2×Dic3—41no—
248C2×Dic3—41no—
249C2×Dic3—41no—
250C2×Dic3—41no—
251C2×Dic3—81no—
252C2×Dic3—81no—
253C3×C8—81no—
254C3⋊C8—42no—
255C24D3×D411yescharacteristic
256C2×Q8—21no—
257C2×Q8—41no—
258C2×Q8—41no—
259C2×Q8(C2×D3)⋊C412yes—
260C22×C4—21no—
261C22×C4—21no—
262C22×C4—41no—
263C22×C4—41no—
264C22×C4—41no—
265C22×C4—61no—
266C22×C4—61no—
267C22×C4—61no—
268C22×C4—61no—
269C22×C4—61no—
270C22×C4—61no—
271C22×C4C2×C4×D311yescharacteristic
272C22×C4C2×D1211yescharacteristic
273C22×C4D3×D411yescharacteristic
274C22×C4#23711yescharacteristic
275C22×C4#23711yescharacteristic
276C22×C4C2×#6711yescharacteristic
277M16—41no—
278M16—41no—
279M16—121no—
280M16—121no—
281C2×C8—41no—
282C2×C8—62no—
283C4⋊C4—22no—
284C4⋊C4—21no—
285C4⋊C4—22no—
286C4⋊C4—61no—
287C4⋊C4—61no—
288C4⋊C4—61no—
289C4⋊C4—61no—
290C4⋊C4—122no—
291C4⋊C4—241no—
292C4⋊C4(C2×D3)⋊C412yes—
293C22⋊C4—22no—
294C22⋊C4—22no—
295C22⋊C4—41no—
296C22⋊C4—41no—
297C22⋊C4—122no—
298C22⋊C4—121no—
299C22⋊C4—121no—
300C22⋊C4—121no—
301C22⋊C4—121no—
302C42—121no—
303C42—121no—
304C42—122no—
305C42—121no—
306C42—121no—
307C42—241no—
308C22×C3—21no—
309C22×C3—21no—
310C22×C3—21no—
311C22×C3—21no—
312C22×C3—21no—
313C22×C3—21no—
314C22×C3—21no—
315C22×C3—21no—
316C22×C3—81no—
317C22×C3—81no—
318C22×C3#38611yescharacteristic
319C22×C3#45811yescharacteristic
320C22×C3#45811yescharacteristic
321C3×C4—21no—
322C3×C4—21no—
323C3×C4—21no—
324C3×C4—21no—
325C3×C4—42no—
326C3×C4—42no—
327C3×C4—161no—
328Dic3—42no—
329Dic3—81no—
330Dic3—81no—
331Dic3—81no—
332Dic3—81no—
333Dic3—81no—
334C23—21no—
335C23—21no—
336C23—41no—
337C23—41no—
338C23#67⋊C411yescharacteristic
339C23#88211yescharacteristic
340C23#93911yescharacteristic
341Q8—42no—
342Q8—81no—
343C2×C4—21no—
344C2×C4—21no—
345C2×C4—21no—
346C2×C4—21no—
347C2×C4—22no—
348C2×C4—22no—
349C2×C4—41no—
350C2×C4—41no—
351C2×C4—41no—
352C2×C4—41no—
353C2×C4—41no—
354C2×C4—41no—
355C2×C4—41no—
356C2×C4—41no—
357C2×C4—62no—
358C2×C4—81no—
359C2×C4—81no—
360C2×C4—81no—
361C2×C4—121no—
362C2×C4—121no—
363C2×C4—121no—
364C2×C4—121no—
365C2×C4—121no—
366C2×C4—121no—
367C2×C4—121no—
368C2×C4—121no—
369C2×C4—121no—
370C2×C4—121no—
371C2×C4—121no—
372C2×C4—121no—
373C2×C4—121no—
374C2×C4—241no—
375C2×C4—241no—
376C2×C4#67⋊C411yescharacteristic
377C2×C4#88611yescharacteristic
378C2×C4C2×(C2×D3)⋊C411yescharacteristic
379C2×C4#93911yescharacteristic
380C8—81no—
381C8—122no—
382C2×C3—21no—
383C2×C3—21no—
384C2×C3—41no—
385C2×C3—41no—
386C2×C3#189011yescharacteristic
387C2×C3#190511yescharacteristic
388C2×C3#190611yescharacteristic
389C22—21no—
390C22—21no—
391C22—21no—
392C22—21no—
393C22—21no—
394C22—21no—
395C22—21no—
396C22—21no—
397C22—81no—
398C22—81no—
399C22#474211yescharacteristic
400C22#474211yescharacteristic
401C22#495811yescharacteristic
402C4—21no—
403C4—21no—
404C4—21no—
405C4—21no—
406C4—42no—
407C4—42no—
408C4—122no—
409C4—161no—
410C4—241no—
411C4—241no—
412C4—241no—
413C4—241no—
414C4—241no—
415C3#952611yescharacteristic, complemented, Sylow
416C2—21no—
417C2—21no—
418C2—41no—
419C2—41no—
420C2#7002311yescharacteristic
421C2#7033411yescharacteristic
422C2#7039511yescharacteristic
subgroupG/H
centerC22#4958
commutatorC22×C3×C4C22×C4
FrattiniC23×C4C22×D3
FittingC3×#1511C2
socleC22×C3#386
prime radicalC24×C3C2×D4
cosocleC23×C3×C4C23
hypercenter#1511D3
hypocenterC3#9526
LCSG ▷ C22×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1511

Complements

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

NH ≅ G/Ntype
19. #5044C4semidirect
20. #5043C4semidirect
415. C3#9526semidirect

Maximal subgroups

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

subgroupindex
#827462
#827272
C3×#15112
#699752
#695342
#676862
#676282
#95263