Finite Group Databasebeta

← #10692241

Subgroups — #10692241

Order 768.

444 conjugacy classes of subgroups; only the 99 normal subgroups are shown.

normal subgroups only

This group has 444 conjugacy classes of subgroups — too many to draw. Only its 99 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#82220C211yesmaximal, characteristic
2#82210C211yesmaximal, characteristic
3C3×#1825C211yesmaximal, characteristic
4#69923C211yesmaximal, characteristic
5#69510C211yesmaximal, characteristic
6#67680C211yesmaximal, characteristic
7#67626C211yesmaximal, characteristic
8#9346—31nomaximal, Sylow
9C2×C3×#117C2211yescharacteristic
10C2×#924C2211yescharacteristic
11C2×C3×C22⋊C8C2211yescharacteristic
12C3×#380C2211yescharacteristic
13#5207C2211yescharacteristic
14#5207C2211yescharacteristic
15C2×#830C2211yescharacteristic
16#5004C411yescharacteristic, complemented
17#5004C411yescharacteristic, complemented
18#5000—22no—
19#4999—21no—
20#4997—21no—
21#4996—21no—
22#4996—21no—
23#4995C411yescharacteristic, complemented
24#4995C411yescharacteristic, complemented
25#4971—21no—
26#4970—21no—
27#4926—21no—
28#4925—21no—
29C3×#340—21no—
30C3×#340—21no—
31#4469—22no—
32#4452—21no—
33#4452—21no—
34#3095—31no—
35#2924—31no—
36#1825D311yescharacteristic
37#1520—31no—
38#1472—31no—
39#1305—31no—
40#1298—31no—
41C23×C3×C4C2311yescharacteristic
42C22×Dic6C2×C411yescharacteristic
43C22×C3×C8—21no—
44C3×#117—21no—
45C3×#117—21no—
46C3×#117C2×C411yescharacteristic
47C3×#117C2×C411yescharacteristic
48C3×#117D411yescharacteristic
49C3×#117D411yescharacteristic
50C2×C3×C4⋊C4—21no—
51C2×C3×C4⋊C4D411yescharacteristic
52C2×C3×C4⋊C4D411yescharacteristic
53C2×C3×C22⋊C4—21no—
54C2×C3×C22⋊C4—21no—
55C2×C3×C42—21no—
56C2×#217—21no—
57C2×C4⋊Dic3C2×C411yescharacteristic
58#924—21no—
59#924—21no—
60#924—21no—
61#924—21no—
62C2×Dic3⋊C4—21no—
63C2×#208—21no—
64C2×#208C2×C411yescharacteristic
65C2×#208C2×C411yescharacteristic
66C22×C3⋊C8—21no—
67C3×C22⋊C8—21no—
68C3×C22⋊C8—21no—
69C3×#95—21no—
70#830—21no—
71#830—21no—
72#830—21no—
73#830—21no—
74#830D411yescharacteristic
75#830D411yescharacteristic
76#830D412yes—
77#810—21no—
78#810—21no—
79#810—21no—
80#810—21no—
81#808—21no—
82#808—21no—
83#807—21no—
84#807—21no—
85C4⋊(C3⋊C8)—22no—
86C4⋊(C3⋊C8)—22no—
87#803—22no—
88C4×C3⋊C8—22no—
89C2×C22⋊Q8—31no—
90C2×#117C2×D311yescharacteristic
91#485—61no—
92#485—61no—
93#484—61no—
94#483—61no—
95#433—62no—
96#432—61no—
97#431—61no—
98#428—31no—
99#428—31no—
100#424—61no—
101#419—31no—
102#419—31no—
103#407—31no—
104#407—31no—
105C2×C22⋊C8—31no—
106C2×C22⋊C8C2×D311yescharacteristic
107#405—61no—
108#380C2×D311yescharacteristic
109#340—21no—
110#340—21no—
111#328—61no—
112#328—61no—
113#324—62no—
114C24×C3#4111yescharacteristic
115C22×C3×C4—21no—
116C22×C3×C4—21no—
117C22×C3×C4—41no—
118C22×C3×C4—41no—
119C22×C3×C4—41no—
120C22×C3×C4C22×C411yescharacteristic
121C22×C3×C4C2×D411yescharacteristic
122C22×C3×C4C2×D411yescharacteristic
123C22×C3×C4C2×D411yescharacteristic
124C22×C3×C4C2×D411yescharacteristic
125C22×C3×C4#4111yescharacteristic
126C22×Dic3—21no—
127C22×Dic3—21no—
128C2×Dic6—21no—
129C2×Dic6—21no—
130C2×Dic6—41no—
131C2×Dic6—41no—
132C2×C3×C8—41no—
133C2×C3×C8—41no—
134C2×C3×C8—41no—
135C3×C4⋊C4—21no—
136C3×C4⋊C4—21no—
137C3×C4⋊C4—41no—
138C3×C4⋊C4—41no—
139C3×C4⋊C4C22⋊C411yescharacteristic
140C3×C4⋊C4C22⋊C411yescharacteristic
141C3×C4⋊C4C22⋊C411yescharacteristic
142C3×C4⋊C4C22⋊C411yescharacteristic
143C3×C22⋊C4—21no—
144C3×C22⋊C4—21no—
145C3×C22⋊C4—21no—
146C3×C22⋊C4—21no—
147C3×C22⋊C4—81no—
148C3×C42—21no—
149C3×C42—21no—
150C3×C42—21no—
151C3×C42—21no—
152#217—41no—
153#217—41no—
154C4⋊Dic3—21no—
155C4⋊Dic3—21no—
156Dic3⋊C4—41no—
157Dic3⋊C4—41no—
158#208—41no—
159#208—41no—
160#208—41no—
161#208—41no—
162C2×C3⋊C8—21no—
163C2×C3⋊C8—21no—
164C2×C3⋊C8—21no—
165C2×C3⋊C8—22no—
166C2×C3⋊C8—21no—
167C2×C3⋊C8—41no—
168C2×C3⋊C8—41no—
169C22×Q8—31no—
170C23×C4C22×D311yescharacteristic
171C2×M16—31no—
172C2×M16—31no—
173C2×M16—61no—
174C22×C8—21no—
175C22×C8—61no—
176C22⋊Q8—61no—
177C22⋊Q8—61no—
178C22⋊Q8—61no—
179C22⋊Q8—61no—
180#117—21no—
181#117—21no—
182#117C4×D311yescharacteristic
183#117C4×D311yescharacteristic
184#117D1211yescharacteristic
185#117D1211yescharacteristic
186C2×C4⋊C4—21no—
187C2×C4⋊C4—31no—
188C2×C4⋊C4—61no—
189C2×C4⋊C4#6711yescharacteristic
190C2×C4⋊C4#6711yescharacteristic
191C2×C22⋊C4—21no—
192C2×C22⋊C4—21no—
193C2×C22⋊C4—61no—
194C2×C42—21no—
195#107—61no—
196#107—61no—
197#106—61no—
198#106—61no—
199C4⋊C8—62no—
200C4⋊C8—62no—
201#103—61no—
202#103—61no—
203#103—61no—
204#103—61no—
205C22⋊C8—21no—
206C22⋊C8—21no—
207C22⋊C8—32no—
208C22⋊C8—31no—
209C22⋊C8—31no—
210C22⋊C8—61no—
211C22⋊C8—61no—
212C22⋊C8—61no—
213C22⋊C8—61no—
214#97—62no—
215C4×C8—62no—
216#95—21no—
217C23×C3—21no—
218C23×C3—21no—
219C23×C3—41no—
220C23×C3—41no—
221C23×C3C4×D411yescharacteristic
222C23×C3#12111yescharacteristic
223C23×C3#12311yescharacteristic
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—21no—
231C2×C3×C4—21no—
232C2×C3×C4—21no—
233C2×C3×C4—21no—
234C2×C3×C4—41no—
235C2×C3×C4—41no—
236C2×C3×C4—41no—
237C2×C3×C4—41no—
238C2×C3×C4—41no—
239C2×C3×C4—81no—
240C2×C3×C4—81no—
241C2×C3×C4—81no—
242C2×C3×C4—81no—
243C2×C3×C4C2×C22⋊C411yescharacteristic
244C2×C3×C4C4×D411yescharacteristic
245C2×C3×C4#12011yescharacteristic
246C2×C3×C4#12111yescharacteristic
247C2×Dic3—42no—
248C2×Dic3—41no—
249C2×Dic3—41no—
250C2×Dic3—81no—
251Dic6—41no—
252Dic6—41no—
253Dic6—81no—
254C3×C8—81no—
255C3⋊C8—41no—
256C3⋊C8—41no—
257C3⋊C8—42no—
258C3⋊C8—41no—
259C3⋊C8—41no—
260C24#23711yescharacteristic
261C2×Q8—61no—
262C2×Q8—61no—
263C2×Q8—121no—
264C2×Q8—121no—
265C22×C4—21no—
266C22×C4—21no—
267C22×C4—41no—
268C22×C4—41no—
269C22×C4—41no—
270C22×C4—61no—
271C22×C4—61no—
272C22×C4C2×C4×D311yescharacteristic
273C22×C4C2×D1211yescharacteristic
274C22×C4D3×D411yescharacteristic
275C22×C4D3×D411yescharacteristic
276C22×C4#23711yescharacteristic
277C22×C4C2×#6711yescharacteristic
278M16—121no—
279M16—121no—
280M16—121no—
281M16—121no—
282C2×C8—41no—
283C2×C8—41no—
284C2×C8—41no—
285C2×C8—61no—
286C2×C8—62no—
287C2×C8—61no—
288C2×C8—61no—
289C2×C8—61no—
290C2×C8—121no—
291C2×C8—121no—
292C4⋊C4—21no—
293C4⋊C4—21no—
294C4⋊C4—41no—
295C4⋊C4—41no—
296C4⋊C4—61no—
297C4⋊C4—61no—
298C4⋊C4—121no—
299C4⋊C4—121no—
300C4⋊C4(C2×D3)⋊C411yescharacteristic
301C4⋊C4(C2×D3)⋊C411yescharacteristic
302C4⋊C4(C2×D3)⋊C411yescharacteristic
303C4⋊C4(C2×D3)⋊C411yescharacteristic
304C22⋊C4—21no—
305C22⋊C4—21no—
306C22⋊C4—21no—
307C22⋊C4—21no—
308C22⋊C4—81no—
309C22⋊C4—121no—
310C22⋊C4—121no—
311C42—21no—
312C42—21no—
313C42—21no—
314C42—21no—
315C22×C3—21no—
316C22×C3—21no—
317C22×C3—21no—
318C22×C3—21no—
319C22×C3—21no—
320C22×C3—21no—
321C22×C3—21no—
322C22×C3—21no—
323C22×C3—81no—
324C22×C3—81no—
325C22×C3#38611yescharacteristic
326C22×C3#47111yescharacteristic
327C22×C3#47311yescharacteristic
328C3×C4—21no—
329C3×C4—21no—
330C3×C4—21no—
331C3×C4—41no—
332C3×C4—41no—
333C3×C4—41no—
334C3×C4—41no—
335C3×C4—161no—
336C3×C4#41011yescharacteristic
337C3×C4#41011yescharacteristic
338Dic3—81no—
339Dic3—81no—
340C23—21no—
341C23—21no—
342C23—41no—
343C23—41no—
344C23#67⋊C411yescharacteristic
345C23#88611yescharacteristic
346C23#93911yescharacteristic
347Q8—121no—
348Q8—121no—
349Q8—241no—
350C2×C4—21no—
351C2×C4—21no—
352C2×C4—21no—
353C2×C4—21no—
354C2×C4—21no—
355C2×C4—21no—
356C2×C4—21no—
357C2×C4—21no—
358C2×C4—21no—
359C2×C4—21no—
360C2×C4—41no—
361C2×C4—41no—
362C2×C4—41no—
363C2×C4—41no—
364C2×C4—41no—
365C2×C4—81no—
366C2×C4—81no—
367C2×C4—81no—
368C2×C4—81no—
369C2×C4—122no—
370C2×C4—121no—
371C2×C4—121no—
372C2×C4—241no—
373C2×C4#67⋊C411yescharacteristic
374C2×C4#88211yescharacteristic
375C2×C4C2×(C2×D3)⋊C411yescharacteristic
376C2×C4#93911yescharacteristic
377C8—81no—
378C8—121no—
379C8—121no—
380C8—122no—
381C8—121no—
382C8—121no—
383C2×C3—21no—
384C2×C3—21no—
385C2×C3—41no—
386C2×C3—41no—
387C2×C3#188111yescharacteristic
388C2×C3#189911yescharacteristic
389C2×C3#190811yescharacteristic
390C22—21no—
391C22—21no—
392C22—21no—
393C22—21no—
394C22—21no—
395C22—21no—
396C22—21no—
397C22—21no—
398C22—81no—
399C22—81no—
400C22#495811yescharacteristic
401C22#520011yescharacteristic
402C22#520411yescharacteristic
403C4—21no—
404C4—21no—
405C4—21no—
406C4—41no—
407C4—41no—
408C4—41no—
409C4—41no—
410C4—161no—
411C4—241no—
412C4—241no—
413C4#500611yescharacteristic
414C4#500611yescharacteristic
415C3#934611yescharacteristic, complemented, Sylow
416C2—21no—
417C2—21no—
418C2—41no—
419C2—41no—
420C2#6953511yescharacteristic
421C2#7029311yescharacteristic
422C2#7030411yescharacteristic
subgroupG/H
centerC22#4958
commutatorC22×C3×C4C22×C4
FrattiniC23×C4C22×D3
FittingC3×#1825C2
socleC22×C3#386
prime radicalC24×C3#41
cosocleC23×C3×C4C23
hypercenter#1825D3
hypocenterC3#9346
LCSG ▷ C22×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1825

Complements

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

NH ≅ G/Ntype
16. #5004C4semidirect
17. #5004C4semidirect
23. #4995C4semidirect
24. #4995C4semidirect
415. C3#9346semidirect

Maximal subgroups

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

subgroupindex
#822202
#822102
C3×#18252
#699232
#695102
#676802
#676262
#93463