Finite Group Databasebeta

← #10697471

Subgroups — #10697471

Order 768.

488 conjugacy classes of subgroups; only the 97 normal subgroups are shown.

normal subgroups only

This group has 488 conjugacy classes of subgroups — too many to draw. Only its 97 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×#3226C211yesmaximal, characteristic
2#83282C211yesmaximal, characteristic
3#83223C211yesmaximal, characteristic, complemented
4#71814C211yesmaximal, characteristic
5#71786C211yesmaximal, characteristic, complemented
6#70793C211yesmaximal, characteristic, complemented
7#67741C211yesmaximal, characteristic, complemented
8#10226—31nomaximal, Sylow
9C3×#532C2211yescharacteristic
10C2×C3×C4⋊Q8C2211yescharacteristic
11#5470C2211yescharacteristic
12C3×#500—21no—
13C3×#486—21no—
14C3×#480—22no—
15C3×#474—22no—
16C3×#431C2211yescharacteristic
17#5233—22no—
18C2×#835—22no—
19#5094—22no—
20#5093—21no—
21#5082—21no—
22#5080—21no—
23#5071—21no—
24#5055—21no—
25#5055—21no—
26#4971—22no—
27#4936C2211yescharacteristic
28#4925C2211yescharacteristic
29C2×#803C2211yescharacteristic
30#4537—22no—
31#4467—22no—
32#3229—31no—
33#3226D311yescharacteristic
34#3135—31no—
35#1960—31no—
36#1518—31no—
37#1517—31no—
38#1281—31no—
39C2×C3×#41—21no—
40C22×C3×Q8—21no—
41C3×C4⋊Q8—21no—
42C3×C4⋊Q8—21no—
43C3×C4⋊Q8—21no—
44C3×C4⋊Q8D411yescharacteristic
45C3×C4⋊Q8D412yes—
46C3×#127D411yescharacteristic
47C3×#124—21no—
48C3×#121—21no—
49C3×#121—21no—
50C3×C4×D4—21no—
51C3×C4×D4—21no—
52C2×C3×C4⋊C4—21no—
53C2×C3×C4⋊C4—21no—
54C2×C3×C42C2311yescharacteristic
55#926—21no—
56C2×C4⋊Dic3—21no—
57#924—21no—
58#924—21no—
59C22×C3⋊C8D412yes—
60#870—21no—
61#869D411yescharacteristic
62#869D411yescharacteristic
63C4×Dic6—21no—
64C4×Dic6—21no—
65C3×#107—24no—
66C3×C4⋊C8—21no—
67C3×#102—24no—
68C3×C22⋊C8D412yes—
69C3×C4×C8—21no—
70#835—22no—
71#835—22no—
72#835—44no—
73#832—22no—
74#832—22no—
75C3⋊#95—21no—
76C3⋊#95—21no—
77#830D412yes—
78#810—24no—
79#808—24no—
80C4⋊(C3⋊C8)—21no—
81#803—21no—
82#803—21no—
83C4×C3⋊C8—21no—
84#533—31no—
85#532C2×D311yescharacteristic
86C2×C4⋊Q8C2×D311yescharacteristic
87#500—21no—
88#498—61no—
89#491—61no—
90#490—61no—
91#489—61no—
92#489—61no—
93#487—61no—
94#486—21no—
95#483—62no—
96#480—22no—
97#475—62no—
98#474—22no—
99#431—31no—
100#431C2×D311yescharacteristic
101#416—62no—
102C2×#103—62no—
103C2×#97—31no—
104#382—31no—
105#328—62no—
106#327—62no—
107C3×#41—42no—
108C3×#41—42no—
109C2×C3×Q8—21no—
110C2×C3×Q8—22no—
111C2×C3×Q8—41no—
112C2×C3×Q8—42no—
113C2×C3×Q8—42no—
114C2×C3×D4—21no—
115C2×C3×D4—21no—
116C2×C3×D4—21no—
117C2×C3×D4—41no—
118C22×C3×C4—41no—
119C22×C3×C4—41no—
120C22×C3×C4C2×D411yescharacteristic
121C22×C3×C4C2×D411yescharacteristic
122C22×C3×C4C2×D411yescharacteristic
123C22×Dic3—41no—
124C2×Dic6—21no—
125C2×Dic6—21no—
126C2×C3×C8—22no—
127C3×C4⋊C4—22no—
128C3×C4⋊C4—22no—
129C3×C4⋊C4—21no—
130C3×C4⋊C4—21no—
131C3×C4⋊C4—41no—
132C3×C4⋊C4—41no—
133C3×C22⋊C4—41no—
134C3×C22⋊C4—41no—
135C3×C42C2×D411yescharacteristic
136C3×C42C2×D411yescharacteristic
137C3×C42C2×D411yescharacteristic
138C3×C42#4111yescharacteristic
139#217—41no—
140C4⋊Dic3—21no—
141C4⋊Dic3—21no—
142C4⋊Dic3—21no—
143C4⋊Dic3—21no—
144C4⋊Dic3—42no—
145Dic3⋊C4—41no—
146Dic3⋊C4—41no—
147C4×Dic3—41no—
148C2×C3⋊C8—22no—
149C2×C3⋊C8—22no—
150C2×C3⋊C8—22no—
151C2×C3⋊C8—22no—
152C2×#41—21no—
153C22×Q8—21no—
154C22×C8—32no—
155C4⋊Q8—21no—
156C4⋊Q8—21no—
157C4⋊Q8—21no—
158C4⋊Q8—31no—
159C4⋊Q8—31no—
160C4⋊Q8#6712yes—
161C4⋊Q8#6711yescharacteristic
162#127#6711yescharacteristic
163#125—61no—
164#124—21no—
165C22⋊Q8—61no—
166C22⋊Q8—61no—
167#121—21no—
168#121—21no—
169C4×Q8—61no—
170C4×Q8—61no—
171C4×D4—21no—
172C4×D4—21no—
173#117—61no—
174C2×C4⋊C4—21no—
175C2×C4⋊C4—21no—
176C2×C4⋊C4—61no—
177C2×C42C22×D311yescharacteristic
178#107—24no—
179#106—64no—
180C4⋊C8—21no—
181C4⋊C8—61no—
182#103—64no—
183#103—62no—
184#103—62no—
185#103—124no—
186#102—24no—
187#102—62no—
188#102—62no—
189C22⋊C8—32no—
190C22⋊C8#6712yes—
191#97—61no—
192#97—61no—
193C4×C8—21no—
194C4×C8—61no—
195#95—61no—
196#95—61no—
197C23×C3—41no—
198C23×C3#12711yescharacteristic
199C3×Q8—42no—
200C3×Q8—44no—
201C3×Q8—82no—
202C3×D4—42no—
203C3×D4—42no—
204C3×D4—81no—
205C3×D4—81no—
206C3×D4—81no—
207C2×C3×C4—41no—
208C2×C3×C4—41no—
209C2×C3×C4—41no—
210C2×C3×C4—42no—
211C2×C3×C4—81no—
212C2×C3×C4—81no—
213C2×C3×C4—82no—
214C2×C3×C4#12011yescharacteristic
215C2×C3×C4#12011yescharacteristic
216C2×C3×C4#12011yescharacteristic
217C2×C3×C4#12111yescharacteristic
218C2×C3×C4#12111yescharacteristic
219C2×C3×C4#12111yescharacteristic
220C2×C3×C4#13512yes—
221C2×C3×C4AGL(1,ℤ/8)12yes—
222C2×Dic3—41no—
223C2×Dic3—41no—
224C2×Dic3—81no—
225C2×Dic3—81no—
226C2×Dic3—81no—
227Dic6—42no—
228Dic6—81no—
229C3×C8—42no—
230C3⋊C8—42no—
231C3⋊C8—42no—
232#41—42no—
233#41—42no—
234C2×Q8—21no—
235C2×Q8—22no—
236C2×Q8—41no—
237C2×Q8—42no—
238C2×Q8—42no—
239C2×Q8—61no—
240C2×Q8—61no—
241C2×D4—21no—
242C2×D4—21no—
243C2×D4—21no—
244C2×D4—41no—
245C22×C4—41no—
246C22×C4—41no—
247C22×C4—121no—
248C22×C4D3×D411yescharacteristic
249C22×C4D3×D411yescharacteristic
250C22×C4C2×#6711yescharacteristic
251C2×C8—22no—
252C2×C8—62no—
253C2×C8—62no—
254C2×C8—62no—
255C2×C8—62no—
256C4⋊C4—22no—
257C4⋊C4—21no—
258C4⋊C4—22no—
259C4⋊C4—21no—
260C4⋊C4—41no—
261C4⋊C4—41no—
262C4⋊C4—61no—
263C4⋊C4—61no—
264C4⋊C4—61no—
265C4⋊C4—61no—
266C4⋊C4—121no—
267C4⋊C4—121no—
268C4⋊C4—122no—
269C22⋊C4—41no—
270C22⋊C4—41no—
271C22⋊C4—121no—
272C42—121no—
273C42D3×D411yescharacteristic
274C42#23711yescharacteristic
275C42C2×#6711yescharacteristic
276C42C2×#6711yescharacteristic
277C22×C3—21no—
278C22×C3—21no—
279C22×C3—81no—
280C22×C3—81no—
281C22×C3—81no—
282C22×C3#39211yescharacteristic
283C22×C3#49511yescharacteristic
284C22×C3#49611yescharacteristic
285C3×C4—21no—
286C3×C4—21no—
287C3×C4—21no—
288C3×C4—21no—
289C3×C4—81no—
290C3×C4—82no—
291C3×C4#44912yes—
292C3×C4#46312yes—
293Dic3—81no—
294Dic3—81no—
295Dic3—161no—
296C23—41no—
297C23#94011yescharacteristic
298Q8—44no—
299Q8—42no—
300Q8—82no—
301Q8—122no—
302Q8—241no—
303D4—42no—
304D4—42no—
305D4—81no—
306D4—81no—
307D4—81no—
308C2×C4—41no—
309C2×C4—41no—
310C2×C4—41no—
311C2×C4—42no—
312C2×C4—81no—
313C2×C4—81no—
314C2×C4—82no—
315C2×C4—121no—
316C2×C4—121no—
317C2×C4—241no—
318C2×C4—241no—
319C2×C4—241no—
320C2×C4#91112yes—
321C2×C4#91212yes—
322C2×C4#93711yescharacteristic
323C2×C4#93711yescharacteristic
324C2×C4#93811yescharacteristic
325C2×C4#93911yescharacteristic
326C2×C4#93911yescharacteristic
327C2×C4#95311yescharacteristic
328C8—42no—
329C8—122no—
330C8—122no—
331C2×C3—21no—
332C2×C3—21no—
333C2×C3—161no—
334C2×C3#200911yescharacteristic
335C2×C3#200911yescharacteristic
336C2×C3#201211yescharacteristic
337C22—21no—
338C22—21no—
339C22—81no—
340C22—81no—
341C22—81no—
342C22#515411yescharacteristic
343C22#515511yescharacteristic
344C22#522311yescharacteristic
345C4—21no—
346C4—21no—
347C4—21no—
348C4—21no—
349C4—81no—
350C4—82no—
351C4—241no—
352C4—241no—
353C4—481no—
354C4#515312yes—
355C4#515812yes—
356C3#1022611yescharacteristic, complemented, Sylow
357C2—21no—
358C2—21no—
359C2—161no—
360C2#7186911yescharacteristic
361C2#7204511yescharacteristic
362C2#7211111yescharacteristic
subgroupG/H
centerC22#5223
commutator, cosocleC2×C3×C42C23
FrattiniC2×C42C22×D3
FittingC3×#3226C2
socleC22×C3#392
prime radicalC3×#532C22
hypercenter#3226D3
hypocenterC3#10226
LCSG ▷ C2×C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3226

Complements

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

NH ≅ G/Ntype
3. #83223C2semidirect
5. #71786C2semidirect
6. #70793C2semidirect
7. #67741C2semidirect
356. C3#10226semidirect

Maximal subgroups

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

subgroupindex
C3×#32262
#832822
#832232
#718142
#717862
#707932
#677412
#102263