Finite Group Databasebeta

← #10696713

Subgroups — #10696713

Order 768.

488 conjugacy classes of subgroups; only the 87 normal subgroups are shown.

normal subgroups only

This group has 488 conjugacy classes of subgroups — too many to draw. Only its 87 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#83321C211yesmaximal, characteristic
2#83317C211yesmaximal, characteristic
3#78537C211yesmaximal, characteristic, complemented
4C3×#1525C211yesmaximal, characteristic
5#69879C211yesmaximal, characteristic, complemented
6#69856C211yesmaximal, characteristic, complemented
7#67669C211yesmaximal, characteristic, complemented
8#10114—31nomaximal, Sylow
9C3×#532C2211yescharacteristic
10#5470C2211yescharacteristic
11C2×#869C2211yescharacteristic
12C3×C4×M16C2211yescharacteristic
13C3×C4×M16C2211yescharacteristic
14#5120—22no—
15#5093—21no—
16#5091—21no—
17#5076—21no—
18#5075—21no—
19#5042—22no—
20#5040—22no—
21#5019—21no—
22#5019—22no—
23#5007—21no—
24#4925C2211yescharacteristic
25#4924C2211yescharacteristic
26#4703—22no—
27#4681—21no—
28#4681—21no—
29C3×#329—22no—
30#4554—22no—
31#4456—22no—
32#3254—31no—
33#3206—31no—
34#3139—31no—
35#1997—31no—
36#1525D311yescharacteristic
37#1485—31no—
38#1299—31no—
39C2×C3×#41—21no—
40C22×Dic6D412yes—
41C2×C3×M16—21no—
42C2×C3×M16—21no—
43C3×C4⋊Q8D411yescharacteristic
44C3×#127D411yescharacteristic
45C3×#124—21no—
46C3×#121—21no—
47C3×#121—21no—
48C3×C4×D4—21no—
49C3×C4×D4—21no—
50C2×C3×C42C2311yescharacteristic
51C2×#214—24no—
52C2×#214—22no—
53#926—21no—
54C2×C4⋊Dic3—21no—
55C2×C4⋊Dic3—21no—
56#924—21no—
57#924—21no—
58C2×#208D412yes—
59#870—21no—
60#869—21no—
61#869—21no—
62#869—21no—
63#869—21no—
64#869D411yescharacteristic
65#869D411yescharacteristic
66C4×Dic6—21no—
67C4×Dic6—21no—
68C3×#97—21no—
69C3×#97D412yes—
70C3×C4×C8—21no—
71C3×C4×C8—21no—
72#832—22no—
73#830—21no—
74#816—24no—
75#816—24no—
76#810—24no—
77#810—22no—
78#807—22no—
79C4⋊(C3⋊C8)—21no—
80C4⋊(C3⋊C8)—21no—
81C4⋊(C3⋊C8)D412yes—
82C4×C3⋊C8—21no—
83#533—31no—
84#532C2×D311yescharacteristic
85C2×C4⋊Q8—31no—
86#492—61no—
87#491—62no—
88#487—61no—
89#487—61no—
90#487—61no—
91#477—61no—
92#474—61no—
93#468—62no—
94#461—62no—
95#461—61no—
96#460—61no—
97#448—62no—
98#431—31no—
99#423—31no—
100#419—62no—
101C4×M16C2×D311yescharacteristic
102C4×M16C2×D311yescharacteristic
103#343—62no—
104#329—22no—
105Q8⋊C8—62no—
106C3×#41—82no—
107C2×C3×Q8—21no—
108C2×C3×D4—21no—
109C2×C3×D4—21no—
110C2×C3×D4—21no—
111C2×C3×D4—41no—
112C22×C3×C4—41no—
113C22×C3×C4C2×D411yescharacteristic
114C22×C3×C4C2×D411yescharacteristic
115C22×C3×C4#4111yescharacteristic
116C22×Dic3—22no—
117C2×Dic6—21no—
118C2×Dic6—22no—
119C2×Dic6—22no—
120C2×Dic6—21no—
121C2×Dic6—44no—
122C3×M16—22no—
123C3×M16—42no—
124C3×M16—41no—
125C2×C3×C8—22no—
126C2×C3×C8—41no—
127C3×C4⋊C4—21no—
128C3×C4⋊C4—21no—
129C3×C22⋊C4—41no—
130C3×C22⋊C4—41no—
131C3×C42C2×D411yescharacteristic
132C3×C42C2×D411yescharacteristic
133C3×C42C2×D411yescharacteristic
134C3×C42C2×D411yescharacteristic
135#217—41no—
136#214—82no—
137#214—84no—
138C4⋊Dic3—21no—
139C4⋊Dic3—21no—
140C4⋊Dic3—21no—
141C4⋊Dic3—21no—
142C4⋊Dic3—41no—
143C4⋊Dic3—41no—
144C4⋊Dic3—41no—
145C4⋊Dic3—41no—
146Dic3⋊C4—41no—
147Dic3⋊C4—41no—
148C4×Dic3—41no—
149#208—42no—
150C2×C3⋊C8—22no—
151C2×C3⋊C8—22no—
152C2×#41—21no—
153C22×Q8—32no—
154C2×SD16—64no—
155C2×SD16—62no—
156C2×M16—21no—
157C2×M16—21no—
158C2×M16—32no—
159C4⋊Q8—31no—
160C4⋊Q8—31no—
161C4⋊Q8—61no—
162C4⋊Q8—61no—
163C4⋊Q8—61no—
164C4⋊Q8—61no—
165C4⋊Q8D1211yescharacteristic
166#127D1211yescharacteristic
167#125—61no—
168#124—21no—
169C22⋊Q8—61no—
170C22⋊Q8—61no—
171#121—21no—
172#121—21no—
173C4×Q8—61no—
174C4×Q8—61no—
175C4×D4—21no—
176C4×D4—21no—
177#117—61no—
178C2×C4⋊C4—61no—
179C2×C4⋊C4—61no—
180C2×C42C22×D311yescharacteristic
181#107—62no—
182C4⋊C8—32no—
183C4⋊C8—61no—
184C4⋊C8—61no—
185#103—64no—
186#103—62no—
187#103—64no—
188#103—64no—
189#102—62no—
190C22⋊C8—61no—
191#97—21no—
192#97#6712yes—
193C4×C8—21no—
194C4×C8—21no—
195C4×C8—61no—
196C23×C3—41no—
197C23×C3#12111yescharacteristic
198C3×Q8—81no—
199C3×D4—42no—
200C3×D4—41no—
201C3×D4—41no—
202C3×D4—81no—
203C3×D4—82no—
204C2×C3×C4—21no—
205C2×C3×C4—22no—
206C2×C3×C4—21no—
207C2×C3×C4—22no—
208C2×C3×C4—41no—
209C2×C3×C4—82no—
210C2×C3×C4#12011yescharacteristic
211C2×C3×C4#12011yescharacteristic
212C2×C3×C4#12011yescharacteristic
213C2×C3×C4#12111yescharacteristic
214C2×C3×C4#12111yescharacteristic
215C2×C3×C4#12711yescharacteristic
216C2×Dic3—41no—
217C2×Dic3—41no—
218C2×Dic3—42no—
219C2×Dic3—82no—
220Dic6—44no—
221Dic6—41no—
222Dic6—44no—
223Dic6—41no—
224Dic6—42no—
225C3×C8—42no—
226C3×C8—42no—
227C3⋊C8—42no—
228C3⋊C8—42no—
229#41—82no—
230C2×Q8—21no—
231C2×Q8—61no—
232C2×Q8—62no—
233C2×Q8—61no—
234C2×Q8—62no—
235C2×Q8—124no—
236C2×D4—21no—
237C2×D4—21no—
238C2×D4—21no—
239C2×D4—41no—
240C22×C4—41no—
241C22×C4—62no—
242C22×C4#23511yescharacteristic
243C22×C4D3×D411yescharacteristic
244C22×C4D3×D411yescharacteristic
245SD16—242no—
246SD16—244no—
247M16—22no—
248M16—42no—
249M16—41no—
250M16—122no—
251C2×C8—22no—
252C2×C8—41no—
253C2×C8—62no—
254C2×C8—62no—
255C4⋊C4—21no—
256C4⋊C4—21no—
257C4⋊C4—61no—
258C4⋊C4—61no—
259C4⋊C4—61no—
260C4⋊C4—61no—
261C4⋊C4—121no—
262C4⋊C4—121no—
263C4⋊C4—121no—
264C4⋊C4—121no—
265C4⋊C4—121no—
266C4⋊C4—121no—
267C22⋊C4—41no—
268C22⋊C4—41no—
269C22⋊C4—121no—
270C42—121no—
271C42C2×D1211yescharacteristic
272C42D3×D411yescharacteristic
273C42D3×D411yescharacteristic
274C42C2×#6711yescharacteristic
275C22×C3—21no—
276C22×C3—21no—
277C22×C3—21no—
278C22×C3—81no—
279C22×C3—81no—
280C22×C3—81no—
281C22×C3#39211yescharacteristic
282C3×C4—21no—
283C3×C4—21no—
284C3×C4—21no—
285C3×C4—81no—
286C3×C4#45512yes—
287C3×C4#45511yescharacteristic
288C3×C4#45611yescharacteristic
289C3×C4#47112yes—
290Dic3—82no—
291Dic3—81no—
292Dic3—81no—
293C23—41no—
294C23#88411yescharacteristic
295Q8—81no—
296Q8—124no—
297Q8—122no—
298Q8—121no—
299Q8—124no—
300Q8—121no—
301D4—42no—
302D4—41no—
303D4—41no—
304D4—81no—
305D4—82no—
306C2×C4—22no—
307C2×C4—21no—
308C2×C4—22no—
309C2×C4—21no—
310C2×C4—41no—
311C2×C4—82no—
312C2×C4—121no—
313C2×C4—121no—
314C2×C4—122no—
315C2×C4—242no—
316C2×C4#88211yescharacteristic
317C2×C4#88211yescharacteristic
318C2×C4#88411yescharacteristic
319C2×C4#93011yescharacteristic
320C2×C4#93711yescharacteristic
321C2×C4#94011yescharacteristic
322C8—42no—
323C8—42no—
324C8—122no—
325C8—122no—
326C2×C3—41no—
327C2×C3—161no—
328C2×C3#200411yescharacteristic
329C2×C3#200711yescharacteristic
330C2×C3#201911yescharacteristic
331C22—21no—
332C22—21no—
333C22—21no—
334C22—81no—
335C22—81no—
336C22—81no—
337C22#496011yescharacteristic
338C4—21no—
339C4—21no—
340C4—21no—
341C4—81no—
342C4—241no—
343C4—242no—
344C4—241no—
345C4#474812yes—
346C4#482512yes—
347C4#482511yescharacteristic
348C4#482611yescharacteristic
349C3#1011411yescharacteristic, complemented, Sylow
350C2—41no—
351C2—161no—
352C2#7009111yescharacteristic
353C2#7038611yescharacteristic
354C2#7039311yescharacteristic
subgroupG/H
centerC22#4960
commutator, cosocleC2×C3×C42C23
FrattiniC2×C42C22×D3
FittingC3×#1525C2
socleC22×C3#392
prime radicalC3×#532C22
hypercenter#1525D3
hypocenterC3#10114
LCSG ▷ C2×C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1525

Complements

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

NH ≅ G/Ntype
3. #78537C2semidirect
5. #69879C2semidirect
6. #69856C2semidirect
7. #67669C2semidirect
349. C3#10114semidirect

Maximal subgroups

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

subgroupindex
#833212
#833172
#785372
C3×#15252
#698792
#698562
#676692
#101143