Finite Group Databasebeta

← #10696711

Subgroups — #10696711

Order 768.

488 conjugacy classes of subgroups; only the 89 normal subgroups are shown.

normal subgroups only

This group has 488 conjugacy classes of subgroups — too many to draw. Only its 89 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×#3206C211yesmaximal, characteristic
2#83321C211yesmaximal, characteristic
3#83198C211yesmaximal, characteristic, complemented
4#71797C211yesmaximal, characteristic
5#71790C211yesmaximal, characteristic, complemented
6#70769C211yesmaximal, characteristic, complemented
7#67743C211yesmaximal, characteristic, complemented
8#10114—31nomaximal, Sylow
9C3×#532C2211yescharacteristic
10C2×C3×C4⋊Q8C2211yescharacteristic
11#5470C2211yescharacteristic
12C3×#492—21no—
13C3×#487—21no—
14C3×#461—22no—
15C3×#448—22no—
16C3×#431C2211yescharacteristic
17#5225—22no—
18#5091—21no—
19#5082—22no—
20#5079—21no—
21#5079—21no—
22#5075—21no—
23#5042—22no—
24#5019—21no—
25#5007—21no—
26#4924C2211yescharacteristic
27C4×#208C2211yescharacteristic
28C4×#208C2211yescharacteristic
29#4540—22no—
30#4530—22no—
31#4484—22no—
32#3254—31no—
33#3206D311yescharacteristic
34#3139—31no—
35#1997—31no—
36#1525—31no—
37#1485—31no—
38#1299—31no—
39C2×C3×#41—21no—
40C22×C3×Q8D412yes—
41C2×C3×SD16—24no—
42C3×C4⋊Q8—21no—
43C3×C4⋊Q8—21no—
44C3×C4⋊Q8—21no—
45C3×C4⋊Q8—21no—
46C3×C4⋊Q8D411yescharacteristic
47C3×#127D411yescharacteristic
48C3×#124—21no—
49C3×#121—21no—
50C3×#121—21no—
51C3×C4×D4—21no—
52C3×C4×D4—21no—
53C2×C3×C4⋊C4—21no—
54C2×C3×C4⋊C4—21no—
55C2×C3×C42C2311yescharacteristic
56C2×#214—22no—
57#926—21no—
58#924—21no—
59#924—21no—
60C2×#208—21no—
61C2×#208—21no—
62C2×#208D412yes—
63#870—21no—
64#869D411yescharacteristic
65#869D411yescharacteristic
66C4×Dic6—21no—
67C4×Dic6—21no—
68C3×C4⋊C8D412yes—
69C3×#103—24no—
70C3×C22⋊C8—21no—
71C3×C4×C8—21no—
72#835—24no—
73#835—24no—
74#832—22no—
75#810—22no—
76#807—22no—
77C4⋊(C3⋊C8)—21no—
78C4⋊(C3⋊C8)—21no—
79#803—21no—
80#803D412yes—
81C4×C3⋊C8—21no—
82C4×C3⋊C8—21no—
83#533—31no—
84#532C2×D311yescharacteristic
85C2×C4⋊Q8C2×D311yescharacteristic
86#492—21no—
87#491—62no—
88#487—21no—
89#487—61no—
90#487—61no—
91#477—61no—
92#474—61no—
93#468—62no—
94#461—22no—
95#461—61no—
96#460—61no—
97#448—22no—
98#431C2×D311yescharacteristic
99#423—31no—
100#419—62no—
101C4×M16—31no—
102C4×M16—31no—
103#343—62no—
104#329—62no—
105Q8⋊C8—62no—
106C3×#41—82no—
107C2×C3×Q8—22no—
108C2×C3×Q8—22no—
109C2×C3×Q8—21no—
110C2×C3×Q8—44no—
111C2×C3×D4—21no—
112C2×C3×D4—21no—
113C2×C3×D4—21no—
114C2×C3×D4—41no—
115C22×C3×C4—22no—
116C22×C3×C4—41no—
117C22×C3×C4C2×D411yescharacteristic
118C22×C3×C4C2×D411yescharacteristic
119C22×C3×C4#4111yescharacteristic
120C2×Dic6—21no—
121C2×Dic6—21no—
122C3×SD16—84no—
123C2×C3×C8—22no—
124C3×C4⋊C4—21no—
125C3×C4⋊C4—21no—
126C3×C4⋊C4—41no—
127C3×C4⋊C4—41no—
128C3×C4⋊C4—41no—
129C3×C4⋊C4—41no—
130C3×C22⋊C4—41no—
131C3×C22⋊C4—41no—
132C3×C42C2×D411yescharacteristic
133C3×C42C2×D411yescharacteristic
134C3×C42C2×D411yescharacteristic
135C3×C42C2×D411yescharacteristic
136#217—41no—
137#214—82no—
138C4⋊Dic3—21no—
139C4⋊Dic3—21no—
140C4⋊Dic3—21no—
141C4⋊Dic3—21no—
142Dic3⋊C4—41no—
143Dic3⋊C4—41no—
144C4×Dic3—41no—
145#208—22no—
146#208—42no—
147#208—41no—
148#208—42no—
149C2×C3⋊C8—22no—
150C2×C3⋊C8—22no—
151C2×C3⋊C8—41no—
152C2×#41—21no—
153C22×Q8#6712yes—
154C2×SD16—24no—
155C2×SD16—62no—
156C2×M16—32no—
157C2×M16—61no—
158C2×M16—61no—
159C4⋊Q8—21no—
160C4⋊Q8—21no—
161C4⋊Q8—21no—
162C4⋊Q8—21no—
163C4⋊Q8—31no—
164C4⋊Q8—31no—
165C4⋊Q8#6711yescharacteristic
166#127#6711yescharacteristic
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—21no—
179C2×C4⋊C4—21no—
180C2×C42C22×D311yescharacteristic
181#107—62no—
182C4⋊C8—61no—
183C4⋊C8—61no—
184C4⋊C8#6712yes—
185#103—24no—
186#103—62no—
187#103—64no—
188#103—64no—
189#102—62no—
190C22⋊C8—21no—
191#97—32no—
192#97—61no—
193C4×C8—21no—
194C4×C8—61no—
195C4×C8—61no—
196C23×C3—41no—
197C23×C3#12111yescharacteristic
198C3×Q8—44no—
199C3×Q8—44no—
200C3×Q8—81no—
201C3×D4—41no—
202C3×D4—42no—
203C3×D4—41no—
204C3×D4—81no—
205C3×D4—82no—
206C2×C3×C4—21no—
207C2×C3×C4—22no—
208C2×C3×C4—21no—
209C2×C3×C4—22no—
210C2×C3×C4—41no—
211C2×C3×C4—42no—
212C2×C3×C4—82no—
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#12711yescharacteristic
220C2×Dic3—41no—
221C2×Dic3—41no—
222Dic6—41no—
223Dic6—41no—
224Dic6—42no—
225C3×C8—42no—
226C3⋊C8—42no—
227C3⋊C8—42no—
228C3⋊C8—42no—
229#41—82no—
230C2×Q8—22no—
231C2×Q8—21no—
232C2×Q8—22no—
233C2×Q8—44no—
234C2×Q8—61no—
235C2×Q8—61no—
236C2×D4—21no—
237C2×D4—21no—
238C2×D4—21no—
239C2×D4—41no—
240C22×C4—22no—
241C22×C4—41no—
242C22×C4D3×D411yescharacteristic
243C22×C4#23711yescharacteristic
244C22×C4C2×#6711yescharacteristic
245SD16—84no—
246SD16—242no—
247M16—62no—
248M16—122no—
249M16—121no—
250M16—122no—
251C2×C8—22no—
252C2×C8—62no—
253C2×C8—62no—
254C2×C8—121no—
255C4⋊C4—21no—
256C4⋊C4—21no—
257C4⋊C4—41no—
258C4⋊C4—41no—
259C4⋊C4—41no—
260C4⋊C4—41no—
261C4⋊C4—61no—
262C4⋊C4—61no—
263C4⋊C4—61no—
264C4⋊C4—61no—
265C4⋊C4—121no—
266C4⋊C4—121no—
267C22⋊C4—41no—
268C22⋊C4—41no—
269C22⋊C4—121no—
270C42—121no—
271C42D3×D411yescharacteristic
272C42D3×D411yescharacteristic
273C42C2×#6711yescharacteristic
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—82no—
286C3×C4—81no—
287C3×C4#45511yescharacteristic
288C3×C4#45512yes—
289C3×C4#45611yescharacteristic
290C3×C4#47112yes—
291Dic3—81no—
292Dic3—81no—
293C23—41no—
294C23#93911yescharacteristic
295Q8—44no—
296Q8—44no—
297Q8—81no—
298Q8—122no—
299Q8—121no—
300Q8—121no—
301D4—41no—
302D4—41no—
303D4—42no—
304D4—81no—
305D4—82no—
306C2×C4—22no—
307C2×C4—21no—
308C2×C4—21no—
309C2×C4—22no—
310C2×C4—41no—
311C2×C4—42no—
312C2×C4—82no—
313C2×C4—82no—
314C2×C4—121no—
315C2×C4—121no—
316C2×C4#93711yescharacteristic
317C2×C4#93711yescharacteristic
318C2×C4#93811yescharacteristic
319C2×C4#93911yescharacteristic
320C2×C4#94011yescharacteristic
321C2×C4#95311yescharacteristic
322C8—42no—
323C8—122no—
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#522311yescharacteristic
338C4—21no—
339C4—21no—
340C4—21no—
341C4—81no—
342C4—82no—
343C4—241no—
344C4—241no—
345C4#517712yes—
346C4#520211yescharacteristic
347C4#520511yescharacteristic
348C4#524212yes—
349C3#1011411yescharacteristic, complemented, Sylow
350C2—41no—
351C2—161no—
352C2#7185711yescharacteristic
353C2#7205911yescharacteristic
354C2#7211011yescharacteristic
subgroupG/H
centerC22#5223
commutator, cosocleC2×C3×C42C23
FrattiniC2×C42C22×D3
FittingC3×#3206C2
socleC22×C3#392
prime radicalC3×#532C22
hypercenter#3206D3
hypocenterC3#10114
LCSG ▷ C2×C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3206

Complements

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

NH ≅ G/Ntype
3. #83198C2semidirect
5. #71790C2semidirect
6. #70769C2semidirect
7. #67743C2semidirect
349. C3#10114semidirect

Maximal subgroups

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

subgroupindex
C3×#32062
#833212
#831982
#717972
#717902
#707692
#677432
#101143