Finite Group Databasebeta

← #10699212

Subgroups — #10699212

Order 768.

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

normal subgroups only

This group has 374 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
1C3×#3230C211yesmaximal, characteristic
2#71841C211yesmaximal, characteristic
3#71830C211yesmaximal, characteristic
4#71790C211yesmaximal, characteristic
5#71779C211yesmaximal, characteristic
6#70746C211yesmaximal, characteristic
7#67746C211yesmaximal, characteristic
8#10506—31nomaximal, Sylow
9C3×#533C2211yescharacteristic
10C2×C3×C4⋊Q8C2211yescharacteristic
11C3×#500—21no—
12C3×#487—21no—
13C3×#484—22no—
14C3×#477—21no—
15C3×#477—21no—
16C3×#431C2211yescharacteristic
17C4×C4⋊Dic3C2211yescharacteristic
18#4924C2211yescharacteristic
19C2×C4⋊(C3⋊C8)C2211yescharacteristic
20C4×#208C2211yescharacteristic
21#4547—21no—
22#4546—21no—
23#4532—21no—
24#4532—21no—
25#4530—21no—
26#4530—21no—
27#4529—22no—
28#4480—21no—
29#4479—21no—
30#3230D311yescharacteristic
31#1997—31no—
32#1987—31no—
33#1986—31no—
34#1565—31no—
35#1550—31no—
36#1303—31no—
37C22×C3×Q8—21no—
38C3×C4⋊Q8—21no—
39C3×C4⋊Q8—21no—
40C3×C4⋊Q8—21no—
41C3×C4⋊Q8—21no—
42C3×C4⋊Q8D411yescharacteristic
43C3×C4⋊Q8D411yescharacteristic
44C3×#125—21no—
45C3×C22⋊Q8—21no—
46C3×C22⋊Q8—21no—
47C3×C4×Q8—21no—
48C3×C4×Q8—21no—
49C3×#117—21no—
50C2×C3×C4⋊C4—21no—
51C2×C3×C4⋊C4D412yes—
52C2×C3×C42C2311yescharacteristic
53C2×C4⋊Dic3—21no—
54C2×C4×Dic3—21no—
55C2×#208—21no—
56C2×#208D411yescharacteristic
57C2×#208D411yescharacteristic
58C22×C3⋊C8—21no—
59C3×#107—22no—
60C3×#107—22no—
61C3×C4⋊C8D411yescharacteristic
62C3×C4⋊C8D411yescharacteristic
63C3×#103—22no—
64C3×#103—22no—
65C3×C22⋊C8—21no—
66C3×C4×C8—21no—
67C3⋊#95—21no—
68C4⋊(C3⋊C8)—21no—
69C4⋊(C3⋊C8)—21no—
70C4⋊(C3⋊C8)—21no—
71C4⋊(C3⋊C8)—21no—
72#803—21no—
73C4×C3⋊C8—21no—
74#533C2×D311yescharacteristic
75C2×C4⋊Q8C2×D311yescharacteristic
76#500—21no—
77#487—21no—
78#484—22no—
79#477—21no—
80#477—21no—
81#431C2×D311yescharacteristic
82#423—31no—
83C2×C4⋊C8—31no—
84C4×M16—31no—
85C4×C4⋊C4—31no—
86#343—61no—
87#343—61no—
88#340—62no—
89#339—61no—
90#339—61no—
91#335—61no—
92#334—61no—
93#333—61no—
94#332—61no—
95C2×C3×Q8—21no—
96C2×C3×Q8—21no—
97C2×C3×Q8—41no—
98C2×C3×Q8—41no—
99C2×C3×Q8—82no—
100C22×C3×C4—22no—
101C22×C3×C4C2×D411yescharacteristic
102C22×C3×C4C2×D411yescharacteristic
103C22×C3×C4#4111yescharacteristic
104C22×Dic3—41no—
105C2×C3×C8—21no—
106C2×C3×C8—21no—
107C3×C4⋊C4—22no—
108C3×C4⋊C4—21no—
109C3×C4⋊C4—21no—
110C3×C4⋊C4—21no—
111C3×C4⋊C4—22no—
112C3×C4⋊C4—21no—
113C3×C4⋊C4—41no—
114C3×C4⋊C4—41no—
115C3×C4⋊C4—41no—
116C3×C4⋊C4—41no—
117C3×C22⋊C4—41no—
118C3×C42—41no—
119C3×C42C2×D411yescharacteristic
120C3×C42C2×D411yescharacteristic
121C3×C42#4111yescharacteristic
122C3×C42#4111yescharacteristic
123C4⋊Dic3—22no—
124C4⋊Dic3—41no—
125C4×Dic3—41no—
126C4×Dic3—41no—
127#208—21no—
128#208—21no—
129#208—21no—
130#208—41no—
131#208—41no—
132#208Q1612yes—
133C2×C3⋊C8—21no—
134C2×C3⋊C8—21no—
135C2×C3⋊C8—42no—
136C2×C3⋊C8—41no—
137C2×C3⋊C8—41no—
138C22×Q8—21no—
139C2×M16—31no—
140C2×M16—31no—
141C2×M16—61no—
142C22×C8—61no—
143C4⋊Q8—21no—
144C4⋊Q8—21no—
145C4⋊Q8—21no—
146C4⋊Q8—21no—
147C4⋊Q8#6711yescharacteristic
148C4⋊Q8#6711yescharacteristic
149#125—21no—
150C22⋊Q8—21no—
151C22⋊Q8—21no—
152C4×Q8—21no—
153C4×Q8—21no—
154#117—21no—
155C2×C4⋊C4—21no—
156C2×C4⋊C4—61no—
157C2×C4⋊C4#6712yes—
158C2×C42—61no—
159C2×C42C22×D311yescharacteristic
160#107—22no—
161#107—22no—
162C4⋊C8—61no—
163C4⋊C8—61no—
164C4⋊C8—61no—
165C4⋊C8—61no—
166C4⋊C8#6711yescharacteristic
167C4⋊C8#6711yescharacteristic
168#103—22no—
169#103—22no—
170C22⋊C8—21no—
171#97—61no—
172C4×C8—21no—
173C4×C8—61no—
174#95—61no—
175C23×C3#12111yescharacteristic
176C3×Q8—41no—
177C3×Q8—41no—
178C3×Q8—81no—
179C3×Q8—82no—
180C3×Q8—82no—
181C2×C3×C4—21no—
182C2×C3×C4—21no—
183C2×C3×C4—41no—
184C2×C3×C4—41no—
185C2×C3×C4—41no—
186C2×C3×C4—42no—
187C2×C3×C4—82no—
188C2×C3×C4#12011yescharacteristic
189C2×C3×C4#12111yescharacteristic
190C2×C3×C4#12111yescharacteristic
191C2×C3×C4#12311yescharacteristic
192C2×C3×C4#12311yescharacteristic
193C2×C3×C4#12411yescharacteristic
194C2×C3×C4C2×Q1611yescharacteristic
195C2×C3×C4#13511yescharacteristic
196C2×C3×C4#13711yescharacteristic
197C2×C3×C4#13711yescharacteristic
198C2×Dic3—42no—
199C2×Dic3—81no—
200C2×Dic3—81no—
201C3×C8—41no—
202C3×C8—41no—
203C3⋊C8—22no—
204C3⋊C8—41no—
205C3⋊C8—81no—
206C3⋊C8—81no—
207C2×Q8—21no—
208C2×Q8—21no—
209C2×Q8—41no—
210C2×Q8—41no—
211C2×Q8—82no—
212C22×C4—22no—
213C22×C4—121no—
214C22×C4D3×D411yescharacteristic
215C22×C4#23711yescharacteristic
216C22×C4C2×#6711yescharacteristic
217M16—32no—
218M16—61no—
219M16—61no—
220M16—61no—
221M16—121no—
222M16—121no—
223C2×C8—21no—
224C2×C8—21no—
225C2×C8—61no—
226C2×C8—61no—
227C2×C8—122no—
228C2×C8—121no—
229C2×C8—121no—
230C4⋊C4—22no—
231C4⋊C4—21no—
232C4⋊C4—21no—
233C4⋊C4—21no—
234C4⋊C4—22no—
235C4⋊C4—21no—
236C4⋊C4—41no—
237C4⋊C4—41no—
238C4⋊C4—41no—
239C4⋊C4—41no—
240C4⋊C4—62no—
241C4⋊C4—121no—
242C22⋊C4—41no—
243C42—41no—
244C42—121no—
245C42—121no—
246C42#23711yescharacteristic
247C42#23711yescharacteristic
248C42C2×#6711yescharacteristic
249C42C2×#6711yescharacteristic
250C22×C3—21no—
251C22×C3—21no—
252C22×C3#39411yescharacteristic
253C22×C3#46211yescharacteristic
254C22×C3#46411yescharacteristic
255C3×C4—21no—
256C3×C4—21no—
257C3×C4—21no—
258C3×C4—82no—
259C3×C4—81no—
260C3×C4—81no—
261C3×C4#45411yescharacteristic
262C3×C4#45611yescharacteristic
263C3×C4#46211yescharacteristic
264C3×C4#46411yescharacteristic
265C3×C4#46711yescharacteristic
266C3×C4#47011yescharacteristic
267Dic3—82no—
268C23#93911yescharacteristic
269Q8—41no—
270Q8—41no—
271Q8—81no—
272Q8—82no—
273Q8—82no—
274C2×C4—21no—
275C2×C4—21no—
276C2×C4—41no—
277C2×C4—41no—
278C2×C4—41no—
279C2×C4—42no—
280C2×C4—82no—
281C2×C4—122no—
282C2×C4—241no—
283C2×C4—241no—
284C2×C4D3×Q1611yescharacteristic
285C2×C4#91811yescharacteristic
286C2×C4#91811yescharacteristic
287C2×C4#91911yescharacteristic
288C2×C4#93511yescharacteristic
289C2×C4#93511yescharacteristic
290C2×C4#93611yescharacteristic
291C2×C4#93811yescharacteristic
292C2×C4#93911yescharacteristic
293C2×C4#95311yescharacteristic
294C8—41no—
295C8—41no—
296C8—62no—
297C8—121no—
298C8—241no—
299C8—241no—
300C2×C3—21no—
301C2×C3—21no—
302C2×C3#203811yescharacteristic
303C2×C3#204811yescharacteristic
304C2×C3#205611yescharacteristic
305C22—21no—
306C22—21no—
307C22#518211yescharacteristic
308C22#518511yescharacteristic
309C22#522211yescharacteristic
310C4—21no—
311C4—21no—
312C4—21no—
313C4—82no—
314C4—81no—
315C4—81no—
316C4—242no—
317C4#518211yescharacteristic
318C4#518511yescharacteristic
319C4#518811yescharacteristic
320C4#518911yescharacteristic
321C4#524511yescharacteristic
322C4#524611yescharacteristic
323C3#1050611yescharacteristic, complemented, Sylow
324C2—21no—
325C2—21no—
326C2#7186311yescharacteristic
327C2#7206811yescharacteristic
328C2#7209211yescharacteristic
subgroupG/H
centerC22#5222
commutator, cosocleC2×C3×C42C23
FrattiniC2×C42C22×D3
FittingC3×#3230C2
socleC22×C3#394
prime radicalC23×C3#121
hypercenter#3230D3
hypocenterC3#10506
LCSG ▷ C2×C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3230

Complements

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

NH ≅ G/Ntype
323. C3#10506semidirect

Maximal subgroups

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

subgroupindex
C3×#32302
#718412
#718302
#717902
#717792
#707462
#677462
#105063