Finite Group Databasebeta

← #10699215

Subgroups — #10699215

Order 768.

374 conjugacy classes of subgroups; only the 95 normal subgroups are shown.

normal subgroups only

This group has 374 conjugacy classes of subgroups — too many to draw. Only its 95 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#83224C211yesmaximal, characteristic
2C3×#1987C211yesmaximal, characteristic
3#71790C211yesmaximal, characteristic
4#70438C211yesmaximal, characteristic
5#70420C211yesmaximal, characteristic
6#69493C211yesmaximal, characteristic
7#67620C211yesmaximal, characteristic
8#10506—31nomaximal, Sylow
9C2×C3×C4⋊Q8C2211yescharacteristic
10#5470C2211yescharacteristic
11C2×C3×C4⋊C8C2211yescharacteristic
12C3×C4×C4⋊C4C2211yescharacteristic
13#5093—21no—
14#5091—21no—
15#5091—21no—
16#5078—21no—
17#4970—22no—
18#4925C2211yescharacteristic
19#4924C2211yescharacteristic
20C4×#208C2211yescharacteristic
21C3×#340—22no—
22#4530—21no—
23#4530—21no—
24#4523—21no—
25#4523—21no—
26#4482—21no—
27#4481—21no—
28#4480—21no—
29#4479—21no—
30#3230—31no—
31#1997—31no—
32#1987D311yescharacteristic
33#1986—31no—
34#1565—31no—
35#1550—31no—
36#1303—31no—
37C22×C3×Q8—21no—
38C22×C3×C8—21no—
39C3×C4⋊Q8—21no—
40C3×C4⋊Q8—21no—
41C3×C4⋊Q8—21no—
42C3×C4⋊Q8—21no—
43C2×C3×C4⋊C4—21no—
44C2×C3×C4⋊C4—21no—
45C2×C3×C4⋊C4D412yes—
46C2×C3×C42—21no—
47C2×C3×C42C2311yescharacteristic
48#926—21no—
49#924—21no—
50#924—21no—
51C2×#208—21no—
52C2×#208D411yescharacteristic
53C2×#208D411yescharacteristic
54#870—21no—
55#869D411yescharacteristic
56#869D411yescharacteristic
57C4×Dic6—21no—
58C4×Dic6—21no—
59C3×C4⋊C8—21no—
60C3×C4⋊C8—21no—
61C3×#95—21no—
62#830—21no—
63#810—22no—
64#810—22no—
65#807—22no—
66#807—22no—
67C4⋊(C3⋊C8)—21no—
68C4⋊(C3⋊C8)—21no—
69C4⋊(C3⋊C8)D411yescharacteristic
70C4⋊(C3⋊C8)D411yescharacteristic
71#803—21no—
72C4×C3⋊C8—21no—
73C4×C3⋊C8—21no—
74#533—31no—
75C2×C4⋊Q8C2×D311yescharacteristic
76#500—61no—
77#487—61no—
78#484—62no—
79#477—61no—
80#477—61no—
81#431—31no—
82#423—31no—
83C2×C4⋊C8C2×D311yescharacteristic
84C4×M16—31no—
85C4×C4⋊C4C2×D311yescharacteristic
86#343—61no—
87#343—61no—
88#340—22no—
89#339—61no—
90#339—61no—
91#335—61no—
92#334—61no—
93#333—61no—
94#332—61no—
95C2×C3×Q8—41no—
96C2×C3×Q8—41no—
97C2×C3×Q8—82no—
98C22×C3×C4—22no—
99C22×C3×C4—41no—
100C22×C3×C4C2×D411yescharacteristic
101C22×C3×C4C2×D411yescharacteristic
102C22×C3×C4#4111yescharacteristic
103C2×Dic6—21no—
104C2×Dic6—21no—
105C2×C3×C8—42no—
106C2×C3×C8—41no—
107C3×C4⋊C4—22no—
108C3×C4⋊C4—22no—
109C3×C4⋊C4—22no—
110C3×C4⋊C4—41no—
111C3×C4⋊C4—41no—
112C3×C4⋊C4—41no—
113C3×C42—41no—
114C3×C42—41no—
115C3×C42C2×D411yescharacteristic
116C3×C42C2×D411yescharacteristic
117C3×C42#4111yescharacteristic
118C3×C42#4111yescharacteristic
119#217—41no—
120C4⋊Dic3—21no—
121C4⋊Dic3—21no—
122C4⋊Dic3—21no—
123C4⋊Dic3—21no—
124Dic3⋊C4—41no—
125Dic3⋊C4—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—21no—
136C2×C3⋊C8—21no—
137C2×C3⋊C8—41no—
138C22×Q8—21no—
139C2×M16—31no—
140C2×M16—31no—
141C2×M16—61no—
142C22×C8—21no—
143C4⋊Q8—21no—
144C4⋊Q8—21no—
145C4⋊Q8—21no—
146C4⋊Q8—21no—
147C4⋊Q8—31no—
148C4⋊Q8—31no—
149#125—61no—
150C22⋊Q8—61no—
151C22⋊Q8—61no—
152C4×Q8—61no—
153C4×Q8—61no—
154#117—61no—
155C2×C4⋊C4—21no—
156C2×C4⋊C4—21no—
157C2×C4⋊C4#6712yes—
158C2×C42—21no—
159C2×C42C22×D311yescharacteristic
160#107—62no—
161#107—62no—
162C4⋊C8—21no—
163C4⋊C8—21no—
164C4⋊C8—31no—
165C4⋊C8—31no—
166C4⋊C8—61no—
167C4⋊C8—61no—
168#103—62no—
169#103—62no—
170C22⋊C8—61no—
171#97—61no—
172C4×C8—61no—
173C4×C8—61no—
174#95—21no—
175C23×C3#12111yescharacteristic
176C3×Q8—82no—
177C3×Q8—82no—
178C2×C3×C4—21no—
179C2×C3×C4—21no—
180C2×C3×C4—41no—
181C2×C3×C4—42no—
182C2×C3×C4—42no—
183C2×C3×C4—81no—
184C2×C3×C4—81no—
185C2×C3×C4—82no—
186C2×C3×C4#12011yescharacteristic
187C2×C3×C4#12111yescharacteristic
188C2×C3×C4#12111yescharacteristic
189C2×C3×C4#12311yescharacteristic
190C2×C3×C4#12311yescharacteristic
191C2×C3×C4#12411yescharacteristic
192C2×C3×C4C2×Q1611yescharacteristic
193C2×C3×C4#13511yescharacteristic
194C2×C3×C4#13711yescharacteristic
195C2×C3×C4#13711yescharacteristic
196C2×Dic3—41no—
197C2×Dic3—41no—
198Dic6—41no—
199Dic6—41no—
200Dic6—81no—
201C3×C8—81no—
202C3⋊C8—22no—
203C3⋊C8—41no—
204C3⋊C8—41no—
205C3⋊C8—41no—
206C3⋊C8—81no—
207C2×Q8—41no—
208C2×Q8—41no—
209C2×Q8—61no—
210C2×Q8—61no—
211C2×Q8—82no—
212C22×C4—22no—
213C22×C4—41no—
214C22×C4D3×D411yescharacteristic
215C22×C4#23711yescharacteristic
216C22×C4C2×#6711yescharacteristic
217M16—32no—
218M16—61no—
219M16—61no—
220M16—61no—
221M16—121no—
222M16—121no—
223C2×C8—42no—
224C2×C8—41no—
225C2×C8—61no—
226C2×C8—61no—
227C2×C8—61no—
228C2×C8—61no—
229C2×C8—121no—
230C4⋊C4—22no—
231C4⋊C4—22no—
232C4⋊C4—22no—
233C4⋊C4—41no—
234C4⋊C4—41no—
235C4⋊C4—41no—
236C4⋊C4—61no—
237C4⋊C4—61no—
238C4⋊C4—61no—
239C4⋊C4—61no—
240C4⋊C4—121no—
241C4⋊C4—121no—
242C22⋊C4—121no—
243C42—41no—
244C42—41no—
245C42—121no—
246C42#23511yescharacteristic
247C42#23511yescharacteristic
248C42D3×D411yescharacteristic
249C42D3×D411yescharacteristic
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—82no—
260C3×C4#45411yescharacteristic
261C3×C4#45611yescharacteristic
262C3×C4#46211yescharacteristic
263C3×C4#46411yescharacteristic
264C3×C4#46711yescharacteristic
265C3×C4#47011yescharacteristic
266Dic3—81no—
267Dic3—81no—
268C23#93911yescharacteristic
269Q8—82no—
270Q8—82no—
271Q8—121no—
272Q8—121no—
273Q8—241no—
274C2×C4—21no—
275C2×C4—21no—
276C2×C4—41no—
277C2×C4—42no—
278C2×C4—42no—
279C2×C4—81no—
280C2×C4—81no—
281C2×C4—82no—
282C2×C4—121no—
283C2×C4—121no—
284C2×C4#88311yescharacteristic
285C2×C4#88411yescharacteristic
286C2×C4#88411yescharacteristic
287C2×C4#88511yescharacteristic
288C2×C4#91511yescharacteristic
289C2×C4#91611yescharacteristic
290C2×C4D3×Q1611yescharacteristic
291C2×C4#91811yescharacteristic
292C2×C4#92911yescharacteristic
293C2×C4#93711yescharacteristic
294C8—62no—
295C8—81no—
296C8—121no—
297C8—121no—
298C8—121no—
299C8—241no—
300C2×C3—21no—
301C2×C3—21no—
302C2×C3#203811yescharacteristic
303C2×C3#204811yescharacteristic
304C2×C3#205611yescharacteristic
305C22—21no—
306C22—21no—
307C22#495911yescharacteristic
308C22#516611yescharacteristic
309C22#518211yescharacteristic
310C4—21no—
311C4—21no—
312C4—21no—
313C4—82no—
314C4—82no—
315C4—241no—
316C4—241no—
317C4#479511yescharacteristic
318C4#479811yescharacteristic
319C4#481311yescharacteristic
320C4#481611yescharacteristic
321C4#506211yescharacteristic
322C4#509511yescharacteristic
323C3#1050611yescharacteristic, complemented, Sylow
324C2—21no—
325C2—21no—
326C2#6969511yescharacteristic
327C2#7028511yescharacteristic
328C2#7033211yescharacteristic
subgroupG/H
centerC22#4959
commutator, cosocleC2×C3×C42C23
FrattiniC2×C42C22×D3
FittingC3×#1987C2
socleC22×C3#394
prime radicalC23×C3#121
hypercenter#1987D3
hypocenterC3#10506
LCSG ▷ C2×C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1987

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
#832242
C3×#19872
#717902
#704382
#704202
#694932
#676202
#105063