Finite Group Databasebeta

← #10698908

Subgroups — #10698908

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
1#82431C211yesmaximal, characteristic
2C2×C3×Q8⋊C8C211yesmaximal, characteristic
3#71637C211yesmaximal, characteristic
4#70438C211yesmaximal, characteristic
5#69738C211yesmaximal, characteristic
6#69730C211yesmaximal, characteristic
7#67705C211yesmaximal, characteristic
8#10462—31nomaximal, Sylow
9C2×C3×C4×Q8C2211yescharacteristic
10#5470C2211yescharacteristic
11C2×C3×C4⋊C8C2211yescharacteristic
12C2×C3×C4×C8C2211yescharacteristic
13C2×#836—21no—
14#5225—21no—
15#5022—22no—
16#5021—21no—
17#5020—21no—
18#4964—21no—
19#4924C2211yescharacteristic
20#4924C2211yescharacteristic
21#4924C2211yescharacteristic
22C3×Q8⋊C8—21no—
23C3×Q8⋊C8—21no—
24#4530—21no—
25#4482—21no—
26#4481—21no—
27#4457—22no—
28#4456—21no—
29#4456—21no—
30#3078—31no—
31#1801—31no—
32#1566—31no—
33#1550—31no—
34#1493—31no—
35C2×Q8⋊C8D311yescharacteristic
36#1292—31no—
37C22×C3×Q8D411yescharacteristic
38C22×C3×C8—21no—
39C22×C3×C8—21no—
40C3×C4×Q8—21no—
41C3×C4×Q8—21no—
42C3×C4×Q8—21no—
43C3×C4×Q8—21no—
44C2×C3×C4⋊C4—21no—
45C2×C3×C4⋊C4D411yescharacteristic
46C2×C3×C42—21no—
47C2×C3×C42C2311yescharacteristic
48#926—21no—
49#924—21no—
50#924—21no—
51C2×#208—21no—
52C2×#208—21no—
53C2×#208—21no—
54#870—21no—
55#869D411yescharacteristic
56#869D411yescharacteristic
57C4×Dic6—21no—
58C4×Dic6—21no—
59C3×C4⋊C8—21no—
60C3×C4⋊C8—21no—
61C3×C4×C8—21no—
62C3×C4×C8D411yescharacteristic
63C3×C4×C8D411yescharacteristic
64#836—42no—
65#835—22no—
66#835—22no—
67#808—22no—
68#807—22no—
69C4⋊(C3⋊C8)—21no—
70C4⋊(C3⋊C8)—21no—
71C4⋊(C3⋊C8)—21no—
72C4⋊(C3⋊C8)—21no—
73C4⋊(C3⋊C8)—21no—
74C4⋊(C3⋊C8)D412yes—
75#533—31no—
76C2×C4×Q8C2×D311yescharacteristic
77#479—62no—
78#477—61no—
79#475—61no—
80#428—61no—
81#423—31no—
82#423—31no—
83#423—31no—
84C2×C4⋊C8C2×D311yescharacteristic
85#419—61no—
86C2×#101—61no—
87C2×C4×C8C2×D311yescharacteristic
88#343—61no—
89#334—62no—
90#333—61no—
91#332—61no—
92Q8⋊C8—21no—
93Q8⋊C8—21no—
94Q8⋊C8—61no—
95Q8⋊C8—61no—
96C2×C3×Q8—21no—
97C2×C3×Q8—21no—
98C2×C3×Q8—21no—
99C2×C3×Q8—21no—
100C2×C3×Q8—21no—
101C2×C3×Q8SD1612yes—
102C22×C3×C4—21no—
103C22×C3×C4—21no—
104C22×C3×C4C2×D411yescharacteristic
105C22×C3×C4#4111yescharacteristic
106C22×C3×C4#4111yescharacteristic
107C2×Dic6—21no—
108C2×Dic6—21no—
109C2×C3×C8—21no—
110C2×C3×C8—21no—
111C2×C3×C8—42no—
112C2×C3×C8—41no—
113C2×C3×C8—81no—
114C3×C4⋊C4—21no—
115C3×C4⋊C4—21no—
116C3×C4⋊C4—41no—
117C3×C4⋊C4—41no—
118C3×C42—41no—
119C3×C42—41no—
120C3×C42C2×D411yescharacteristic
121C3×C42C2×D411yescharacteristic
122C3×C42C2×D411yescharacteristic
123C3×C42#4111yescharacteristic
124#217—41no—
125C4⋊Dic3—21no—
126C4⋊Dic3—21no—
127C4⋊Dic3—21no—
128C4⋊Dic3—21no—
129Dic3⋊C4—41no—
130Dic3⋊C4—41no—
131C4×Dic3—41no—
132#208—41no—
133#208—41no—
134#208—41no—
135#208—41no—
136#208—41no—
137#208—41no—
138C2×C3⋊C8—22no—
139C2×C3⋊C8—41no—
140C2×C3⋊C8—41no—
141C22×Q8D1211yescharacteristic
142C2×M16—61no—
143C2×M16—61no—
144C2×M16—61no—
145C22×C8—21no—
146C22×C8—21no—
147C4⋊Q8—31no—
148C4⋊Q8—31no—
149#125—61no—
150C22⋊Q8—61no—
151C22⋊Q8—61no—
152C4×Q8—21no—
153C4×Q8—21no—
154C4×Q8—21no—
155C4×Q8—21no—
156C4×Q8—61no—
157C4×Q8—61no—
158#117—61no—
159C2×C4⋊C4—21no—
160C2×C4⋊C4D1211yescharacteristic
161C2×C42—21no—
162C2×C42C22×D311yescharacteristic
163#107—62no—
164#106—62no—
165C4⋊C8—21no—
166C4⋊C8—21no—
167C4⋊C8—32no—
168C4⋊C8—61no—
169C4⋊C8—61no—
170C4⋊C8—61no—
171C4⋊C8—61no—
172C4⋊C8—61no—
173#103—62no—
174#103—62no—
175#101—122no—
176C4×C8—21no—
177C4×C8#6711yescharacteristic
178C4×C8#6711yescharacteristic
179C23×C3#12311yescharacteristic
180C3×Q8—22no—
181C3×Q8—22no—
182C3×Q8—41no—
183C3×Q8—41no—
184C2×C3×C4—21no—
185C2×C3×C4—21no—
186C2×C3×C4—22no—
187C2×C3×C4—41no—
188C2×C3×C4—41no—
189C2×C3×C4—41no—
190C2×C3×C4—41no—
191C2×C3×C4—81no—
192C2×C3×C4#12011yescharacteristic
193C2×C3×C4#12111yescharacteristic
194C2×C3×C4#12111yescharacteristic
195C2×C3×C4#12111yescharacteristic
196C2×C3×C4#12311yescharacteristic
197C2×C3×C4#12411yescharacteristic
198C2×C3×C4C2×SD1611yescharacteristic
199C2×C3×C4#13511yescharacteristic
200C2×C3×C4#13711yescharacteristic
201C2×C3×C4#13711yescharacteristic
202C2×Dic3—41no—
203C2×Dic3—41no—
204Dic6—41no—
205Dic6—41no—
206Dic6—81no—
207C3×C8—22no—
208C3×C8—41no—
209C3×C8—81no—
210C3⋊C8—42no—
211C3⋊C8—81no—
212C3⋊C8—81no—
213C2×Q8—21no—
214C2×Q8—21no—
215C2×Q8—21no—
216C2×Q8—21no—
217C2×Q8—21no—
218C2×Q8—61no—
219C2×Q8—61no—
220C2×Q8#20412yes—
221C22×C4—21no—
222C22×C4—21no—
223C22×C4C2×D1211yescharacteristic
224C22×C4#23711yescharacteristic
225C22×C4#23711yescharacteristic
226M16—121no—
227M16—121no—
228M16—121no—
229M16—121no—
230M16—121no—
231M16—121no—
232C2×C8—21no—
233C2×C8—21no—
234C2×C8—42no—
235C2×C8—41no—
236C2×C8—62no—
237C2×C8—81no—
238C2×C8—121no—
239C2×C8—121no—
240C4⋊C4—21no—
241C4⋊C4—21no—
242C4⋊C4—41no—
243C4⋊C4—41no—
244C4⋊C4—61no—
245C4⋊C4—61no—
246C4⋊C4—61no—
247C4⋊C4—61no—
248C4⋊C4—121no—
249C4⋊C4—121no—
250C22⋊C4—121no—
251C42—41no—
252C42—41no—
253C42—121no—
254C42#23511yescharacteristic
255C42D3×D411yescharacteristic
256C42D3×D411yescharacteristic
257C42C2×#6711yescharacteristic
258C22×C3—21no—
259C22×C3—21no—
260C22×C3#39411yescharacteristic
261C22×C3#48311yescharacteristic
262C22×C3#48511yescharacteristic
263C3×C4—21no—
264C3×C4—21no—
265C3×C4—21no—
266C3×C4—42no—
267C3×C4—81no—
268C3×C4#44811yescharacteristic
269C3×C4#45211yescharacteristic
270C3×C4#46511yescharacteristic
271C3×C4#47011yescharacteristic
272C3×C4#47111yescharacteristic
273C3×C4#47311yescharacteristic
274Dic3—81no—
275Dic3—81no—
276C23#88611yescharacteristic
277Q8—22no—
278Q8—22no—
279Q8—41no—
280Q8—41no—
281Q8—121no—
282Q8—121no—
283Q8—241no—
284C2×C4—21no—
285C2×C4—21no—
286C2×C4—22no—
287C2×C4—41no—
288C2×C4—41no—
289C2×C4—41no—
290C2×C4—41no—
291C2×C4—81no—
292C2×C4—121no—
293C2×C4—121no—
294C2×C4#88211yescharacteristic
295C2×C4#88311yescharacteristic
296C2×C4#88511yescharacteristic
297C2×C4C2×#20411yescharacteristic
298C2×C4#90411yescharacteristic
299C2×C4#90911yescharacteristic
300C2×C4#90911yescharacteristic
301C2×C4#93011yescharacteristic
302C2×C4#93811yescharacteristic
303C2×C4#93911yescharacteristic
304C8—22no—
305C8—41no—
306C8—81no—
307C8—122no—
308C8—241no—
309C8—241no—
310C2×C3—21no—
311C2×C3—21no—
312C2×C3#203511yescharacteristic
313C2×C3#205011yescharacteristic
314C2×C3#205211yescharacteristic
315C22—21no—
316C22—21no—
317C22#472111yescharacteristic
318C22#472311yescharacteristic
319C22#496111yescharacteristic
320C4—21no—
321C4—21no—
322C4—21no—
323C4—42no—
324C4—81no—
325C4—241no—
326C4—241no—
327C4#473311yescharacteristic
328C4#473811yescharacteristic
329C4#511411yescharacteristic
330C4#513711yescharacteristic
331C4#517711yescharacteristic
332C4#519211yescharacteristic
333C3#1046211yescharacteristic, complemented, Sylow
334C2—21no—
335C2—21no—
336C2#6993611yescharacteristic
337C2#7026411yescharacteristic
338C2#7037911yescharacteristic
subgroupG/H
centerC22#4961
commutator, cosocleC2×C3×C42C23
FrattiniC2×C42C22×D3
FittingC2×C3×Q8⋊C8C2
socleC22×C3#394
prime radicalC23×C3#123
hypercenterC2×Q8⋊C8D3
hypocenterC3#10462
LCSG ▷ C2×C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ C2×Q8⋊C8

Complements

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

NH ≅ G/Ntype
333. C3#10462semidirect

Maximal subgroups

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

subgroupindex
#824312
C2×C3×Q8⋊C82
#716372
#704382
#697382
#697302
#677052
#104623