Finite Group Databasebeta

← #10694597

Subgroups — #10694597

Order 768.

418 conjugacy classes of subgroups; only the 97 normal subgroups are shown.

normal subgroups only

This group has 418 conjugacy classes of subgroups — too many to draw. Only its 97 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#83344C211yesmaximal, characteristic
2C3×#1531C211yesmaximal, characteristic
3#71836C211yesmaximal, characteristic
4#70836C211yesmaximal, characteristic
5#70794C211yesmaximal, characteristic
6#69804C211yesmaximal, characteristic
7#69801C211yesmaximal, characteristic
8#9763—31nomaximal, Sylow
9C3×#533C2211yescharacteristic
10#5470C2211yescharacteristic
11C3×#432C2211yescharacteristic
12C2×C3×#97C2211yescharacteristic
13C2×#815—21no—
14C2×C8×Dic3—21no—
15#5093C411yescharacteristic, complemented
16#5092C411yescharacteristic, complemented
17#5079C411yescharacteristic, complemented
18#5078C411yescharacteristic, complemented
19#5051—21no—
20#5051—21no—
21C4×#216—21no—
22C4×#216—21no—
23#4932C2211yescharacteristic
24#4926C2211yescharacteristic
25C2×C4×C3⋊C8C2211yescharacteristic
26C3×#330—22no—
27#4538—22no—
28#4480—21no—
29#4479—21no—
30#4467—22no—
31#4465—22no—
32#3170—31no—
33#1726—31no—
34#1568—31no—
35#1531—31no—
36#1531D311yescharacteristic
37#1506—31no—
38#1506—31no—
39C22×C3×C8—21no—
40C3×C4⋊Q8C2×C411yescharacteristic
41C3×C4⋊Q8C2×C411yescharacteristic
42C3×#125—21no—
43C3×C22⋊Q8—21no—
44C3×C22⋊Q8—21no—
45C3×C4×Q8—21no—
46C3×C4×Q8—21no—
47C3×#117—21no—
48C2×C3×C42C2311yescharacteristic
49C2×#216—21no—
50C2×#216—21no—
51#926—21no—
52#924—21no—
53#924—21no—
54C2×C4×Dic3—21no—
55C22×C3⋊C8D411yescharacteristic
56C22×C3⋊C8D411yescharacteristic
57#870—21no—
58#869C2×C411yescharacteristic
59#869C2×C411yescharacteristic
60C4×Dic6—21no—
61C4×Dic6—21no—
62C3×C4⋊C8—21no—
63C3×C22⋊C8—21no—
64C3×#97—21no—
65C3×#97—21no—
66C3×#97D412yes—
67#835—21no—
68#835—21no—
69C3⋊#95—21no—
70C3⋊#95D412yes—
71#830—21no—
72#815—24no—
73C8×Dic3—24no—
74#810—21no—
75#810—21no—
76#807—21no—
77#807—21no—
78C4⋊(C3⋊C8)—21no—
79#803—21no—
80C4×C3⋊C8C2×C411yescharacteristic
81C4×C3⋊C8C2×C411yescharacteristic
82C4×C3⋊C8D411yescharacteristic
83C4×C3⋊C8D411yescharacteristic
84#533—31no—
85#533C2×D311yescharacteristic
86#500—31no—
87#494—31no—
88#487—31no—
89#487—31no—
90#467—61no—
91#467—61no—
92C4×Q16—61no—
93C4×Q16—61no—
94#432—31no—
95#432C2×D311yescharacteristic
96C2×#97—61no—
97C2×#97C2×D311yescharacteristic
98C2×C4×C8—31no—
99C2×C4×C8—61no—
100#376—31no—
101#335—61no—
102#334—61no—
103#330—22no—
104#330—62no—
105#328—62no—
106#328—62no—
107C2×C3×Q8—21no—
108C2×C3×Q8—21no—
109C22×C3×C4C2×D411yescharacteristic
110C22×C3×C4C2×D411yescharacteristic
111C22×C3×C4#4111yescharacteristic
112C22×Dic3—22no—
113C2×Dic6—21no—
114C2×Dic6—21no—
115C2×C3×C8—22no—
116C2×C3×C8—22no—
117C2×C3×C8—22no—
118C2×C3×C8—41no—
119C3×C4⋊C4—21no—
120C3×C4⋊C4—21no—
121C3×C4⋊C4—21no—
122C3×C4⋊C4—21no—
123C3×C4⋊C4—41no—
124C3×C4⋊C4—41no—
125C3×C22⋊C4—41no—
126C3×C42—41no—
127C3×C42C22×C411yescharacteristic
128C3×C42C2×D411yescharacteristic
129C3×C42C2×D411yescharacteristic
130C3×C42#4111yescharacteristic
131#217—41no—
132#216—42no—
133#216—42no—
134C4⋊Dic3—21no—
135C4⋊Dic3—21no—
136C4⋊Dic3—21no—
137C4⋊Dic3—21no—
138Dic3⋊C4—41no—
139Dic3⋊C4—41no—
140C4×Dic3—22no—
141C4×Dic3—22no—
142C4×Dic3—41no—
143C2×C3⋊C8—22no—
144C2×C3⋊C8—22no—
145C2×C3⋊C8—41no—
146C2×C3⋊C8C22⋊C411yescharacteristic
147C2×C3⋊C8C22⋊C411yescharacteristic
148C2×C3⋊C8C22⋊C411yescharacteristic
149C2×C3⋊C8C22⋊C411yescharacteristic
150C2×Q16—61no—
151C2×Q16—61no—
152C22×C8—21no—
153C22×C8—31no—
154C22×C8—31no—
155C4⋊Q8—31no—
156C4⋊Q8—31no—
157C4⋊Q8C4×D311yescharacteristic
158C4⋊Q8C4×D311yescharacteristic
159#125—21no—
160#125—61no—
161C22⋊Q8—21no—
162C22⋊Q8—21no—
163C22⋊Q8—61no—
164C22⋊Q8—61no—
165C4×Q8—21no—
166C4×Q8—21no—
167C4×Q8—61no—
168C4×Q8—61no—
169#117—21no—
170#117—61no—
171C2×C42—61no—
172C2×C42C22×D311yescharacteristic
173#107—61no—
174#107—61no—
175C4⋊C8—21no—
176C4⋊C8—61no—
177#103—61no—
178#103—61no—
179#103—61no—
180#103—61no—
181C22⋊C8—21no—
182C22⋊C8—61no—
183#97—21no—
184#97—21no—
185#97—64no—
186#97—61no—
187#97#6712yes—
188C4×C8—31no—
189C4×C8—31no—
190C4×C8—31no—
191C4×C8—31no—
192C4×C8—64no—
193#95—32no—
194#95—61no—
195C23×C3#12111yescharacteristic
196C3×Q8—41no—
197C3×Q8—41no—
198C3×Q8—81no—
199C2×C3×C4—22no—
200C2×C3×C4—21no—
201C2×C3×C4—21no—
202C2×C3×C4—41no—
203C2×C3×C4—41no—
204C2×C3×C4—41no—
205C2×C3×C4C2×C22⋊C411yescharacteristic
206C2×C3×C4C4×D411yescharacteristic
207C2×C3×C4C4×D411yescharacteristic
208C2×C3×C4#12111yescharacteristic
209C2×C3×C4#12411yescharacteristic
210C2×C3×C4#12711yescharacteristic
211C2×Dic3—24no—
212C2×Dic3—42no—
213C2×Dic3—42no—
214C2×Dic3—41no—
215C2×Dic3—41no—
216Dic6—41no—
217Dic6—41no—
218Dic6—81no—
219C3×C8—24no—
220C3×C8—81no—
221C3⋊C8—21no—
222C3⋊C8—21no—
223C3⋊C8—81no—
224C3⋊C8#10412yes—
225C3⋊C8#10412yes—
226C2×Q8—21no—
227C2×Q8—21no—
228C2×Q8—61no—
229C2×Q8—61no—
230C22×C4—62no—
231C22×C4#23511yescharacteristic
232C22×C4D3×D411yescharacteristic
233C22×C4D3×D411yescharacteristic
234Q16—122no—
235Q16—122no—
236C2×C8—22no—
237C2×C8—22no—
238C2×C8—22no—
239C2×C8—31no—
240C2×C8—31no—
241C2×C8—31no—
242C2×C8—31no—
243C2×C8—41no—
244C2×C8—62no—
245C2×C8—62no—
246C2×C8—121no—
247C4⋊C4—21no—
248C4⋊C4—21no—
249C4⋊C4—21no—
250C4⋊C4—21no—
251C4⋊C4—41no—
252C4⋊C4—41no—
253C4⋊C4—61no—
254C4⋊C4—61no—
255C4⋊C4—61no—
256C4⋊C4—61no—
257C4⋊C4—121no—
258C4⋊C4—121no—
259C22⋊C4—41no—
260C22⋊C4—121no—
261C42—41no—
262C42—62no—
263C42—62no—
264C42—121no—
265C42C2×C4×D311yescharacteristic
266C42D3×D411yescharacteristic
267C42#23711yescharacteristic
268C42C2×#6711yescharacteristic
269C22×C3—21no—
270C22×C3—21no—
271C22×C3#39011yescharacteristic
272C22×C3#47312yes—
273C3×C4—21no—
274C3×C4—21no—
275C3×C4—21no—
276C3×C4—21no—
277C3×C4—21no—
278C3×C4—81no—
279C3×C4—81no—
280C3×C4C2×#10411yescharacteristic
281C3×C4C2×#10411yescharacteristic
282Dic3—44no—
283Dic3—81no—
284Dic3—81no—
285C23#88411yescharacteristic
286Q8—41no—
287Q8—41no—
288Q8—81no—
289Q8—121no—
290Q8—121no—
291Q8—241no—
292C2×C4—22no—
293C2×C4—21no—
294C2×C4—21no—
295C2×C4—41no—
296C2×C4—41no—
297C2×C4—41no—
298C2×C4—64no—
299C2×C4—122no—
300C2×C4—122no—
301C2×C4—121no—
302C2×C4—121no—
303C2×C4D3×C22⋊C411yescharacteristic
304C2×C4#67⋊C411yescharacteristic
305C2×C4#88511yescharacteristic
306C2×C4C4×#6711yescharacteristic
307C2×C4#93911yescharacteristic
308C2×C4#94011yescharacteristic
309C8—24no—
310C8—32no—
311C8—32no—
312C8—61no—
313C8—61no—
314C8—81no—
315C8—241no—
316C2×C3—22no—
317C2×C3#196211yescharacteristic
318C2×C3#197611yescharacteristic
319C2×C3#197911yescharacteristic
320C22—21no—
321C22—21no—
322C22#475512yes—
323C22#495711yescharacteristic
324C4—21no—
325C4—21no—
326C4—21no—
327C4—21no—
328C4—21no—
329C4—81no—
330C4—81no—
331C4—124no—
332C4—241no—
333C4—241no—
334C4D3×#10411yescharacteristic
335C4D3×#10411yescharacteristic
336C3#976311yescharacteristic, complemented, Sylow
337C2—22no—
338C2#7010811yescharacteristic
339C2#7035611yescharacteristic
340C2#7039711yescharacteristic
subgroupG/H
centerC22#4957
commutatorC3×C42C22×C4
FrattiniC2×C42C22×D3
FittingC3×#1531C2
socleC22×C3#390
prime radicalC23×C3#121
cosocleC2×C3×C42C23
hypercenter#1531D3
hypocenterC3#9763
LCSG ▷ C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1531

Complements

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

NH ≅ G/Ntype
15. #5093C4semidirect
16. #5092C4semidirect
17. #5079C4semidirect
18. #5078C4semidirect
336. C3#9763semidirect

Maximal subgroups

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

subgroupindex
#833442
C3×#15312
#718362
#708362
#707942
#698042
#698012
#97633