Finite Group Databasebeta

← #10696562

Subgroups — #10696562

Order 768.

496 conjugacy classes of subgroups; only the 85 normal subgroups are shown.

normal subgroups only

This group has 496 conjugacy classes of subgroups — too many to draw. Only its 85 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×#3234C211yesmaximal, characteristic
2#83283C211yesmaximal, characteristic
3#83044C211yesmaximal, characteristic, complemented
4#71812C211yesmaximal, characteristic
5#71735C211yesmaximal, characteristic, complemented
6#70822C211yesmaximal, characteristic, complemented
7#67748C211yesmaximal, characteristic, complemented
8#10093—31nomaximal, Sylow
9C3×#532C2211yescharacteristic
10C2×C3×#125C2211yescharacteristic
11C2×#869C2211yescharacteristic
12C3×#501—21no—
13C3×#490—21no—
14C3×#481—21no—
15C3×#480—21no—
16C3×#478—22no—
17C3×#432C2211yescharacteristic
18#5232—22no—
19#5079—21no—
20#5070—21no—
21#5055—22no—
22#4981—22no—
23#4979—22no—
24#4977—21no—
25#4975—21no—
26#4932C2211yescharacteristic
27#4926C2211yescharacteristic
28C4×#208C2211yescharacteristic
29#4543—22no—
30#4537—22no—
31#4486—22no—
32#3234D311yescharacteristic
33#3141—31no—
34#2660—31no—
35#1556—31no—
36#1528—31no—
37#1505—31no—
38#1300—31no—
39C2×C3×#41—21no—
40C22×Dic6—21no—
41C3×C4⋊Q8D411yescharacteristic
42C3×#127D411yescharacteristic
43C3×#125—21no—
44C3×#125—21no—
45C3×#125—21no—
46C3×#125—21no—
47C3×#124—21no—
48C3×#121—21no—
49C3×#121—21no—
50C3×C4×D4—21no—
51C3×C4×D4—21no—
52C2×C3×C4⋊C4—21no—
53C2×C3×C4⋊C4—21no—
54C2×C3×C4⋊C4D411yescharacteristic
55C2×C3×C4⋊C4D411yescharacteristic
56C2×C3×C42C2311yescharacteristic
57C2×C4⋊Dic3—21no—
58C2×C4⋊Dic3D412yes—
59C2×Dic3⋊C4—24no—
60C2×C4×Dic3—21no—
61C2×#208—21no—
62#869—21no—
63#869—21no—
64#869—21no—
65#869—21no—
66C3×#107—22no—
67C3×#106—22no—
68C3×C4⋊C8D412yes—
69C3×#102—22no—
70C3×#102—22no—
71C3×C22⋊C8—21no—
72C3×#97—21no—
73#835—24no—
74#832—24no—
75C3⋊#95—24no—
76C3⋊#95—21no—
77C3⋊#95D412yes—
78#830—21no—
79C4⋊(C3⋊C8)—21no—
80#803—21no—
81#803D412yes—
82C4×C3⋊C8—21no—
83#532C2×D311yescharacteristic
84C2×C4⋊Q8—31no—
85C2×#125C2×D311yescharacteristic
86#501—21no—
87#490—21no—
88#489—62no—
89#487—61no—
90#486—61no—
91#481—21no—
92#480—21no—
93#478—22no—
94#432—31no—
95#432C2×D311yescharacteristic
96#417—62no—
97C4×M16—31no—
98#398—62no—
99#395—62no—
100#384—61no—
101#382—61no—
102#376—31no—
103#332—62no—
104#330—62no—
105#327—62no—
106C3×#41—44no—
107C2×C3×Q8—21no—
108C2×C3×D4—21no—
109C2×C3×D4—21no—
110C2×C3×D4—21no—
111C2×C3×D4—41no—
112C22×C3×C4—21no—
113C22×C3×C4—21no—
114C22×C3×C4—41no—
115C22×C3×C4C2×D411yescharacteristic
116C22×C3×C4C2×D411yescharacteristic
117C22×C3×C4C2×D411yescharacteristic
118C22×Dic3—22no—
119C22×Dic3—22no—
120C2×Dic6—41no—
121C2×Dic6—41no—
122C2×Dic6—82no—
123C2×C3×C8—22no—
124C3×C4⋊C4—21no—
125C3×C4⋊C4—21no—
126C3×C4⋊C4—21no—
127C3×C4⋊C4—21no—
128C3×C4⋊C4—21no—
129C3×C4⋊C4—21no—
130C3×C4⋊C4—41no—
131C3×C4⋊C4—41no—
132C3×C4⋊C4—41no—
133C3×C4⋊C4—41no—
134C3×C22⋊C4—41no—
135C3×C22⋊C4—41no—
136C3×C42C2×D411yescharacteristic
137C3×C42C2×D411yescharacteristic
138C3×C42C2×D411yescharacteristic
139C3×C42#4111yescharacteristic
140C4⋊Dic3—22no—
141C4⋊Dic3—22no—
142C4⋊Dic3—41no—
143C4⋊Dic3—41no—
144Dic3⋊C4—84no—
145C4×Dic3—24no—
146#208—42no—
147C2×C3⋊C8—22no—
148C2×C3⋊C8—41no—
149C2×#41—21no—
150C22×Q8—61no—
151C2×M16—61no—
152C4⋊Q8—61no—
153C4⋊Q8—61no—
154C4⋊Q8—61no—
155C4⋊Q8—61no—
156C4⋊Q8#6711yescharacteristic
157#127#6711yescharacteristic
158#125—21no—
159#125—21no—
160#125—21no—
161#125—21no—
162#124—21no—
163#121—21no—
164#121—21no—
165C4×D4—21no—
166C4×D4—21no—
167C2×C4⋊C4—21no—
168C2×C4⋊C4—21no—
169C2×C4⋊C4—32no—
170C2×C4⋊C4—61no—
171C2×C4⋊C4—64no—
172C2×C4⋊C4#6711yescharacteristic
173C2×C4⋊C4#6711yescharacteristic
174C2×C42—61no—
175C2×C42C22×D311yescharacteristic
176#107—22no—
177#106—22no—
178C4⋊C8—61no—
179C4⋊C8#6712yes—
180#103—64no—
181#102—22no—
182#102—22no—
183#102—64no—
184C22⋊C8—21no—
185C22⋊C8—61no—
186#97—21no—
187#97—32no—
188#97—61no—
189C4×C8—61no—
190#95—32no—
191#95—64no—
192#95—61no—
193C23×C3—41no—
194C23×C3#12011yescharacteristic
195C3×Q8—42no—
196C3×D4—42no—
197C3×D4—42no—
198C3×D4—81no—
199C3×D4—82no—
200C2×C3×C4—22no—
201C2×C3×C4—22no—
202C2×C3×C4—41no—
203C2×C3×C4—42no—
204C2×C3×C4—41no—
205C2×C3×C4—41no—
206C2×C3×C4—41no—
207C2×C3×C4—42no—
208C2×C3×C4—82no—
209C2×C3×C4#12011yescharacteristic
210C2×C3×C4#12011yescharacteristic
211C2×C3×C4#12111yescharacteristic
212C2×C3×C4#12111yescharacteristic
213C2×C3×C4#12111yescharacteristic
214C2×C3×C4#12711yescharacteristic
215C2×Dic3—24no—
216C2×Dic3—42no—
217C2×Dic3—42no—
218C2×Dic3—44no—
219C2×Dic3—42no—
220Dic6—82no—
221Dic6—82no—
222C3×C8—42no—
223C3⋊C8—42no—
224C3⋊C8—81no—
225#41—44no—
226C2×Q8—21no—
227C2×Q8—121no—
228C2×Q8—121no—
229C2×Q8—242no—
230C2×D4—21no—
231C2×D4—21no—
232C2×D4—21no—
233C2×D4—41no—
234C22×C4—21no—
235C22×C4—21no—
236C22×C4—41no—
237C22×C4—62no—
238C22×C4—62no—
239C22×C4D3×D411yescharacteristic
240C22×C4D3×D411yescharacteristic
241C22×C4C2×#6711yescharacteristic
242M16—122no—
243C2×C8—22no—
244C2×C8—62no—
245C2×C8—121no—
246C4⋊C4—21no—
247C4⋊C4—21no—
248C4⋊C4—21no—
249C4⋊C4—21no—
250C4⋊C4—21no—
251C4⋊C4—21no—
252C4⋊C4—41no—
253C4⋊C4—41no—
254C4⋊C4—41no—
255C4⋊C4—41no—
256C4⋊C4—62no—
257C4⋊C4—62no—
258C4⋊C4—121no—
259C4⋊C4—121no—
260C4⋊C4—244no—
261C22⋊C4—41no—
262C22⋊C4—41no—
263C42—64no—
264C42D3×D411yescharacteristic
265C42#23711yescharacteristic
266C42C2×#6711yescharacteristic
267C42C2×#6711yescharacteristic
268C22×C3—21no—
269C22×C3—21no—
270C22×C3—81no—
271C22×C3—81no—
272C22×C3—81no—
273C22×C3#39211yescharacteristic
274C22×C3#45412yes—
275C3×C4—21no—
276C3×C4—21no—
277C3×C4—21no—
278C3×C4—21no—
279C3×C4—21no—
280C3×C4—81no—
281C3×C4—81no—
282C3×C4—81no—
283C3×C4#45612yes—
284Dic3—44no—
285Dic3—82no—
286C23—41no—
287C23#93711yescharacteristic
288Q8—42no—
289Q8—242no—
290Q8—242no—
291D4—42no—
292D4—42no—
293D4—81no—
294D4—82no—
295C2×C4—22no—
296C2×C4—22no—
297C2×C4—41no—
298C2×C4—42no—
299C2×C4—42no—
300C2×C4—41no—
301C2×C4—41no—
302C2×C4—41no—
303C2×C4—64no—
304C2×C4—82no—
305C2×C4—122no—
306C2×C4—122no—
307C2×C4—124no—
308C2×C4—122no—
309C2×C4#93711yescharacteristic
310C2×C4#93811yescharacteristic
311C2×C4#93911yescharacteristic
312C2×C4#93911yescharacteristic
313C2×C4#94011yescharacteristic
314C2×C4#95311yescharacteristic
315C8—42no—
316C8—122no—
317C8—241no—
318C2×C3—22no—
319C2×C3—161no—
320C2×C3#200311yescharacteristic
321C2×C3#200511yescharacteristic
322C2×C3#200811yescharacteristic
323C22—21no—
324C22—21no—
325C22—81no—
326C22—81no—
327C22—81no—
328C22#520112yes—
329C22#522311yescharacteristic
330C4—21no—
331C4—21no—
332C4—21no—
333C4—21no—
334C4—21no—
335C4—81no—
336C4—81no—
337C4—81no—
338C4—124no—
339C4—242no—
340C4#524612yes—
341C3#1009311yescharacteristic, complemented, Sylow
342C2—22no—
343C2—161no—
344C2#7186811yescharacteristic
345C2#7209611yescharacteristic
346C2#7211911yescharacteristic
subgroupG/H
centerC22#5223
commutator, cosocleC2×C3×C42C23
FrattiniC2×C42C22×D3
FittingC3×#3234C2
socleC22×C3#392
prime radicalC3×#532C22
hypercenter#3234D3
hypocenterC3#10093
LCSG ▷ C2×C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3234

Complements

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

NH ≅ G/Ntype
3. #83044C2semidirect
5. #71735C2semidirect
6. #70822C2semidirect
7. #67748C2semidirect
341. C3#10093semidirect

Maximal subgroups

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

subgroupindex
C3×#32342
#832832
#830442
#718122
#717352
#708222
#677482
#100933