Finite Group Databasebeta

← #10696561

Subgroups — #10696561

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×#3141C211yesmaximal, characteristic
2#83273C211yesmaximal, characteristic
3#83246C211yesmaximal, characteristic, complemented
4#71812C211yesmaximal, characteristic
5#71776C211yesmaximal, characteristic, complemented
6#69863C211yesmaximal, characteristic, complemented
7#67665C211yesmaximal, characteristic, complemented
8#10093—31nomaximal, Sylow
9C3×#532C2211yescharacteristic
10C2×C3×C4⋊Q8C2211yescharacteristic
11C2×#870C2211yescharacteristic
12C3×#489—22no—
13C3×#487—21no—
14C3×#486—21no—
15C3×#417—22no—
16C3×C4×M16C2211yescharacteristic
17#5230—22no—
18#5081—21no—
19#5071—21no—
20#5034—21no—
21#5033—21no—
22#5009—22no—
23#4983—22no—
24#4977—21no—
25#4975—21no—
26#4932C2211yescharacteristic
27#4926C2211yescharacteristic
28#4926C2211yescharacteristic
29#4547—22no—
30#4537—22no—
31#4465—22no—
32#3234—31no—
33#3141D311yescharacteristic
34#2660—31no—
35#1556—31no—
36#1528—31no—
37#1505—31no—
38#1300—31no—
39C2×C3×#41—21no—
40C22×C3×Q8—21no—
41C2×C3×M16—21no—
42C3×C4⋊Q8—21no—
43C3×C4⋊Q8—21no—
44C3×C4⋊Q8—21no—
45C3×C4⋊Q8—21no—
46C3×C4⋊Q8D411yescharacteristic
47C3×#127D411yescharacteristic
48C3×#124—21no—
49C3×#121—21no—
50C3×#121—21no—
51C3×C4×D4—21no—
52C3×C4×D4—21no—
53C2×C3×C4⋊C4—21no—
54C2×C3×C4⋊C4D412yes—
55C2×C3×C42C2311yescharacteristic
56C2×C4⋊Dic3D411yescharacteristic
57C2×C4⋊Dic3D411yescharacteristic
58C2×Dic3⋊C4—21no—
59C2×Dic3⋊C4—21no—
60C2×Dic3⋊C4—24no—
61C2×C4×Dic3—21no—
62#870—21no—
63#870—21no—
64#870—21no—
65#870—21no—
66C3×#103—24no—
67C3×#102—24no—
68C3×#97D412yes—
69C3×C4×C8—21no—
70#832—22no—
71#832—22no—
72C3⋊#95—24no—
73C3⋊#95—21no—
74C3⋊#95D412yes—
75#830—21no—
76#830—21no—
77#808—22no—
78#807—22no—
79C4⋊(C3⋊C8)—21no—
80C4⋊(C3⋊C8)D412yes—
81#803—21no—
82#803—21no—
83#532C2×D311yescharacteristic
84C2×C4⋊Q8C2×D311yescharacteristic
85C2×#125—31no—
86#501—61no—
87#490—61no—
88#489—22no—
89#487—21no—
90#486—21no—
91#481—61no—
92#480—61no—
93#478—62no—
94#432—31no—
95#432—31no—
96#417—22no—
97C4×M16C2×D311yescharacteristic
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×Q8—41no—
109C2×C3×Q8—41no—
110C2×C3×Q8—82no—
111C2×C3×D4—21no—
112C2×C3×D4—21no—
113C2×C3×D4—21no—
114C2×C3×D4—41no—
115C22×C3×C4—22no—
116C22×C3×C4—41no—
117C22×C3×C4C2×D411yescharacteristic
118C22×C3×C4C2×D411yescharacteristic
119C22×C3×C4C2×D411yescharacteristic
120C22×Dic3—22no—
121C22×Dic3—21no—
122C22×Dic3—21no—
123C3×M16—42no—
124C2×C3×C8—22no—
125C3×C4⋊C4—21no—
126C3×C4⋊C4—22no—
127C3×C4⋊C4—22no—
128C3×C4⋊C4—21no—
129C3×C4⋊C4—41no—
130C3×C4⋊C4—41no—
131C3×C22⋊C4—41no—
132C3×C22⋊C4—41no—
133C3×C42C2×D411yescharacteristic
134C3×C42C2×D411yescharacteristic
135C3×C42C2×D411yescharacteristic
136C3×C42#4111yescharacteristic
137C4⋊Dic3—21no—
138C4⋊Dic3—21no—
139C4⋊Dic3—21no—
140C4⋊Dic3—21no—
141Dic3⋊C4—41no—
142Dic3⋊C4—41no—
143Dic3⋊C4—41no—
144Dic3⋊C4—41no—
145Dic3⋊C4—84no—
146C4×Dic3—24no—
147C2×C3⋊C8—22no—
148C2×C3⋊C8—41no—
149C2×#41—21no—
150C22×Q8—21no—
151C2×M16—21no—
152C4⋊Q8—21no—
153C4⋊Q8—21no—
154C4⋊Q8—21no—
155C4⋊Q8—21no—
156C4⋊Q8#6711yescharacteristic
157#127#6711yescharacteristic
158#125—61no—
159#125—61no—
160#125—61no—
161#125—61no—
162#124—21no—
163#121—21no—
164#121—21no—
165C4×D4—21no—
166C4×D4—21no—
167C2×C4⋊C4—21no—
168C2×C4⋊C4—31no—
169C2×C4⋊C4—31no—
170C2×C4⋊C4—61no—
171C2×C4⋊C4—61no—
172C2×C4⋊C4—64no—
173C2×C4⋊C4#6712yes—
174C2×C42—61no—
175C2×C42C22×D311yescharacteristic
176#107—62no—
177#106—62no—
178C4⋊C8—32no—
179C4⋊C8—61no—
180#103—24no—
181#102—24no—
182#102—62no—
183#102—62no—
184C22⋊C8—61no—
185C22⋊C8—61no—
186#97—61no—
187#97—61no—
188#97#6712yes—
189C4×C8—21no—
190#95—32no—
191#95—64no—
192#95—61no—
193C23×C3—41no—
194C23×C3#12011yescharacteristic
195C3×Q8—42no—
196C3×Q8—82no—
197C3×Q8—82no—
198C3×D4—42no—
199C3×D4—42no—
200C3×D4—81no—
201C3×D4—82no—
202C2×C3×C4—22no—
203C2×C3×C4—22no—
204C2×C3×C4—41no—
205C2×C3×C4—44no—
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—42no—
219C2×Dic3—41no—
220C2×Dic3—41no—
221C2×Dic3—42no—
222C3×C8—42no—
223C3⋊C8—42no—
224C3⋊C8—81no—
225#41—44no—
226C2×Q8—21no—
227C2×Q8—41no—
228C2×Q8—41no—
229C2×Q8—82no—
230C2×D4—21no—
231C2×D4—21no—
232C2×D4—21no—
233C2×D4—41no—
234C22×C4—22no—
235C22×C4—41no—
236C22×C4—62no—
237C22×C4—61no—
238C22×C4—61no—
239C22×C4D3×D411yescharacteristic
240C22×C4D3×D411yescharacteristic
241C22×C4C2×#6711yescharacteristic
242M16—42no—
243C2×C8—22no—
244C2×C8—62no—
245C2×C8—121no—
246C4⋊C4—21no—
247C4⋊C4—22no—
248C4⋊C4—21no—
249C4⋊C4—22no—
250C4⋊C4—41no—
251C4⋊C4—41no—
252C4⋊C4—61no—
253C4⋊C4—61no—
254C4⋊C4—61no—
255C4⋊C4—61no—
256C4⋊C4—121no—
257C4⋊C4—121no—
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—82no—
282C3×C4#45612yes—
283Dic3—44no—
284Dic3—81no—
285Dic3—81no—
286C23—41no—
287C23#93711yescharacteristic
288Q8—42no—
289Q8—82no—
290Q8—82no—
291D4—42no—
292D4—42no—
293D4—81no—
294D4—82no—
295C2×C4—22no—
296C2×C4—22no—
297C2×C4—41no—
298C2×C4—44no—
299C2×C4—41no—
300C2×C4—42no—
301C2×C4—64no—
302C2×C4—82no—
303C2×C4—122no—
304C2×C4—122no—
305C2×C4—122no—
306C2×C4—121no—
307C2×C4—121no—
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#506212yes—
329C22#522311yescharacteristic
330C4—21no—
331C4—21no—
332C4—21no—
333C4—21no—
334C4—21no—
335C4—81no—
336C4—82no—
337C4—124no—
338C4—241no—
339C4—241no—
340C4#520512yes—
341C3#1009311yescharacteristic, complemented, Sylow
342C2—22no—
343C2—161no—
344C2#7173011yescharacteristic
345C2#7209911yescharacteristic
346C2#7210811yescharacteristic
subgroupG/H
centerC22#5223
commutator, cosocleC2×C3×C42C23
FrattiniC2×C42C22×D3
FittingC3×#3141C2
socleC22×C3#392
prime radicalC3×#532C22
hypercenter#3141D3
hypocenterC3#10093
LCSG ▷ C2×C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3141

Complements

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

NH ≅ G/Ntype
3. #83246C2semidirect
5. #71776C2semidirect
6. #69863C2semidirect
7. #67665C2semidirect
341. C3#10093semidirect

Maximal subgroups

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

subgroupindex
C3×#31412
#832732
#832462
#718122
#717762
#698632
#676652
#100933