Finite Group Databasebeta

← #10697884

Subgroups — #10697884

Order 768.

652 conjugacy classes of subgroups; only the 105 normal subgroups are shown.

normal subgroups only

This group has 652 conjugacy classes of subgroups — too many to draw. Only its 105 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×#2858C211yesmaximal, characteristic
2#83222C211yesmaximal, characteristic, complemented
3#83169C211yesmaximal, characteristic
4#71774C211yesmaximal, characteristic, complemented
5#71759C211yesmaximal, characteristic
6#69396C211yesmaximal, characteristic, complemented
7#67611C211yesmaximal, characteristic, complemented
8#10291—31nomaximal, Sylow
9C2×C3×C4⋊Q8C2211yescharacteristic
10C2×C3×#127C2211yescharacteristic
11C2×C3×#124—22no—
12#5470C2211yescharacteristic
13C3×#390—22no—
14C3×#390—22no—
15C3×C43C2211yescharacteristic
16#5228—22no—
17#5218—22no—
18#5092—21no—
19#5090—22no—
20#5079—21no—
21#5076—21no—
22#5075—22no—
23#5070—21no—
24#4925C2211yescharacteristic
25#4925C2211yescharacteristic
26#4924C2211yescharacteristic
27#4546—22no—
28#4544—22no—
29#4523—22no—
30#3209—31no—
31#3208—31no—
32#2858D311yescharacteristic
33#1994—31no—
34#1548—31no—
35#1545—31no—
36#1276—31no—
37C22×C3×Q8D412yes—
38C22×C3×D4—21no—
39C22×C3×D4—21no—
40C22×C3×D4D412yes—
41C3×C4⋊Q8—21no—
42C3×C4⋊Q8—21no—
43C3×C4⋊Q8—21no—
44C3×C4⋊Q8—21no—
45C3×#127—21no—
46C3×#127—21no—
47C3×#127—21no—
48C3×#127—21no—
49C3×#124—48no—
50C2×C3×C4⋊C4—21no—
51C2×C3×C4⋊C4—21no—
52C2×C3×C22⋊C4—28no—
53C2×C3×C42—21no—
54C2×C3×C42—21no—
55C2×C3×C42D412yes—
56C2×C3×C42C2311yescharacteristic
57C2×#216—24no—
58C2×#214—24no—
59#926—21no—
60#924—21no—
61#924—21no—
62C2×#208—21no—
63#870—21no—
64#869D411yescharacteristic
65#869D411yescharacteristic
66C4×Dic6—21no—
67C4×Dic6—21no—
68#835—24no—
69#832—24no—
70#830—21no—
71#830—21no—
72C4⋊(C3⋊C8)—21no—
73C4⋊(C3⋊C8)—21no—
74C4⋊(C3⋊C8)D412yes—
75C4⋊(C3⋊C8)D412yes—
76C4×C3⋊C8—21no—
77C4×C3⋊C8—21no—
78#533—31no—
79C2×C4⋊Q8C2×D311yescharacteristic
80C2×#127C2×D311yescharacteristic
81C2×#124—22no—
82#494—61no—
83#492—61no—
84#487—61no—
85#486—61no—
86#462—62no—
87#460—62no—
88#451—62no—
89#450—62no—
90#431—31no—
91#431—31no—
92#423—31no—
93#390—22no—
94#390—22no—
95C43C2×D311yescharacteristic
96#339—62no—
97#333—62no—
98#331—62no—
99C24×C3—22no—
100C2×C3×Q8—22no—
101C2×C3×Q8—22no—
102C2×C3×Q8—44no—
103C2×C3×D4—22no—
104C2×C3×D4—22no—
105C2×C3×D4—44no—
106C2×C3×D4—44no—
107C2×C3×D4—41no—
108C2×C3×D4—41no—
109C2×C3×D4—41no—
110C2×C3×D4—41no—
111C2×C3×D4—42no—
112C2×C3×D4—42no—
113C22×C3×C4—22no—
114C22×C3×C4—22no—
115C22×C3×C4C2×D411yescharacteristic
116C22×C3×C4C2×D411yescharacteristic
117C22×C3×C4C2×D411yescharacteristic
118C2×Dic6—21no—
119C2×Dic6—21no—
120C3×C4⋊C4—41no—
121C3×C4⋊C4—41no—
122C3×C4⋊C4—41no—
123C3×C4⋊C4—41no—
124C3×C22⋊C4—416no—
125C3×C42—24no—
126C3×C42—22no—
127C3×C42—22no—
128C3×C42—42no—
129C3×C42C2×D411yescharacteristic
130C3×C42C2×D411yescharacteristic
131C3×C42C2×D411yescharacteristic
132C3×C42#4111yescharacteristic
133#217—41no—
134#216—84no—
135#214—84no—
136C4⋊Dic3—21no—
137C4⋊Dic3—21no—
138C4⋊Dic3—21no—
139C4⋊Dic3—21no—
140Dic3⋊C4—41no—
141Dic3⋊C4—41no—
142C4×Dic3—41no—
143#208—42no—
144C2×C3⋊C8—22no—
145C2×C3⋊C8—22no—
146C2×C3⋊C8—41no—
147C22×Q8#6712yes—
148C22×D4—21no—
149C22×D4—21no—
150C22×D4#6712yes—
151C2×Q16—64no—
152C2×SD16—64no—
153C2×M16—61no—
154C4⋊Q8—21no—
155C4⋊Q8—21no—
156C4⋊Q8—21no—
157C4⋊Q8—21no—
158C4⋊Q8—31no—
159C4⋊Q8—31no—
160#127—21no—
161#127—21no—
162#127—21no—
163#127—21no—
164#125—61no—
165#124—48no—
166C22⋊Q8—61no—
167C22⋊Q8—61no—
168C4×Q8—61no—
169C4×Q8—61no—
170#117—61no—
171C2×C4⋊C4—21no—
172C2×C4⋊C4—21no—
173C2×C22⋊C4—28no—
174C2×C42—21no—
175C2×C42—21no—
176C2×C42#6712yes—
177C2×C42C22×D311yescharacteristic
178C4⋊C8—32no—
179C4⋊C8—32no—
180C4⋊C8—61no—
181C4⋊C8—61no—
182#103—64no—
183#102—64no—
184C22⋊C8—61no—
185C22⋊C8—61no—
186C4×C8—61no—
187C4×C8—61no—
188C23×C3—44no—
189C23×C3—44no—
190C23×C3—42no—
191C23×C3—44no—
192C23×C3#12011yescharacteristic
193C3×Q8—44no—
194C3×Q8—82no—
195C3×D4—44no—
196C3×D4—82no—
197C3×D4—82no—
198C3×D4—82no—
199C3×D4—82no—
200C3×D4—81no—
201C3×D4—81no—
202C2×C3×C4—22no—
203C2×C3×C4—22no—
204C2×C3×C4—24no—
205C2×C3×C4—42no—
206C2×C3×C4—42no—
207C2×C3×C4—44no—
208C2×C3×C4—42no—
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×C3×C4#13512yes—
216C2×C3×C4#13712yes—
217C2×Dic3—41no—
218C2×Dic3—41no—
219Dic6—42no—
220Dic6—42no—
221C3⋊C8—42no—
222C3⋊C8—42no—
223C3⋊C8—81no—
224C24—22no—
225C2×Q8—22no—
226C2×Q8—22no—
227C2×Q8—44no—
228C2×Q8—61no—
229C2×Q8—61no—
230C2×D4—22no—
231C2×D4—22no—
232C2×D4—44no—
233C2×D4—44no—
234C2×D4—41no—
235C2×D4—41no—
236C2×D4—41no—
237C2×D4—41no—
238C2×D4—42no—
239C2×D4—42no—
240C22×C4—22no—
241C22×C4—22no—
242C22×C4C2×#6711yescharacteristic
243C22×C4C2×#6711yescharacteristic
244C22×C4C2×#6711yescharacteristic
245Q16—244no—
246SD16—244no—
247M16—122no—
248C2×C8—62no—
249C2×C8—62no—
250C2×C8—121no—
251C4⋊C4—41no—
252C4⋊C4—41no—
253C4⋊C4—41no—
254C4⋊C4—41no—
255C4⋊C4—61no—
256C4⋊C4—61no—
257C4⋊C4—61no—
258C4⋊C4—61no—
259C4⋊C4—121no—
260C4⋊C4—121no—
261C22⋊C4—416no—
262C22⋊C4—121no—
263C42—24no—
264C42—22no—
265C42—22no—
266C42—42no—
267C42—121no—
268C42D3×D411yescharacteristic
269C42D3×D411yescharacteristic
270C42D3×D411yescharacteristic
271C42#23711yescharacteristic
272C22×C3—21no—
273C22×C3—21no—
274C22×C3—88no—
275C22×C3—82no—
276C22×C3—82no—
277C22×C3—82no—
278C22×C3#39211yescharacteristic
279C22×C3#45212yes—
280C3×C4—21no—
281C3×C4—21no—
282C3×C4—44no—
283C3×C4—82no—
284C3×C4#45312yes—
285C3×C4#45512yes—
286C3×C4#46412yes—
287C3×C4#49511yescharacteristic
288C3×C4#49711yescharacteristic
289Dic3—81no—
290Dic3—81no—
291C23—44no—
292C23—44no—
293C23—42no—
294C23—44no—
295C23#95311yescharacteristic
296Q8—44no—
297Q8—82no—
298Q8—122no—
299Q8—122no—
300D4—44no—
301D4—82no—
302D4—82no—
303D4—82no—
304D4—81no—
305D4—81no—
306D4—82no—
307C2×C4—22no—
308C2×C4—22no—
309C2×C4—24no—
310C2×C4—42no—
311C2×C4—42no—
312C2×C4—44no—
313C2×C4—42no—
314C2×C4—121no—
315C2×C4—121no—
316C2×C4#93711yescharacteristic
317C2×C4#93711yescharacteristic
318C2×C4#93811yescharacteristic
319C2×C4#93911yescharacteristic
320C2×C4#93911yescharacteristic
321C2×C4#94011yescharacteristic
322C2×C4#95012yes—
323C2×C4#95112yes—
324C8—122no—
325C8—122no—
326C8—241no—
327C2×C3—22no—
328C2×C3—162no—
329C2×C3#200211yescharacteristic
330C2×C3#200411yescharacteristic
331C2×C3#201511yescharacteristic
332C22—21no—
333C22—21no—
334C22—88no—
335C22—82no—
336C22—82no—
337C22—82no—
338C22#522311yescharacteristic
339C22#523912yes—
340C4—21no—
341C4—21no—
342C4—44no—
343C4—82no—
344C4—241no—
345C4—241no—
346C4#505412yes—
347C4#505911yescharacteristic
348C4#506011yescharacteristic
349C4#506112yes—
350C4#507712yes—
351C3#1029111yescharacteristic, complemented, Sylow
352C2—22no—
353C2—162no—
354C2#7172911yescharacteristic
355C2#7173111yescharacteristic
356C2#7176911yescharacteristic
subgroupG/H
centerC22#5223
commutator, cosocleC2×C3×C42C23
FrattiniC2×C42C22×D3
FittingC3×#2858C2
socleC22×C3#392
prime radicalC2×C3×#127C22
hypercenter#2858D3
hypocenterC3#10291
LCSG ▷ C2×C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2858

Complements

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

NH ≅ G/Ntype
2. #83222C2semidirect
4. #71774C2semidirect
6. #69396C2semidirect
7. #67611C2semidirect
351. C3#10291semidirect

Maximal subgroups

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

subgroupindex
C3×#28582
#832222
#831692
#717742
#717592
#693962
#676112
#102913