Finite Group Databasebeta

← #10696795

Subgroups — #10696795

Order 768.

454 conjugacy classes of subgroups; only the 81 normal subgroups are shown.

normal subgroups only

This group has 454 conjugacy classes of subgroups — too many to draw. Only its 81 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#83319C211yesmaximal, characteristic
3#83015C211yesmaximal, characteristic, complemented
4#71801C211yesmaximal, characteristic
5#71737C211yesmaximal, characteristic, complemented
6#70803C211yesmaximal, characteristic, complemented
7#67753C211yesmaximal, characteristic, complemented
8#10126—31nomaximal, Sylow
9C3×#532C2211yescharacteristic
10C2×C3×#125C2211yescharacteristic
11#5470C2211yescharacteristic
12C3×#501—21no—
13C3×#490—21no—
14C3×#481—21no—
15C3×#480—21no—
16C3×#478—22no—
17C3×#432C2211yescharacteristic
18#5239—22no—
19#5090—21no—
20#5075—21no—
21#5065—22no—
22#5064—21no—
23#5064—21no—
24#5060—21no—
25#5055—21no—
26#4964—21no—
27#4964—21no—
28#4926C2211yescharacteristic
29#4925C2211yescharacteristic
30C4×#208C2211yescharacteristic
31D4⋊(C3⋊C8)—22no—
32#4528—22no—
33#4484—22no—
34#3234D311yescharacteristic
35#3216—31no—
36#3166—31no—
37#1497—31no—
38#1484—31no—
39#1470—31no—
40#1291—31no—
41C2×C3×#41—21no—
42C3×C4⋊Q8D411yescharacteristic
43C3×#127D411yescharacteristic
44C3×#125—21no—
45C3×#125—21no—
46C3×#125—21no—
47C3×#125—21no—
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⋊C4—21no—
55C2×C3×C4⋊C4D411yescharacteristic
56C2×C3×C4⋊C4D411yescharacteristic
57C2×C3×C42C2311yescharacteristic
58C2×#216—22no—
59C2×#214—22no—
60#926—21no—
61#924—21no—
62#924—21no—
63C2×#208—21no—
64#870—21no—
65#869D411yescharacteristic
66#869D411yescharacteristic
67C4×Dic6—21no—
68C4×Dic6—21no—
69C3×#107—22no—
70C3×#106—22no—
71C3×C4⋊C8D412yes—
72C3×#102—22no—
73C3×#102—22no—
74C3×C22⋊C8—21no—
75C3×#97—21no—
76#835—22no—
77#832—22no—
78#830—21no—
79#830D412yes—
80#808—22no—
81#808—22no—
82#807—22no—
83#807—22no—
84C4⋊(C3⋊C8)—21no—
85C4⋊(C3⋊C8)—21no—
86#803—21no—
87#803—21no—
88C4×C3⋊C8—21no—
89C4×C3⋊C8D412yes—
90#533—31no—
91#532C2×D311yescharacteristic
92C2×#125C2×D311yescharacteristic
93#501—21no—
94#501—61no—
95#501—61no—
96#499—62no—
97#497—61no—
98#490—21no—
99#489—61no—
100#481—21no—
101#480—21no—
102#478—22no—
103#462—61no—
104#460—61no—
105#452—62no—
106#432—31no—
107#432C2×D311yescharacteristic
108#431—31no—
109#428—61no—
110#428—61no—
111C4×M16—31no—
112Q8⋊C8—62no—
113D4⋊C8—62no—
114#324—62no—
115C3×#41—82no—
116C2×C3×Q8—21no—
117C2×C3×D4—21no—
118C2×C3×D4—21no—
119C2×C3×D4—21no—
120C2×C3×D4—41no—
121C22×C3×C4—21no—
122C22×C3×C4—21no—
123C22×C3×C4—41no—
124C22×C3×C4C2×D411yescharacteristic
125C22×C3×C4C2×D411yescharacteristic
126C22×C3×C4#4111yescharacteristic
127C2×Dic6—21no—
128C2×Dic6—21no—
129C2×C3×C8—22no—
130C3×C4⋊C4—21no—
131C3×C4⋊C4—21no—
132C3×C4⋊C4—21no—
133C3×C4⋊C4—21no—
134C3×C4⋊C4—21no—
135C3×C4⋊C4—21no—
136C3×C4⋊C4—41no—
137C3×C4⋊C4—41no—
138C3×C4⋊C4—41no—
139C3×C4⋊C4—41no—
140C3×C22⋊C4—41no—
141C3×C22⋊C4—41no—
142C3×C42C2×D411yescharacteristic
143C3×C42C2×D411yescharacteristic
144C3×C42C2×D411yescharacteristic
145C3×C42C2×D411yescharacteristic
146#217—41no—
147#216—82no—
148#214—82no—
149C4⋊Dic3—21no—
150C4⋊Dic3—21no—
151C4⋊Dic3—21no—
152C4⋊Dic3—21no—
153Dic3⋊C4—41no—
154Dic3⋊C4—41no—
155C4×Dic3—41no—
156#208—24no—
157C2×C3⋊C8—22no—
158C2×C3⋊C8—22no—
159C2×C3⋊C8—41no—
160C2×#41—21no—
161C2×Q16—62no—
162C2×SD16—62no—
163C2×M16—61no—
164C4⋊Q8—31no—
165C4⋊Q8—31no—
166C4⋊Q8#6711yescharacteristic
167#127#6711yescharacteristic
168#125—21no—
169#125—21no—
170#125—21no—
171#125—21no—
172#125—61no—
173#124—21no—
174C22⋊Q8—61no—
175C22⋊Q8—61no—
176#121—21no—
177#121—21no—
178C4×Q8—61no—
179C4×Q8—61no—
180C4×D4—21no—
181C4×D4—21no—
182#117—61no—
183C2×C4⋊C4—21no—
184C2×C4⋊C4—21no—
185C2×C4⋊C4#6711yescharacteristic
186C2×C4⋊C4#6711yescharacteristic
187C2×C42C22×D311yescharacteristic
188#107—22no—
189#107—62no—
190#107—62no—
191#106—22no—
192#106—62no—
193#106—62no—
194C4⋊C8—61no—
195C4⋊C8—61no—
196C4⋊C8#6712yes—
197#103—62no—
198#102—22no—
199#102—22no—
200#102—62no—
201C22⋊C8—21no—
202C22⋊C8—32no—
203C22⋊C8—61no—
204#97—21no—
205#97—61no—
206#97—61no—
207C4×C8—32no—
208C4×C8—61no—
209C23×C3—41no—
210C23×C3#12111yescharacteristic
211C3×Q8—42no—
212C3×D4—42no—
213C3×D4—42no—
214C3×D4—42no—
215C3×D4—82no—
216C2×C3×C4—22no—
217C2×C3×C4—41no—
218C2×C3×C4—41no—
219C2×C3×C4—42no—
220C2×C3×C4—41no—
221C2×C3×C4—41no—
222C2×C3×C4—41no—
223C2×C3×C4—82no—
224C2×C3×C4—81no—
225C2×C3×C4#12011yescharacteristic
226C2×C3×C4#12011yescharacteristic
227C2×C3×C4#12011yescharacteristic
228C2×C3×C4#12111yescharacteristic
229C2×C3×C4#12111yescharacteristic
230C2×C3×C4#12711yescharacteristic
231C2×Dic3—41no—
232C2×Dic3—41no—
233Dic6—41no—
234Dic6—41no—
235Dic6—42no—
236C3×C8—42no—
237C3⋊C8—24no—
238C3⋊C8—42no—
239C3⋊C8—81no—
240#41—82no—
241C2×Q8—21no—
242C2×Q8—61no—
243C2×Q8—61no—
244C2×D4—21no—
245C2×D4—21no—
246C2×D4—21no—
247C2×D4—41no—
248C22×C4—21no—
249C22×C4—21no—
250C22×C4—41no—
251C22×C4D3×D411yescharacteristic
252C22×C4#23711yescharacteristic
253C22×C4C2×#6711yescharacteristic
254Q16—242no—
255SD16—242no—
256M16—64no—
257C2×C8—22no—
258C2×C8—62no—
259C2×C8—62no—
260C2×C8—121no—
261C4⋊C4—21no—
262C4⋊C4—21no—
263C4⋊C4—21no—
264C4⋊C4—21no—
265C4⋊C4—21no—
266C4⋊C4—21no—
267C4⋊C4—41no—
268C4⋊C4—41no—
269C4⋊C4—41no—
270C4⋊C4—41no—
271C4⋊C4—61no—
272C4⋊C4—61no—
273C4⋊C4—61no—
274C4⋊C4—61no—
275C4⋊C4—121no—
276C4⋊C4—121no—
277C22⋊C4—41no—
278C22⋊C4—41no—
279C22⋊C4—121no—
280C42—121no—
281C42D3×D411yescharacteristic
282C42D3×D411yescharacteristic
283C42C2×#6711yescharacteristic
284C42C2×#6711yescharacteristic
285C22×C3—21no—
286C22×C3—21no—
287C22×C3—21no—
288C22×C3—81no—
289C22×C3—81no—
290C22×C3—81no—
291C22×C3#39211yescharacteristic
292C3×C4—21no—
293C3×C4—21no—
294C3×C4—21no—
295C3×C4—21no—
296C3×C4—21no—
297C3×C4—81no—
298C3×C4—81no—
299C3×C4—81no—
300C3×C4#45612yes—
301Dic3—81no—
302Dic3—81no—
303C23—41no—
304C23#93911yescharacteristic
305Q8—42no—
306Q8—121no—
307Q8—121no—
308Q8—122no—
309D4—42no—
310D4—42no—
311D4—42no—
312D4—82no—
313C2×C4—22no—
314C2×C4—41no—
315C2×C4—41no—
316C2×C4—42no—
317C2×C4—41no—
318C2×C4—41no—
319C2×C4—41no—
320C2×C4—82no—
321C2×C4—81no—
322C2×C4—121no—
323C2×C4—121no—
324C2×C4#93711yescharacteristic
325C2×C4#93711yescharacteristic
326C2×C4#93811yescharacteristic
327C2×C4#93911yescharacteristic
328C2×C4#94011yescharacteristic
329C2×C4#95311yescharacteristic
330C8—42no—
331C8—64no—
332C8—122no—
333C8—241no—
334C2×C3—41no—
335C2×C3—161no—
336C2×C3#200511yescharacteristic
337C2×C3#200811yescharacteristic
338C2×C3#201711yescharacteristic
339C22—21no—
340C22—21no—
341C22—21no—
342C22—81no—
343C22—81no—
344C22—81no—
345C22#522311yescharacteristic
346C4—21no—
347C4—21no—
348C4—21no—
349C4—21no—
350C4—21no—
351C4—81no—
352C4—81no—
353C4—81no—
354C4—241no—
355C4—241no—
356C4#524612yes—
357C3#1012611yescharacteristic, complemented, Sylow
358C2—41no—
359C2—161no—
360C2#7186811yescharacteristic
361C2#7209911yescharacteristic
362C2#7212111yescharacteristic
subgroupG/H
centerC22#5223
commutator, cosocleC2×C3×C42C23
FrattiniC2×C42C22×D3
FittingC3×#3234C2
socleC22×C3#392
prime radicalC3×#532C22
hypercenter#3234D3
hypocenterC3#10126
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. #83015C2semidirect
5. #71737C2semidirect
6. #70803C2semidirect
7. #67753C2semidirect
357. C3#10126semidirect

Maximal subgroups

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

subgroupindex
C3×#32342
#833192
#830152
#718012
#717372
#708032
#677532
#101263