Finite Group Databasebeta

← #10696797

Subgroups — #10696797

Order 768.

454 conjugacy classes of subgroups; only the 79 normal subgroups are shown.

normal subgroups only

This group has 454 conjugacy classes of subgroups — too many to draw. Only its 79 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#83319C211yesmaximal, characteristic
2#83273C211yesmaximal, characteristic
3#78507C211yesmaximal, characteristic, complemented
4C3×#1484C211yesmaximal, characteristic
5#70787C211yesmaximal, characteristic, complemented
6#70771C211yesmaximal, characteristic, complemented
7#67670C211yesmaximal, characteristic, complemented
8#10126—31nomaximal, Sylow
9C3×#532C2211yescharacteristic
10#5470C2211yescharacteristic
11C2×#870C2211yescharacteristic
12C3×#431C2211yescharacteristic
13C3×C4×M16C2211yescharacteristic
14#5239—22no—
15#5121—21no—
16#5121—21no—
17#5090—21no—
18#5081—21no—
19#5075—21no—
20#5071—21no—
21#5060—21no—
22#5055—21no—
23#5034—21no—
24#5033—21no—
25#5009—22no—
26#4926C2211yescharacteristic
27#4926C2211yescharacteristic
28#4702—21no—
29#4702—21no—
30#4683—22no—
31C3×D4⋊C8—22no—
32#4469—22no—
33#4456—22no—
34#3234—31no—
35#3216—31no—
36#3166—31no—
37#1497—31no—
38#1484D311yescharacteristic
39#1470—31no—
40#1291—31no—
41C2×C3×#41—21no—
42C2×C3×M16—21no—
43C3×C4⋊Q8D411yescharacteristic
44C3×#127D411yescharacteristic
45C3×#124—21no—
46C3×#121—21no—
47C3×#121—21no—
48C3×C4×D4—21no—
49C3×C4×D4—21no—
50C2×C3×C42C2311yescharacteristic
51C2×#216—22no—
52C2×#214—22no—
53#926—21no—
54C2×C4⋊Dic3D411yescharacteristic
55C2×C4⋊Dic3D411yescharacteristic
56#924—21no—
57#924—21no—
58C2×Dic3⋊C4—21no—
59C2×Dic3⋊C4—21no—
60#870—21no—
61#870—21no—
62#870—21no—
63#870—21no—
64#870—21no—
65#869D411yescharacteristic
66#869D411yescharacteristic
67C4×Dic6—21no—
68C4×Dic6—21no—
69C3×C4⋊C8—21no—
70C3×C22⋊C8—21no—
71C3×#97—21no—
72C3×C4×C8—21no—
73C3×C4×C8D412yes—
74#835—22no—
75#832—22no—
76#832—22no—
77#832—22no—
78#830—21no—
79#830D412yes—
80#818—22no—
81#818—22no—
82#817—22no—
83#817—22no—
84#808—22no—
85#807—22no—
86C4⋊(C3⋊C8)—21no—
87C4⋊(C3⋊C8)D412yes—
88#803—21no—
89#803—21no—
90#533—31no—
91#532C2×D311yescharacteristic
92C2×#125—31no—
93#501—61no—
94#501—61no—
95#501—61no—
96#499—62no—
97#497—61no—
98#490—61no—
99#489—61no—
100#481—61no—
101#480—61no—
102#478—62no—
103#462—61no—
104#460—61no—
105#452—62no—
106#432—31no—
107#432—31no—
108#431C2×D311yescharacteristic
109#428—61no—
110#428—61no—
111C4×M16C2×D311yescharacteristic
112Q8⋊C8—62no—
113D4⋊C8—22no—
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—41no—
122C22×C3×C4C2×D411yescharacteristic
123C22×C3×C4C2×D411yescharacteristic
124C22×C3×C4#4111yescharacteristic
125C22×Dic3—21no—
126C22×Dic3—21no—
127C2×Dic6—21no—
128C2×Dic6—21no—
129C3×M16—24no—
130C2×C3×C8—22no—
131C2×C3×C8—41no—
132C3×C4⋊C4—21no—
133C3×C4⋊C4—21no—
134C3×C22⋊C4—41no—
135C3×C22⋊C4—41no—
136C3×C42C2×D411yescharacteristic
137C3×C42C2×D411yescharacteristic
138C3×C42C2×D411yescharacteristic
139C3×C42C2×D411yescharacteristic
140#217—41no—
141#216—82no—
142#214—82no—
143C4⋊Dic3—21no—
144C4⋊Dic3—21no—
145C4⋊Dic3—21no—
146C4⋊Dic3—21no—
147C4⋊Dic3—21no—
148C4⋊Dic3—21no—
149C4⋊Dic3—21no—
150C4⋊Dic3—21no—
151Dic3⋊C4—41no—
152Dic3⋊C4—41no—
153Dic3⋊C4—41no—
154Dic3⋊C4—41no—
155Dic3⋊C4—41no—
156Dic3⋊C4—41no—
157C4×Dic3—41no—
158C2×C3⋊C8—22no—
159C2×C3⋊C8—22no—
160C2×#41—21no—
161C2×Q16—62no—
162C2×SD16—62no—
163C2×M16—21no—
164C4⋊Q8—31no—
165C4⋊Q8—31no—
166C4⋊Q8D1211yescharacteristic
167#127D1211yescharacteristic
168#125—61no—
169#125—61no—
170#125—61no—
171#125—61no—
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—31no—
184C2×C4⋊C4—31no—
185C2×C4⋊C4—61no—
186C2×C4⋊C4—61no—
187C2×C42C22×D311yescharacteristic
188#107—62no—
189#107—62no—
190#107—62no—
191#106—62no—
192#106—62no—
193#106—62no—
194C4⋊C8—21no—
195C4⋊C8—32no—
196C4⋊C8—61no—
197#103—62no—
198#102—62no—
199#102—62no—
200#102—62no—
201C22⋊C8—21no—
202C22⋊C8—32no—
203C22⋊C8—61no—
204#97—21no—
205#97—61no—
206#97—61no—
207C4×C8—21no—
208C4×C8#6712yes—
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—41no—
220C2×C3×C4—82no—
221C2×C3×C4#12011yescharacteristic
222C2×C3×C4#12011yescharacteristic
223C2×C3×C4#12011yescharacteristic
224C2×C3×C4#12111yescharacteristic
225C2×C3×C4#12111yescharacteristic
226C2×C3×C4#12711yescharacteristic
227C2×Dic3—42no—
228C2×Dic3—41no—
229C2×Dic3—41no—
230C2×Dic3—41no—
231C2×Dic3—41no—
232C2×Dic3—81no—
233Dic6—41no—
234Dic6—41no—
235Dic6—42no—
236C3×C8—24no—
237C3×C8—81no—
238C3⋊C8—42no—
239C3⋊C8—42no—
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—41no—
249C22×C4—61no—
250C22×C4—61no—
251C22×C4#23511yescharacteristic
252C22×C4D3×D411yescharacteristic
253C22×C4D3×D411yescharacteristic
254Q16—242no—
255SD16—242no—
256M16—24no—
257C2×C8—22no—
258C2×C8—41no—
259C2×C8—62no—
260C2×C8—62no—
261C4⋊C4—21no—
262C4⋊C4—21no—
263C4⋊C4—61no—
264C4⋊C4—61no—
265C4⋊C4—61no—
266C4⋊C4—61no—
267C4⋊C4—61no—
268C4⋊C4—61no—
269C4⋊C4—61no—
270C4⋊C4—61no—
271C4⋊C4—121no—
272C4⋊C4—121no—
273C4⋊C4—121no—
274C4⋊C4—121no—
275C4⋊C4—121no—
276C4⋊C4—121no—
277C22⋊C4—41no—
278C22⋊C4—41no—
279C22⋊C4—121no—
280C42—121no—
281C42C2×D1211yescharacteristic
282C42D3×D411yescharacteristic
283C42D3×D411yescharacteristic
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#45612yes—
299Dic3—81no—
300Dic3—81no—
301Dic3—81no—
302Dic3—81no—
303C23—41no—
304C23#88411yescharacteristic
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—41no—
317C2×C4—82no—
318C2×C4—122no—
319C2×C4—121no—
320C2×C4—121no—
321C2×C4—121no—
322C2×C4—121no—
323C2×C4—241no—
324C2×C4#88211yescharacteristic
325C2×C4#88211yescharacteristic
326C2×C4#88411yescharacteristic
327C2×C4#93011yescharacteristic
328C2×C4#93711yescharacteristic
329C2×C4#94011yescharacteristic
330C8—24no—
331C8—81no—
332C8—122no—
333C8—122no—
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#496011yescharacteristic
346C4—21no—
347C4—21no—
348C4—21no—
349C4—21no—
350C4—21no—
351C4—81no—
352C4—241no—
353C4—241no—
354C4—241no—
355C4—241no—
356C4#482612yes—
357C3#1012611yescharacteristic, complemented, Sylow
358C2—41no—
359C2—161no—
360C2#7001311yescharacteristic
361C2#7033111yescharacteristic
362C2#7038911yescharacteristic
subgroupG/H
centerC22#4960
commutator, cosocleC2×C3×C42C23
FrattiniC2×C42C22×D3
FittingC3×#1484C2
socleC22×C3#392
prime radicalC3×#532C22
hypercenter#1484D3
hypocenterC3#10126
LCSG ▷ C2×C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1484

Complements

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

NH ≅ G/Ntype
3. #78507C2semidirect
5. #70787C2semidirect
6. #70771C2semidirect
7. #67670C2semidirect
357. C3#10126semidirect

Maximal subgroups

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

subgroupindex
#833192
#832732
#785072
C3×#14842
#707872
#707712
#676702
#101263