Finite Group Databasebeta

← #10692913

Subgroups — #10692913

Order 768.

528 conjugacy classes of subgroups; only the 93 normal subgroups are shown.

normal subgroups only

This group has 528 conjugacy classes of subgroups — too many to draw. Only its 93 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#82775C211yesmaximal, characteristic
2#74054C211yesmaximal, characteristic
3C3×#1511C211yesmaximal, characteristic
4#71689C211yesmaximal, characteristic
5#69973C211yesmaximal, characteristic
6#67682C211yesmaximal, characteristic
7#67628C211yesmaximal, characteristic
8#9461—31nomaximal, Sylow
9C2×C3×C22⋊Q8C2211yescharacteristic
10C22×C4⋊Dic3C2211yescharacteristic
11C3×#407C2211yescharacteristic
12C3×#381C2211yescharacteristic
13#5229C412yescomplemented
14#5211C2211yescharacteristic
15C4×#217C2211yescharacteristic
16#5207C2211yescharacteristic
17#4981—22no—
18#4975C412yescomplemented
19#4946—21no—
20#4946—21no—
21#4942—21no—
22Dic3×C22⋊C4—21no—
23#4654—22no—
24#4653—22no—
25C3×#328—22no—
26#4554—22no—
27#4538—22no—
28#2502—31no—
29#2313—31no—
30#1821—31no—
31#1511—31no—
32#1511D311yescharacteristic
33#1284—31no—
34#1284—31no—
35C22×C3×Q8C2×C411yescharacteristic
36C23×C3×C4C2311yescharacteristic
37C23×Dic3D411yescharacteristic
38C23×Dic3D411yescharacteristic
39C2×C3×M16—21no—
40C3×C22⋊Q8—21no—
41C3×C22⋊Q8—21no—
42C3×C22⋊Q8—21no—
43C3×C22⋊Q8—21no—
44C2×C3×C4⋊C4—21no—
45C2×C3×C4⋊C4C2×C411yescharacteristic
46C2×C3×C22⋊C4—21no—
47C2×C3×C22⋊C4—21no—
48C2×#217—21no—
49C2×#217—21no—
50C2×C4⋊Dic3—21no—
51C2×C4⋊Dic3—21no—
52C2×C4⋊Dic3—21no—
53C2×C4⋊Dic3—21no—
54C2×C4⋊Dic3C2×C412yes—
55C2×C4⋊Dic3D412yes—
56C2×Dic3⋊C4—22no—
57C2×Dic3⋊C4—22no—
58C2×Dic3⋊C4—22no—
59C2×C4×Dic3—22no—
60C2×C4×Dic3C2×C412yes—
61C2×#208—21no—
62C3×C22⋊C8—21no—
63C3×C22⋊C8—21no—
64C3×#95—21no—
65C3×#95D412yes—
66C3⋊#95—22no—
67C3⋊#95—22no—
68C3⋊#95—21no—
69C3⋊#95—21no—
70C3⋊#95—22no—
71C3⋊#95—22no—
72C3⋊#95D412yes—
73#830—21no—
74#830—21no—
75C2×C22⋊Q8C2×D311yescharacteristic
76C22×C4⋊C4—31no—
77#407—31no—
78#407C2×D311yescharacteristic
79#398—62no—
80#391—32no—
81#387—62no—
82#387—62no—
83#385—61no—
84#385—61no—
85#384—32no—
86#381—31no—
87#381C2×D311yescharacteristic
88#380—61no—
89C4×C22⋊C4—31no—
90C4×C22⋊C4—61no—
91#343—62no—
92#328—22no—
93#328—62no—
94C24×C3C2×D411yescharacteristic
95C2×C3×Q8—21no—
96C2×C3×Q8—21no—
97C2×C3×Q8—42no—
98C22×C3×C4—21no—
99C22×C3×C4—21no—
100C22×C3×C4—22no—
101C22×C3×C4—42no—
102C22×C3×C4C22×C411yescharacteristic
103C22×C3×C4C2×D411yescharacteristic
104C22×C3×C4C2×D411yescharacteristic
105C22×C3×C4C2×D411yescharacteristic
106C22×C3×C4#4111yescharacteristic
107C22×C3×C4#4111yescharacteristic
108C22×Dic3—22no—
109C22×Dic3—21no—
110C22×Dic3—21no—
111C22×Dic3—21no—
112C22×Dic3—21no—
113C22×Dic3—22no—
114C22×Dic3—81no—
115C22×Dic3—81no—
116C22×Dic3C22⋊C412yes—
117C22×Dic3C22⋊C412yes—
118C3×M16—42no—
119C2×C3×C8—41no—
120C3×C4⋊C4—21no—
121C3×C4⋊C4—21no—
122C3×C4⋊C4—41no—
123C3×C4⋊C4—41no—
124C3×C22⋊C4—42no—
125C3×C22⋊C4—41no—
126C3×C22⋊C4—41no—
127#217—24no—
128#217—42no—
129C4⋊Dic3—22no—
130C4⋊Dic3—22no—
131C4⋊Dic3—22no—
132C4⋊Dic3—22no—
133Dic3⋊C4—82no—
134Dic3⋊C4—82no—
135Dic3⋊C4—82no—
136C4×Dic3—42no—
137C4×Dic3—82no—
138#208—42no—
139C2×C3⋊C8—41no—
140C22×Q8C4×D311yescharacteristic
141C23×C4—31no—
142C23×C4—31no—
143C23×C4C22×D311yescharacteristic
144C2×M16—21no—
145C2×M16—61no—
146C22⋊Q8—21no—
147C22⋊Q8—21no—
148C22⋊Q8—21no—
149C22⋊Q8—21no—
150C2×C4⋊C4—21no—
151C2×C4⋊C4—32no—
152C2×C4⋊C4—32no—
153C2×C4⋊C4—61no—
154C2×C4⋊C4—61no—
155C2×C4⋊C4—61no—
156C2×C4⋊C4—61no—
157C2×C4⋊C4—62no—
158C2×C4⋊C4—62no—
159C2×C4⋊C4—62no—
160C2×C4⋊C4C4×D311yescharacteristic
161C2×C22⋊C4—21no—
162C2×C22⋊C4—21no—
163C2×C22⋊C4—61no—
164C2×C22⋊C4—61no—
165C2×C42—32no—
166C2×C42—62no—
167C22⋊C8—21no—
168C22⋊C8—21no—
169C22⋊C8—61no—
170C22⋊C8—61no—
171#95—21no—
172#95—32no—
173#95—62no—
174#95—62no—
175#95—62no—
176#95—62no—
177#95—61no—
178#95—61no—
179#95#6712yes—
180C23×C3—22no—
181C23×C3—22no—
182C23×C3—41no—
183C23×C3C2×C22⋊C411yescharacteristic
184C23×C3#12011yescharacteristic
185C23×C3#12311yescharacteristic
186C3×Q8—81no—
187C3×Q8—81no—
188C2×C3×C4—22no—
189C2×C3×C4—22no—
190C2×C3×C4—41no—
191C2×C3×C4—41no—
192C2×C3×C4—41no—
193C2×C3×C4—41no—
194C2×C3×C4—42no—
195C2×C3×C4—42no—
196C2×C3×C4—42no—
197C2×C3×C4—41no—
198C2×C3×C4—41no—
199C2×C3×C4—42no—
200C2×C3×C4—42no—
201C2×C3×C4C4×D411yescharacteristic
202C2×C3×C4C4×D411yescharacteristic
203C2×C3×C4#12111yescharacteristic
204C2×C3×C4#12111yescharacteristic
205C2×Dic3—24no—
206C2×Dic3—22no—
207C2×Dic3—22no—
208C2×Dic3—42no—
209C2×Dic3—42no—
210C2×Dic3—42no—
211C2×Dic3—42no—
212C2×Dic3—42no—
213C2×Dic3—42no—
214C2×Dic3—42no—
215C2×Dic3—42no—
216C2×Dic3—42no—
217C2×Dic3—81no—
218C2×Dic3—81no—
219C3×C8—81no—
220C3⋊C8—81no—
221C24D3×D411yescharacteristic
222C2×Q8—21no—
223C2×Q8—21no—
224C2×Q8—42no—
225C22×C4—21no—
226C22×C4—21no—
227C22×C4—22no—
228C22×C4—32no—
229C22×C4—32no—
230C22×C4—42no—
231C22×C4—62no—
232C22×C4—61no—
233C22×C4—61no—
234C22×C4—61no—
235C22×C4—61no—
236C22×C4—62no—
237C22×C4—241no—
238C22×C4—241no—
239C22×C4C2×C4×D311yescharacteristic
240C22×C4#23511yescharacteristic
241C22×C4D3×D411yescharacteristic
242C22×C4D3×D411yescharacteristic
243C22×C4#23711yescharacteristic
244C22×C4C2×#6711yescharacteristic
245M16—42no—
246M16—122no—
247C2×C8—41no—
248C2×C8—121no—
249C4⋊C4—21no—
250C4⋊C4—21no—
251C4⋊C4—41no—
252C4⋊C4—41no—
253C4⋊C4—62no—
254C4⋊C4—62no—
255C4⋊C4—62no—
256C4⋊C4—62no—
257C4⋊C4—242no—
258C4⋊C4—242no—
259C4⋊C4—242no—
260C22⋊C4—42no—
261C22⋊C4—41no—
262C22⋊C4—41no—
263C22⋊C4—64no—
264C22⋊C4—122no—
265C42—122no—
266C42—242no—
267C22×C3—21no—
268C22×C3—21no—
269C22×C3—21no—
270C22×C3—21no—
271C22×C3—21no—
272C22×C3—21no—
273C22×C3—21no—
274C22×C3—44no—
275C22×C3#38611yescharacteristic
276C22×C3#41012yes—
277C22×C3#45612yes—
278C3×C4—21no—
279C3×C4—21no—
280C3×C4—21no—
281C3×C4—21no—
282C3×C4—82no—
283C3×C4—81no—
284C3×C4—81no—
285Dic3—44no—
286Dic3—42no—
287Dic3—42no—
288Dic3—82no—
289C23—22no—
290C23—22no—
291C23—41no—
292C23D3×C22⋊C411yescharacteristic
293C23#88311yescharacteristic
294C23#93711yescharacteristic
295Q8—81no—
296Q8—81no—
297C2×C4—22no—
298C2×C4—22no—
299C2×C4—41no—
300C2×C4—41no—
301C2×C4—41no—
302C2×C4—41no—
303C2×C4—42no—
304C2×C4—42no—
305C2×C4—42no—
306C2×C4—42no—
307C2×C4—41no—
308C2×C4—41no—
309C2×C4—42no—
310C2×C4—64no—
311C2×C4—62no—
312C2×C4—62no—
313C2×C4—122no—
314C2×C4—122no—
315C2×C4—122no—
316C2×C4—122no—
317C2×C4—122no—
318C2×C4—122no—
319C2×C4—122no—
320C2×C4—122no—
321C2×C4—122no—
322C2×C4—241no—
323C2×C4—241no—
324C2×C4#67⋊C411yescharacteristic
325C2×C4#88411yescharacteristic
326C2×C4C4×#6711yescharacteristic
327C2×C4#93811yescharacteristic
328C8—81no—
329C8—241no—
330C2×C3—22no—
331C2×C3—22no—
332C2×C3—41no—
333C2×C3#188611yescharacteristic
334C2×C3#189911yescharacteristic
335C2×C3#191111yescharacteristic
336C22—21no—
337C22—21no—
338C22—21no—
339C22—21no—
340C22—21no—
341C22—21no—
342C22—21no—
343C22—44no—
344C22#474012yes—
345C22#495611yescharacteristic
346C22#520512yes—
347C4—21no—
348C4—21no—
349C4—21no—
350C4—21no—
351C4—82no—
352C4—81no—
353C4—81no—
354C4—124no—
355C4—122no—
356C4—122no—
357C4—242no—
358C3#946111yescharacteristic, complemented, Sylow
359C2—22no—
360C2—22no—
361C2—41no—
362C2#7002911yescharacteristic
363C2#7034511yescharacteristic
364C2#7040811yescharacteristic
subgroupG/H
centerC22#4956
commutatorC22×C3×C4C22×C4
FrattiniC23×C4C22×D3
FittingC3×#1511C2
socleC22×C3#386
prime radicalC24×C3C2×D4
cosocleC23×C3×C4C23
hypercenter#1511D3
hypocenterC3#9461
LCSG ▷ C22×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1511

Complements

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

NH ≅ G/Ntype
13. #5229C4semidirect
18. #4975C4semidirect
358. C3#9461semidirect

Maximal subgroups

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

subgroupindex
#827752
#740542
C3×#15112
#716892
#699732
#676822
#676282
#94613