Finite Group Databasebeta

← #10692810

Subgroups — #10692810

Order 768.

434 conjugacy classes of subgroups; only the 89 normal subgroups are shown.

normal subgroups only

This group has 434 conjugacy classes of subgroups — too many to draw. Only its 89 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#78507C211yesmaximal, characteristic
2#73394C211yesmaximal, characteristic
3#70836C211yesmaximal, characteristic
4#70814C211yesmaximal, characteristic
5#69433C211yesmaximal, characteristic
6C3×#1300C211yesmaximal, characteristic
7#67665C211yesmaximal, characteristic
8#9440—31nomaximal, Sylow
9#5470C2211yescharacteristic
10C2×#870C2211yescharacteristic
11C3×#432C2211yescharacteristic
12C3×C4×M16C2211yescharacteristic
13C3×#376C2211yescharacteristic
14#5121—21no—
15#5121—21no—
16#4933—21no—
17#4932C2211yescharacteristic
18C4×Dic3⋊C4—21no—
19#4926C2211yescharacteristic
20#4702C412yescomplemented
21#4702C412yescomplemented
22#4683—22no—
23#4654—21no—
24#4653—21no—
25#4652—21no—
26#4649—21no—
27#4467—21no—
28#4467—21no—
29#4465—22no—
30#4452—21no—
31#4452—21no—
32#3166—31no—
33#2371—31no—
34#1534—31no—
35#1506—31no—
36#1506—31no—
37#1300—31no—
38#1300D311yescharacteristic
39C2×C3×M16—21no—
40C2×C3×C42—21no—
41C2×C3×C42C2311yescharacteristic
42#926—21no—
43C2×C4⋊Dic3D411yescharacteristic
44C2×C4⋊Dic3D411yescharacteristic
45#924—21no—
46#924—21no—
47C2×Dic3⋊C4—21no—
48C2×Dic3⋊C4—21no—
49C2×Dic3⋊C4—21no—
50C2×Dic3⋊C4—21no—
51C2×C4×Dic3—21no—
52C2×C4×Dic3—21no—
53C2×C4×Dic3—21no—
54#870—21no—
55#870—21no—
56#870—21no—
57#870C2×C412yes—
58#870D412yes—
59#869C2×C411yescharacteristic
60#869C2×C411yescharacteristic
61C4×Dic6—21no—
62C4×Dic6—21no—
63C3×C4⋊C8—21no—
64C3×C22⋊C8—21no—
65C3×#97—21no—
66C3×#97C2×C412yes—
67C3×C4×C8—21no—
68C3×#95—21no—
69C3×#95D411yescharacteristic
70C3×#95D411yescharacteristic
71C3⋊#95—21no—
72C3⋊#95—21no—
73C3⋊#95—21no—
74C3⋊#95—21no—
75C3⋊#95—21no—
76C3⋊#95D411yescharacteristic
77C3⋊#95D411yescharacteristic
78#830—21no—
79#818—22no—
80#818—22no—
81#817—22no—
82#817—22no—
83C4⋊(C3⋊C8)—21no—
84#803—21no—
85#533—31no—
86C2×#125—31no—
87#501—32no—
88#501—32no—
89#499—62no—
90#432—31no—
91#432C2×D311yescharacteristic
92#428—61no—
93#428—61no—
94C4×M16C2×D311yescharacteristic
95#387—61no—
96#387—61no—
97#387—61no—
98#387—61no—
99C4×C4⋊C4—61no—
100C4×C4⋊C4—61no—
101#376—31no—
102#376C2×D311yescharacteristic
103#330—62no—
104#328—61no—
105#328—61no—
106#328—61no—
107#328—61no—
108C22×C3×C4—21no—
109C22×C3×C4—21no—
110C22×C3×C4C2×D411yescharacteristic
111C22×C3×C4C2×D411yescharacteristic
112C22×C3×C4C2×D411yescharacteristic
113C22×Dic3—21no—
114C22×Dic3—21no—
115C22×Dic3—21no—
116C22×Dic3—21no—
117C2×Dic6—21no—
118C2×Dic6—21no—
119C3×M16—22no—
120C3×M16—22no—
121C2×C3×C8—22no—
122C2×C3×C8—41no—
123C3×C42—42no—
124C3×C42C22×C411yescharacteristic
125C3×C42C2×D411yescharacteristic
126C3×C42#4111yescharacteristic
127C3×C42#4111yescharacteristic
128#217—41no—
129C4⋊Dic3—21no—
130C4⋊Dic3—21no—
131C4⋊Dic3—21no—
132C4⋊Dic3—21no—
133C4⋊Dic3—21no—
134C4⋊Dic3—21no—
135C4⋊Dic3C22⋊C412yes—
136C4⋊Dic3C22⋊C412yes—
137Dic3⋊C4—22no—
138Dic3⋊C4—22no—
139Dic3⋊C4—22no—
140Dic3⋊C4—22no—
141Dic3⋊C4—42no—
142Dic3⋊C4—42no—
143Dic3⋊C4—41no—
144Dic3⋊C4—41no—
145C4×Dic3—42no—
146C4×Dic3—42no—
147C4×Dic3—42no—
148C4×Dic3—41no—
149C2×C3⋊C8—41no—
150C2×M16—21no—
151C4⋊Q8—31no—
152C4⋊Q8—31no—
153#125—32no—
154#125—32no—
155#125—61no—
156#125—61no—
157#125—61no—
158C22⋊Q8—61no—
159C22⋊Q8—61no—
160C4×Q8—61no—
161C4×Q8—61no—
162#117—61no—
163C2×C4⋊C4—31no—
164C2×C4⋊C4—31no—
165C2×C4⋊C4—61no—
166C2×C4⋊C4—61no—
167C2×C4⋊C4—61no—
168C2×C4⋊C4—61no—
169C2×C42—21no—
170C2×C42—61no—
171C2×C42—61no—
172C2×C42—61no—
173C2×C42C22×D311yescharacteristic
174#107—62no—
175#107—62no—
176#106—62no—
177#106—62no—
178C4⋊C8—21no—
179C4⋊C8—61no—
180C22⋊C8—21no—
181C22⋊C8—61no—
182#97—21no—
183#97—61no—
184#97C4×D312yes—
185C4×C8—21no—
186#95—21no—
187#95—31no—
188#95—31no—
189#95—61no—
190#95—61no—
191#95—61no—
192#95—61no—
193#95—61no—
194#95D1211yescharacteristic
195#95D1211yescharacteristic
196C23×C3#12011yescharacteristic
197C2×C3×C4—21no—
198C2×C3×C4—21no—
199C2×C3×C4—22no—
200C2×C3×C4—41no—
201C2×C3×C4—41no—
202C2×C3×C4—41no—
203C2×C3×C4—41no—
204C2×C3×C4—41no—
205C2×C3×C4—41no—
206C2×C3×C4—41no—
207C2×C3×C4C2×C22⋊C411yescharacteristic
208C2×C3×C4C4×D411yescharacteristic
209C2×C3×C4C4×D411yescharacteristic
210C2×C3×C4#12111yescharacteristic
211C2×C3×C4#12111yescharacteristic
212C2×C3×C4#12311yescharacteristic
213C2×Dic3—22no—
214C2×Dic3—22no—
215C2×Dic3—22no—
216C2×Dic3—41no—
217C2×Dic3—41no—
218C2×Dic3—41no—
219C2×Dic3—41no—
220C2×Dic3—41no—
221C2×Dic3—41no—
222C2×Dic3—41no—
223C2×Dic3—41no—
224C2×Dic3—41no—
225C2×Dic3—41no—
226C2×Dic3—41no—
227Dic6—81no—
228Dic6—81no—
229C3×C8—24no—
230C3×C8—81no—
231C3⋊C8—81no—
232C2×Q8—61no—
233C2×Q8—61no—
234C22×C4—21no—
235C22×C4—21no—
236C22×C4—61no—
237C22×C4—61no—
238C22×C4—61no—
239C22×C4—61no—
240C22×C4C2×D1211yescharacteristic
241C22×C4D3×D411yescharacteristic
242C22×C4D3×D411yescharacteristic
243M16—22no—
244M16—22no—
245C2×C8—22no—
246C2×C8—41no—
247C2×C8—121no—
248C4⋊C4—32no—
249C4⋊C4—32no—
250C4⋊C4—62no—
251C4⋊C4—62no—
252C4⋊C4—62no—
253C4⋊C4—61no—
254C4⋊C4—61no—
255C4⋊C4—62no—
256C4⋊C4—61no—
257C4⋊C4—61no—
258C4⋊C4—61no—
259C4⋊C4—61no—
260C4⋊C4—122no—
261C4⋊C4—122no—
262C4⋊C4—121no—
263C4⋊C4—121no—
264C22⋊C4—121no—
265C42—42no—
266C42—122no—
267C42—122no—
268C42—122no—
269C42—121no—
270C42C2×C4×D311yescharacteristic
271C42#23511yescharacteristic
272C42D3×D411yescharacteristic
273C42#23911yescharacteristic
274C22×C3—21no—
275C22×C3—21no—
276C22×C3#38611yescharacteristic
277C22×C3#45611yescharacteristic
278C22×C3#45611yescharacteristic
279C3×C4—21no—
280C3×C4—21no—
281C3×C4—21no—
282C3×C4—21no—
283C3×C4—21no—
284C3×C4—21no—
285C3×C4—42no—
286C3×C4—81no—
287Dic3—42no—
288Dic3—42no—
289Dic3—42no—
290Dic3—81no—
291Dic3—81no—
292Dic3—81no—
293C23#88211yescharacteristic
294Q8—241no—
295Q8—241no—
296C2×C4—21no—
297C2×C4—21no—
298C2×C4—22no—
299C2×C4—41no—
300C2×C4—41no—
301C2×C4—41no—
302C2×C4—41no—
303C2×C4—41no—
304C2×C4—41no—
305C2×C4—41no—
306C2×C4—62no—
307C2×C4—62no—
308C2×C4—62no—
309C2×C4—121no—
310C2×C4—121no—
311C2×C4—121no—
312C2×C4—121no—
313C2×C4—121no—
314C2×C4—121no—
315C2×C4—121no—
316C2×C4—121no—
317C2×C4—121no—
318C2×C4—121no—
319C2×C4—121no—
320C2×C4C4×D1211yescharacteristic
321C2×C4D3×C22⋊C411yescharacteristic
322C2×C4#88411yescharacteristic
323C2×C4#89311yescharacteristic
324C2×C4#89411yescharacteristic
325C2×C4#89511yescharacteristic
326C8—24no—
327C8—81no—
328C8—241no—
329C2×C3—21no—
330C2×C3—21no—
331C2×C3#188611yescharacteristic
332C2×C3#190611yescharacteristic
333C2×C3#190611yescharacteristic
334C22—21no—
335C22—21no—
336C22#466511yescharacteristic
337C22#482611yescharacteristic
338C22#482611yescharacteristic
339C4—21no—
340C4—21no—
341C4—21no—
342C4—21no—
343C4—21no—
344C4—21no—
345C4—42no—
346C4—81no—
347C4—122no—
348C4—122no—
349C4—122no—
350C4—241no—
351C4—241no—
352C4—241no—
353C3#944011yescharacteristic, complemented, Sylow
354C2—21no—
355C2—21no—
356C2#6876511yescharacteristic
357C2#6877811yescharacteristic
358C2#6903911yescharacteristic
subgroupG/H
centerC22#4665
commutatorC3×C42C22×C4
FrattiniC2×C42C22×D3
FittingC3×#1300C2
socleC22×C3#386
prime radicalC23×C3#120
cosocleC2×C3×C42C23
hypercenter#1300D3
hypocenterC3#9440
LCSG ▷ C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1300

Complements

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

NH ≅ G/Ntype
20. #4702C4semidirect
21. #4702C4semidirect
353. C3#9440semidirect

Maximal subgroups

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

subgroupindex
#785072
#733942
#708362
#708142
#694332
C3×#13002
#676652
#94403