Finite Group Databasebeta

← #10698356

Subgroups — #10698356

Order 768.

422 conjugacy classes of subgroups; only the 91 normal subgroups are shown.

normal subgroups only

This group has 422 conjugacy classes of subgroups — too many to draw. Only its 91 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#73539C211yesmaximal, characteristic
2C3×#1996C211yesmaximal, characteristic
3#71690C211yesmaximal, characteristic
4C2×#4538C211yesmaximal, characteristic
5#67625C211yesmaximal, characteristic
6#67624C211yesmaximal, characteristic
7#67617C211yesmaximal, characteristic
8#10377—31nomaximal, Sylow
9C2×C3×C22⋊Q8C2211yescharacteristic
10C3×#380C2211yescharacteristic
11C3×C4×C22⋊C4C2211yescharacteristic
12#5212C2211yescharacteristic
13C2×C3⋊#95C2211yescharacteristic
14C2×#830C2211yescharacteristic
15C2×#830C2211yescharacteristic
16#4946—21no—
17#4942—21no—
18#4933—21no—
19#4933—21no—
20#4660—21no—
21#4657—21no—
22C3×#342—21no—
23C3×#342—21no—
24#4538—21no—
25#4538—21no—
26#4538—21no—
27#4538—21no—
28#2569—31no—
29#1996D311yescharacteristic
30#1520—31no—
31C2×#328—31no—
32#1294—31no—
33#1289—31no—
34#1282—31no—
35C22×C3×Q8Q811yescharacteristic
36C23×C3×C4C2311yescharacteristic
37C23×Dic3—21no—
38C3×C22⋊Q8—21no—
39C3×C22⋊Q8—21no—
40C3×C22⋊Q8—21no—
41C3×C22⋊Q8—21no—
42C2×C3×C4⋊C4—21no—
43C2×C3×C4⋊C4—21no—
44C2×C3×C4⋊C4Q811yescharacteristic
45C2×C3×C22⋊C4—21no—
46C2×C3×C22⋊C4—21no—
47C2×C3×C22⋊C4D411yescharacteristic
48C2×C3×C22⋊C4D411yescharacteristic
49C2×C3×C42—21no—
50C2×#217—21no—
51C2×C4⋊Dic3—21no—
52C2×Dic3⋊C4—21no—
53C2×Dic3⋊C4—21no—
54C2×Dic3⋊C4—21no—
55C2×Dic3⋊C4—21no—
56C22×C3⋊C8—21no—
57C22×C3⋊C8—21no—
58C3×#95—21no—
59C3×#95—21no—
60C3⋊#95—21no—
61C3⋊#95—21no—
62C3⋊#95—21no—
63C3⋊#95—21no—
64C3⋊#95—21no—
65C3⋊#95D411yescharacteristic
66C3⋊#95D411yescharacteristic
67C3⋊#95D411yescharacteristic
68C3⋊#95D411yescharacteristic
69#830—21no—
70#830—21no—
71#830—21no—
72#830—21no—
73C2×C22⋊Q8C2×D311yescharacteristic
74C2×C22⋊C8—31no—
75C2×C22⋊C8—31no—
76#400—61no—
77#398—61no—
78#387—61no—
79#387—61no—
80#385—61no—
81#380—31no—
82#380—61no—
83#380C2×D311yescharacteristic
84C4×C22⋊C4C2×D311yescharacteristic
85C2×#95—31no—
86#342—21no—
87#342—21no—
88#328—61no—
89#328—61no—
90#328—61no—
91#328—61no—
92C24×C3C2×D411yescharacteristic
93C2×C3×Q8—21no—
94C2×C3×Q8—21no—
95C2×C3×Q8—81no—
96C22×C3×C4—21no—
97C22×C3×C4—21no—
98C22×C3×C4—21no—
99C22×C3×C4—21no—
100C22×C3×C4—41no—
101C22×C3×C4—81no—
102C22×C3×C4C2×D411yescharacteristic
103C22×C3×C4C2×D411yescharacteristic
104C22×C3×C4C2×Q811yescharacteristic
105C22×C3×C4#4111yescharacteristic
106C22×C3×C4#4111yescharacteristic
107C22×C3×C4#4111yescharacteristic
108C22×Dic3—21no—
109C22×Dic3—21no—
110C22×Dic3—21no—
111C22×Dic3—21no—
112C22×Dic3—42no—
113C22×Dic3—41no—
114C22×Dic3—41no—
115C22×Dic3—81no—
116C3×C4⋊C4—21no—
117C3×C4⋊C4—21no—
118C3×C4⋊C4—41no—
119C3×C4⋊C4—41no—
120C3×C4⋊C4—81no—
121C3×C22⋊C4—21no—
122C3×C22⋊C4—21no—
123C3×C22⋊C4—21no—
124C3×C22⋊C4—21no—
125C3×C22⋊C4—41no—
126C3×C22⋊C4—41no—
127C3×C22⋊C4—41no—
128C3×C22⋊C4—41no—
129C3×C42—81no—
130#217—41no—
131#217—41no—
132C4⋊Dic3—81no—
133Dic3⋊C4—42no—
134Dic3⋊C4—42no—
135Dic3⋊C4—81no—
136Dic3⋊C4—81no—
137C2×C3⋊C8—41no—
138C2×C3⋊C8—41no—
139C2×C3⋊C8—81no—
140C2×C3⋊C8—81no—
141C22×Q8Dic611yescharacteristic
142C23×C4—61no—
143C23×C4C22×D311yescharacteristic
144C22×C8—61no—
145C22×C8—61no—
146C22⋊Q8—21no—
147C22⋊Q8—21no—
148C22⋊Q8—21no—
149C22⋊Q8—21no—
150C2×C4⋊C4—21no—
151C2×C4⋊C4—21no—
152C2×C4⋊C4—61no—
153C2×C4⋊C4—61no—
154C2×C4⋊C4—61no—
155C2×C4⋊C4—61no—
156C2×C4⋊C4—61no—
157C2×C4⋊C4Dic611yescharacteristic
158C2×C22⋊C4—21no—
159C2×C22⋊C4—21no—
160C2×C22⋊C4—61no—
161C2×C22⋊C4#6711yescharacteristic
162C2×C22⋊C4#6711yescharacteristic
163C2×C42—21no—
164C22⋊C8—61no—
165C22⋊C8—61no—
166C22⋊C8—61no—
167C22⋊C8—61no—
168#95—21no—
169#95—21no—
170#95—31no—
171#95—31no—
172#95—31no—
173#95—31no—
174#95—61no—
175#95—61no—
176#95—61no—
177#95—61no—
178#95—61no—
179C23×C3—41no—
180C23×C3—41no—
181C23×C3#12011yescharacteristic
182C23×C3C22⋊Q811yescharacteristic
183C23×C3#12411yescharacteristic
184C23×C3#13511yescharacteristic
185C23×C3#13511yescharacteristic
186C23×C3#13711yescharacteristic
187C23×C3#13711yescharacteristic
188C3×Q8—81no—
189C3×Q8—81no—
190C2×C3×C4—41no—
191C2×C3×C4—41no—
192C2×C3×C4—41no—
193C2×C3×C4—41no—
194C2×C3×C4—41no—
195C2×C3×C4—41no—
196C2×C3×C4—41no—
197C2×C3×C4—41no—
198C2×C3×C4—41no—
199C2×C3×C4—41no—
200C2×C3×C4—41no—
201C2×C3×C4—41no—
202C2×C3×C4—41no—
203C2×C3×C4—41no—
204C2×C3×C4—81no—
205C2×C3×C4—81no—
206C2×C3×C4—81no—
207C2×C3×C4—81no—
208C2×C3×C4—81no—
209C2×C3×C4C22⋊Q811yescharacteristic
210C2×C3×C4C22⋊Q811yescharacteristic
211C2×C3×C4#12411yescharacteristic
212C2×C3×C4#12411yescharacteristic
213C2×Dic3—22no—
214C2×Dic3—41no—
215C2×Dic3—41no—
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—82no—
226C2×Dic3—82no—
227C3⋊C8—81no—
228C3⋊C8—81no—
229C24C2×#6711yescharacteristic
230C2×Q8—21no—
231C2×Q8—21no—
232C2×Q8—81no—
233C22×C4—21no—
234C22×C4—21no—
235C22×C4—21no—
236C22×C4—21no—
237C22×C4—41no—
238C22×C4—61no—
239C22×C4—61no—
240C22×C4—61no—
241C22×C4—61no—
242C22×C4—81no—
243C22×C4—122no—
244C22×C4—121no—
245C22×C4—121no—
246C22×C4—241no—
247C22×C4C2×Dic611yescharacteristic
248C22×C4#23511yescharacteristic
249C22×C4D3×D411yescharacteristic
250C22×C4D3×D411yescharacteristic
251C22×C4#23711yescharacteristic
252C22×C4#23711yescharacteristic
253C2×C8—121no—
254C2×C8—121no—
255C2×C8—241no—
256C2×C8—241no—
257C4⋊C4—21no—
258C4⋊C4—21no—
259C4⋊C4—41no—
260C4⋊C4—41no—
261C4⋊C4—81no—
262C4⋊C4—122no—
263C4⋊C4—122no—
264C4⋊C4—241no—
265C4⋊C4—241no—
266C4⋊C4—241no—
267C22⋊C4—21no—
268C22⋊C4—21no—
269C22⋊C4—21no—
270C22⋊C4—21no—
271C22⋊C4—41no—
272C22⋊C4—41no—
273C22⋊C4—41no—
274C22⋊C4—41no—
275C22⋊C4—121no—
276C22⋊C4—121no—
277C42—81no—
278C22×C3—21no—
279C22×C3—21no—
280C22×C3—21no—
281C22×C3—21no—
282C22×C3—21no—
283C22×C3—21no—
284C22×C3—41no—
285C22×C3—41no—
286C22×C3—41no—
287C22×C3—41no—
288C22×C3#39311yescharacteristic
289C22×C3#45211yescharacteristic
290C22×C3#45211yescharacteristic
291C22×C3#47911yescharacteristic
292C22×C3#47911yescharacteristic
293C22×C3#48811yescharacteristic
294C22×C3#49111yescharacteristic
295C3×C4—21no—
296C3×C4—21no—
297C3×C4—21no—
298C3×C4—21no—
299C3×C4—81no—
300C3×C4—81no—
301C3×C4—81no—
302C3×C4—81no—
303C3×C4—161no—
304Dic3—42no—
305Dic3—81no—
306Dic3—81no—
307Dic3—81no—
308C23—41no—
309C23—41no—
310C23#92411yescharacteristic
311C23#93611yescharacteristic
312C23#93711yescharacteristic
313C23#94211yescharacteristic
314C23#94211yescharacteristic
315C23#95011yescharacteristic
316C23#95011yescharacteristic
317Q8—81no—
318Q8—81no—
319C2×C4—41no—
320C2×C4—41no—
321C2×C4—41no—
322C2×C4—41no—
323C2×C4—41no—
324C2×C4—41no—
325C2×C4—41no—
326C2×C4—41no—
327C2×C4—41no—
328C2×C4—41no—
329C2×C4—41no—
330C2×C4—41no—
331C2×C4—41no—
332C2×C4—41no—
333C2×C4—62no—
334C2×C4—81no—
335C2×C4—81no—
336C2×C4—81no—
337C2×C4—81no—
338C2×C4—81no—
339C2×C4—121no—
340C2×C4—121no—
341C2×C4—121no—
342C2×C4—121no—
343C2×C4—121no—
344C2×C4—121no—
345C2×C4—121no—
346C2×C4—121no—
347C2×C4—121no—
348C2×C4—121no—
349C2×C4—121no—
350C2×C4—242no—
351C2×C4—242no—
352C2×C4#87811yescharacteristic
353C2×C4#87811yescharacteristic
354C2×C4#88511yescharacteristic
355C2×C4#88511yescharacteristic
356C8—241no—
357C8—241no—
358C2×C3—21no—
359C2×C3—21no—
360C2×C3—21no—
361C2×C3—21no—
362C2×C3—21no—
363C2×C3—21no—
364C2×C3#202611yescharacteristic
365C2×C3#203111yescharacteristic
366C2×C3#203211yescharacteristic
367C22—21no—
368C22—21no—
369C22—21no—
370C22—21no—
371C22—21no—
372C22—21no—
373C22—41no—
374C22—41no—
375C22—41no—
376C22—41no—
377C22#494711yescharacteristic
378C22#502211yescharacteristic
379C22#502211yescharacteristic
380C22#504711yescharacteristic
381C22#504711yescharacteristic
382C22#505611yescharacteristic
383C22#508211yescharacteristic
384C4—21no—
385C4—21no—
386C4—21no—
387C4—21no—
388C4—81no—
389C4—81no—
390C4—81no—
391C4—81no—
392C4—122no—
393C4—161no—
394C4—241no—
395C4—241no—
396C4—241no—
397C3#1037711yescharacteristic, complemented, Sylow
398C2—21no—
399C2—21no—
400C2—21no—
401C2—21no—
402C2—21no—
403C2—21no—
404C2#6966911yescharacteristic
405C2#7002511yescharacteristic
406C2#7003811yescharacteristic
subgroupG/H
centerC22#4947
commutator, cosocleC23×C3×C4C23
FrattiniC23×C4C22×D3
FittingC3×#1996C2
socleC22×C3#393
prime radicalC24×C3C2×D4
hypercenter#1996D3
hypocenterC3#10377
LCSG ▷ C23×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1996

Complements

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

NH ≅ G/Ntype
397. C3#10377semidirect

Maximal subgroups

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

subgroupindex
#735392
C3×#19962
#716902
C2×#45382
#676252
#676242
#676172
#103773