Finite Group Databasebeta

← #10698400

Subgroups — #10698400

Order 768.

404 conjugacy classes of subgroups; only the 75 normal subgroups are shown.

normal subgroups only

This group has 404 conjugacy classes of subgroups — too many to draw. Only its 75 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#73557C211yesmaximal, characteristic
2C3×#1508C211yesmaximal, characteristic
3#71701C211yesmaximal, characteristic
4#71689C211yesmaximal, characteristic
5#67684C211yesmaximal, characteristic
6#67682C211yesmaximal, characteristic
7#67618C211yesmaximal, characteristic
8#10383—31nomaximal, Sylow
9C2×C3×C22⋊Q8C2211yescharacteristic
10C3×#407C2211yescharacteristic
11C3×C4×C22⋊C4C2211yescharacteristic
12#5212C2211yescharacteristic
13#5211C2211yescharacteristic
14C4×#217C2211yescharacteristic
15#5207C2211yescharacteristic
16#4948—21no—
17#4947—21no—
18#4933—21no—
19#4929—21no—
20#4660—21no—
21#4656—21no—
22C3×#328—21no—
23C3×#328—21no—
24#4538—21no—
25#4538—21no—
26#4526—21no—
27#4526—21no—
28#2665—31no—
29#1996—31no—
30#1511—31no—
31#1508D311yescharacteristic
32#1305—31no—
33#1297—31no—
34#1284—31no—
35C22×C3×Q8Q811yescharacteristic, complemented
36C23×C3×C4C2311yescharacteristic
37C22×Dic6—21no—
38C2×C3×M16—21no—
39C3×C22⋊Q8—21no—
40C3×C22⋊Q8—21no—
41C3×C22⋊Q8—21no—
42C3×C22⋊Q8—21no—
43C2×C3×C4⋊C4—21no—
44C2×C3×C4⋊C4Q811yescharacteristic, complemented
45C2×C3×C22⋊C4—21no—
46C2×C3×C22⋊C4—21no—
47C2×C3×C42—21no—
48C2×#217—21no—
49C2×#217—21no—
50C2×#217D411yescharacteristic
51C2×#217D411yescharacteristic
52C2×C4⋊Dic3—21no—
53C2×C4⋊Dic3—21no—
54C2×Dic3⋊C4—21no—
55C2×Dic3⋊C4—21no—
56C2×C4×Dic3—21no—
57C2×#208—21no—
58C3×C22⋊C8—21no—
59C3×C22⋊C8—21no—
60C3×#95D411yescharacteristic
61C3×#95D411yescharacteristic
62C3⋊#95—21no—
63C3⋊#95—21no—
64C3⋊#95—21no—
65C3⋊#95—21no—
66C3⋊#95—21no—
67C3⋊#95—21no—
68C3⋊#95—21no—
69C3⋊#95D411yescharacteristic
70C3⋊#95D411yescharacteristic
71#830—21no—
72#830—21no—
73C2×C22⋊Q8C2×D311yescharacteristic
74#407—31no—
75#407C2×D311yescharacteristic
76#400—61no—
77#396—61no—
78#395—61no—
79#393—61no—
80#391—61no—
81#387—61no—
82#381—31no—
83#380—31no—
84C4×C22⋊C4—31no—
85C4×C22⋊C4C2×D311yescharacteristic
86#342—61no—
87#342—61no—
88#328—21no—
89#328—21no—
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—81no—
101C22×C3×C4C2×D411yescharacteristic
102C22×C3×C4C2×D411yescharacteristic
103C22×C3×C4C2×Q811yescharacteristic
104C22×C3×C4#4111yescharacteristic
105C22×C3×C4#4111yescharacteristic
106C22×C3×C4#4111yescharacteristic
107C22×Dic3—21no—
108C22×Dic3—21no—
109C22×Dic3—21no—
110C22×Dic3—21no—
111C22×Dic3—41no—
112C2×Dic6—42no—
113C2×Dic6—81no—
114C2×Dic6—81no—
115C3×M16—81no—
116C2×C3×C8—41no—
117C3×C4⋊C4—21no—
118C3×C4⋊C4—21no—
119C3×C4⋊C4—41no—
120C3×C4⋊C4—41no—
121C3×C22⋊C4—42no—
122C3×C22⋊C4—41no—
123C3×C22⋊C4—41no—
124C3×C42—81no—
125#217—22no—
126#217—41no—
127#217—41no—
128#217—41no—
129#217—81no—
130C4⋊Dic3—81no—
131C4⋊Dic3—81no—
132Dic3⋊C4—42no—
133Dic3⋊C4—81no—
134C4×Dic3—81no—
135#208—81no—
136C2×C3⋊C8—41no—
137C22×Q8—61no—
138C22×Q8Dic611yescharacteristic, complemented
139C23×C4C22×D311yescharacteristic
140C2×M16—21no—
141C2×M16—61no—
142C22⋊Q8—21no—
143C22⋊Q8—21no—
144C22⋊Q8—21no—
145C22⋊Q8—21no—
146C2×C4⋊C4—21no—
147C2×C4⋊C4—61no—
148C2×C4⋊C4—61no—
149C2×C4⋊C4—61no—
150C2×C4⋊C4—61no—
151C2×C4⋊C4Dic611yescharacteristic, complemented
152C2×C22⋊C4—21no—
153C2×C22⋊C4—21no—
154C2×C22⋊C4—31no—
155C2×C22⋊C4—31no—
156C2×C22⋊C4—61no—
157C2×C22⋊C4—61no—
158C2×C42—21no—
159C2×C42—61no—
160C22⋊C8—21no—
161C22⋊C8—21no—
162C22⋊C8—61no—
163C22⋊C8—61no—
164#95—31no—
165#95—31no—
166#95—61no—
167#95—61no—
168#95—61no—
169#95—61no—
170#95—61no—
171#95—61no—
172#95—61no—
173#95#6711yescharacteristic
174#95#6711yescharacteristic
175C23×C3—21no—
176C23×C3—21no—
177C23×C3—41no—
178C23×C3—41no—
179C23×C3#12011yescharacteristic
180C23×C3C22⋊Q811yescharacteristic
181C23×C3#12411yescharacteristic
182C3×Q8—81no—
183C3×Q8—81no—
184C2×C3×C4—22no—
185C2×C3×C4—41no—
186C2×C3×C4—41no—
187C2×C3×C4—41no—
188C2×C3×C4—41no—
189C2×C3×C4—41no—
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—81no—
199C2×C3×C4—81no—
200C2×C3×C4C22⋊Q811yescharacteristic
201C2×C3×C4C22⋊Q811yescharacteristic
202C2×C3×C4#12411yescharacteristic
203C2×C3×C4#12411yescharacteristic
204C2×Dic3—41no—
205C2×Dic3—41no—
206C2×Dic3—41no—
207C2×Dic3—41no—
208C2×Dic3—41no—
209C2×Dic3—42no—
210C2×Dic3—41no—
211C2×Dic3—41no—
212C2×Dic3—41no—
213C2×Dic3—41no—
214C2×Dic3—41no—
215C2×Dic3—41no—
216C2×Dic3—41no—
217C2×Dic3—81no—
218C2×Dic3—81no—
219Dic6—84no—
220C3×C8—81no—
221C3⋊C8—81no—
222C24D3×D411yescharacteristic
223C2×Q8—21no—
224C2×Q8—21no—
225C2×Q8—81no—
226C2×Q8—122no—
227C2×Q8—241no—
228C2×Q8—241no—
229C22×C4—21no—
230C22×C4—21no—
231C22×C4—21no—
232C22×C4—21no—
233C22×C4—61no—
234C22×C4—61no—
235C22×C4—61no—
236C22×C4—61no—
237C22×C4—81no—
238C22×C4—121no—
239C22×C4C2×Dic611yescharacteristic
240C22×C4#23511yescharacteristic
241C22×C4D3×D411yescharacteristic
242C22×C4#23711yescharacteristic
243C22×C4#23711yescharacteristic
244C22×C4C2×#6711yescharacteristic
245M16—81no—
246M16—241no—
247C2×C8—41no—
248C2×C8—121no—
249C4⋊C4—21no—
250C4⋊C4—21no—
251C4⋊C4—41no—
252C4⋊C4—41no—
253C4⋊C4—122no—
254C4⋊C4—241no—
255C4⋊C4—241no—
256C4⋊C4—241no—
257C22⋊C4—42no—
258C22⋊C4—41no—
259C22⋊C4—41no—
260C22⋊C4—62no—
261C22⋊C4—121no—
262C22⋊C4—121no—
263C22⋊C4—121no—
264C22⋊C4—241no—
265C42—81no—
266C42—241no—
267C22×C3—21no—
268C22×C3—21no—
269C22×C3—21no—
270C22×C3—21no—
271C22×C3—21no—
272C22×C3—21no—
273C22×C3—21no—
274C22×C3—21no—
275C22×C3—81no—
276C22×C3—81no—
277C22×C3#39311yescharacteristic
278C22×C3#45511yescharacteristic
279C22×C3#45611yescharacteristic
280C3×C4—21no—
281C3×C4—21no—
282C3×C4—21no—
283C3×C4—21no—
284C3×C4—42no—
285C3×C4—81no—
286C3×C4—81no—
287C3×C4—81no—
288Dic3—81no—
289Dic3—81no—
290Dic3—81no—
291Dic3—81no—
292Dic3—82no—
293C23—21no—
294C23—21no—
295C23—41no—
296C23—41no—
297C23#87811yescharacteristic
298C23#88511yescharacteristic
299C23#93711yescharacteristic
300Q8—81no—
301Q8—81no—
302Q8—244no—
303C2×C4—22no—
304C2×C4—41no—
305C2×C4—41no—
306C2×C4—41no—
307C2×C4—41no—
308C2×C4—41no—
309C2×C4—41no—
310C2×C4—41no—
311C2×C4—41no—
312C2×C4—41no—
313C2×C4—41no—
314C2×C4—41no—
315C2×C4—41no—
316C2×C4—41no—
317C2×C4—81no—
318C2×C4—81no—
319C2×C4—122no—
320C2×C4—121no—
321C2×C4—121no—
322C2×C4—121no—
323C2×C4—121no—
324C2×C4—121no—
325C2×C4—121no—
326C2×C4—121no—
327C2×C4—121no—
328C2×C4—121no—
329C2×C4—121no—
330C2×C4—121no—
331C2×C4—121no—
332C2×C4—241no—
333C2×C4—241no—
334C2×C4#87811yescharacteristic
335C2×C4#88511yescharacteristic
336C2×C4#92411yescharacteristic
337C2×C4#93611yescharacteristic
338C8—81no—
339C8—241no—
340C2×C3—21no—
341C2×C3—21no—
342C2×C3—41no—
343C2×C3—41no—
344C2×C3#202811yescharacteristic
345C2×C3#203111yescharacteristic
346C2×C3#203211yescharacteristic
347C22—21no—
348C22—21no—
349C22—21no—
350C22—21no—
351C22—21no—
352C22—21no—
353C22—21no—
354C22—21no—
355C22—81no—
356C22—81no—
357C22#494711yescharacteristic
358C22#520211yescharacteristic
359C22#520511yescharacteristic
360C4—21no—
361C4—21no—
362C4—21no—
363C4—21no—
364C4—42no—
365C4—81no—
366C4—81no—
367C4—81no—
368C4—241no—
369C4—241no—
370C4—241no—
371C4—241no—
372C4—242no—
373C3#1038311yescharacteristic, complemented, Sylow
374C2—21no—
375C2—21no—
376C2—41no—
377C2—41no—
378C2#7002511yescharacteristic
379C2#7033311yescharacteristic
380C2#7036811yescharacteristic
subgroupG/H
centerC22#4947
commutator, cosocleC23×C3×C4C23
FrattiniC23×C4C22×D3
FittingC3×#1508C2
socleC22×C3#393
prime radicalC24×C3C2×D4
hypercenter#1508D3
hypocenterC3#10383
LCSG ▷ C23×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1508

Complements

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

NH ≅ G/Ntype
35. C22×C3×Q8Q8semidirect
44. C2×C3×C4⋊C4Q8semidirect
138. C22×Q8Dic6semidirect
151. C2×C4⋊C4Dic6semidirect
373. C3#10383semidirect

Maximal subgroups

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

subgroupindex
#735572
C3×#15082
#717012
#716892
#676842
#676822
#676182
#103833