Finite Group Databasebeta

← #10698422

Subgroups — #10698422

Order 768.

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

normal subgroups only

This group has 414 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#73553C211yesmaximal, characteristic
2C3×#1826C211yesmaximal, characteristic
3#71594C211yesmaximal, characteristic
4#71590C211yesmaximal, characteristic
5#67686C211yesmaximal, characteristic
6#67682C211yesmaximal, characteristic
7#67616C211yesmaximal, characteristic
8#10387—31nomaximal, Sylow
9C2×C3×#117C2211yescharacteristic
10C3×#407C2211yescharacteristic
11C3×C4×C22⋊C4C2211yescharacteristic
12#5211C2211yescharacteristic
13#5211C2211yescharacteristic
14C4×#217C2211yescharacteristic
15#5207C2211yescharacteristic
16#4949—21no—
17#4947—21no—
18#4933—21no—
19#4929—21no—
20#4654—21no—
21#4646—21no—
22C3×#339—21no—
23C3×#339—21no—
24#4532—21no—
25#4532—21no—
26#4526—21no—
27#4526—21no—
28#2607—31no—
29#1826—31no—
30#1826D311yescharacteristic
31#1820—31no—
32#1306—31no—
33#1284—31no—
34#1284—31no—
35C23×C3×C4C2311yescharacteristic
36C22×Dic6—21no—
37C2×C3×M16—21no—
38C3×#117—21no—
39C3×#117—21no—
40C3×#117—21no—
41C3×#117—21no—
42C2×C3×C4⋊C4Q811yescharacteristic, complemented
43C2×C3×C4⋊C4Q811yescharacteristic, complemented
44C2×C3×C22⋊C4—21no—
45C2×C3×C22⋊C4—21no—
46C2×C3×C42—21no—
47C2×C3×C42D411yescharacteristic
48C2×C3×C42D411yescharacteristic
49C2×#217—21no—
50C2×#217—21no—
51C2×#217D411yescharacteristic
52C2×#217D411yescharacteristic
53C2×C4⋊Dic3—21no—
54C2×Dic3⋊C4—21no—
55C2×Dic3⋊C4—21no—
56C2×C4×Dic3D411yescharacteristic
57C2×C4×Dic3D411yescharacteristic
58C2×#208—21no—
59C3×C22⋊C8—21no—
60C3×C22⋊C8—21no—
61C3×#95—21no—
62C3⋊#95—21no—
63C3⋊#95—21no—
64C3⋊#95—21no—
65C3⋊#95—21no—
66C3⋊#95—21no—
67C3⋊#95—21no—
68C3⋊#95—21no—
69C3⋊#95—21no—
70C3⋊#95—21no—
71#830—21no—
72#830—21no—
73C2×#117C2×D311yescharacteristic
74#407—31no—
75#407C2×D311yescharacteristic
76#399—61no—
77#393—61no—
78#391—61no—
79#391—61no—
80#387—61no—
81#387—61no—
82#381—31no—
83#381—31no—
84C4×C22⋊C4—31no—
85C4×C22⋊C4C2×D311yescharacteristic
86#342—61no—
87#342—61no—
88#339—21no—
89#339—21no—
90#339—61no—
91#339—61no—
92C24×C3C2×D411yescharacteristic
93C22×C3×C4—21no—
94C22×C3×C4—21no—
95C22×C3×C4—21no—
96C22×C3×C4—21no—
97C22×C3×C4—41no—
98C22×C3×C4—41no—
99C22×C3×C4C2×D411yescharacteristic
100C22×C3×C4C2×D411yescharacteristic
101C22×C3×C4C2×Q811yescharacteristic
102C22×C3×C4#4111yescharacteristic
103C22×C3×C4#4111yescharacteristic
104C22×C3×C4#4111yescharacteristic
105C22×Dic3—21no—
106C22×Dic3—21no—
107C22×Dic3—21no—
108C22×Dic3—21no—
109C22×Dic3—41no—
110C2×Dic6—42no—
111C2×Dic6—42no—
112C2×Dic6—81no—
113C3×M16—41no—
114C3×M16—41no—
115C2×C3×C8—41no—
116C3×C4⋊C4—21no—
117C3×C4⋊C4—21no—
118C3×C4⋊C4—21no—
119C3×C4⋊C4—21no—
120C3×C22⋊C4—42no—
121C3×C22⋊C4—41no—
122C3×C22⋊C4—41no—
123C3×C42—41no—
124C3×C42—41no—
125C3×C42—41no—
126C3×C42—41no—
127#217—42no—
128#217—41no—
129#217—41no—
130#217—41no—
131#217—41no—
132C4⋊Dic3—42no—
133Dic3⋊C4—42no—
134Dic3⋊C4—81no—
135C4×Dic3—41no—
136C4×Dic3—41no—
137#208—41no—
138#208—41no—
139C2×C3⋊C8—41no—
140C22×Q8—61no—
141C23×C4C22×D311yescharacteristic
142C2×M16—21no—
143C2×M16—61no—
144#117—21no—
145#117—21no—
146#117—21no—
147#117—21no—
148C2×C4⋊C4—61no—
149C2×C4⋊C4—61no—
150C2×C4⋊C4—61no—
151C2×C4⋊C4Dic611yescharacteristic, complemented
152C2×C4⋊C4Dic611yescharacteristic, complemented
153C2×C22⋊C4—21no—
154C2×C22⋊C4—21no—
155C2×C22⋊C4—31no—
156C2×C22⋊C4—31no—
157C2×C22⋊C4—61no—
158C2×C22⋊C4—61no—
159C2×C42—21no—
160C2×C42—31no—
161C2×C42—31no—
162C2×C42#6711yescharacteristic
163C2×C42#6711yescharacteristic
164C22⋊C8—21no—
165C22⋊C8—21no—
166C22⋊C8—61no—
167C22⋊C8—61no—
168#95—21no—
169#95—61no—
170#95—61no—
171#95—61no—
172#95—61no—
173#95—61no—
174#95—61no—
175#95—61no—
176#95—61no—
177#95—61no—
178C23×C3—21no—
179C23×C3—21no—
180C23×C3—41no—
181C23×C3—41no—
182C23×C3#12011yescharacteristic
183C23×C3C22⋊Q811yescharacteristic
184C23×C3#12411yescharacteristic
185C2×C3×C4—21no—
186C2×C3×C4—21no—
187C2×C3×C4—21no—
188C2×C3×C4—21no—
189C2×C3×C4—22no—
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—81no—
202C2×C3×C4—81no—
203C2×C3×C4C22⋊Q811yescharacteristic
204C2×C3×C4C22⋊Q811yescharacteristic
205C2×C3×C4#12411yescharacteristic
206C2×C3×C4#12411yescharacteristic
207C2×Dic3—22no—
208C2×Dic3—41no—
209C2×Dic3—41no—
210C2×Dic3—41no—
211C2×Dic3—41no—
212C2×Dic3—41no—
213C2×Dic3—41no—
214C2×Dic3—41no—
215C2×Dic3—41no—
216C2×Dic3—41no—
217C2×Dic3—41no—
218C2×Dic3—41no—
219C2×Dic3—81no—
220C2×Dic3—81no—
221C2×Dic3—81no—
222Dic6—84no—
223C3×C8—81no—
224C3⋊C8—81no—
225C24D3×D411yescharacteristic
226C2×Q8—122no—
227C2×Q8—122no—
228C2×Q8—241no—
229C22×C4—21no—
230C22×C4—21no—
231C22×C4—21no—
232C22×C4—21no—
233C22×C4—41no—
234C22×C4—41no—
235C22×C4—61no—
236C22×C4—61no—
237C22×C4—61no—
238C22×C4—61no—
239C22×C4—121no—
240C22×C4C2×Dic611yescharacteristic
241C22×C4#23511yescharacteristic
242C22×C4D3×D411yescharacteristic
243C22×C4#23711yescharacteristic
244C22×C4#23711yescharacteristic
245C22×C4C2×#6711yescharacteristic
246M16—41no—
247M16—41no—
248M16—121no—
249M16—121no—
250C2×C8—41no—
251C2×C8—121no—
252C4⋊C4—21no—
253C4⋊C4—21no—
254C4⋊C4—21no—
255C4⋊C4—21no—
256C4⋊C4—122no—
257C4⋊C4—122no—
258C4⋊C4—241no—
259C22⋊C4—42no—
260C22⋊C4—41no—
261C22⋊C4—41no—
262C22⋊C4—122no—
263C22⋊C4—121no—
264C22⋊C4—121no—
265C22⋊C4—121no—
266C22⋊C4—121no—
267C42—41no—
268C42—41no—
269C42—41no—
270C42—41no—
271C42—121no—
272C42—121no—
273C22×C3—21no—
274C22×C3—21no—
275C22×C3—21no—
276C22×C3—21no—
277C22×C3—21no—
278C22×C3—21no—
279C22×C3—21no—
280C22×C3—21no—
281C22×C3—81no—
282C22×C3—81no—
283C22×C3#39311yescharacteristic
284C22×C3#45811yescharacteristic
285C22×C3#45811yescharacteristic
286C3×C4—21no—
287C3×C4—21no—
288C3×C4—21no—
289C3×C4—21no—
290C3×C4—42no—
291C3×C4—81no—
292C3×C4—81no—
293C3×C4—81no—
294Dic3—42no—
295Dic3—81no—
296Dic3—81no—
297Dic3—81no—
298Dic3—161no—
299C23—21no—
300C23—21no—
301C23—41no—
302C23—41no—
303C23#87811yescharacteristic
304C23#88511yescharacteristic
305C23#93711yescharacteristic
306Q8—244no—
307C2×C4—21no—
308C2×C4—21no—
309C2×C4—21no—
310C2×C4—21no—
311C2×C4—22no—
312C2×C4—41no—
313C2×C4—41no—
314C2×C4—41no—
315C2×C4—41no—
316C2×C4—41no—
317C2×C4—41no—
318C2×C4—41no—
319C2×C4—41no—
320C2×C4—41no—
321C2×C4—41no—
322C2×C4—41no—
323C2×C4—62no—
324C2×C4—81no—
325C2×C4—81no—
326C2×C4—121no—
327C2×C4—121no—
328C2×C4—121no—
329C2×C4—121no—
330C2×C4—121no—
331C2×C4—121no—
332C2×C4—121no—
333C2×C4—121no—
334C2×C4—121no—
335C2×C4—121no—
336C2×C4—121no—
337C2×C4—241no—
338C2×C4—241no—
339C2×C4—241no—
340C2×C4#87811yescharacteristic
341C2×C4#88511yescharacteristic
342C2×C4#92411yescharacteristic
343C2×C4#93611yescharacteristic
344C8—81no—
345C8—241no—
346C2×C3—21no—
347C2×C3—21no—
348C2×C3—41no—
349C2×C3—41no—
350C2×C3#203011yescharacteristic
351C2×C3#203111yescharacteristic
352C2×C3#203111yescharacteristic
353C22—21no—
354C22—21no—
355C22—21no—
356C22—21no—
357C22—21no—
358C22—21no—
359C22—21no—
360C22—21no—
361C22—81no—
362C22—81no—
363C22#494711yescharacteristic
364C22#505311yescharacteristic
365C22#505311yescharacteristic
366C4—21no—
367C4—21no—
368C4—21no—
369C4—21no—
370C4—42no—
371C4—81no—
372C4—81no—
373C4—81no—
374C4—122no—
375C4—241no—
376C4—241no—
377C4—241no—
378C4—481no—
379C3#1038711yescharacteristic, complemented, Sylow
380C2—21no—
381C2—21no—
382C2—41no—
383C2—41no—
384C2#6953611yescharacteristic
385C2#7036811yescharacteristic
386C2#7039411yescharacteristic
subgroupG/H
centerC22#4947
commutator, cosocleC23×C3×C4C23
FrattiniC23×C4C22×D3
FittingC3×#1826C2
socleC22×C3#393
prime radicalC24×C3C2×D4
hypercenter#1826D3
hypocenterC3#10387
LCSG ▷ C23×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1826

Complements

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

NH ≅ G/Ntype
42. C2×C3×C4⋊C4Q8semidirect
43. C2×C3×C4⋊C4Q8semidirect
151. C2×C4⋊C4Dic6semidirect
152. C2×C4⋊C4Dic6semidirect
379. C3#10387semidirect

Maximal subgroups

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

subgroupindex
#735532
C3×#18262
#715942
#715902
#676862
#676822
#676162
#103873