Finite Group Databasebeta

← #10708317

Subgroups — #10708317

Order 768.

458 conjugacy classes of subgroups; only the 55 normal subgroups are shown.

normal subgroups only

This group has 458 conjugacy classes of subgroups — too many to draw. Only its 55 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#73842C211yesmaximal, characteristic
2C3×#2029C211yesmaximal, characteristic
3#71891C211yesmaximal, characteristic, complemented
4#71890C211yesmaximal, characteristic, complemented
5#67622C211yesmaximal, characteristic
6#67492C211yesmaximal, characteristic, complemented
7#67491C211yesmaximal, characteristic, complemented
8#12268—31nomaximal, Sylow
9C3×#543C2211yescharacteristic
10C3×#379—21no—
11C3×#379C2211yescharacteristic
12#5247—21no—
13#5044—21no—
14#5043—21no—
15#4953C2211yescharacteristic
16#4952—21no—
17#4951—21no—
18#4949C2211yescharacteristic
19#4948—21no—
20#4945—21no—
21#4944—21no—
22C3×#342—21no—
23C3×#342C2211yescharacteristic
24#4538—21no—
25#4538—21no—
26#4525C2211yescharacteristic
27#4525C2211yescharacteristic
28#2735—31no—
29#2029D311yescharacteristic
30#1614—31no—
31#1608—31no—
32#1352—31no—
33#1344—31no—
34#1316—31no—
35C25×C3D411yescharacteristic
36C3×C22⋊Q8—21no—
37C3×C22⋊Q8—21no—
38C3×C22⋊Q8—41no—
39C3×C22⋊Q8—41no—
40C2×C3×C22⋊C4—21no—
41C2×C3×C22⋊C4—21no—
42C2×C3×C22⋊C4—21no—
43C2×C3×C22⋊C4—21no—
44C2×C3×C22⋊C4—41no—
45C2×C3×C22⋊C4—41no—
46C2×C3×C22⋊C4D411yescharacteristic
47C2×C3×C22⋊C4C2311yescharacteristic
48C2×#217—21no—
49C2×#217—21no—
50C2×#217—21no—
51C2×#217—21no—
52C2×#217—21no—
53C2×#217—21no—
54C2×C4⋊Dic3D411yescharacteristic
55C2×C4⋊Dic3D411yescharacteristic
56C2×Dic3⋊C4—21no—
57C2×Dic3⋊C4—21no—
58C2×C4×Dic3—21no—
59#835—41no—
60#835—41no—
61C3⋊#95—21no—
62C3⋊#95—21no—
63C3⋊#95—21no—
64C3⋊#95D411yescharacteristic
65C3⋊#95D411yescharacteristic
66#830—21no—
67#830—21no—
68#808—41no—
69#807—41no—
70#543C2×D311yescharacteristic
71#484—61no—
72#483—61no—
73#399—31no—
74#397—31no—
75#397—61no—
76#396—61no—
77#388—61no—
78#388—61no—
79#388—61no—
80#379—21no—
81#379—61no—
82#379C2×D311yescharacteristic
83#342—21no—
84#342C2×D311yescharacteristic
85#328—61no—
86#328—61no—
87#323—31no—
88#323—31no—
89C24×C3—21no—
90C24×C3—21no—
91C24×C3—21no—
92C24×C3—41no—
93C24×C3—41no—
94C24×C3—42no—
95C24×C3—81no—
96C24×C3C2×D411yescharacteristic
97C2×C3×Q8—21no—
98C2×C3×Q8—21no—
99C22×C3×C4—21no—
100C22×C3×C4—21no—
101C22×C3×C4—21no—
102C22×C3×C4—41no—
103C22×C3×C4C2×D411yescharacteristic
104C22×C3×C4C2×D411yescharacteristic
105C22×Dic3—21no—
106C22×Dic3—21no—
107C22×Dic3—21no—
108C22×Dic3—21no—
109C3×C4⋊C4—21no—
110C3×C4⋊C4—21no—
111C3×C4⋊C4—41no—
112C3×C4⋊C4—41no—
113C3×C22⋊C4—41no—
114C3×C22⋊C4—42no—
115C3×C22⋊C4—41no—
116C3×C22⋊C4—41no—
117C3×C22⋊C4—41no—
118C3×C22⋊C4—41no—
119C3×C22⋊C4—41no—
120C3×C22⋊C4—81no—
121C3×C22⋊C4—82no—
122C3×C22⋊C4—82no—
123#217—42no—
124#217—41no—
125#217—41no—
126#217—81no—
127#217—81no—
128#217—81no—
129#217—81no—
130C4⋊Dic3—41no—
131C4⋊Dic3—41no—
132Dic3⋊C4—81no—
133Dic3⋊C4—81no—
134C4×Dic3—81no—
135C2×C3⋊C8—41no—
136C2×C3⋊C8—41no—
137C25#6711yescharacteristic
138C22⋊Q8—21no—
139C22⋊Q8—21no—
140C22⋊Q8—41no—
141C22⋊Q8—41no—
142C2×C4⋊C4—31no—
143C2×C4⋊C4—31no—
144C2×C4⋊C4—61no—
145C2×C4⋊C4—61no—
146C2×C22⋊C4—21no—
147C2×C22⋊C4—21no—
148C2×C22⋊C4—21no—
149C2×C22⋊C4—21no—
150C2×C22⋊C4—41no—
151C2×C22⋊C4—41no—
152C2×C22⋊C4—61no—
153C2×C22⋊C4—61no—
154C2×C22⋊C4—61no—
155C2×C22⋊C4—61no—
156C2×C22⋊C4—61no—
157C2×C22⋊C4—61no—
158C2×C22⋊C4#6711yescharacteristic
159C2×C22⋊C4C22×D311yescharacteristic
160C2×C42—61no—
161#107—121no—
162#106—121no—
163#103—121no—
164#103—121no—
165C22⋊C8—61no—
166C22⋊C8—61no—
167#95—31no—
168#95—31no—
169#95—61no—
170#95—61no—
171#95—61no—
172C23×C3—21no—
173C23×C3—21no—
174C23×C3—22no—
175C23×C3—41no—
176C23×C3—41no—
177C23×C3—41no—
178C23×C3—41no—
179C23×C3—41no—
180C23×C3—42no—
181C23×C3—42no—
182C23×C3—42no—
183C23×C3—41no—
184C23×C3—41no—
185C23×C3—41no—
186C23×C3—81no—
187C23×C3—82no—
188C23×C3—81no—
189C23×C3—82no—
190C23×C3—84no—
191C23×C3—81no—
192C23×C3#12011yescharacteristic
193C23×C3#13511yescharacteristic
194C23×C3#13711yescharacteristic
195C3×Q8—81no—
196C3×Q8—81no—
197C2×C3×C4—21no—
198C2×C3×C4—21no—
199C2×C3×C4—41no—
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—81no—
207C2×C3×C4—81no—
208C2×C3×C4—81no—
209C2×C3×C4—81no—
210C2×C3×C4—81no—
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—41no—
220C2×Dic3—41no—
221C2×Dic3—41no—
222C2×Dic3—41no—
223C3⋊C8—81no—
224C3⋊C8—81no—
225C24—21no—
226C24—21no—
227C24—21no—
228C24—41no—
229C24—41no—
230C24—42no—
231C24—81no—
232C24C2×#6711yescharacteristic
233C2×Q8—21no—
234C2×Q8—21no—
235C22×C4—21no—
236C22×C4—21no—
237C22×C4—21no—
238C22×C4—41no—
239C22×C4—61no—
240C22×C4—61no—
241C22×C4—61no—
242C22×C4—61no—
243C22×C4D3×D411yescharacteristic
244C22×C4D3×D411yescharacteristic
245C2×C8—121no—
246C2×C8—121no—
247C4⋊C4—21no—
248C4⋊C4—21no—
249C4⋊C4—41no—
250C4⋊C4—41no—
251C4⋊C4—121no—
252C4⋊C4—121no—
253C4⋊C4—241no—
254C4⋊C4—241no—
255C22⋊C4—41no—
256C22⋊C4—41no—
257C22⋊C4—42no—
258C22⋊C4—41no—
259C22⋊C4—41no—
260C22⋊C4—41no—
261C22⋊C4—41no—
262C22⋊C4—81no—
263C22⋊C4—82no—
264C22⋊C4—82no—
265C22⋊C4—122no—
266C22⋊C4—121no—
267C22⋊C4—121no—
268C22⋊C4—241no—
269C22⋊C4—241no—
270C22⋊C4—241no—
271C22⋊C4—241no—
272C42—241no—
273C22×C3—21no—
274C22×C3—21no—
275C22×C3—21no—
276C22×C3—21no—
277C22×C3—41no—
278C22×C3—41no—
279C22×C3—42no—
280C22×C3—42no—
281C22×C3—41no—
282C22×C3—41no—
283C22×C3—44no—
284C22×C3—42no—
285C22×C3—81no—
286C22×C3—81no—
287C22×C3—81no—
288C22×C3—82no—
289C22×C3—82no—
290C22×C3—82no—
291C22×C3—81no—
292C22×C3—81no—
293C22×C3#45211yescharacteristic
294C22×C3#45511yescharacteristic
295C22×C3B(4,𝔽2)11yescharacteristic
296C3×C4—41no—
297C3×C4—41no—
298C3×C4—81no—
299C3×C4—81no—
300C3×C4—81no—
301C3×C4—161no—
302Dic3—81no—
303Dic3—81no—
304Dic3—81no—
305Dic3—81no—
306C23—21no—
307C23—21no—
308C23—22no—
309C23—41no—
310C23—41no—
311C23—41no—
312C23—41no—
313C23—41no—
314C23—42no—
315C23—42no—
316C23—42no—
317C23—41no—
318C23—41no—
319C23—41no—
320C23—81no—
321C23—82no—
322C23—81no—
323C23—82no—
324C23—84no—
325C23—81no—
326C23#93711yescharacteristic
327C23#94211yescharacteristic
328C23#95011yescharacteristic
329Q8—81no—
330Q8—81no—
331C2×C4—21no—
332C2×C4—21no—
333C2×C4—41no—
334C2×C4—41no—
335C2×C4—41no—
336C2×C4—41no—
337C2×C4—41no—
338C2×C4—41no—
339C2×C4—41no—
340C2×C4—81no—
341C2×C4—81no—
342C2×C4—81no—
343C2×C4—81no—
344C2×C4—81no—
345C2×C4—121no—
346C2×C4—121no—
347C2×C4—121no—
348C2×C4—121no—
349C2×C4—121no—
350C2×C4—121no—
351C2×C4—121no—
352C2×C4—121no—
353C2×C4—121no—
354C2×C4—121no—
355C2×C4—121no—
356C2×C4—121no—
357C8—241no—
358C8—241no—
359C2×C3—21no—
360C2×C3—21no—
361C2×C3—41no—
362C2×C3—41no—
363C2×C3—82no—
364C2×C3#160611yescharacteristic
365C2×C3#219711yescharacteristic
366C2×C3#220011yescharacteristic
367C22—21no—
368C22—21no—
369C22—21no—
370C22—21no—
371C22—41no—
372C22—41no—
373C22—42no—
374C22—42no—
375C22—41no—
376C22—41no—
377C22—44no—
378C22—42no—
379C22—81no—
380C22—81no—
381C22—81no—
382C22—82no—
383C22—82no—
384C22—82no—
385C22—81no—
386C22—81no—
387C22#503211yescharacteristic
388C22#504711yescharacteristic
389C22#506111yescharacteristic
390C4—41no—
391C4—41no—
392C4—81no—
393C4—81no—
394C4—81no—
395C4—161no—
396C4—241no—
397C4—241no—
398C4—241no—
399C4—241no—
400C3#1226811yescharacteristic, complemented, Sylow
401C2—21no—
402C2—21no—
403C2—41no—
404C2—41no—
405C2—82no—
406C2#6970011yescharacteristic
407C2#7099811yescharacteristic
408C2#7099911yescharacteristic
subgroupG/H
centerC22#5032
commutator, cosocleC2×C3×C22⋊C4C23
FrattiniC2×C22⋊C4C22×D3
FittingC3×#2029C2
socleC22×C3B(4,𝔽2)
prime radicalC25×C3D4
hypercenter#2029D3
hypocenterC3#12268
LCSG ▷ C2×C3×C22⋊C4 ▷ C23×C3 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23 ◁ C2×C22⋊C4 ◁ #2029

Complements

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

NH ≅ G/Ntype
3. #71891C2semidirect
4. #71890C2semidirect
6. #67492C2semidirect
7. #67491C2semidirect
400. C3#12268semidirect

Maximal subgroups

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

subgroupindex
#738422
C3×#20292
#718912
#718902
#676222
#674922
#674912
#122683