Finite Group Databasebeta

← #10708367

Subgroups — #10708367

Order 768.

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

normal subgroups only

This group has 372 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#77242C211yesmaximal, characteristic
2C3×#1605C211yesmaximal, characteristic, complemented
3#71905C211yesmaximal, characteristic, complemented
4#71902C211yesmaximal, characteristic
5#67648C211yesmaximal, characteristic, complemented
6#67456C211yesmaximal, characteristic
7#67455C211yesmaximal, characteristic, complemented
8#12275—31nomaximal, Sylow
9C3×#544C2211yescharacteristic
10C3×#451—21no—
11C3×#395C2211yescharacteristic
12#5231—21no—
13#5046—21no—
14#5004—21no—
15#4991—21no—
16#4991C2211yescharacteristic
17#4985—21no—
18#4973C2211yescharacteristic
19#4675—21no—
20#4673—21no—
21#4669—21no—
22C3×#328—21no—
23C3×#324C2211yescharacteristic
24#4538—21no—
25#4532—21no—
26#4530C2211yescharacteristic
27#4528C2211yescharacteristic
28#2832—31no—
29#2028—31no—
30#1605D311yescharacteristic, complemented
31#1599—31no—
32#1354—31no—
33#1345—31no—
34#1319—31no—
35C22×C3×Q8D411yescharacteristic
36C22×D12D411yescharacteristic
37C22×Dic6D411yescharacteristic
38C2×C3×Q16—41no—
39C3×C4⋊Q8—41no—
40C3×#125—41no—
41C3×C22⋊Q8—21no—
42C3×C22⋊Q8—21no—
43C3×#117—21no—
44C2×C3×C4⋊C4—21no—
45C2×C3×C4⋊C4C2311yescharacteristic
46C2×#215—41no—
47C2×(C2×D3)⋊C4—21no—
48C2×(C2×D3)⋊C4—21no—
49C2×(C2×D3)⋊C4—21no—
50C2×(C2×D3)⋊C4—21no—
51C2×(C2×D3)⋊C4—21no—
52C2×(C2×D3)⋊C4—21no—
53C2×Dic3⋊C4—21no—
54C2×C4×Dic3D411yescharacteristic
55C2×#208—21no—
56C3×#103—41no—
57C3×C22⋊C8—21no—
58C3×#95—21no—
59C3×#95D411yescharacteristic
60C3⋊#95—21no—
61C3⋊#95—21no—
62C3⋊#95—21no—
63C3⋊#95—21no—
64C3⋊#95—21no—
65C3⋊#95D411yescharacteristic
66#830—21no—
67#810—41no—
68#810—41no—
69#809—41no—
70#544C2×D311yescharacteristic
71#451—21no—
72#450—61no—
73#419—61no—
74#400—61no—
75#397—61no—
76#395C2×D311yescharacteristic
77#394—31no—
78#394—61no—
79#393—61no—
80#393—61no—
81#391—31no—
82#388—61no—
83#343—31no—
84#339—61no—
85#328—21no—
86#328—61no—
87#324—31no—
88#324C2×D311yescharacteristic
89C23×D3—21no—
90C2×C3×Q8—21no—
91C2×C3×Q8—21no—
92C2×C3×Q8—41no—
93C2×C3×Q8—81no—
94C22×C3×C4—21no—
95C22×C3×C4—21no—
96C22×C3×C4C2×D411yescharacteristic
97C22×C3×C4C2×D411yescharacteristic
98C22×C3×C4C2×D411yescharacteristic
99C22×Dic3—21no—
100C22×Dic3—21no—
101C22×Dic3—21no—
102C2×D12—41no—
103C2×D12—81no—
104C2×Dic6—21no—
105C2×Dic6—21no—
106C2×Dic6—41no—
107C2×Dic6—41no—
108C3×Q16—82no—
109C2×C3×C8—41no—
110C3×C4⋊C4—21no—
111C3×C4⋊C4—21no—
112C3×C4⋊C4—21no—
113C3×C4⋊C4—21no—
114C3×C4⋊C4—41no—
115C3×C4⋊C4—41no—
116C3×C4⋊C4—41no—
117C3×C4⋊C4—81no—
118C3×C22⋊C4—41no—
119C3×C42—41no—
120#215—82no—
121(C2×D3)⋊C4—41no—
122(C2×D3)⋊C4—41no—
123(C2×D3)⋊C4—81no—
124(C2×D3)⋊C4—81no—
125(C2×D3)⋊C4—81no—
126(C2×D3)⋊C4—81no—
127(C2×D3)⋊C4—81no—
128Dic3⋊C4—81no—
129C4×Dic3—41no—
130#208—41no—
131#208—41no—
132C2×C3⋊C8—41no—
133C2×C3⋊C8—41no—
134C22×Q8—31no—
135C22×Q8#6711yescharacteristic
136C22×D4—31no—
137C2×Q16—41no—
138C2×SD16—121no—
139C2×M16—61no—
140C4⋊Q8—41no—
141#125—41no—
142C22⋊Q8—21no—
143C22⋊Q8—21no—
144#117—21no—
145C2×C4⋊C4—21no—
146C2×C4⋊C4—61no—
147C2×C4⋊C4C22×D311yescharacteristic
148C2×C22⋊C4—61no—
149C2×C22⋊C4—61no—
150C2×C22⋊C4—61no—
151C2×C22⋊C4—61no—
152C2×C22⋊C4—61no—
153C2×C22⋊C4—61no—
154C2×C42—31no—
155#103—41no—
156#103—121no—
157#103—121no—
158#102—121no—
159C22⋊C8—21no—
160C22⋊C8—61no—
161#95—21no—
162#95—31no—
163#95—61no—
164#95—61no—
165#95—61no—
166#95—61no—
167#95—61no—
168#95#6711yescharacteristic
169C23×C3#12011yescharacteristic
170C22×D3—41no—
171C22×D3—41no—
172C22×D3—41no—
173C22×D3—41no—
174C22×D3—41no—
175C22×D3—41no—
176C22×D3—41no—
177C3×Q8—42no—
178C3×Q8—81no—
179C3×Q8—81no—
180C3×Q8—81no—
181C2×C3×C4—21no—
182C2×C3×C4—21no—
183C2×C3×C4—21no—
184C2×C3×C4—21no—
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#13511yescharacteristic
196C2×C3×C4#13711yescharacteristic
197C2×Dic3—41no—
198C2×Dic3—41no—
199C2×Dic3—41no—
200C2×Dic3—41no—
201C2×Dic3—41no—
202C2×Dic3—41no—
203C2×Dic3—41no—
204C2×Dic3—41no—
205C2×Dic3—41no—
206D12—42no—
207D12—81no—
208Dic6—42no—
209Dic6—81no—
210C3×C8—81no—
211C3⋊C8—81no—
212C3⋊C8—81no—
213C24—61no—
214C2×Q8—21no—
215C2×Q8—21no—
216C2×Q8—41no—
217C2×Q8—61no—
218C2×Q8—61no—
219C2×Q8—81no—
220C2×Q8—121no—
221C2×Q8—121no—
222C2×D4—121no—
223C2×D4—241no—
224C22×C4—21no—
225C22×C4—21no—
226C22×C4—61no—
227C22×C4—61no—
228C22×C4—61no—
229C22×C4D3×D411yescharacteristic
230C22×C4D3×D411yescharacteristic
231C22×C4C2×#6711yescharacteristic
232Q16—82no—
233SD16—242no—
234M16—121no—
235M16—121no—
236C2×C8—41no—
237C2×C8—121no—
238C2×C8—121no—
239C4⋊C4—21no—
240C4⋊C4—21no—
241C4⋊C4—21no—
242C4⋊C4—21no—
243C4⋊C4—41no—
244C4⋊C4—41no—
245C4⋊C4—41no—
246C4⋊C4—81no—
247C4⋊C4—241no—
248C22⋊C4—41no—
249C22⋊C4—121no—
250C22⋊C4—121no—
251C22⋊C4—241no—
252C22⋊C4—241no—
253C22⋊C4—241no—
254C22⋊C4—241no—
255C22⋊C4—241no—
256C42—41no—
257C42—121no—
258C22×C3—21no—
259C22×C3—21no—
260C22×C3#45211yescharacteristic
261C22×C3#45511yescharacteristic
262C22×C3B(4,𝔽2)11yescharacteristic
263C2×D3—81no—
264C2×D3—81no—
265C2×D3—81no—
266C2×D3—81no—
267C2×D3—81no—
268C2×D3—81no—
269C2×D3—81no—
270C3×C4—21no—
271C3×C4—21no—
272C3×C4—41no—
273C3×C4—41no—
274C3×C4—81no—
275C3×C4—81no—
276C3×C4—81no—
277C3×C4—81no—
278Dic3—81no—
279Dic3—81no—
280Dic3—81no—
281C23—121no—
282C23—121no—
283C23—121no—
284C23—121no—
285C23—121no—
286C23—121no—
287C23—121no—
288C23#93711yescharacteristic
289Q8—42no—
290Q8—81no—
291Q8—81no—
292Q8—81no—
293Q8—122no—
294Q8—241no—
295D4—122no—
296D4—241no—
297C2×C4—21no—
298C2×C4—21no—
299C2×C4—21no—
300C2×C4—21no—
301C2×C4—41no—
302C2×C4—41no—
303C2×C4—41no—
304C2×C4—41no—
305C2×C4—41no—
306C2×C4—41no—
307C2×C4—41no—
308C2×C4—41no—
309C2×C4—41no—
310C2×C4—41no—
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×C4#91811yescharacteristic
321C2×C4#91911yescharacteristic
322C8—81no—
323C8—241no—
324C8—241no—
325C2×C3—21no—
326C2×C3—21no—
327C2×C3#160611yescharacteristic
328C2×C3#219911yescharacteristic
329C2×C3#220311yescharacteristic
330D3—161no—
331C22—21no—
332C22—21no—
333C22—241no—
334C22—241no—
335C22—241no—
336C22—241no—
337C22—241no—
338C22—241no—
339C22—241no—
340C22#503211yescharacteristic
341C22#518711yescharacteristic
342C22#520211yescharacteristic
343C4—21no—
344C4—21no—
345C4—41no—
346C4—41no—
347C4—81no—
348C4—81no—
349C4—81no—
350C4—81no—
351C4—241no—
352C4—241no—
353C4—241no—
354C3#1227511yescharacteristic, complemented, Sylow
355C2—21no—
356C2—21no—
357C2—481no—
358C2#7098111yescharacteristic
359C2#7099311yescharacteristic
360C2#7099711yescharacteristic
subgroupG/H
centerC22#5032
commutator, cosocleC2×C3×C4⋊C4C23
FrattiniC2×C4⋊C4C22×D3
FittingC3×#1605C2
socleC22×C3B(4,𝔽2)
prime radicalC22×D12D4
hypercenter#1605D3
hypocenterC3#12275
LCSG ▷ C2×C3×C4⋊C4 ▷ C23×C3 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23 ◁ C2×C4⋊C4 ◁ #1605

Complements

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

NH ≅ G/Ntype
2. C3×#1605C2semidirect
3. #71905C2semidirect
5. #67648C2semidirect
7. #67455C2semidirect
30. #1605D3semidirect
354. C3#12275semidirect

Maximal subgroups

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

subgroupindex
#772422
C3×#16052
#719052
#719022
#676482
#674562
#674552
#122753