Finite Group Databasebeta

← #10708129

Subgroups — #10708129

Order 768.

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

normal subgroups only

This group has 378 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#77070C211yesmaximal, characteristic
2C3×#1599C211yesmaximal, characteristic
3#71904C211yesmaximal, characteristic, complemented
4#71901C211yesmaximal, characteristic, complemented
5#67647C211yesmaximal, characteristic
6#67457C211yesmaximal, characteristic, complemented
7#67452C211yesmaximal, characteristic, complemented
8#12239—31nomaximal, Sylow
9C3×#544C2211yescharacteristic
10C3×#450—21no—
11C3×#394C2211yescharacteristic
12#5222—21no—
13#5050—21no—
14#4995—21no—
15#4979C2211yescharacteristic
16#4977C2211yescharacteristic
17#4951—21no—
18#4947—21no—
19#4945—21no—
20#4944—21no—
21#4650—21no—
22C3×#328—21no—
23C3×#324C2211yescharacteristic
24#4538—21no—
25#4532—21no—
26#4530C2211yescharacteristic
27#4528C2211yescharacteristic
28#2785—31no—
29#2061—31no—
30#1605—31no—
31#1599D311yescharacteristic
32#1354—31no—
33#1345—31no—
34#1324—31no—
35C22×C3×D4D411yescharacteristic
36C22×Dic6D411yescharacteristic
37C2×C3×SD16—41no—
38C3×C4⋊Q8—41no—
39C3×#125—41no—
40C3×C22⋊Q8—21no—
41C3×C22⋊Q8—21no—
42C3×#117—21no—
43C2×C3×C4⋊C4C2311yescharacteristic
44C2×C3×C22⋊C4—21no—
45C2×C3×C22⋊C4—21no—
46C2×#217—21no—
47C2×#217—21no—
48C2×#217—21no—
49C2×#217—21no—
50C2×#216—41no—
51C2×C4⋊Dic3D411yescharacteristic
52C2×Dic3⋊C4—21no—
53C2×Dic3⋊C4—21no—
54C2×C4×Dic3—21no—
55C2×C4×Dic3D411yescharacteristic
56C2×#208—21no—
57C3×#102—41no—
58C3×C22⋊C8—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#808—41no—
69#807—41no—
70#544C2×D311yescharacteristic
71#451—61no—
72#450—21no—
73#428—61no—
74#395—31no—
75#394—61no—
76#394C2×D311yescharacteristic
77#393—61no—
78#388—61no—
79#388—61no—
80#388—61no—
81#382—31no—
82#382—61no—
83#343—31no—
84#339—61no—
85#328—21no—
86#328—61no—
87#324—31no—
88#324C2×D311yescharacteristic
89C24×C3—21no—
90C2×C3×Q8—21no—
91C2×C3×Q8—21no—
92C2×C3×D4—41no—
93C2×C3×D4—81no—
94C22×C3×C4—21no—
95C22×C3×C4C2×D411yescharacteristic
96C22×C3×C4C2×D411yescharacteristic
97C22×C3×C4C2×D411yescharacteristic
98C22×Dic3—21no—
99C22×Dic3—21no—
100C22×Dic3—21no—
101C22×Dic3—21no—
102C2×Dic6—41no—
103C2×Dic6—81no—
104C3×SD16—82no—
105C2×C3×C8—41no—
106C3×C4⋊C4—21no—
107C3×C4⋊C4—21no—
108C3×C4⋊C4—21no—
109C3×C4⋊C4—21no—
110C3×C4⋊C4—41no—
111C3×C4⋊C4—41no—
112C3×C4⋊C4—41no—
113C3×C22⋊C4—41no—
114C3×C22⋊C4—41no—
115C3×C22⋊C4—41no—
116C3×C22⋊C4—81no—
117C3×C42—41no—
118#217—42no—
119#217—42no—
120#217—81no—
121#217—81no—
122#216—82no—
123C4⋊Dic3—41no—
124Dic3⋊C4—81no—
125Dic3⋊C4—81no—
126C4×Dic3—21no—
127C4×Dic3—21no—
128C4×Dic3—42no—
129#208—22no—
130#208—41no—
131C2×C3⋊C8—41no—
132C2×C3⋊C8—41no—
133C22×Q8—31no—
134C22×D4#6711yescharacteristic
135C2×Q16—121no—
136C2×SD16—41no—
137C2×M16—61no—
138C4⋊Q8—41no—
139#125—41no—
140C22⋊Q8—21no—
141C22⋊Q8—21no—
142#117—21no—
143C2×C4⋊C4—31no—
144C2×C4⋊C4—61no—
145C2×C4⋊C4—61no—
146C2×C4⋊C4C22×D311yescharacteristic
147C2×C22⋊C4—21no—
148C2×C22⋊C4—21no—
149C2×C22⋊C4—61no—
150C2×C22⋊C4—61no—
151C2×C22⋊C4—61no—
152C2×C22⋊C4—61no—
153C2×C42—31no—
154C2×C42—61no—
155#107—121no—
156#106—121no—
157#103—121no—
158#102—41no—
159C22⋊C8—21no—
160C22⋊C8—61no—
161#95—31no—
162#95—61no—
163#95—61no—
164#95—61no—
165#95—61no—
166#95—61no—
167#95#6711yescharacteristic
168C23×C3—41no—
169C23×C3—41no—
170C23×C3—41no—
171C23×C3—41no—
172C23×C3—41no—
173C23×C3—41no—
174C23×C3—41no—
175C23×C3#12011yescharacteristic
176C3×Q8—81no—
177C3×Q8—81no—
178C3×D4—42no—
179C3×D4—81no—
180C2×C3×C4—21no—
181C2×C3×C4—21no—
182C2×C3×C4—21no—
183C2×C3×C4—21no—
184C2×C3×C4—41no—
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#13511yescharacteristic
192C2×C3×C4#13711yescharacteristic
193C2×Dic3—22no—
194C2×Dic3—41no—
195C2×Dic3—41no—
196C2×Dic3—41no—
197C2×Dic3—41no—
198C2×Dic3—41no—
199C2×Dic3—41no—
200C2×Dic3—41no—
201C2×Dic3—41no—
202C2×Dic3—41no—
203C2×Dic3—41no—
204C2×Dic3—41no—
205Dic6—42no—
206Dic6—81no—
207C3×C8—81no—
208C3⋊C8—42no—
209C3⋊C8—81no—
210C24—21no—
211C2×Q8—21no—
212C2×Q8—21no—
213C2×Q8—121no—
214C2×Q8—241no—
215C2×D4—41no—
216C2×D4—81no—
217C22×C4—21no—
218C22×C4—61no—
219C22×C4—61no—
220C22×C4—61no—
221C22×C4—61no—
222C22×C4D3×D411yescharacteristic
223C22×C4D3×D411yescharacteristic
224C22×C4C2×#6711yescharacteristic
225Q16—242no—
226SD16—82no—
227M16—62no—
228M16—121no—
229C2×C8—41no—
230C2×C8—121no—
231C2×C8—121no—
232C4⋊C4—21no—
233C4⋊C4—21no—
234C4⋊C4—21no—
235C4⋊C4—21no—
236C4⋊C4—41no—
237C4⋊C4—41no—
238C4⋊C4—41no—
239C4⋊C4—121no—
240C4⋊C4—241no—
241C4⋊C4—241no—
242C22⋊C4—41no—
243C22⋊C4—41no—
244C22⋊C4—41no—
245C22⋊C4—81no—
246C22⋊C4—122no—
247C22⋊C4—122no—
248C22⋊C4—241no—
249C22⋊C4—241no—
250C42—41no—
251C42—61no—
252C42—61no—
253C42—122no—
254C22×C3—21no—
255C22×C3—21no—
256C22×C3—81no—
257C22×C3—81no—
258C22×C3—81no—
259C22×C3—81no—
260C22×C3—81no—
261C22×C3—81no—
262C22×C3—81no—
263C22×C3#45211yescharacteristic
264C22×C3#45611yescharacteristic
265C22×C3B(4,𝔽2)11yescharacteristic
266C3×C4—21no—
267C3×C4—21no—
268C3×C4—41no—
269C3×C4—41no—
270C3×C4—81no—
271C3×C4—81no—
272C3×C4—81no—
273Dic3—42no—
274Dic3—81no—
275Dic3—81no—
276Dic3—81no—
277C23—41no—
278C23—41no—
279C23—41no—
280C23—41no—
281C23—41no—
282C23—41no—
283C23—41no—
284C23#93711yescharacteristic
285Q8—81no—
286Q8—81no—
287Q8—122no—
288Q8—241no—
289D4—42no—
290D4—81no—
291C2×C4—21no—
292C2×C4—21no—
293C2×C4—21no—
294C2×C4—21no—
295C2×C4—41no—
296C2×C4—41no—
297C2×C4—41no—
298C2×C4—41no—
299C2×C4—41no—
300C2×C4—41no—
301C2×C4—41no—
302C2×C4—62no—
303C2×C4—121no—
304C2×C4—121no—
305C2×C4—121no—
306C2×C4—121no—
307C2×C4—121no—
308C2×C4—121no—
309C2×C4—121no—
310C2×C4—121no—
311C2×C4—121no—
312C2×C4—121no—
313C2×C4—121no—
314C2×C4#91511yescharacteristic
315C2×C4#91611yescharacteristic
316C8—81no—
317C8—122no—
318C8—241no—
319C2×C3—21no—
320C2×C3—21no—
321C2×C3—161no—
322C2×C3#160311yescharacteristic
323C2×C3#219911yescharacteristic
324C2×C3#220411yescharacteristic
325C22—21no—
326C22—21no—
327C22—81no—
328C22—81no—
329C22—81no—
330C22—81no—
331C22—81no—
332C22—81no—
333C22—81no—
334C22#503211yescharacteristic
335C22#517311yescharacteristic
336C22#520511yescharacteristic
337C4—21no—
338C4—21no—
339C4—41no—
340C4—41no—
341C4—81no—
342C4—81no—
343C4—81no—
344C4—122no—
345C4—241no—
346C4—241no—
347C4—241no—
348C3#1223911yescharacteristic, complemented, Sylow
349C2—21no—
350C2—21no—
351C2—161no—
352C2#7097411yescharacteristic
353C2#7098811yescharacteristic
354C2#7099711yescharacteristic
subgroupG/H
centerC22#5032
commutator, cosocleC2×C3×C4⋊C4C23
FrattiniC2×C4⋊C4C22×D3
FittingC3×#1599C2
socleC22×C3B(4,𝔽2)
prime radicalC22×C3×D4D4
hypercenter#1599D3
hypocenterC3#12239
LCSG ▷ C2×C3×C4⋊C4 ▷ C23×C3 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23 ◁ C2×C4⋊C4 ◁ #1599

Complements

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

NH ≅ G/Ntype
3. #71904C2semidirect
4. #71901C2semidirect
6. #67457C2semidirect
7. #67452C2semidirect
348. C3#12239semidirect

Maximal subgroups

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

subgroupindex
#770702
C3×#15992
#719042
#719012
#676472
#674572
#674522
#122393