Finite Group Databasebeta

← #10698947

Subgroups — #10698947

Order 768.

398 conjugacy classes of subgroups; only the 91 normal subgroups are shown.

normal subgroups only

This group has 398 conjugacy classes of subgroups — too many to draw. Only its 91 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#83350C211yesmaximal, characteristic
2C3×#1542C211yesmaximal, characteristic, complemented
3#71838C211yesmaximal, characteristic, complemented
4#70783C211yesmaximal, characteristic
5#70480C211yesmaximal, characteristic
6#70421C211yesmaximal, characteristic, complemented
7#67738C211yesmaximal, characteristic, complemented
8#10468—31nomaximal, Sylow
9C3×#533C2211yescharacteristic
10C2×C4×D12C2211yescharacteristic
11C3×#431C2211yescharacteristic
12C2×C3×C4⋊C8C2211yescharacteristic
13C2×#821—21no—
14#5087—21no—
15#5084—21no—
16#5066—22no—
17#5001—21no—
18#4995—21no—
19#4936C2211yescharacteristic
20#4925C2211yescharacteristic
21#4924C2211yescharacteristic
22C3×#333—21no—
23C3×#332—21no—
24#4551—21no—
25#4538—22no—
26#4480—22no—
27#4468—22no—
28#3077—31no—
29#1804—31no—
30#1570—31no—
31#1542D311yescharacteristic, complemented
32#1518—31no—
33#1460—31no—
34#1286—31no—
35C22×D12D411yescharacteristic
36C22×C4×D3—21no—
37C22×C3×C8—21no—
38C3×C4⋊Q8D411yescharacteristic
39C3×C4⋊Q8D411yescharacteristic
40C3×#125—21no—
41C3×C22⋊Q8—21no—
42C3×C22⋊Q8—21no—
43C3×C4×Q8—21no—
44C3×C4×Q8—21no—
45C3×#117—21no—
46C2×C3×C42C2311yescharacteristic
47C2×(C2×D3)⋊C4—21no—
48C2×C4⋊Dic3—21no—
49C2×C4⋊Dic3D411yescharacteristic
50C2×#208—21no—
51C4×D12—21no—
52C4×D12—21no—
53C4×D12—21no—
54C4×D12—21no—
55C3×C4⋊C8—21no—
56C3×C4⋊C8—21no—
57C3×C4⋊C8—21no—
58C3×C22⋊C8—21no—
59C3×C4×C8—21no—
60C3⋊#95—21no—
61C3⋊#95—21no—
62#830D412yes—
63#821—42no—
64#809—22no—
65#809—22no—
66#808—22no—
67#807—22no—
68C4⋊(C3⋊C8)—21no—
69C4⋊(C3⋊C8)—21no—
70C4⋊(C3⋊C8)D412yes—
71C4×C3⋊C8—21no—
72#533C2×D311yescharacteristic
73C2×C4×D4—31no—
74#478—62no—
75#476—61no—
76#474—61no—
77#431—31no—
78#431C2×D311yescharacteristic
79#428—61no—
80#423—31no—
81C2×C4⋊C8C2×D311yescharacteristic
82#418—61no—
83C2×#102—61no—
84#382—31no—
85#340—61no—
86#334—62no—
87#333—21no—
88#332—21no—
89#328—62no—
90#323—62no—
91C23×D3—21no—
92C2×C3×Q8—21no—
93C2×C3×Q8—21no—
94C22×C3×C4C2×D411yescharacteristic
95C22×C3×C4C2×D411yescharacteristic
96C22×C3×C4#4111yescharacteristic
97C22×Dic3—21no—
98C22×Dic3—41no—
99C2×D12—21no—
100C2×D12—21no—
101C2×D12—21no—
102C2×D12—21no—
103C2×D12—21no—
104C2×D12SD1612yes—
105C2×C4×D3—41no—
106C2×C4×D3—41no—
107C2×C4×D3—82no—
108C2×C3×C8—41no—
109C2×C3×C8—41no—
110C2×C3×C8—41no—
111C2×C3×C8—41no—
112C3×C4⋊C4—21no—
113C3×C4⋊C4—21no—
114C3×C4⋊C4—21no—
115C3×C4⋊C4—21no—
116C3×C4⋊C4—41no—
117C3×C4⋊C4—41no—
118C3×C22⋊C4—41no—
119C3×C42—41no—
120C3×C42C2×D411yescharacteristic
121C3×C42C2×D411yescharacteristic
122C3×C42#4111yescharacteristic
123C3×C42#4111yescharacteristic
124(C2×D3)⋊C4—41no—
125(C2×D3)⋊C4—41no—
126C4⋊Dic3—21no—
127C4⋊Dic3—21no—
128C4⋊Dic3—41no—
129C4⋊Dic3—41no—
130#208—41no—
131#208—41no—
132C2×C3⋊C8—22no—
133C2×C3⋊C8—22no—
134C22×D4—31no—
135C23×C4—61no—
136C2×M16—61no—
137C22×C8—21no—
138C4⋊Q8#6711yescharacteristic
139C4⋊Q8#6711yescharacteristic
140#125—21no—
141C22⋊Q8—21no—
142C22⋊Q8—21no—
143C4×Q8—21no—
144C4×Q8—21no—
145C4×D4—61no—
146C4×D4—61no—
147C4×D4—61no—
148C4×D4—61no—
149#117—21no—
150C2×C4⋊C4—31no—
151C2×C4⋊C4—61no—
152C2×C22⋊C4—61no—
153C2×C42C22×D311yescharacteristic
154#107—62no—
155#106—62no—
156C4⋊C8—21no—
157C4⋊C8—21no—
158C4⋊C8—21no—
159C4⋊C8—32no—
160C4⋊C8—61no—
161C4⋊C8—61no—
162#102—62no—
163#102—62no—
164#102—122no—
165C22⋊C8—21no—
166C22⋊C8—32no—
167C4×C8—21no—
168C4×C8—61no—
169#95—61no—
170#95—61no—
171C23×C3#12111yescharacteristic
172C22×D3—22no—
173C22×D3—41no—
174C22×D3—41no—
175C22×D3—81no—
176C22×D3—81no—
177C3×Q8—81no—
178C3×Q8—81no—
179C2×C3×C4—41no—
180C2×C3×C4—41no—
181C2×C3×C4—41no—
182C2×C3×C4—41no—
183C2×C3×C4#12011yescharacteristic
184C2×C3×C4#12111yescharacteristic
185C2×C3×C4#12111yescharacteristic
186C2×C3×C4#12311yescharacteristic
187C2×C3×C4#12311yescharacteristic
188C2×C3×C4#12411yescharacteristic
189C2×C3×C4C2×SD1611yescharacteristic
190C2×C3×C4#13511yescharacteristic
191C2×C3×C4#13711yescharacteristic
192C2×C3×C4#13711yescharacteristic
193C2×Dic3—41no—
194C2×Dic3—81no—
195C2×Dic3—81no—
196C2×Dic3—81no—
197C2×Dic3—81no—
198D12—22no—
199D12—22no—
200D12—41no—
201D12—41no—
202C4×D3—82no—
203C4×D3—82no—
204C3×C8—81no—
205C3×C8—81no—
206C3⋊C8—42no—
207C3⋊C8—42no—
208C24—61no—
209C2×Q8—21no—
210C2×Q8—21no—
211C2×D4—32no—
212C2×D4—61no—
213C2×D4—61no—
214C2×D4—61no—
215C2×D4—61no—
216C2×D4—61no—
217C22×C4—61no—
218C22×C4—121no—
219C22×C4—121no—
220C22×C4—121no—
221C22×C4—242no—
222C22×C4#23511yescharacteristic
223C22×C4D3×D411yescharacteristic
224C22×C4D3×D411yescharacteristic
225M16—121no—
226M16—121no—
227C2×C8—41no—
228C2×C8—41no—
229C2×C8—41no—
230C2×C8—41no—
231C2×C8—62no—
232C2×C8—62no—
233C4⋊C4—21no—
234C4⋊C4—21no—
235C4⋊C4—21no—
236C4⋊C4—21no—
237C4⋊C4—41no—
238C4⋊C4—41no—
239C4⋊C4—61no—
240C4⋊C4—61no—
241C4⋊C4—121no—
242C4⋊C4—121no—
243C22⋊C4—41no—
244C22⋊C4—121no—
245C22⋊C4—121no—
246C42—41no—
247C42#23511yescharacteristic
248C42D3×D411yescharacteristic
249C42#23711yescharacteristic
250C42C2×#6711yescharacteristic
251C22×C3—21no—
252C22×C3—21no—
253C22×C3#39411yescharacteristic
254C22×C3#46511yescharacteristic
255C22×C3#47011yescharacteristic
256C2×D3—42no—
257C2×D3—82no—
258C2×D3—81no—
259C2×D3—81no—
260C2×D3—82no—
261C3×C4—21no—
262C3×C4—21no—
263C3×C4—21no—
264C3×C4—21no—
265C3×C4—81no—
266C3×C4—81no—
267C3×C4#44811yescharacteristic
268C3×C4#45211yescharacteristic
269C3×C4#48311yescharacteristic
270C3×C4#48511yescharacteristic
271Dic3—81no—
272Dic3—161no—
273C23—62no—
274C23—121no—
275C23—121no—
276C23—241no—
277C23—241no—
278C23#88411yescharacteristic
279Q8—81no—
280Q8—81no—
281D4—62no—
282D4—62no—
283D4—121no—
284D4—121no—
285C2×C4—41no—
286C2×C4—41no—
287C2×C4—41no—
288C2×C4—41no—
289C2×C4—121no—
290C2×C4—242no—
291C2×C4—242no—
292C2×C4—241no—
293C2×C4—241no—
294C2×C4—241no—
295C2×C4—241no—
296C2×C4#88311yescharacteristic
297C2×C4#88411yescharacteristic
298C2×C4#88511yescharacteristic
299C2×C4D3×SD1611yescharacteristic
300C2×C4#91511yescharacteristic
301C2×C4#91811yescharacteristic
302C2×C4#91911yescharacteristic
303C2×C4#92911yescharacteristic
304C2×C4#93711yescharacteristic
305C2×C4#93911yescharacteristic
306C8—81no—
307C8—81no—
308C8—122no—
309C8—122no—
310C2×C3—21no—
311C2×C3—21no—
312C2×C3#203511yescharacteristic
313C2×C3#204811yescharacteristic
314C2×C3#205311yescharacteristic
315D3—82no—
316C22—21no—
317C22—21no—
318C22—122no—
319C22—242no—
320C22—242no—
321C22—241no—
322C22—241no—
323C22#480711yescharacteristic
324C22#481611yescharacteristic
325C22#495911yescharacteristic
326C4—21no—
327C4—21no—
328C4—21no—
329C4—21no—
330C4—81no—
331C4—81no—
332C4—241no—
333C4—481no—
334C4#480511yescharacteristic
335C4#481411yescharacteristic
336C4#517011yescharacteristic
337C4#518711yescharacteristic
338C3#1046811yescharacteristic, complemented, Sylow
339C2—21no—
340C2—21no—
341C2—242no—
342C2#7014011yescharacteristic
343C2#7027411yescharacteristic
344C2#7029611yescharacteristic
subgroupG/H
centerC22#4959
commutator, cosocleC2×C3×C42C23
FrattiniC2×C42C22×D3
FittingC3×#1542C2
socleC22×C3#394
prime radicalC22×D12D4
hypercenter#1542D3
hypocenterC3#10468
LCSG ▷ C2×C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1542

Complements

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

NH ≅ G/Ntype
2. C3×#1542C2semidirect
3. #71838C2semidirect
6. #70421C2semidirect
7. #67738C2semidirect
31. #1542D3semidirect
338. C3#10468semidirect

Maximal subgroups

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

subgroupindex
#833502
C3×#15422
#718382
#707832
#704802
#704212
#677382
#104683