Finite Group Databasebeta

← #10698364

Subgroups — #10698364

Order 768.

398 conjugacy classes of subgroups; only the 73 normal subgroups are shown.

normal subgroups only

This group has 398 conjugacy classes of subgroups — too many to draw. Only its 73 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#73743C211yesmaximal, characteristic, complemented
2#70835C211yesmaximal, characteristic
3#70832C211yesmaximal, characteristic
4#69492C211yesmaximal, characteristic
5C3×#1285C211yesmaximal, characteristic, complemented
6#67752C211yesmaximal, characteristic, complemented
7#67748C211yesmaximal, characteristic, complemented
8#10378—31nomaximal, Sylow
9#5477C2211yescharacteristic
10C3×#432C2211yescharacteristic
11C3×#432C2211yescharacteristic
12C3×C4×C4⋊C4C2211yescharacteristic
13#4986—21no—
14#4979—21no—
15#4937C2211yescharacteristic
16#4933—21no—
17#4932C2211yescharacteristic
18#4929—21no—
19C2×#805—21no—
20C2×#805—21no—
21C4×#208C2211yescharacteristic
22#4654—21no—
23#4646—21no—
24#4470—21no—
25#4470—21no—
26#4470—21no—
27#4470—21no—
28#2605—31no—
29#1980—31no—
30#1514—31no—
31#1505—31no—
32#1300—31no—
33#1288—31no—
34#1285D311yescharacteristic, complemented
35C2×#235—21no—
36C22×Dic6—21no—
37C2×C3×C4⋊C4—21no—
38C2×C3×C42D411yescharacteristic
39C2×C3×C42D411yescharacteristic
40C2×C3×C42C2311yescharacteristic
41#930—21no—
42#930—21no—
43C2×C4⋊Dic3—21no—
44C2×Dic3⋊C4—21no—
45C2×Dic3⋊C4—21no—
46C2×C4×Dic3—21no—
47C2×#208—21no—
48#875—21no—
49#874Q811yescharacteristic, complemented
50C4×D12—21no—
51C4×D12—21no—
52#869Q811yescharacteristic, complemented
53C3×C4⋊C8—21no—
54C3×C4⋊C8—21no—
55C3×C22⋊C8—21no—
56C3×C22⋊C8—21no—
57C3×#97—21no—
58C3×#97—21no—
59C3×#95—21no—
60C3⋊#95—21no—
61C3⋊#95—21no—
62C3⋊#95—21no—
63C3⋊#95—21no—
64C3⋊#95—21no—
65C3⋊#95—21no—
66C3⋊#95—21no—
67C3⋊#95D411yescharacteristic
68C3⋊#95D411yescharacteristic
69C3⋊#95D411yescharacteristic
70C3⋊#95D411yescharacteristic
71#805—42no—
72#805—42no—
73#803—21no—
74C4×C3⋊C8—21no—
75#532—31no—
76#432C2×D311yescharacteristic
77#432C2×D311yescharacteristic
78C2×#104—61no—
79C2×#104—61no—
80C4×M16—31no—
81#400—61no—
82#395—61no—
83#391—61no—
84#391—61no—
85#387—61no—
86#387—61no—
87#383—31no—
88C4×C4⋊C4C2×D311yescharacteristic
89#376—31no—
90#327—61no—
91#327—61no—
92#327—61no—
93#327—61no—
94C22×C3×C4—21no—
95C22×C3×C4—21no—
96C22×C3×C4C2×D411yescharacteristic
97C22×C3×C4C2×D411yescharacteristic
98C22×C3×C4C2×D411yescharacteristic
99C2×#67—41no—
100C22×Dic3—21no—
101C22×Dic3—21no—
102C22×Dic3—21no—
103C22×Dic3—21no—
104#235—41no—
105#235—41no—
106#235—41no—
107#235—41no—
108C2×D12—21no—
109C2×D12—21no—
110C2×D12—21no—
111C2×C4×D3—41no—
112C2×Dic6—21no—
113C2×Dic6—42no—
114C2×Dic6—42no—
115C2×Dic6—81no—
116C2×C3×C8—41no—
117C2×C3×C8—41no—
118C3×C4⋊C4—42no—
119C3×C42—41no—
120C3×C42—41no—
121C3×C42C2×Q811yescharacteristic
122C3×C42#4111yescharacteristic
123C3×C42#4111yescharacteristic
124C3×C42#4111yescharacteristic
125(C2×D3)⋊C4—41no—
126(C2×D3)⋊C4—41no—
127C4⋊Dic3—21no—
128C4⋊Dic3—21no—
129C4⋊Dic3—42no—
130Dic3⋊C4—42no—
131Dic3⋊C4—81no—
132C4×Dic3—42no—
133#208—22no—
134#208—41no—
135C2×C3⋊C8—41no—
136C2×#41—61no—
137C22×Q8—61no—
138C2×M16—61no—
139C4⋊Q8—31no—
140#127—31no—
141#124—61no—
142#121—61no—
143#121—61no—
144C4×D4—61no—
145C4×D4—61no—
146C2×C4⋊C4—21no—
147C2×C4⋊C4—61no—
148C2×C4⋊C4—61no—
149C2×C4⋊C4—61no—
150C2×C42—61no—
151C2×C42D1211yescharacteristic
152C2×C42D1211yescharacteristic
153C2×C42C22×D311yescharacteristic
154C4⋊C8—21no—
155C4⋊C8—21no—
156#104—122no—
157#104—122no—
158C22⋊C8—21no—
159C22⋊C8—21no—
160#97—21no—
161#97—21no—
162#97—61no—
163C4×C8—61no—
164#95—21no—
165#95—31no—
166#95—31no—
167#95—31no—
168#95—31no—
169#95—61no—
170#95—61no—
171#95—61no—
172#95—61no—
173#95—61no—
174#95—61no—
175#95—61no—
176C23×C3#12011yescharacteristic
177C22×D3—41no—
178C2×C3×C4—21no—
179C2×C3×C4—21no—
180C2×C3×C4—22no—
181C2×C3×C4—41no—
182C2×C3×C4—41no—
183C2×C3×C4—41no—
184C2×C3×C4—41no—
185C2×C3×C4—41no—
186C2×C3×C4—41no—
187C2×C3×C4—41no—
188C2×C3×C4C22⋊Q811yescharacteristic
189C2×C3×C4C22⋊Q811yescharacteristic
190C2×C3×C4C22⋊Q811yescharacteristic
191C2×C3×C4#12411yescharacteristic
192C2×C3×C4#12411yescharacteristic
193C2×C3×C4#12411yescharacteristic
194#67—81no—
195#67—81no—
196C2×Dic3—22no—
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—
206C2×Dic3—41no—
207C2×Dic3—41no—
208C2×Dic3—41no—
209D12—41no—
210D12—41no—
211D12—81no—
212D12—81no—
213C4×D3—81no—
214C4×D3—81no—
215Dic6—41no—
216Dic6—41no—
217Dic6—84no—
218C3×C8—81no—
219C3×C8—81no—
220C3⋊C8—42no—
221#41—121no—
222#41—121no—
223#41—121no—
224#41—121no—
225C2×Q8—61no—
226C2×Q8—122no—
227C2×Q8—122no—
228C2×Q8—241no—
229C2×D4—61no—
230C2×D4—61no—
231C2×D4—61no—
232C2×D4—121no—
233C22×C4—21no—
234C22×C4—21no—
235C22×C4—61no—
236C22×C4—61no—
237C22×C4—61no—
238C22×C4—61no—
239C22×C4—121no—
240C22×C4C2×D1211yescharacteristic
241C22×C4D3×D411yescharacteristic
242C22×C4D3×D411yescharacteristic
243M16—62no—
244M16—121no—
245C2×C8—41no—
246C2×C8—41no—
247C2×C8—121no—
248C4⋊C4—42no—
249C4⋊C4—61no—
250C4⋊C4—61no—
251C4⋊C4—122no—
252C4⋊C4—122no—
253C4⋊C4—241no—
254C22⋊C4—121no—
255C22⋊C4—121no—
256C42—41no—
257C42—41no—
258C42—122no—
259C42#23511yescharacteristic
260C42#23511yescharacteristic
261C42#23711yescharacteristic
262C42D3×Q811yescharacteristic
263C22×C3—21no—
264C22×C3—21no—
265C22×C3#39311yescharacteristic
266C22×C3#45311yescharacteristic
267C22×C3#45411yescharacteristic
268C2×D3—81no—
269C2×D3—81no—
270C2×D3—81no—
271C3×C4—21no—
272C3×C4—21no—
273C3×C4—21no—
274C3×C4—21no—
275C3×C4—21no—
276C3×C4—21no—
277C3×C4—42no—
278C3×C4—81no—
279Dic3—42no—
280Dic3—81no—
281Dic3—81no—
282Dic3—81no—
283Dic3—81no—
284C23—121no—
285C23#88211yescharacteristic
286Q8—121no—
287Q8—121no—
288Q8—244no—
289D4—121no—
290D4—121no—
291D4—241no—
292D4—241no—
293D4—241no—
294D4—241no—
295C2×C4—21no—
296C2×C4—21no—
297C2×C4—22no—
298C2×C4—41no—
299C2×C4—41no—
300C2×C4—41no—
301C2×C4—41no—
302C2×C4—41no—
303C2×C4—41no—
304C2×C4—41no—
305C2×C4—62no—
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—121no—
315C2×C4—121no—
316C2×C4—121no—
317C2×C4—121no—
318C2×C4—241no—
319C2×C4—241no—
320C2×C4#87511yescharacteristic
321C2×C4#88511yescharacteristic
322C2×C4#88511yescharacteristic
323C2×C4#89611yescharacteristic
324C2×C4#89611yescharacteristic
325C2×C4#89711yescharacteristic
326C8—81no—
327C8—81no—
328C8—122no—
329C2×C3—21no—
330C2×C3—21no—
331C2×C3#202711yescharacteristic
332C2×C3#203111yescharacteristic
333C2×C3#203111yescharacteristic
334D3—161no—
335C22—21no—
336C22—21no—
337C22—241no—
338C22—241no—
339C22—241no—
340C22#467311yescharacteristic
341C22#474711yescharacteristic
342C22#475311yescharacteristic
343C4—21no—
344C4—21no—
345C4—21no—
346C4—21no—
347C4—21no—
348C4—21no—
349C4—42no—
350C4—81no—
351C4—122no—
352C4—241no—
353C4—241no—
354C4—241no—
355C4—241no—
356C3#1037811yescharacteristic, complemented, Sylow
357C2—21no—
358C2—21no—
359C2—481no—
360C2#6871411yescharacteristic
361C2#6877911yescharacteristic
362C2#6899511yescharacteristic
subgroupG/H
centerC22#4673
commutator, cosocleC2×C3×C42C23
FrattiniC2×C42C22×D3
FittingC3×#1285C2
socleC22×C3#393
prime radical#5477C22
hypercenter#1285D3
hypocenterC3#10378
LCSG ▷ C2×C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1285

Complements

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

NH ≅ G/Ntype
1. #73743C2semidirect
5. C3×#1285C2semidirect
6. #67752C2semidirect
7. #67748C2semidirect
34. #1285D3semidirect
49. #874Q8semidirect
52. #869Q8semidirect
356. C3#10378semidirect

Maximal subgroups

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

subgroupindex
#737432
#708352
#708322
#694922
C3×#12852
#677522
#677482
#103783