Finite Group Databasebeta

← #10698374

Subgroups — #10698374

Order 768.

390 conjugacy classes of subgroups; only the 77 normal subgroups are shown.

normal subgroups only

This group has 390 conjugacy classes of subgroups — too many to draw. Only its 77 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
1C3×#2606C211yesmaximal, characteristic, complemented
2#69564C211yesmaximal, characteristic
3#69492C211yesmaximal, characteristic
4#69432C211yesmaximal, characteristic
5#67633C211yesmaximal, characteristic, complemented
6#67621C211yesmaximal, characteristic, complemented
7#67615C211yesmaximal, characteristic, complemented
8#10380—31nomaximal, Sylow
9#5477C2211yescharacteristic
10C3×#398—21no—
11C3×#398—21no—
12C3×#389—21no—
13C3×#389—21no—
14C3×#387—21no—
15C3×#387—21no—
16C3×#383C2211yescharacteristic
17C3×C4×C4⋊C4C2211yescharacteristic
18C3×#376C2211yescharacteristic
19C2×#805—21no—
20C2×#805—21no—
21#4926C2211yescharacteristic
22#4926C2211yescharacteristic
23C4×#208C2211yescharacteristic
24#4451—21no—
25#4451—21no—
26#4451—21no—
27#4451—21no—
28#2606D311yescharacteristic, complemented
29#1980—31no—
30#1514—31no—
31#1505—31no—
32#1300—31no—
33#1288—31no—
34#1285—31no—
35C2×#235—21no—
36C2×C3×C4⋊C4—21no—
37C2×C3×C4⋊C4—21no—
38C2×C3×C4⋊C4—21no—
39C2×C3×C4⋊C4—21no—
40C2×C3×C4⋊C4—21no—
41C2×C3×C4⋊C4—21no—
42C2×C3×C4⋊C4—21no—
43C2×C3×C42—21no—
44C2×C3×C42D411yescharacteristic
45C2×C3×C42D411yescharacteristic
46C2×C3×C42C2311yescharacteristic
47#930—21no—
48#930—21no—
49C2×#208—21no—
50#875—21no—
51#874Q811yescharacteristic
52C4×D12—21no—
53C4×D12—21no—
54#869Q811yescharacteristic
55C3×#95—21no—
56C3×#95—21no—
57C3×#95—21no—
58C3×#95—21no—
59C3×#95—21no—
60C3×#95—21no—
61C3×#95D411yescharacteristic
62C3×#95D411yescharacteristic
63C3×#95D411yescharacteristic
64C3×#95D411yescharacteristic
65#830—21no—
66#830—21no—
67#805—42no—
68#805—42no—
69C4⋊(C3⋊C8)—21no—
70C4⋊(C3⋊C8)—21no—
71#803—21no—
72#803—21no—
73#803—21no—
74C4×C3⋊C8—21no—
75#532—31no—
76#432—31no—
77#432—31no—
78C2×#104—61no—
79C2×#104—61no—
80C4×M16—31no—
81#398—21no—
82#398—21no—
83#389—21no—
84#389—21no—
85#387—21no—
86#387—21no—
87#383C2×D311yescharacteristic
88C4×C4⋊C4C2×D311yescharacteristic
89#376C2×D311yescharacteristic
90#327—61no—
91#327—61no—
92#327—61no—
93#327—61no—
94C22×C3×C4—21no—
95C22×C3×C4—21no—
96C22×C3×C4—21no—
97C22×C3×C4—21no—
98C22×C3×C4—21no—
99C22×C3×C4—21no—
100C22×C3×C4C2×D411yescharacteristic
101C22×C3×C4C2×D411yescharacteristic
102C22×C3×C4C2×D411yescharacteristic
103C2×#67—41no—
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—
113C3×C4⋊C4—42no—
114C3×C4⋊C4—42no—
115C3×C4⋊C4—42no—
116C3×C4⋊C4—42no—
117C3×C4⋊C4—42no—
118C3×C4⋊C4—81no—
119C3×C4⋊C4—81no—
120C3×C42—41no—
121C3×C42—41no—
122C3×C42—42no—
123C3×C42C2×Q811yescharacteristic
124C3×C42#4111yescharacteristic
125C3×C42#4111yescharacteristic
126C3×C42#4111yescharacteristic
127(C2×D3)⋊C4—41no—
128(C2×D3)⋊C4—41no—
129C4⋊Dic3—21no—
130C4⋊Dic3—21no—
131#208—22no—
132#208—41no—
133C2×C3⋊C8—41no—
134C2×C3⋊C8—41no—
135C2×C3⋊C8—41no—
136C2×#41—61no—
137C2×M16—61no—
138C4⋊Q8—31no—
139#127—31no—
140#124—61no—
141#121—61no—
142#121—61no—
143C4×D4—61no—
144C4×D4—61no—
145C2×C4⋊C4—21no—
146C2×C4⋊C4—21no—
147C2×C4⋊C4—21no—
148C2×C4⋊C4—21no—
149C2×C4⋊C4—21no—
150C2×C4⋊C4—21no—
151C2×C4⋊C4—21no—
152C2×C42—21no—
153C2×C42#6711yescharacteristic
154C2×C42#6711yescharacteristic
155C2×C42C22×D311yescharacteristic
156C4⋊C8—61no—
157C4⋊C8—61no—
158#104—122no—
159#104—122no—
160C22⋊C8—61no—
161C22⋊C8—61no—
162#97—61no—
163#97—61no—
164#97—61no—
165C4×C8—61no—
166#95—21no—
167#95—21no—
168#95—21no—
169#95—21no—
170#95—21no—
171#95—21no—
172#95#6711yescharacteristic
173#95#6711yescharacteristic
174#95#6711yescharacteristic
175#95#6711yescharacteristic
176C23×C3#12011yescharacteristic
177C22×D3—41no—
178C2×C3×C4—21no—
179C2×C3×C4—21no—
180C2×C3×C4—22no—
181C2×C3×C4—22no—
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×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—41no—
196C2×C3×C4—41no—
197C2×C3×C4—41no—
198C2×C3×C4—41no—
199C2×C3×C4—41no—
200C2×C3×C4C22⋊Q811yescharacteristic
201C2×C3×C4C22⋊Q811yescharacteristic
202C2×C3×C4C22⋊Q811yescharacteristic
203C2×C3×C4#12411yescharacteristic
204C2×C3×C4#12411yescharacteristic
205C2×C3×C4#12411yescharacteristic
206#67—81no—
207#67—81no—
208C2×Dic3—41no—
209D12—41no—
210D12—41no—
211D12—81no—
212D12—81no—
213C4×D3—81no—
214C4×D3—81no—
215Dic6—41no—
216Dic6—41no—
217C3⋊C8—42no—
218C3⋊C8—81no—
219C3⋊C8—81no—
220#41—121no—
221#41—121no—
222#41—121no—
223#41—121no—
224C2×Q8—61no—
225C2×D4—61no—
226C2×D4—61no—
227C2×D4—61no—
228C2×D4—121no—
229C22×C4—21no—
230C22×C4—21no—
231C22×C4—21no—
232C22×C4—21no—
233C22×C4—21no—
234C22×C4—21no—
235C22×C4—121no—
236C22×C4C2×#6711yescharacteristic
237C22×C4C2×#6711yescharacteristic
238C22×C4C2×#6711yescharacteristic
239M16—62no—
240M16—121no—
241C2×C8—121no—
242C2×C8—121no—
243C2×C8—121no—
244C4⋊C4—42no—
245C4⋊C4—42no—
246C4⋊C4—42no—
247C4⋊C4—42no—
248C4⋊C4—42no—
249C4⋊C4—61no—
250C4⋊C4—61no—
251C4⋊C4—81no—
252C4⋊C4—81no—
253C22⋊C4—121no—
254C22⋊C4—121no—
255C42—41no—
256C42—41no—
257C42—42no—
258C42D3×Q811yescharacteristic
259C42#23911yescharacteristic
260C42#23911yescharacteristic
261C42#23911yescharacteristic
262C22×C3—21no—
263C22×C3—21no—
264C22×C3#39311yescharacteristic
265C22×C3#45311yescharacteristic
266C22×C3#45411yescharacteristic
267C2×D3—81no—
268C2×D3—81no—
269C2×D3—81no—
270C3×C4—21no—
271C3×C4—21no—
272C3×C4—21no—
273C3×C4—21no—
274C3×C4—21no—
275C3×C4—21no—
276C3×C4—42no—
277C3×C4—42no—
278C3×C4—81no—
279C3×C4—81no—
280C3×C4—81no—
281C3×C4—81no—
282Dic3—81no—
283C23—121no—
284C23#95311yescharacteristic
285Q8—121no—
286Q8—121no—
287D4—121no—
288D4—121no—
289D4—241no—
290D4—241no—
291D4—241no—
292D4—241no—
293C2×C4—21no—
294C2×C4—21no—
295C2×C4—22no—
296C2×C4—22no—
297C2×C4—41no—
298C2×C4—41no—
299C2×C4—41no—
300C2×C4—41no—
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—41no—
312C2×C4—41no—
313C2×C4—41no—
314C2×C4—41no—
315C2×C4—121no—
316C2×C4—241no—
317C2×C4—241no—
318C2×C4#94611yescharacteristic
319C2×C4#94611yescharacteristic
320C2×C4#94611yescharacteristic
321C2×C4#94711yescharacteristic
322C2×C4#94711yescharacteristic
323C2×C4#94711yescharacteristic
324C8—122no—
325C8—241no—
326C8—241no—
327C2×C3—21no—
328C2×C3—21no—
329C2×C3#202711yescharacteristic
330C2×C3#203111yescharacteristic
331C2×C3#203111yescharacteristic
332D3—161no—
333C22—21no—
334C22—21no—
335C22—241no—
336C22—241no—
337C22—241no—
338C22#523111yescharacteristic
339C22#524111yescharacteristic
340C22#524511yescharacteristic
341C4—21no—
342C4—21no—
343C4—21no—
344C4—21no—
345C4—21no—
346C4—21no—
347C4—42no—
348C4—42no—
349C4—81no—
350C4—81no—
351C4—81no—
352C4—81no—
353C4—241no—
354C3#1038011yescharacteristic, complemented, Sylow
355C2—21no—
356C2—21no—
357C2—481no—
358C2#7184211yescharacteristic
359C2#7184211yescharacteristic
360C2#7190311yescharacteristic
subgroupG/H
centerC22#5231
commutator, cosocleC2×C3×C42C23
FrattiniC2×C42C22×D3
FittingC3×#2606C2
socleC22×C3#393
prime radical#5477C22
hypercenter#2606D3
hypocenterC3#10380
LCSG ▷ C2×C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2606

Complements

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

NH ≅ G/Ntype
1. C3×#2606C2semidirect
5. #67633C2semidirect
6. #67621C2semidirect
7. #67615C2semidirect
28. #2606D3semidirect
354. C3#10380semidirect

Maximal subgroups

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

subgroupindex
C3×#26062
#695642
#694922
#694322
#676332
#676212
#676152
#103803