Finite Group Databasebeta

← #10698375

Subgroups — #10698375

Order 768.

390 conjugacy classes of subgroups; only the 75 normal subgroups are shown.

normal subgroups only

This group has 390 conjugacy classes of subgroups — too many to draw. Only its 75 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#74276C211yesmaximal, characteristic, complemented
2C3×#1514C211yesmaximal, characteristic
3#71818C211yesmaximal, characteristic
4#71812C211yesmaximal, characteristic
5#67749C211yesmaximal, characteristic, complemented
6#67748C211yesmaximal, characteristic, complemented
7#67633C211yesmaximal, characteristic, complemented
8#10380—31nomaximal, Sylow
9C3×#532C2211yescharacteristic
10C3×#432C2211yescharacteristic
11C3×#383C2211yescharacteristic
12C2×#837—21no—
13C2×#837—21no—
14C4×C4⋊Dic3C2211yescharacteristic
15#4932C2211yescharacteristic
16#4926C2211yescharacteristic
17C4×#208C2211yescharacteristic
18#4661—21no—
19#4657—21no—
20#4655—21no—
21#4654—21no—
22#4647—21no—
23#4647—21no—
24C3×#327—21no—
25C3×#327—21no—
26#4537—21no—
27#4537—21no—
28#2606—31no—
29#1980—31no—
30#1514D311yescharacteristic
31#1505—31no—
32#1300—31no—
33#1288—31no—
34#1285—31no—
35C2×C3×#41—21no—
36C3×C4⋊Q8Q811yescharacteristic
37C3×#127Q811yescharacteristic
38C3×#124—21no—
39C3×#121—21no—
40C3×#121—21no—
41C3×C4×D4—21no—
42C3×C4×D4—21no—
43C2×C3×C42C2311yescharacteristic
44C2×C4⋊Dic3—21no—
45C2×C4⋊Dic3—21no—
46C2×C4⋊Dic3—21no—
47C2×Dic3⋊C4—21no—
48C2×Dic3⋊C4—21no—
49C2×Dic3⋊C4—21no—
50C2×Dic3⋊C4—21no—
51C2×C4×Dic3—21no—
52C2×C4×Dic3D411yescharacteristic
53C2×C4×Dic3D411yescharacteristic
54C2×#208—21no—
55C3×C4⋊C8—21no—
56C3×C22⋊C8—21no—
57C3×#97—21no—
58C3×#95—21no—
59C3×#95—21no—
60C3×#95D411yescharacteristic
61C3×#95D411yescharacteristic
62#837—42no—
63#837—42no—
64C3⋊#95—21no—
65C3⋊#95—21no—
66C3⋊#95—21no—
67C3⋊#95—21no—
68C3⋊#95D411yescharacteristic
69C3⋊#95D411yescharacteristic
70#830—21no—
71C4⋊(C3⋊C8)—21no—
72#803—21no—
73#803—21no—
74C4×C3⋊C8—21no—
75#532C2×D311yescharacteristic
76#432—31no—
77#432C2×D311yescharacteristic
78C2×#104—61no—
79C2×#104—61no—
80C4×M16—31no—
81#398—61no—
82#398—61no—
83#389—61no—
84#389—61no—
85#387—61no—
86#387—61no—
87#383C2×D311yescharacteristic
88C4×C4⋊C4—31no—
89#376—31no—
90#327—21no—
91#327—21no—
92#327—61no—
93#327—61no—
94C3×#41—41no—
95C3×#41—41no—
96C3×#41—41no—
97C3×#41—41no—
98C2×C3×Q8—21no—
99C2×C3×D4—21no—
100C2×C3×D4—21no—
101C2×C3×D4—21no—
102C2×C3×D4—41no—
103C22×C3×C4—21no—
104C22×C3×C4—21no—
105C22×C3×C4—41no—
106C22×C3×C4C2×D411yescharacteristic
107C22×C3×C4C2×D411yescharacteristic
108C22×C3×C4C2×D411yescharacteristic
109C22×Dic3—21no—
110C22×Dic3—21no—
111C22×Dic3—21no—
112C22×Dic3—21no—
113C2×C3×C8—41no—
114C3×C4⋊C4—21no—
115C3×C4⋊C4—21no—
116C3×C22⋊C4—41no—
117C3×C22⋊C4—41no—
118C3×C42C2×Q811yescharacteristic
119C3×C42#4111yescharacteristic
120C3×C42#4111yescharacteristic
121C3×C42#4111yescharacteristic
122C4⋊Dic3—42no—
123C4⋊Dic3—42no—
124C4⋊Dic3—81no—
125Dic3⋊C4—42no—
126Dic3⋊C4—42no—
127Dic3⋊C4—42no—
128Dic3⋊C4—81no—
129C4×Dic3—42no—
130C4×Dic3—41no—
131C4×Dic3—41no—
132#208—22no—
133#208—41no—
134C2×C3⋊C8—41no—
135C2×C3⋊C8—41no—
136C2×#41—21no—
137C2×M16—61no—
138C4⋊Q8Dic611yescharacteristic
139#127Dic611yescharacteristic
140#124—21no—
141#121—21no—
142#121—21no—
143C4×D4—21no—
144C4×D4—21no—
145C2×C4⋊C4—61no—
146C2×C4⋊C4—61no—
147C2×C4⋊C4—61no—
148C2×C4⋊C4—61no—
149C2×C4⋊C4—61no—
150C2×C4⋊C4—61no—
151C2×C4⋊C4—61no—
152C2×C42—31no—
153C2×C42—31no—
154C2×C42—61no—
155C2×C42C22×D311yescharacteristic
156C4⋊C8—21no—
157C4⋊C8—61no—
158#104—122no—
159#104—122no—
160C22⋊C8—21no—
161C22⋊C8—61no—
162#97—21no—
163#97—61no—
164#97—61no—
165C4×C8—61no—
166#95—21no—
167#95—21no—
168#95—31no—
169#95—31no—
170#95—61no—
171#95—61no—
172#95—61no—
173#95—61no—
174#95#6711yescharacteristic
175#95#6711yescharacteristic
176C23×C3—41no—
177C23×C3#12011yescharacteristic
178C3×Q8—41no—
179C3×Q8—41no—
180C3×D4—41no—
181C3×D4—41no—
182C3×D4—81no—
183C3×D4—81no—
184C3×D4—81no—
185C3×D4—81no—
186C2×C3×C4—21no—
187C2×C3×C4—21no—
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—81no—
198C2×C3×C4—81no—
199C2×C3×C4C22⋊Q811yescharacteristic
200C2×C3×C4C22⋊Q811yescharacteristic
201C2×C3×C4C22⋊Q811yescharacteristic
202C2×C3×C4#12411yescharacteristic
203C2×C3×C4#12411yescharacteristic
204C2×C3×C4#12411yescharacteristic
205C2×Dic3—22no—
206C2×Dic3—22no—
207C2×Dic3—41no—
208C2×Dic3—41no—
209C2×Dic3—41no—
210C2×Dic3—41no—
211C2×Dic3—41no—
212C2×Dic3—41no—
213C2×Dic3—41no—
214C2×Dic3—41no—
215C2×Dic3—41no—
216C2×Dic3—41no—
217C3×C8—81no—
218C3⋊C8—42no—
219C3⋊C8—81no—
220#41—41no—
221#41—41no—
222#41—41no—
223#41—41no—
224C2×Q8—21no—
225C2×D4—21no—
226C2×D4—21no—
227C2×D4—21no—
228C2×D4—41no—
229C22×C4—21no—
230C22×C4—21no—
231C22×C4—41no—
232C22×C4—61no—
233C22×C4—61no—
234C22×C4—61no—
235C22×C4—61no—
236C22×C4D3×D411yescharacteristic
237C22×C4D3×D411yescharacteristic
238C22×C4C2×#6711yescharacteristic
239M16—62no—
240M16—121no—
241C2×C8—41no—
242C2×C8—121no—
243C2×C8—121no—
244C4⋊C4—21no—
245C4⋊C4—21no—
246C4⋊C4—122no—
247C4⋊C4—122no—
248C4⋊C4—122no—
249C4⋊C4—122no—
250C4⋊C4—122no—
251C4⋊C4—241no—
252C4⋊C4—241no—
253C22⋊C4—41no—
254C22⋊C4—41no—
255C42—121no—
256C42—121no—
257C42—122no—
258C42C2×Dic611yescharacteristic
259C42#23511yescharacteristic
260C42#23711yescharacteristic
261C42#23711yescharacteristic
262C22×C3—21no—
263C22×C3—21no—
264C22×C3—81no—
265C22×C3—81no—
266C22×C3—81no—
267C22×C3#39311yescharacteristic
268C22×C3#45311yescharacteristic
269C22×C3#45411yescharacteristic
270C3×C4—21no—
271C3×C4—21no—
272C3×C4—21no—
273C3×C4—21no—
274C3×C4—21no—
275C3×C4—21no—
276C3×C4—81no—
277C3×C4—81no—
278C3×C4—81no—
279Dic3—42no—
280Dic3—42no—
281Dic3—81no—
282Dic3—81no—
283C23—41no—
284C23#93711yescharacteristic
285Q8—41no—
286Q8—41no—
287D4—41no—
288D4—41no—
289D4—81no—
290D4—81no—
291D4—81no—
292D4—81no—
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—41no—
303C2×C4—41no—
304C2×C4—62no—
305C2×C4—62no—
306C2×C4—81no—
307C2×C4—81no—
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#87811yescharacteristic
319C2×C4#87811yescharacteristic
320C2×C4#88511yescharacteristic
321C2×C4#88511yescharacteristic
322C2×C4#92411yescharacteristic
323C2×C4#93611yescharacteristic
324C8—81no—
325C8—122no—
326C8—241no—
327C2×C3—21no—
328C2×C3—21no—
329C2×C3—161no—
330C2×C3#202711yescharacteristic
331C2×C3#203111yescharacteristic
332C2×C3#203111yescharacteristic
333C22—21no—
334C22—21no—
335C22—81no—
336C22—81no—
337C22—81no—
338C22#494711yescharacteristic
339C22#519811yescharacteristic
340C22#520111yescharacteristic
341C4—21no—
342C4—21no—
343C4—21no—
344C4—21no—
345C4—21no—
346C4—21no—
347C4—81no—
348C4—81no—
349C4—81no—
350C4—122no—
351C4—122no—
352C4—241no—
353C4—241no—
354C3#1038011yescharacteristic, complemented, Sylow
355C2—21no—
356C2—21no—
357C2—161no—
358C2#7003811yescharacteristic
359C2#7031311yescharacteristic
360C2#7039411yescharacteristic
subgroupG/H
centerC22#4947
commutator, cosocleC2×C3×C42C23
FrattiniC2×C42C22×D3
FittingC3×#1514C2
socleC22×C3#393
prime radicalC3×#532C22
hypercenter#1514D3
hypocenterC3#10380
LCSG ▷ C2×C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1514

Complements

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

NH ≅ G/Ntype
1. #74276C2semidirect
5. #67749C2semidirect
6. #67748C2semidirect
7. #67633C2semidirect
354. C3#10380semidirect

Maximal subgroups

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

subgroupindex
#742762
C3×#15142
#718182
#718122
#677492
#677482
#676332
#103803