Finite Group Databasebeta

← #10696572

Subgroups — #10696572

Order 768.

590 conjugacy classes of subgroups; only the 83 normal subgroups are shown.

normal subgroups only

This group has 590 conjugacy classes of subgroups — too many to draw. Only its 83 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#83264C211yesmaximal, characteristic, complemented
2#83199C211yesmaximal, characteristic, complemented
3#79349C211yesmaximal, characteristic
4C3×#1556C211yesmaximal, characteristic, complemented
5#70832C211yesmaximal, characteristic, complemented
6#70804C211yesmaximal, characteristic, complemented
7#67748C211yesmaximal, characteristic, complemented
8#10094—31nomaximal, Sylow
9C2×C3×C4⋊Q8C2211yescharacteristic
10C2×#947—21no—
11C2×#944—21no—
12#5477C2211yescharacteristic
13C2×#875C2211yescharacteristic
14C3×#432C2211yescharacteristic
15C3×#432C2211yescharacteristic
16#5093—21no—
17#5089—22no—
18#5088—21no—
19#4993—22no—
20#4993—22no—
21#4966—22no—
22#4932C2211yescharacteristic
23C4×#208C2211yescharacteristic
24#4844—22no—
25#4738—21no—
26#4732—21no—
27#4714—21no—
28#4712—21no—
29C3×#332—22no—
30#4470—22no—
31#4464—22no—
32#3213—31no—
33#3141—31no—
34#2658—31no—
35#1556D311yescharacteristic, complemented
36#1527—31no—
37#1505—31no—
38#1300—31no—
39C22×C3×Q8—21no—
40C2×#235—21no—
41C22×D12D411yescharacteristic
42C22×Dic6D411yescharacteristic
43C3×C4⋊Q8—21no—
44C3×C4⋊Q8—21no—
45C3×C4⋊Q8—21no—
46C3×C4⋊Q8—21no—
47C2×C3×C4⋊C4—21no—
48C2×C3×C4⋊C4D412yes—
49C2×C3×C42C2311yescharacteristic
50#947—44no—
51#944—44no—
52#930—21no—
53#930—21no—
54C2×(C2×D3)⋊C4—24no—
55C2×(C2×D3)⋊C4—21no—
56C2×(C2×D3)⋊C4—21no—
57C2×Dic3⋊C4—24no—
58C2×C4×Dic3—21no—
59C2×#208—21no—
60C2×D24—22no—
61C2×#204—22no—
62#875—21no—
63#875—21no—
64#875—21no—
65#875—21no—
66#875—21no—
67#874D411yescharacteristic
68C4×D12—21no—
69C4×D12—21no—
70#869D411yescharacteristic
71C3×C4⋊C8—21no—
72C3×C4⋊C8D412yes—
73C3×C22⋊C8—21no—
74C3×C22⋊C8—21no—
75C3×#97—21no—
76C3×#97—21no—
77C3⋊#95—21no—
78C3⋊#95D412yes—
79#821—22no—
80#816—22no—
81#810—24no—
82#809—24no—
83#803D412yes—
84C4×C3⋊C8—21no—
85#532—31no—
86C2×C4⋊Q8—61no—
87C2×C4⋊Q8C2×D311yescharacteristic
88C2×#124—31no—
89C2×#124—61no—
90#496—61no—
91#489—61no—
92#489—62no—
93#487—61no—
94#486—61no—
95#463—62no—
96#448—61no—
97#447—61no—
98#432C2×D311yescharacteristic
99#432C2×D311yescharacteristic
100#417—62no—
101C4×M16—31no—
102#399—62no—
103#399—62no—
104#376—31no—
105#332—22no—
106#329—62no—
107#327—62no—
108C23×D3—21no—
109C2×C3×Q8—42no—
110C2×C3×Q8—42no—
111C2×C3×Q8—41no—
112C2×C3×Q8—41no—
113C22×C3×C4—22no—
114C22×C3×C4C2×D411yescharacteristic
115C22×C3×C4C2×D411yescharacteristic
116C22×C3×C4C2×D411yescharacteristic
117C2×#67—41no—
118C22×Dic3—22no—
119C22×Dic3—21no—
120#235—44no—
121C2×D12—21no—
122C2×D12—21no—
123C2×D12—21no—
124C2×D12—21no—
125C2×D12—21no—
126C2×D12—42no—
127C2×C4×D3—41no—
128C2×Dic6—21no—
129C2×Dic6—21no—
130C2×Dic6—21no—
131C2×Dic6—42no—
132C2×C3×C8—22no—
133C2×C3×C8—41no—
134C3×C4⋊C4—22no—
135C3×C4⋊C4—22no—
136C3×C4⋊C4—41no—
137C3×C4⋊C4—41no—
138C3×C42C2×D411yescharacteristic
139C3×C42C2×D411yescharacteristic
140C3×C42C2×D411yescharacteristic
141C3×C42#4111yescharacteristic
142(C2×D3)⋊C4—48no—
143(C2×D3)⋊C4—41no—
144(C2×D3)⋊C4—41no—
145(C2×D3)⋊C4—41no—
146(C2×D3)⋊C4—41no—
147(C2×D3)⋊C4—41no—
148(C2×D3)⋊C4—41no—
149C4⋊Dic3—21no—
150C4⋊Dic3—21no—
151Dic3⋊C4—48no—
152C4×Dic3—24no—
153#208—42no—
154C2×C3⋊C8—22no—
155D24—82no—
156#204—82no—
157C2×#41—61no—
158C22×Q8—21no—
159C22×Q8—31no—
160C22×D4—31no—
161C2×SD16—62no—
162C2×D8—62no—
163C2×M16—61no—
164C4⋊Q8—21no—
165C4⋊Q8—21no—
166C4⋊Q8—21no—
167C4⋊Q8—21no—
168C4⋊Q8—31no—
169C4⋊Q8—124no—
170#127—31no—
171#124—61no—
172#124—61no—
173#124—61no—
174#124—61no—
175#124—61no—
176#124—124no—
177#121—61no—
178#121—61no—
179C4×D4—61no—
180C4×D4—61no—
181C2×C4⋊C4—21no—
182C2×C4⋊C4—64no—
183C2×C4⋊C4D1212yes—
184C2×C22⋊C4—64no—
185C2×C22⋊C4—61no—
186C2×C22⋊C4—61no—
187C2×C42—61no—
188C2×C42C22×D311yescharacteristic
189C4⋊C8—21no—
190C4⋊C8#6712yes—
191#103—64no—
192#103—62no—
193#102—64no—
194#102—62no—
195C22⋊C8—21no—
196C22⋊C8—21no—
197#97—21no—
198#97—21no—
199#97—32no—
200C4×C8—61no—
201#95—32no—
202#95—61no—
203C23×C3#12011yescharacteristic
204C22×D3—42no—
205C22×D3—42no—
206C22×D3—41no—
207C22×D3—41no—
208C22×D3—42no—
209C3×Q8—82no—
210C3×Q8—82no—
211C2×C3×C4—22no—
212C2×C3×C4—22no—
213C2×C3×C4—41no—
214C2×C3×C4—42no—
215C2×C3×C4—42no—
216C2×C3×C4—42no—
217C2×C3×C4#12011yescharacteristic
218C2×C3×C4#12011yescharacteristic
219C2×C3×C4#12111yescharacteristic
220C2×C3×C4#12111yescharacteristic
221C2×C3×C4#12111yescharacteristic
222C2×C3×C4#12711yescharacteristic
223#67—82no—
224C2×Dic3—24no—
225C2×Dic3—42no—
226C2×Dic3—42no—
227C2×Dic3—42no—
228C2×Dic3—41no—
229C2×Dic3—41no—
230D12—42no—
231D12—42no—
232D12—42no—
233D12—81no—
234D12—81no—
235C4×D3—82no—
236Dic6—42no—
237Dic6—42no—
238Dic6—81no—
239C3×C8—42no—
240C3×C8—81no—
241C3⋊C8—42no—
242C24—61no—
243#41—124no—
244C2×Q8—42no—
245C2×Q8—42no—
246C2×Q8—41no—
247C2×Q8—41no—
248C2×Q8—61no—
249C2×Q8—61no—
250C2×Q8—61no—
251C2×Q8—122no—
252C2×D4—61no—
253C2×D4—61no—
254C2×D4—61no—
255C2×D4—61no—
256C2×D4—61no—
257C2×D4—122no—
258C2×D4—121no—
259C22×C4—22no—
260C22×C4—62no—
261C22×C4—61no—
262C22×C4—121no—
263C22×C4C2×D1211yescharacteristic
264C22×C4D3×D411yescharacteristic
265C22×C4D3×D411yescharacteristic
266SD16—242no—
267D8—242no—
268M16—122no—
269C2×C8—22no—
270C2×C8—41no—
271C2×C8—62no—
272C4⋊C4—22no—
273C4⋊C4—22no—
274C4⋊C4—41no—
275C4⋊C4—41no—
276C4⋊C4—61no—
277C4⋊C4—61no—
278C4⋊C4—128no—
279C22⋊C4—128no—
280C22⋊C4—121no—
281C22⋊C4—121no—
282C22⋊C4—121no—
283C22⋊C4—121no—
284C22⋊C4—121no—
285C22⋊C4—121no—
286C42—64no—
287C42#23511yescharacteristic
288C42D3×D411yescharacteristic
289C42D3×D411yescharacteristic
290C42C2×#6711yescharacteristic
291C22×C3—21no—
292C22×C3—21no—
293C22×C3#39211yescharacteristic
294C22×C3#45412yes—
295C2×D3—84no—
296C2×D3—81no—
297C2×D3—81no—
298C2×D3—81no—
299C2×D3—81no—
300C2×D3—81no—
301C2×D3—81no—
302C3×C4—21no—
303C3×C4—21no—
304C3×C4—21no—
305C3×C4—21no—
306C3×C4—21no—
307C3×C4—82no—
308C3×C4#45312yes—
309Dic3—44no—
310Dic3—81no—
311Dic3—81no—
312C23—122no—
313C23—122no—
314C23—121no—
315C23—121no—
316C23—122no—
317C23#88211yescharacteristic
318Q8—82no—
319Q8—82no—
320Q8—122no—
321Q8—122no—
322Q8—241no—
323D4—122no—
324D4—122no—
325D4—122no—
326D4—241no—
327D4—241no—
328D4—242no—
329C2×C4—22no—
330C2×C4—22no—
331C2×C4—41no—
332C2×C4—42no—
333C2×C4—42no—
334C2×C4—42no—
335C2×C4—64no—
336C2×C4—122no—
337C2×C4—122no—
338C2×C4—122no—
339C2×C4—121no—
340C2×C4—121no—
341C2×C4—242no—
342C2×C4#88211yescharacteristic
343C2×C4#88411yescharacteristic
344C2×C4#88411yescharacteristic
345C2×C4#93011yescharacteristic
346C2×C4#93711yescharacteristic
347C2×C4#94011yescharacteristic
348C8—42no—
349C8—81no—
350C8—122no—
351C2×C3—22no—
352C2×C3#200211yescharacteristic
353C2×C3#200311yescharacteristic
354C2×C3#200911yescharacteristic
355D3—161no—
356D3—161no—
357C22—21no—
358C22—21no—
359C22—244no—
360C22—241no—
361C22—241no—
362C22—241no—
363C22—241no—
364C22—241no—
365C22—241no—
366C22#475312yes—
367C22#496011yescharacteristic
368C4—21no—
369C4—21no—
370C4—21no—
371C4—21no—
372C4—21no—
373C4—82no—
374C4—124no—
375C4—241no—
376C4—241no—
377C4#519812yes—
378C3#1009411yescharacteristic, complemented, Sylow
379C2—22no—
380C2—481no—
381C2—481no—
382C2#7011611yescharacteristic
383C2#7032411yescharacteristic
384C2#7035111yescharacteristic
subgroupG/H
centerC22#4960
commutator, cosocleC2×C3×C42C23
FrattiniC2×C42C22×D3
FittingC3×#1556C2
socleC22×C3#392
prime radical#79349C2
hypercenter#1556D3
hypocenterC3#10094
LCSG ▷ C2×C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1556

Complements

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

NH ≅ G/Ntype
1. #83264C2semidirect
2. #83199C2semidirect
4. C3×#1556C2semidirect
5. #70832C2semidirect
6. #70804C2semidirect
7. #67748C2semidirect
35. #1556D3semidirect
378. C3#10094semidirect

Maximal subgroups

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

subgroupindex
#832642
#831992
#793492
C3×#15562
#708322
#708042
#677482
#100943