Finite Group Databasebeta

← #10692093

Subgroups — #10692093

Order 768.

650 conjugacy classes of subgroups; only the 97 normal subgroups are shown.

normal subgroups only

This group has 650 conjugacy classes of subgroups — too many to draw. Only its 97 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#83300C211yesmaximal, characteristic
2#73412C211yesmaximal, characteristic, complemented
3C3×#1505C211yesmaximal, characteristic, complemented
4#71818C211yesmaximal, characteristic, complemented
5#70799C211yesmaximal, characteristic, complemented
6#67749C211yesmaximal, characteristic, complemented
7#67615C211yesmaximal, characteristic, complemented
8#9325—31nomaximal, Sylow
9C3×#532C2211yescharacteristic
10C2×C4×D12—21no—
11C2×C4×D12C2211yescharacteristic
12C3×#432C2211yescharacteristic
13C3×#376C2211yescharacteristic
14C2×#837—22no—
15#5084C412yescomplemented
16#5072C412yescomplemented
17#5036—21no—
18#5034—21no—
19#5014—22no—
20C4×C4⋊Dic3—21no—
21C4×C4⋊Dic3C2211yescharacteristic
22#4926C2211yescharacteristic
23C4×#208C2211yescharacteristic
24#4666—22no—
25#4666—21no—
26D3×#95—21no—
27C3×#327—21no—
28C3×#327—21no—
29#4483—22no—
30#3109—31no—
31#2275—31no—
32#1980—31no—
33#1505D311yescharacteristic, complemented
34#1490—31no—
35#1300—31no—
36#1285—31no—
37C2×C3×#41—21no—
38C22×D12D411yescharacteristic
39C22×C4×D3—22no—
40C22×C4×D3—21no—
41C3×C4⋊Q8C2×C411yescharacteristic, complemented
42C3×#127C2×C411yescharacteristic, complemented
43C3×#124—21no—
44C3×#121—21no—
45C3×#121—21no—
46C3×C4×D4—21no—
47C3×C4×D4—21no—
48C2×C3×C42—21no—
49C2×C3×C42C2311yescharacteristic
50C2×#213—22no—
51C2×(C2×D3)⋊C4—22no—
52C2×(C2×D3)⋊C4—21no—
53C2×C4⋊Dic3—21no—
54C2×C4⋊Dic3D411yescharacteristic
55C2×C4×Dic3—22no—
56C2×C4×Dic3D412yes—
57C2×#208—21no—
58C4×D12—28no—
59C4×D12—21no—
60C4×D12—21no—
61C4×D12C2×C412yes—
62C4×D12D412yes—
63C3×C4⋊C8—21no—
64C3×C22⋊C8—21no—
65C3×#97—21no—
66C3×#95—21no—
67C3×#95D411yescharacteristic
68C3×#95D411yescharacteristic
69#837—44no—
70#832—22no—
71C3⋊#95—22no—
72C3⋊#95—21no—
73#830—21no—
74#809—22no—
75#808—22no—
76C4⋊(C3⋊C8)C2×C412yescomplemented
77#803—21no—
78#803—21no—
79C4×C3⋊C8—21no—
80#532C2×D311yescharacteristic
81C2×C4×D4—31no—
82C2×C4×D4—61no—
83#481—61no—
84#476—32no—
85#459—32no—
86#447—61no—
87#442—62no—
88#432—31no—
89#432C2×D311yescharacteristic
90C2×#104—62no—
91C4×M16—31no—
92C4×C4⋊C4—31no—
93C4×C4⋊C4—61no—
94C4×C22⋊C4—62no—
95C4×C22⋊C4—61no—
96#376C2×D311yescharacteristic
97C2×#95—61no—
98#327—21no—
99#327—21no—
100D4⋊C8—62no—
101C23×D3—21no—
102C3×#41—42no—
103C3×#41—42no—
104C2×C3×Q8—21no—
105C2×C3×D4—21no—
106C2×C3×D4—21no—
107C2×C3×D4—21no—
108C2×C3×D4—41no—
109C22×C3×C4—21no—
110C22×C3×C4—21no—
111C22×C3×C4—41no—
112C22×C3×C4C2×D411yescharacteristic
113C22×C3×C4C2×D411yescharacteristic
114C22×C3×C4C2×D411yescharacteristic
115C22×Dic3—21no—
116C22×Dic3—22no—
117C2×D12—21no—
118C2×D12—24no—
119C2×D12C22⋊C412yes—
120C2×C4×D3—28no—
121C2×C4×D3—22no—
122C2×C4×D3—22no—
123C2×C4×D3—82no—
124C2×C4×D3—82no—
125C2×C4×D3—82no—
126C2×C3×C8—41no—
127C3×C4⋊C4—21no—
128C3×C4⋊C4—21no—
129C3×C22⋊C4—41no—
130C3×C22⋊C4—41no—
131C3×C42—24no—
132C3×C42C22×C411yescharacteristic
133C3×C42C2×D411yescharacteristic
134C3×C42#4111yescharacteristic
135C3×C42#4111yescharacteristic
136#213—44no—
137(C2×D3)⋊C4—28no—
138(C2×D3)⋊C4—22no—
139(C2×D3)⋊C4—22no—
140C4⋊Dic3—24no—
141C4⋊Dic3—21no—
142C4⋊Dic3C22⋊C412yes—
143C4×Dic3—42no—
144C4×Dic3—82no—
145#208—22no—
146#208—41no—
147C2×C3⋊C8—22no—
148C2×C3⋊C8—41no—
149C2×#41—21no—
150C22×D4—31no—
151C23×C4—62no—
152C23×C4—61no—
153C2×D8—62no—
154C2×M16—61no—
155C4⋊Q8C4×D311yescharacteristic, complemented
156#127C4×D311yescharacteristic, complemented
157#124—21no—
158#121—21no—
159#121—21no—
160C4×D4—21no—
161C4×D4—21no—
162C4×D4—32no—
163C4×D4—32no—
164C4×D4—68no—
165C4×D4—61no—
166C4×D4—61no—
167C2×C4⋊C4—31no—
168C2×C4⋊C4—61no—
169C2×C22⋊C4—62no—
170C2×C22⋊C4—61no—
171C2×C42—21no—
172C2×C42—32no—
173C2×C42—62no—
174C2×C42C22×D311yescharacteristic
175#106—62no—
176C4⋊C8—21no—
177C4⋊C8—32no—
178#104—124no—
179#102—62no—
180#102—62no—
181C22⋊C8—21no—
182C22⋊C8—61no—
183#97—21no—
184#97—61no—
185#97—61no—
186C4×C8—61no—
187#95—21no—
188#95—62no—
189#95—61no—
190#95#6711yescharacteristic
191#95#6711yescharacteristic
192C23×C3—41no—
193C23×C3#12011yescharacteristic
194C22×D3—24no—
195C22×D3—22no—
196C22×D3—42no—
197C22×D3—41no—
198C22×D3—41no—
199C3×Q8—42no—
200C3×D4—42no—
201C3×D4—42no—
202C3×D4—81no—
203C3×D4—81no—
204C3×D4—81no—
205C2×C3×C4—21no—
206C2×C3×C4—21no—
207C2×C3×C4—22no—
208C2×C3×C4—22no—
209C2×C3×C4—22no—
210C2×C3×C4—41no—
211C2×C3×C4—41no—
212C2×C3×C4—41no—
213C2×C3×C4—41no—
214C2×C3×C4—41no—
215C2×C3×C4—41no—
216C2×C3×C4—82no—
217C2×C3×C4C2×C22⋊C411yescharacteristic
218C2×C3×C4C4×D411yescharacteristic
219C2×C3×C4C4×D411yescharacteristic
220C2×C3×C4#12111yescharacteristic
221C2×C3×C4#12111yescharacteristic
222C2×C3×C4#12311yescharacteristic
223C2×Dic3—24no—
224C2×Dic3—22no—
225C2×Dic3—42no—
226C2×Dic3—42no—
227C2×Dic3—42no—
228D12—24no—
229D12—22no—
230D12#10414yescomplemented
231C4×D3—82no—
232C4×D3—82no—
233C4×D3—88no—
234C3×C8—81no—
235C3⋊C8—42no—
236C3⋊C8—42no—
237C24—61no—
238#41—42no—
239#41—42no—
240C2×Q8—21no—
241C2×D4—21no—
242C2×D4—21no—
243C2×D4—21no—
244C2×D4—32no—
245C2×D4—41no—
246C2×D4—64no—
247C2×D4—61no—
248C22×C4—21no—
249C22×C4—21no—
250C22×C4—41no—
251C22×C4—68no—
252C22×C4—62no—
253C22×C4—61no—
254C22×C4—62no—
255C22×C4—62no—
256C22×C4—242no—
257C22×C4—242no—
258C22×C4—242no—
259C22×C4D3×D411yescharacteristic
260C22×C4D3×D411yescharacteristic
261C22×C4C2×#6711yescharacteristic
262D8—124no—
263M16—62no—
264M16—121no—
265C2×C8—41no—
266C2×C8—62no—
267C2×C8—121no—
268C4⋊C4—21no—
269C4⋊C4—21no—
270C4⋊C4—32no—
271C4⋊C4—64no—
272C4⋊C4—61no—
273C22⋊C4—41no—
274C22⋊C4—41no—
275C22⋊C4—68no—
276C22⋊C4—62no—
277C22⋊C4—62no—
278C42—24no—
279C42—122no—
280C42—242no—
281C42C2×C4×D311yescharacteristic
282C42#23511yescharacteristic
283C42D3×D411yescharacteristic
284C42#23711yescharacteristic
285C22×C3—21no—
286C22×C3—21no—
287C22×C3—81no—
288C22×C3—81no—
289C22×C3—81no—
290C22×C3#38611yescharacteristic
291C22×C3#45311yescharacteristic
292C22×C3#45411yescharacteristic
293C2×D3—24no—
294C2×D3—44no—
295C2×D3—44no—
296C2×D3—42no—
297C2×D3—42no—
298C3×C4—21no—
299C3×C4—21no—
300C3×C4—21no—
301C3×C4—21no—
302C3×C4—21no—
303C3×C4—42no—
304C3×C4—42no—
305C3×C4—81no—
306C3×C4C2×#10412yes—
307Dic3—44no—
308Dic3—42no—
309C23—41no—
310C23—64no—
311C23—62no—
312C23—122no—
313C23—121no—
314C23—121no—
315C23#93711yescharacteristic
316Q8—42no—
317D4—34no—
318D4—42no—
319D4—42no—
320D4—64no—
321D4—62no—
322D4—81no—
323D4—81no—
324D4—81no—
325C2×C4—21no—
326C2×C4—21no—
327C2×C4—22no—
328C2×C4—22no—
329C2×C4—22no—
330C2×C4—41no—
331C2×C4—41no—
332C2×C4—41no—
333C2×C4—41no—
334C2×C4—41no—
335C2×C4—41no—
336C2×C4—64no—
337C2×C4—62no—
338C2×C4—82no—
339C2×C4—122no—
340C2×C4—122no—
341C2×C4—122no—
342C2×C4—242no—
343C2×C4—242no—
344C2×C4—248no—
345C2×C4D3×C22⋊C411yescharacteristic
346C2×C4#67⋊C411yescharacteristic
347C2×C4#88311yescharacteristic
348C2×C4#88411yescharacteristic
349C2×C4C4×#6711yescharacteristic
350C2×C4#93811yescharacteristic
351C8—81no—
352C8—122no—
353C8—122no—
354C2×C3—21no—
355C2×C3—21no—
356C2×C3—161no—
357C2×C3#188311yescharacteristic
358C2×C3#189611yescharacteristic
359C2×C3#190411yescharacteristic
360D3—44no—
361C22—21no—
362C22—21no—
363C22—64no—
364C22—81no—
365C22—81no—
366C22—81no—
367C22—124no—
368C22—124no—
369C22—122no—
370C22—122no—
371C22#495611yescharacteristic
372C22#519811yescharacteristic
373C22#520111yescharacteristic
374C4—21no—
375C4—21no—
376C4—21no—
377C4—21no—
378C4—21no—
379C4—42no—
380C4—42no—
381C4—81no—
382C4—124no—
383C4—122no—
384C4D3×#10412yes—
385C3#932511yescharacteristic, complemented, Sylow
386C2—21no—
387C2—21no—
388C2—124no—
389C2—161no—
390C2#7004911yescharacteristic
391C2#7032811yescharacteristic
392C2#7038211yescharacteristic
subgroupG/H
centerC22#4956
commutatorC3×C42C22×C4
FrattiniC2×C42C22×D3
FittingC3×#1505C2
socleC22×C3#386
prime radical#83300C2
cosocleC2×C3×C42C23
hypercenter#1505D3
hypocenterC3#9325
LCSG ▷ C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1505

Complements

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

NH ≅ G/Ntype
2. #73412C2semidirect
3. C3×#1505C2semidirect
4. #71818C2semidirect
5. #70799C2semidirect
6. #67749C2semidirect
7. #67615C2semidirect
15. #5084C4semidirect
16. #5072C4semidirect
33. #1505D3semidirect
41. C3×C4⋊Q8C2×C4semidirect
42. C3×#127C2×C4semidirect
76. C4⋊(C3⋊C8)C2×C4semidirect
155. C4⋊Q8C4×D3semidirect
156. #127C4×D3semidirect
230. D12#104semidirect
385. C3#9325semidirect

Maximal subgroups

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

subgroupindex
#833002
#734122
C3×#15052
#718182
#707992
#677492
#676152
#93253