Finite Group Databasebeta

← #10691097

Subgroups — #10691097

Order 768.

566 conjugacy classes of subgroups.

The subgroup lattice has 392 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#83350C211yesmaximal, characteristic
2#73770C211yesmaximal, characteristic
3C3×#1503C211yesmaximal, characteristic, complemented
4#71838C211yesmaximal, characteristic, complemented
5#70756C211yesmaximal, characteristic
6#67738C211yesmaximal, characteristic, complemented
7#67620C211yesmaximal, characteristic, complemented
8#9175—31nomaximal, Sylow
9C3×#533C2211yescharacteristic
10C2×C4×D12—21no—
11C2×C4×D12C2211yescharacteristic
12C3×#431C2211yescharacteristic
13C3×C4×C4⋊C4C2211yescharacteristic
14#5087C412yescomplemented
15#5084C412yescomplemented
16#5066—22no—
17#5001—21no—
18#4995—21no—
19#4989—22no—
20#4936C2211yescharacteristic
21#4928—21no—
22#4925C2211yescharacteristic
23#4924C2211yescharacteristic
24#4667—21no—
25D3×#95—21no—
26C3×#328—21no—
27C3×#328—21no—
28#4538—22no—
29#4483—22no—
30#3077—31no—
31#2349—31no—
32#1518—31no—
33#1503D311yescharacteristic, complemented
34#1478—31no—
35#1303—31no—
36#1286—31no—
37C22×D12D411yescharacteristic
38C22×C4×D3—21no—
39C22×C4×D3—22no—
40C3×C4⋊Q8C2×C411yescharacteristic, complemented
41C3×C4⋊Q8C2×C411yescharacteristic, complemented
42C3×#125—21no—
43C3×C22⋊Q8—21no—
44C3×C22⋊Q8—21no—
45C3×C4×Q8—21no—
46C3×C4×Q8—21no—
47C3×#117—21no—
48C2×C3×C4⋊C4—21no—
49C2×C3×C42—21no—
50C2×C3×C42C2311yescharacteristic
51C2×(C2×D3)⋊C4—21no—
52C2×(C2×D3)⋊C4—22no—
53C2×C4⋊Dic3—21no—
54C2×C4⋊Dic3D411yescharacteristic
55C2×Dic3⋊C4—22no—
56C2×#208—21no—
57C4×D12—21no—
58C4×D12—21no—
59C4×D12—42no—
60C4×D12—42no—
61C4×D12C2×C412yes—
62C4×D12D412yes—
63C3×C4⋊C8—21no—
64C3×C22⋊C8—21no—
65C3×C4×C8—21no—
66C3×#95D411yescharacteristic
67C3×#95D411yescharacteristic
68C3⋊#95—22no—
69C3⋊#95—21no—
70C3⋊#95D412yes—
71#830—21no—
72#809—22no—
73#809—22no—
74#808—22no—
75#807—22no—
76C4⋊(C3⋊C8)—21no—
77C4⋊(C3⋊C8)—21no—
78C4⋊(C3⋊C8)C2×C412yescomplemented
79C4×C3⋊C8—21no—
80#533C2×D311yescharacteristic
81C2×C4×D4—31no—
82C2×C4×D4—61no—
83#478—62no—
84#476—32no—
85#474—32no—
86#431—31no—
87#431C2×D311yescharacteristic
88#428—61no—
89#423—31no—
90#418—61no—
91#389—61no—
92#385—62no—
93#382—31no—
94#380—61no—
95C4×C4⋊C4C2×D311yescharacteristic
96C2×#95—61no—
97#328—21no—
98#328—21no—
99#328—62no—
100D4⋊C8—62no—
101C23×D3—21no—
102C2×C3×Q8—21no—
103C2×C3×Q8—21no—
104C22×C3×C4—21no—
105C22×C3×C4—21no—
106C22×C3×C4C2×D411yescharacteristic
107C22×C3×C4C2×D411yescharacteristic
108C22×C3×C4C2×D411yescharacteristic
109C22×Dic3—21no—
110C22×Dic3—22no—
111C2×D12—21no—
112C2×D12—21no—
113C2×D12—21no—
114C2×D12C22⋊C412yes—
115C2×D12SD1612yes—
116C2×D12Q1612yes—
117C2×C4×D3—22no—
118C2×C4×D3—22no—
119C2×C4×D3—44no—
120C2×C4×D3—44no—
121C2×C4×D3—82no—
122C2×C4×D3—82no—
123C2×C3×C8—41no—
124C3×C4⋊C4—21no—
125C3×C4⋊C4—21no—
126C3×C4⋊C4—21no—
127C3×C4⋊C4—21no—
128C3×C4⋊C4—42no—
129C3×C4⋊C4—41no—
130C3×C4⋊C4—41no—
131C3×C22⋊C4—41no—
132C3×C42—41no—
133C3×C42—41no—
134C3×C42—41no—
135C3×C42C22×C411yescharacteristic
136C3×C42C2×D411yescharacteristic
137C3×C42#4111yescharacteristic
138C3×C42#4111yescharacteristic
139(C2×D3)⋊C4—22no—
140(C2×D3)⋊C4—22no—
141(C2×D3)⋊C4—44no—
142C4⋊Dic3—21no—
143C4⋊Dic3—41no—
144C4⋊Dic3—41no—
145C4⋊Dic3C22⋊C412yes—
146Dic3⋊C4—82no—
147#208—41no—
148#208—41no—
149C2×C3⋊C8—22no—
150C2×C3⋊C8—41no—
151C22×D4—31no—
152C23×C4—62no—
153C23×C4—61no—
154C2×M16—61no—
155C4⋊Q8C4×D311yescharacteristic, complemented
156C4⋊Q8C4×D311yescharacteristic, complemented
157#125—21no—
158C22⋊Q8—21no—
159C22⋊Q8—21no—
160C4×Q8—21no—
161C4×Q8—21no—
162C4×D4—32no—
163C4×D4—32no—
164C4×D4—61no—
165C4×D4—61no—
166C4×D4—122no—
167C4×D4—122no—
168#117—21no—
169C2×C4⋊C4—21no—
170C2×C4⋊C4—31no—
171C2×C4⋊C4—61no—
172C2×C4⋊C4—62no—
173C2×C22⋊C4—61no—
174C2×C22⋊C4—62no—
175C2×C42—21no—
176C2×C42C22×D311yescharacteristic
177#107—62no—
178#106—62no—
179C4⋊C8—21no—
180C4⋊C8—32no—
181C4⋊C8—61no—
182C4⋊C8—61no—
183#102—62no—
184#102—62no—
185C22⋊C8—21no—
186C22⋊C8—61no—
187C4×C8—21no—
188C4×C8—61no—
189#95—32no—
190#95—62no—
191#95—61no—
192#95#6711yescharacteristic
193#95#6711yescharacteristic
194C23×C3#12011yescharacteristic
195C22×D3—22no—
196C22×D3—22no—
197C22×D3—22no—
198C22×D3—42no—
199C22×D3—42no—
200C3×Q8—81no—
201C3×Q8—81no—
202C2×C3×C4—22no—
203C2×C3×C4—41no—
204C2×C3×C4—41no—
205C2×C3×C4—41no—
206C2×C3×C4—41no—
207C2×C3×C4—41no—
208C2×C3×C4—41no—
209C2×C3×C4—41no—
210C2×C3×C4—41no—
211C2×C3×C4—41no—
212C2×C3×C4C2×C22⋊C411yescharacteristic
213C2×C3×C4C4×D411yescharacteristic
214C2×C3×C4C4×D411yescharacteristic
215C2×C3×C4#12111yescharacteristic
216C2×C3×C4#12111yescharacteristic
217C2×C3×C4#12311yescharacteristic
218C2×C3×C4C2×SD1611yescharacteristic
219C2×C3×C4C2×Q1611yescharacteristic
220C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
221C2×C3×C4#13711yescharacteristic
222C2×Dic3—22no—
223C2×Dic3—42no—
224C2×Dic3—42no—
225C2×Dic3—42no—
226C2×Dic3—42no—
227D12—22no—
228D12—22no—
229D12—22no—
230D12#10314yes—
231C4×D3—82no—
232C4×D3—82no—
233C4×D3—88no—
234C3×C8—81no—
235C3⋊C8—42no—
236C3⋊C8—81no—
237C24—61no—
238C2×Q8—21no—
239C2×Q8—21no—
240C2×D4—32no—
241C2×D4—32no—
242C2×D4—32no—
243C2×D4—61no—
244C2×D4—61no—
245C2×D4—61no—
246C22×C4—21no—
247C22×C4—21no—
248C22×C4—61no—
249C22×C4—62no—
250C22×C4—62no—
251C22×C4—62no—
252C22×C4—124no—
253C22×C4—124no—
254C22×C4—242no—
255C22×C4—242no—
256C22×C4D3×D411yescharacteristic
257C22×C4D3×D411yescharacteristic
258C22×C4C2×#6711yescharacteristic
259M16—121no—
260M16—121no—
261C2×C8—41no—
262C2×C8—62no—
263C2×C8—121no—
264C4⋊C4—21no—
265C4⋊C4—21no—
266C4⋊C4—21no—
267C4⋊C4—21no—
268C4⋊C4—32no—
269C4⋊C4—42no—
270C4⋊C4—41no—
271C4⋊C4—41no—
272C4⋊C4—61no—
273C4⋊C4—121no—
274C4⋊C4—121no—
275C4⋊C4—242no—
276C22⋊C4—41no—
277C22⋊C4—62no—
278C22⋊C4—62no—
279C22⋊C4—124no—
280C42—41no—
281C42—41no—
282C42—41no—
283C42C2×C4×D311yescharacteristic
284C42#23511yescharacteristic
285C42D3×D411yescharacteristic
286C42#23711yescharacteristic
287C22×C3—21no—
288C22×C3—21no—
289C22×C3#38611yescharacteristic
290C22×C3#45011yescharacteristic
291C22×C3#45111yescharacteristic
292C2×D3—24no—
293C2×D3—44no—
294C2×D3—44no—
295C2×D3—42no—
296C2×D3—42no—
297C3×C4—21no—
298C3×C4—21no—
299C3×C4—21no—
300C3×C4—21no—
301C3×C4—42no—
302C3×C4—81no—
303C3×C4—81no—
304C3×C4—81no—
305C3×C4C2×#10311yescharacteristic
306C3×C4#41711yescharacteristic
307C3×C4#48311yescharacteristic
308C3×C4#48411yescharacteristic
309Dic3—42no—
310Dic3—82no—
311C23—62no—
312C23—62no—
313C23—62no—
314C23—122no—
315C23—122no—
316C23#93711yescharacteristic
317Q8—81no—
318Q8—81no—
319D4—34no—
320D4—62no—
321D4—62no—
322D4—62no—
323C2×C4—22no—
324C2×C4—41no—
325C2×C4—41no—
326C2×C4—41no—
327C2×C4—41no—
328C2×C4—41no—
329C2×C4—41no—
330C2×C4—41no—
331C2×C4—41no—
332C2×C4—41no—
333C2×C4—62no—
334C2×C4—122no—
335C2×C4—122no—
336C2×C4—122no—
337C2×C4—122no—
338C2×C4—242no—
339C2×C4—242no—
340C2×C4—248no—
341C2×C4D3×C22⋊C411yescharacteristic
342C2×C4#67⋊C411yescharacteristic
343C2×C4#88311yescharacteristic
344C2×C4#88411yescharacteristic
345C2×C4D3×SD1611yescharacteristic
346C2×C4#91411yescharacteristic
347C2×C4D3×Q1611yescharacteristic
348C2×C4#91811yescharacteristic
349C2×C4C4×#6711yescharacteristic
350C2×C4#93811yescharacteristic
351C8—81no—
352C8—122no—
353C8—241no—
354C2×C3—21no—
355C2×C3—21no—
356C2×C3#187211yescharacteristic
357C2×C3#190411yescharacteristic
358C2×C3#190611yescharacteristic
359D3—44no—
360C22—21no—
361C22—21no—
362C22—64no—
363C22—124no—
364C22—124no—
365C22—122no—
366C22—122no—
367C22#495611yescharacteristic
368C22#516911yescharacteristic
369C22#518611yescharacteristic
370C4—21no—
371C4—21no—
372C4—21no—
373C4—21no—
374C4—42no—
375C4—81no—
376C4—81no—
377C4—81no—
378C4—122no—
379C4—242no—
380C4D3×#10311yescharacteristic
381C4#480311yescharacteristic
382C4#480511yescharacteristic
383C4#481111yescharacteristic
384C3#917511yescharacteristic, complemented, Sylow
385C2—21no—
386C2—21no—
387C2—124no—
388C2#7004911yescharacteristic
389C2#7026811yescharacteristic
390C2#7039111yescharacteristic
subgroupG/H
centerC22#4956
commutatorC3×C42C22×C4
FrattiniC2×C42C22×D3
FittingC3×#1503C2
socleC22×C3#386
prime radicalC22×D12D4
cosocleC2×C3×C42C23
hypercenter#1503D3
hypocenterC3#9175
LCSG ▷ C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1503

Complements

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

NH ≅ G/Ntype
3. C3×#1503C2semidirect
4. #71838C2semidirect
6. #67738C2semidirect
7. #67620C2semidirect
14. #5087C4semidirect
15. #5084C4semidirect
33. #1503D3semidirect
40. C3×C4⋊Q8C2×C4semidirect
41. C3×C4⋊Q8C2×C4semidirect
78. C4⋊(C3⋊C8)C2×C4semidirect
155. C4⋊Q8C4×D3semidirect
156. C4⋊Q8C4×D3semidirect
384. C3#9175semidirect

Maximal subgroups

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

subgroupindex
#833502
#737702
C3×#15032
#718382
#707562
#677382
#676202
#91753