Finite Group Databasebeta

← #10691092

Subgroups — #10691092

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#82333C211yesmaximal, characteristic
2#82296C211yesmaximal, characteristic
3C3×#1478C211yesmaximal, characteristic
4#70793C211yesmaximal, characteristic, complemented
5#70792C211yesmaximal, characteristic, complemented
6#67746C211yesmaximal, characteristic, complemented
7#67738C211yesmaximal, characteristic, complemented
8#9175—31nomaximal, Sylow
9C2×C3×C4×D4C2211yescharacteristic
10C2×D4×Dic3—21no—
11#5470C2211yescharacteristic
12C3×#431C2211yescharacteristic
13C3×#431C2211yescharacteristic
14#5215—21no—
15#5212—21no—
16C2×C3⋊#95—21no—
17#5009—22no—
18#5008C412yescomplemented
19#5007C412yescomplemented
20#4984—21no—
21#4964—21no—
22#4950—22no—
23#4936C2211yescharacteristic
24C4×C4⋊Dic3C2211yescharacteristic
25#4924C2211yescharacteristic
26C3×D4⋊C8—22no—
27#4467—21no—
28#4467—21no—
29#4467—22no—
30#3077—31no—
31#2349—31no—
32#1518—31no—
33#1503—31no—
34#1478D311yescharacteristic
35#1303—31no—
36#1286—31no—
37C22×C3×D4D411yescharacteristic
38C23×C3×C4—21no—
39C23×Dic3—22no—
40C3×C4×D4—21no—
41C3×C4×D4—21no—
42C3×C4×D4C2×C412yes—
43C3×C4×D4D412yes—
44C2×C3×C4⋊C4D411yescharacteristic
45C2×C3×C22⋊C4—21no—
46C2×C3×C42C2311yescharacteristic
47C2×#217—22no—
48D4×Dic3—42no—
49D4×Dic3—42no—
50#926—21no—
51C2×C4⋊Dic3—21no—
52C2×C4⋊Dic3—21no—
53C2×C4⋊Dic3—22no—
54#924—21no—
55#924—21no—
56C2×C4×Dic3—21no—
57C2×#208—21no—
58#870—21no—
59#869C2×C411yescharacteristic, complemented
60#869C2×C411yescharacteristic, complemented
61C4×Dic6—21no—
62C4×Dic6—21no—
63C3×C4⋊C8—21no—
64C3×C4⋊C8—21no—
65C3×C22⋊C8—21no—
66C3×C22⋊C8—21no—
67C3×C4×C8—21no—
68C3×C4×C8—21no—
69#832—22no—
70#832—22no—
71C3⋊#95—22no—
72C3⋊#95—21no—
73C3⋊#95D411yescharacteristic
74C3⋊#95D411yescharacteristic
75C3⋊#95D412yes—
76#808—22no—
77#807—22no—
78C4⋊(C3⋊C8)—21no—
79C4⋊(C3⋊C8)C2×C412yescomplemented
80#533—31no—
81C2×C4×D4—61no—
82C2×C4×D4C2×D311yescharacteristic
83#478—62no—
84#476—32no—
85#474—32no—
86#431C2×D311yescharacteristic
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⋊C4—31no—
96C2×#95—61no—
97#328—61no—
98#328—61no—
99#328—62no—
100D4⋊C8—22no—
101C24×C3—21no—
102C2×C3×D4—21no—
103C2×C3×D4—21no—
104C2×C3×D4—21no—
105C2×C3×D4C22⋊C412yes—
106C2×C3×D4SD1612yes—
107C2×C3×D4Q1612yes—
108C22×C3×C4—21no—
109C22×C3×C4—22no—
110C22×C3×C4—22no—
111C22×C3×C4—82no—
112C22×C3×C4C2×D411yescharacteristic
113C22×C3×C4C2×D411yescharacteristic
114C22×C3×C4C2×D411yescharacteristic
115C22×Dic3—21no—
116C22×Dic3—22no—
117C22×Dic3—21no—
118C22×Dic3—44no—
119C22×Dic3—44no—
120C22×Dic3—82no—
121C2×Dic6—21no—
122C2×Dic6—21no—
123C2×C3×C8—41no—
124C2×C3×C8—41no—
125C3×C4⋊C4—21no—
126C3×C4⋊C4C22⋊C412yes—
127C3×C22⋊C4—22no—
128C3×C22⋊C4—22no—
129C3×C42C22×C411yescharacteristic
130C3×C42C2×D411yescharacteristic
131C3×C42#4111yescharacteristic
132C3×C42#4111yescharacteristic
133#217—44no—
134#217—41no—
135C4⋊Dic3—21no—
136C4⋊Dic3—21no—
137C4⋊Dic3—21no—
138C4⋊Dic3—21no—
139C4⋊Dic3—42no—
140C4⋊Dic3—41no—
141C4⋊Dic3—41no—
142C4⋊Dic3—82no—
143Dic3⋊C4—41no—
144Dic3⋊C4—41no—
145C4×Dic3—41no—
146C4×Dic3—41no—
147C4×Dic3—41no—
148#208—41no—
149#208—41no—
150C2×C3⋊C8—22no—
151C22×D4D1211yescharacteristic
152C23×C4—21no—
153C23×C4—62no—
154C2×M16—61no—
155C4⋊Q8—31no—
156C4⋊Q8—31no—
157#125—61no—
158C22⋊Q8—61no—
159C22⋊Q8—61no—
160C4×Q8—61no—
161C4×Q8—61no—
162C4×D4—21no—
163C4×D4—21no—
164C4×D4—122no—
165C4×D4—122no—
166C4×D4C4×D312yes—
167C4×D4#6712yes—
168#117—61no—
169C2×C4⋊C4—61no—
170C2×C4⋊C4—61no—
171C2×C4⋊C4—62no—
172C2×C4⋊C4D1211yescharacteristic
173C2×C22⋊C4—21no—
174C2×C22⋊C4—62no—
175C2×C42—61no—
176C2×C42C22×D311yescharacteristic
177#107—62no—
178#106—62no—
179C4⋊C8—21no—
180C4⋊C8—21no—
181C4⋊C8—32no—
182C4⋊C8—61no—
183#102—62no—
184#102—62no—
185C22⋊C8—21no—
186C22⋊C8—21no—
187C4×C8—21no—
188C4×C8—21no—
189#95—31no—
190#95—31no—
191#95—32no—
192#95—62no—
193#95—61no—
194C23×C3—22no—
195C23×C3—22no—
196C23×C3—22no—
197C23×C3—42no—
198C23×C3—42no—
199C23×C3#12011yescharacteristic
200C3×D4—22no—
201C3×D4—22no—
202C3×D4—22no—
203C3×D4#10314yes—
204C2×C3×C4—22no—
205C2×C3×C4—41no—
206C2×C3×C4—41no—
207C2×C3×C4—42no—
208C2×C3×C4—82no—
209C2×C3×C4—82no—
210C2×C3×C4C2×C22⋊C411yescharacteristic
211C2×C3×C4C4×D411yescharacteristic
212C2×C3×C4C4×D411yescharacteristic
213C2×C3×C4#12111yescharacteristic
214C2×C3×C4#12111yescharacteristic
215C2×C3×C4#12311yescharacteristic
216C2×C3×C4C2×SD1611yescharacteristic
217C2×C3×C4C2×Q1611yescharacteristic
218C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
219C2×C3×C4#13711yescharacteristic
220C2×Dic3—22no—
221C2×Dic3—41no—
222C2×Dic3—41no—
223C2×Dic3—41no—
224C2×Dic3—41no—
225C2×Dic3—42no—
226C2×Dic3—42no—
227C2×Dic3—41no—
228C2×Dic3—41no—
229C2×Dic3—41no—
230C2×Dic3—42no—
231C2×Dic3—88no—
232Dic6—81no—
233Dic6—81no—
234C3×C8—81no—
235C3×C8—81no—
236C3⋊C8—42no—
237C24—21no—
238C2×Q8—61no—
239C2×Q8—61no—
240C2×D4—21no—
241C2×D4—21no—
242C2×D4—21no—
243C2×D4#20412yes—
244C2×D4Dic1212yes—
245C2×D4(C2×D3)⋊C412yes—
246C22×C4—21no—
247C22×C4—22no—
248C22×C4—22no—
249C22×C4—61no—
250C22×C4—62no—
251C22×C4—61no—
252C22×C4—82no—
253C22×C4—124no—
254C22×C4—124no—
255C22×C4—242no—
256C22×C4C2×D1211yescharacteristic
257C22×C4D3×D411yescharacteristic
258C22×C4D3×D411yescharacteristic
259M16—121no—
260M16—121no—
261C2×C8—41no—
262C2×C8—41no—
263C2×C8—62no—
264C4⋊C4—21no—
265C4⋊C4—61no—
266C4⋊C4—61no—
267C4⋊C4—61no—
268C4⋊C4—61no—
269C4⋊C4—122no—
270C4⋊C4—121no—
271C4⋊C4—121no—
272C4⋊C4—121no—
273C4⋊C4—121no—
274C4⋊C4—242no—
275C4⋊C4(C2×D3)⋊C412yes—
276C22⋊C4—22no—
277C22⋊C4—22no—
278C22⋊C4—124no—
279C22⋊C4—121no—
280C42—121no—
281C42—121no—
282C42—121no—
283C42C2×C4×D311yescharacteristic
284C42#23711yescharacteristic
285C42#23711yescharacteristic
286C42C2×#6711yescharacteristic
287C22×C3—21no—
288C22×C3—21no—
289C22×C3—24no—
290C22×C3—44no—
291C22×C3—44no—
292C22×C3—42no—
293C22×C3—42no—
294C22×C3#38611yescharacteristic
295C22×C3#45011yescharacteristic
296C22×C3#45111yescharacteristic
297C3×C4—21no—
298C3×C4—21no—
299C3×C4—21no—
300C3×C4—21no—
301C3×C4—42no—
302C3×C4C2×#10311yescharacteristic
303C3×C4#41711yescharacteristic
304C3×C4#48311yescharacteristic
305C3×C4#48411yescharacteristic
306Dic3—42no—
307Dic3—81no—
308Dic3—82no—
309Dic3—81no—
310Dic3—81no—
311C23—22no—
312C23—22no—
313C23—22no—
314C23—42no—
315C23—42no—
316C23#88211yescharacteristic
317Q8—241no—
318Q8—241no—
319D4—22no—
320D4—22no—
321D4—22no—
322D4#81614yes—
323C2×C4—22no—
324C2×C4—41no—
325C2×C4—41no—
326C2×C4—42no—
327C2×C4—62no—
328C2×C4—82no—
329C2×C4—82no—
330C2×C4—121no—
331C2×C4—121no—
332C2×C4—121no—
333C2×C4—121no—
334C2×C4—122no—
335C2×C4—122no—
336C2×C4—121no—
337C2×C4—121no—
338C2×C4—121no—
339C2×C4—122no—
340C2×C4—248no—
341C2×C4#67⋊C411yescharacteristic
342C2×C4#67⋊C411yescharacteristic
343C2×C4#88611yescharacteristic
344C2×C4C2×#20411yescharacteristic
345C2×C4C2×Dic1211yescharacteristic
346C2×C4#90811yescharacteristic
347C2×C4#90911yescharacteristic
348C2×C4C2×(C2×D3)⋊C411yescharacteristic
349C2×C4#93911yescharacteristic
350C2×C4#93911yescharacteristic
351C8—81no—
352C8—81no—
353C8—122no—
354C2×C3—21no—
355C2×C3—21no—
356C2×C3—44no—
357C2×C3#187211yescharacteristic
358C2×C3#190411yescharacteristic
359C2×C3#190611yescharacteristic
360C22—21no—
361C22—21no—
362C22—24no—
363C22—44no—
364C22—44no—
365C22—42no—
366C22—42no—
367C22#473111yescharacteristic
368C22#473911yescharacteristic
369C22#495811yescharacteristic
370C4—21no—
371C4—21no—
372C4—21no—
373C4—21no—
374C4—42no—
375C4—122no—
376C4—242no—
377C4—241no—
378C4—241no—
379C4—241no—
380C4#472111yescharacteristic
381C4#472211yescharacteristic
382C4C2×#81611yescharacteristic
383C4#513311yescharacteristic
384C3#917511yescharacteristic, complemented, Sylow
385C2—21no—
386C2—21no—
387C2—44no—
388C2#6993011yescharacteristic
389C2#7031411yescharacteristic
390C2#7039511yescharacteristic
subgroupG/H
centerC22#4958
commutatorC3×C42C22×C4
FrattiniC2×C42C22×D3
FittingC3×#1478C2
socleC22×C3#386
prime radicalC22×C3×D4D4
cosocleC2×C3×C42C23
hypercenter#1478D3
hypocenterC3#9175
LCSG ▷ C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1478

Complements

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

NH ≅ G/Ntype
4. #70793C2semidirect
5. #70792C2semidirect
6. #67746C2semidirect
7. #67738C2semidirect
18. #5008C4semidirect
19. #5007C4semidirect
59. #869C2×C4semidirect
60. #869C2×C4semidirect
79. C4⋊(C3⋊C8)C2×C4semidirect
384. C3#9175semidirect

Maximal subgroups

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

subgroupindex
#823332
#822962
C3×#14782
#707932
#707922
#677462
#677382
#91753