Finite Group Databasebeta

← #10690610

Subgroups — #10690610

Order 768.

564 conjugacy classes of subgroups.

The subgroup lattice has 390 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#82301C211yesmaximal, characteristic, complemented
2C2×#5014C211yesmaximal, characteristic
3C3×#1478C211yesmaximal, characteristic, complemented
4#70783C211yesmaximal, characteristic, complemented
5#70756C211yesmaximal, characteristic, complemented
6#67741C211yesmaximal, characteristic, complemented
7#67740C211yesmaximal, characteristic, complemented
8#9089—31nomaximal, Sylow
9C2×C3×C4×D4C2211yescharacteristic
10C22×#213C411yescharacteristic
11C2×C4×D12C2211yescharacteristic
12C3×#431C2211yescharacteristic
13C3×#431C2211yescharacteristic
14C2×#832C411yescharacteristic
15C2×#830—22no—
16#5014—22no—
17#5014—22no—
18D4×C3⋊C8—22no—
19D4×C3⋊C8—22no—
20C2×#809C411yescharacteristic
21C2×#808C411yescharacteristic
22#4925C2211yescharacteristic
23#4925C2211yescharacteristic
24C2×#803C2211yescharacteristic
25C3×D4⋊C8—22no—
26#4483—22no—
27#4468—22no—
28#2972—31no—
29C2×#442—31no—
30#1478—31no—
31#1478D311yescharacteristic, complemented
32#1460—31no—
33#1281—31no—
34#1275—31no—
35C22×C3×D4C2×C411yescharacteristic
36C23×C3×C4—21no—
37C22×D12C2×C411yescharacteristic
38C22×C4×D3—21no—
39C3×C4×D4—21no—
40C3×C4×D4—21no—
41C3×C4×D4D412yes—
42C3×C4×D4D412yes—
43C2×C3×C4⋊C4C2×C411yescharacteristic
44C2×C3×C22⋊C4—21no—
45C2×C3×C42C2311yescharacteristic
46C2×#213—22no—
47C2×#213—22no—
48C2×#213C814yescomplemented
49C2×(C2×D3)⋊C4—21no—
50C2×C4⋊Dic3C2×C411yescharacteristic
51C22×C3⋊C8—24no—
52C22×C3⋊C8C2×C411yescharacteristic
53C22×C3⋊C8C2×C411yescharacteristic
54C4×D12—21no—
55C4×D12—21no—
56C4×D12—21no—
57C4×D12—21no—
58C3×C4⋊C8—21no—
59C3×C4⋊C8—21no—
60C3×C22⋊C8—21no—
61C3×C22⋊C8—21no—
62C3×C4×C8—21no—
63C3×C4×C8—21no—
64#832—22no—
65#830—24no—
66#830—21no—
67#830D412yes—
68#809—22no—
69#808—22no—
70C4⋊(C3⋊C8)—21no—
71C4⋊(C3⋊C8)—21no—
72#803—21no—
73#803—21no—
74C4×C3⋊C8—21no—
75C4×C3⋊C8D412yes—
76C22×D8—31no—
77C2×C4×D4—31no—
78C2×C4×D4C2×D311yescharacteristic
79#442—62no—
80#442—62no—
81C8×D4—62no—
82C8×D4—62no—
83#431—31no—
84#431—31no—
85#431C2×D311yescharacteristic
86#431C2×D311yescharacteristic
87C2×#106—31no—
88C2×#102—31no—
89C2×#102—31no—
90C2×C22⋊C8—62no—
91C2×#97—31no—
92D4⋊C8—22no—
93D4⋊C8—62no—
94#323—62no—
95C24×C3—21no—
96C23×D3—21no—
97C2×C3×D4—21no—
98C2×C3×D4—21no—
99C2×C3×D4—21no—
100C2×C3×D4C22⋊C412yes—
101C2×C3×D4C2×C812yes—
102C2×C3×D4M1612yes—
103C22×C3×C4—21no—
104C22×C3×C4—22no—
105C22×C3×C4—22no—
106C22×C3×C4—82no—
107C22×C3×C4C22×C411yescharacteristic
108C22×C3×C4C2×D411yescharacteristic
109C22×C3×C4#4111yescharacteristic
110C22×Dic3—21no—
111C2×D12—21no—
112C2×D12—21no—
113C2×D12—21no—
114C2×D12—21no—
115C2×D12—21no—
116C2×D12C2×C812yescomplemented
117C2×C4×D3—41no—
118C2×C4×D3—41no—
119C2×C4×D3—82no—
120C2×C3×C8—41no—
121C2×C3×C8—41no—
122C3×C4⋊C4—21no—
123C3×C4⋊C4C22⋊C412yes—
124C3×C22⋊C4—22no—
125C3×C22⋊C4—22no—
126C3×C42C2×D411yescharacteristic
127C3×C42C2×D411yescharacteristic
128C3×C42C2×D411yescharacteristic
129C3×C42#4111yescharacteristic
130#213—28no—
131(C2×D3)⋊C4—41no—
132(C2×D3)⋊C4—41no—
133C4⋊Dic3—21no—
134C4⋊Dic3—21no—
135C2×C3⋊C8—21no—
136C2×C3⋊C8—21no—
137C2×C3⋊C8—21no—
138C2×C3⋊C8—22no—
139C2×C3⋊C8—22no—
140C2×C3⋊C8—21no—
141C2×C3⋊C8—21no—
142C2×C3⋊C8—84no—
143C2×C3⋊C8C2×C812yescomplemented
144C22×D4—31no—
145C22×D4C4×D311yescharacteristic
146C23×C4—21no—
147C23×C4—61no—
148C2×D8—34no—
149C2×D8—62no—
150C2×D8—62no—
151C22×C8—31no—
152C22×C8—31no—
153C22×C8—64no—
154C4×D4—21no—
155C4×D4—21no—
156C4×D4—61no—
157C4×D4—61no—
158C4×D4—61no—
159C4×D4—61no—
160C4×D4D1212yes—
161C4×D4#6712yes—
162C2×C4⋊C4—31no—
163C2×C4⋊C4C4×D311yescharacteristic
164C2×C22⋊C4—21no—
165C2×C22⋊C4—61no—
166C2×C42C22×D311yescharacteristic
167#106—62no—
168C4⋊C8—21no—
169C4⋊C8—21no—
170C4⋊C8—61no—
171C4⋊C8—61no—
172#102—62no—
173#102—62no—
174C22⋊C8—21no—
175C22⋊C8—21no—
176C22⋊C8—32no—
177C22⋊C8—64no—
178C22⋊C8—61no—
179#97—61no—
180#97—61no—
181C4×C8—21no—
182C4×C8—21no—
183C4×C8—32no—
184C4×C8—61no—
185C23×C3—22no—
186C23×C3—22no—
187C23×C3—22no—
188C23×C3—81no—
189C23×C3—81no—
190C23×C3C4×D411yescharacteristic
191C22×D3—22no—
192C22×D3—41no—
193C22×D3—41no—
194C22×D3—81no—
195C22×D3—81no—
196C3×D4—22no—
197C3×D4—22no—
198C3×D4—22no—
199C3×D4C22⋊C814yes—
200C2×C3×C4—22no—
201C2×C3×C4—41no—
202C2×C3×C4—41no—
203C2×C3×C4—44no—
204C2×C3×C4—82no—
205C2×C3×C4—81no—
206C2×C3×C4C2×C22⋊C411yescharacteristic
207C2×C3×C4C4×D411yescharacteristic
208C2×C3×C4#12011yescharacteristic
209C2×C3×C4#12111yescharacteristic
210C2×C3×C4#12111yescharacteristic
211C2×C3×C4#12311yescharacteristic
212C2×C3×C4C22×C811yescharacteristic
213C2×C3×C4C2×M1611yescharacteristic
214C2×C3×C4#13111yescharacteristic
215C2×C3×C4#13111yescharacteristic
216C2×Dic3—41no—
217C2×Dic3—81no—
218D12—22no—
219D12—22no—
220D12—41no—
221D12—41no—
222C4×D3—82no—
223C4×D3—82no—
224C3×C8—81no—
225C3×C8—81no—
226C3⋊C8—22no—
227C3⋊C8—42no—
228C3⋊C8—42no—
229C3⋊C8—41no—
230C24—21no—
231C24—61no—
232C2×D4—21no—
233C2×D4—21no—
234C2×D4—21no—
235C2×D4—32no—
236C2×D4—61no—
237C2×D4—61no—
238C2×D4—61no—
239C2×D4—61no—
240C2×D4—61no—
241C2×D4D3×C812yes—
242C2×D4ΓL(1,𝔽25)12yes—
243C2×D4(C2×D3)⋊C412yes—
244C22×C4—21no—
245C22×C4—22no—
246C22×C4—22no—
247C22×C4—61no—
248C22×C4—82no—
249C22×C4—121no—
250C22×C4—121no—
251C22×C4—242no—
252C22×C4C2×C4×D311yescharacteristic
253C22×C4D3×D411yescharacteristic
254C22×C4#23711yescharacteristic
255D8—68no—
256C2×C8—32no—
257C2×C8—41no—
258C2×C8—41no—
259C2×C8—61no—
260C2×C8—61no—
261C2×C8—62no—
262C2×C8—62no—
263C2×C8—61no—
264C2×C8—61no—
265C2×C8—61no—
266C2×C8—244no—
267C4⋊C4—21no—
268C4⋊C4—61no—
269C4⋊C4—61no—
270C4⋊C4(C2×D3)⋊C412yes—
271C22⋊C4—22no—
272C22⋊C4—22no—
273C22⋊C4—121no—
274C22⋊C4—121no—
275C42C2×D1211yescharacteristic
276C42D3×D411yescharacteristic
277C42#23711yescharacteristic
278C42C2×#6711yescharacteristic
279C22×C3—21no—
280C22×C3—21no—
281C22×C3—24no—
282C22×C3—42no—
283C22×C3—42no—
284C22×C3—82no—
285C22×C3—82no—
286C22×C3#38611yescharacteristic
287C22×C3C8×D411yescharacteristic
288C22×C3#43511yescharacteristic
289C2×D3—42no—
290C2×D3—82no—
291C2×D3—81no—
292C2×D3—81no—
293C2×D3—82no—
294C3×C4—21no—
295C3×C4—21no—
296C3×C4—21no—
297C3×C4—21no—
298C3×C4—42no—
299C3×C4C2×C22⋊C811yescharacteristic
300C3×C4#40811yescharacteristic
301C3×C4C8×D411yescharacteristic
302C3×C4#43511yescharacteristic
303Dic3—81no—
304C23—22no—
305C23—22no—
306C23—22no—
307C23—62no—
308C23—81no—
309C23—81no—
310C23—121no—
311C23—121no—
312C23—241no—
313C23—241no—
314C23#67⋊C411yescharacteristic
315D4—22no—
316D4—22no—
317D4—22no—
318D4—62no—
319D4—62no—
320D4—121no—
321D4—121no—
322D4(C2×D3)⋊C814yes—
323C2×C4—22no—
324C2×C4—41no—
325C2×C4—41no—
326C2×C4—44no—
327C2×C4—82no—
328C2×C4—81no—
329C2×C4—121no—
330C2×C4—242no—
331C2×C4—242no—
332C2×C4—241no—
333C2×C4#67⋊C411yescharacteristic
334C2×C4#88211yescharacteristic
335C2×C4#88611yescharacteristic
336C2×C4C2×D3×C811yescharacteristic
337C2×C4C2×ΓL(1,𝔽25)11yescharacteristic
338C2×C4#90711yescharacteristic
339C2×C4#90711yescharacteristic
340C2×C4C2×(C2×D3)⋊C411yescharacteristic
341C2×C4#93911yescharacteristic
342C2×C4#93911yescharacteristic
343C8—62no—
344C8—81no—
345C8—81no—
346C8—122no—
347C8—122no—
348C8—121no—
349C2×C3—21no—
350C2×C3—21no—
351C2×C3—44no—
352C2×C3#186911yescharacteristic
353C2×C3#190311yescharacteristic
354C2×C3#190411yescharacteristic
355D3—82no—
356C22—21no—
357C22—21no—
358C22—24no—
359C22—42no—
360C22—42no—
361C22—82no—
362C22—82no—
363C22—122no—
364C22—242no—
365C22—242no—
366C22—241no—
367C22—241no—
368C22#67⋊C811yescharacteristic
369C22#473011yescharacteristic
370C22#495811yescharacteristic
371C4—21no—
372C4—21no—
373C4—21no—
374C4—21no—
375C4—42no—
376C4—241no—
377C4#67⋊C811yescharacteristic
378C4#473011yescharacteristic
379C4C2×(C2×D3)⋊C811yescharacteristic
380C4#512911yescharacteristic
381C3#908911yescharacteristic, complemented, Sylow
382C2—21no—
383C2—21no—
384C2—44no—
385C2—242no—
386C2#6995211yescharacteristic
387C2#7031411yescharacteristic
388C2#7040011yescharacteristic
subgroupG/H
centerC22#4958
commutatorC2×C3×C4C22×C8
FrattiniC2×C42C22×D3
FittingC3×#1478C2
socleC22×C3#386
prime radicalC22×#213C4
cosocleC2×C3×C42C23
hypercenter#1478D3
hypocenterC3#9089
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1478

Complements

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

NH ≅ G/Ntype
1. #82301C2semidirect
3. C3×#1478C2semidirect
4. #70783C2semidirect
5. #70756C2semidirect
6. #67741C2semidirect
7. #67740C2semidirect
31. #1478D3semidirect
48. C2×#213C8semidirect
116. C2×D12C2×C8semidirect
143. C2×C3⋊C8C2×C8semidirect
381. C3#9089semidirect

Maximal subgroups

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

subgroupindex
#823012
C2×#50142
C3×#14782
#707832
#707562
#677412
#677402
#90893