Finite Group Databasebeta

← #10690617

Subgroups — #10690617

Order 768.

512 conjugacy classes of subgroups.

The subgroup lattice has 368 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#5026C211yesmaximal, characteristic
2#82435C211yesmaximal, characteristic, complemented
3C3×#1479C211yesmaximal, characteristic, complemented
4#70783C211yesmaximal, characteristic
5#70756C211yesmaximal, characteristic
6#67741C211yesmaximal, characteristic, complemented
7#67740C211yesmaximal, characteristic, complemented
8#9090—31nomaximal, Sylow
9C2×C3×C4×Q8C2211yescharacteristic
10C22×#215C411yescharacteristic
11C2×C4×D12C2211yescharacteristic
12C3×#431C2211yescharacteristic
13C3×#431C2211yescharacteristic
14C2×#835C411yescharacteristic
15#5026—22no—
16#5026—22no—
17Q8×C3⋊C8—22no—
18Q8×C3⋊C8—22no—
19C2×#809C411yescharacteristic
20C2×#807C411yescharacteristic
21#4925—22no—
22#4925C2211yescharacteristic
23#4925C2211yescharacteristic
24C2×#803C2211yescharacteristic
25C3×Q8⋊C8—22no—
26#4483—22no—
27#4468—22no—
28#2982—31no—
29C2×#440—31no—
30#1479D311yescharacteristic, complemented
31#1478—31no—
32#1460—31no—
33#1281—31no—
34#1275—31no—
35C22×C3×Q8C2×C411yescharacteristic
36C22×D12C2×C411yescharacteristic
37C22×C4×D3—21no—
38C3×C4×Q8—21no—
39C3×C4×Q8—21no—
40C3×C4×Q8D412yes—
41C3×C4×Q8D412yes—
42C2×C3×C4⋊C4—21no—
43C2×C3×C4⋊C4C2×C411yescharacteristic
44C2×C3×C42—21no—
45C2×C3×C42C2311yescharacteristic
46C2×#215—22no—
47C2×#215—22no—
48C2×#215C814yescomplemented
49C2×(C2×D3)⋊C4—21no—
50C2×C4⋊Dic3C2×C411yescharacteristic
51C22×C3⋊C8C2×C411yescharacteristic
52C22×C3⋊C8C2×C411yescharacteristic
53C4×D12—21no—
54C4×D12—21no—
55C4×D12—21no—
56C4×D12—21no—
57C3×C4⋊C8—21no—
58C3×C4⋊C8—21no—
59C3×C22⋊C8—21no—
60C3×C22⋊C8—21no—
61C3×C4×C8—21no—
62C3×C4×C8—21no—
63#835—22no—
64#830—21no—
65#830D412yes—
66#809—22no—
67#807—22no—
68C4⋊(C3⋊C8)—24no—
69C4⋊(C3⋊C8)—21no—
70C4⋊(C3⋊C8)—21no—
71#803—21no—
72#803—21no—
73C4×C3⋊C8—24no—
74C4×C3⋊C8—21no—
75C4×C3⋊C8D412yes—
76C22×SD16—31no—
77C2×C4×Q8C2×D311yescharacteristic
78C2×C4×D4—31no—
79C8×Q8—62no—
80C8×Q8—62no—
81#440—62no—
82#440—62no—
83#431—31no—
84#431—31no—
85#431—62no—
86#431C2×D311yescharacteristic
87#431C2×D311yescharacteristic
88C2×#107—31no—
89C2×#103—31no—
90C2×#102—31no—
91C2×#97—31no—
92Q8⋊C8—22no—
93D4⋊C8—62no—
94#323—62no—
95C23×D3—21no—
96C2×C3×Q8—21no—
97C2×C3×Q8—21no—
98C2×C3×Q8—21no—
99C2×C3×Q8C22⋊C412yes—
100C2×C3×Q8C2×C812yes—
101C2×C3×Q8M1612yes—
102C22×C3×C4—21no—
103C22×C3×C4—21no—
104C22×C3×C4C22×C411yescharacteristic
105C22×C3×C4C2×D411yescharacteristic
106C22×C3×C4#4111yescharacteristic
107C22×Dic3—21no—
108C2×D12—21no—
109C2×D12—21no—
110C2×D12—21no—
111C2×D12—21no—
112C2×D12—21no—
113C2×D12C2×C812yes—
114C2×C4×D3—41no—
115C2×C4×D3—41no—
116C2×C4×D3—82no—
117C2×C3×C8—41no—
118C2×C3×C8—41no—
119C3×C4⋊C4—21no—
120C3×C4⋊C4—22no—
121C3×C4⋊C4—22no—
122C3×C4⋊C4C22⋊C412yes—
123C3×C42—22no—
124C3×C42—22no—
125C3×C42C2×D411yescharacteristic
126C3×C42C2×D411yescharacteristic
127C3×C42C2×D411yescharacteristic
128C3×C42#4111yescharacteristic
129#215—28no—
130(C2×D3)⋊C4—41no—
131(C2×D3)⋊C4—41no—
132C4⋊Dic3—21no—
133C4⋊Dic3—21no—
134C2×C3⋊C8—21no—
135C2×C3⋊C8—21no—
136C2×C3⋊C8—21no—
137C2×C3⋊C8—22no—
138C2×C3⋊C8—22no—
139C2×C3⋊C8—21no—
140C2×C3⋊C8—21no—
141C2×C3⋊C8C2×C812yes—
142C22×Q8C4×D311yescharacteristic
143C22×D4—31no—
144C23×C4—61no—
145C2×SD16—34no—
146C2×SD16—62no—
147C2×SD16—62no—
148C22×C8—31no—
149C22×C8—31no—
150C4×Q8—21no—
151C4×Q8—21no—
152C4×Q8D1212yes—
153C4×Q8#6712yes—
154C4×D4—61no—
155C4×D4—61no—
156C4×D4—61no—
157C4×D4—61no—
158C2×C4⋊C4—21no—
159C2×C4⋊C4—31no—
160C2×C4⋊C4C4×D311yescharacteristic
161C2×C22⋊C4—61no—
162C2×C42—21no—
163C2×C42C22×D311yescharacteristic
164#107—62no—
165C4⋊C8—21no—
166C4⋊C8—21no—
167C4⋊C8—64no—
168C4⋊C8—61no—
169C4⋊C8—61no—
170#103—62no—
171#102—62no—
172C22⋊C8—21no—
173C22⋊C8—21no—
174C22⋊C8—32no—
175C22⋊C8—61no—
176#97—61no—
177#97—61no—
178C4×C8—21no—
179C4×C8—21no—
180C4×C8—32no—
181C4×C8—64no—
182C4×C8—61no—
183C23×C3C4×D411yescharacteristic
184C22×D3—22no—
185C22×D3—41no—
186C22×D3—41no—
187C22×D3—81no—
188C22×D3—81no—
189C3×Q8—22no—
190C3×Q8—22no—
191C3×Q8—22no—
192C3×Q8C22⋊C814yes—
193C2×C3×C4—22no—
194C2×C3×C4—22no—
195C2×C3×C4—22no—
196C2×C3×C4—22no—
197C2×C3×C4—41no—
198C2×C3×C4—41no—
199C2×C3×C4—81no—
200C2×C3×C4C2×C22⋊C411yescharacteristic
201C2×C3×C4C4×D411yescharacteristic
202C2×C3×C4#12011yescharacteristic
203C2×C3×C4#12111yescharacteristic
204C2×C3×C4#12111yescharacteristic
205C2×C3×C4#12311yescharacteristic
206C2×C3×C4C22×C811yescharacteristic
207C2×C3×C4C2×M1611yescharacteristic
208C2×C3×C4#13111yescharacteristic
209C2×C3×C4#13111yescharacteristic
210C2×Dic3—41no—
211C2×Dic3—81no—
212D12—22no—
213D12—22no—
214D12—41no—
215D12—41no—
216C4×D3—82no—
217C4×D3—82no—
218C3×C8—81no—
219C3×C8—81no—
220C3⋊C8—22no—
221C3⋊C8—42no—
222C3⋊C8—42no—
223C3⋊C8—41no—
224C24—61no—
225C2×Q8—21no—
226C2×Q8—21no—
227C2×Q8—21no—
228C2×Q8D3×C812yes—
229C2×Q8ΓL(1,𝔽25)12yes—
230C2×Q8(C2×D3)⋊C412yes—
231C2×D4—32no—
232C2×D4—61no—
233C2×D4—61no—
234C2×D4—61no—
235C2×D4—61no—
236C2×D4—61no—
237C22×C4—21no—
238C22×C4—21no—
239C22×C4—61no—
240C22×C4—121no—
241C22×C4—121no—
242C22×C4—242no—
243C22×C4C2×C4×D311yescharacteristic
244C22×C4D3×D411yescharacteristic
245C22×C4#23711yescharacteristic
246SD16—68no—
247C2×C8—32no—
248C2×C8—41no—
249C2×C8—41no—
250C2×C8—61no—
251C2×C8—61no—
252C2×C8—62no—
253C2×C8—62no—
254C2×C8—61no—
255C2×C8—61no—
256C2×C8—61no—
257C4⋊C4—21no—
258C4⋊C4—22no—
259C4⋊C4—22no—
260C4⋊C4—61no—
261C4⋊C4—61no—
262C4⋊C4(C2×D3)⋊C412yes—
263C22⋊C4—121no—
264C22⋊C4—121no—
265C42—22no—
266C42—22no—
267C42C2×D1211yescharacteristic
268C42D3×D411yescharacteristic
269C42#23711yescharacteristic
270C42C2×#6711yescharacteristic
271C22×C3—21no—
272C22×C3—21no—
273C22×C3#38611yescharacteristic
274C22×C3C8×D411yescharacteristic
275C22×C3#43511yescharacteristic
276C2×D3—42no—
277C2×D3—82no—
278C2×D3—81no—
279C2×D3—81no—
280C2×D3—82no—
281C3×C4—24no—
282C3×C4—21no—
283C3×C4—21no—
284C3×C4—21no—
285C3×C4—21no—
286C3×C4—42no—
287C3×C4C2×C22⋊C811yescharacteristic
288C3×C4#40811yescharacteristic
289C3×C4C8×D411yescharacteristic
290C3×C4#43511yescharacteristic
291Dic3—81no—
292C23—62no—
293C23—121no—
294C23—121no—
295C23—241no—
296C23—241no—
297C23#67⋊C411yescharacteristic
298Q8—22no—
299Q8—22no—
300Q8—22no—
301Q8(C2×D3)⋊C814yes—
302D4—62no—
303D4—62no—
304D4—121no—
305D4—121no—
306C2×C4—22no—
307C2×C4—22no—
308C2×C4—22no—
309C2×C4—22no—
310C2×C4—41no—
311C2×C4—41no—
312C2×C4—81no—
313C2×C4—121no—
314C2×C4—242no—
315C2×C4—242no—
316C2×C4—241no—
317C2×C4#67⋊C411yescharacteristic
318C2×C4#88211yescharacteristic
319C2×C4#88611yescharacteristic
320C2×C4C2×D3×C811yescharacteristic
321C2×C4C2×ΓL(1,𝔽25)11yescharacteristic
322C2×C4#90711yescharacteristic
323C2×C4#90711yescharacteristic
324C2×C4C2×(C2×D3)⋊C411yescharacteristic
325C2×C4#93911yescharacteristic
326C2×C4#93911yescharacteristic
327C8—62no—
328C8—81no—
329C8—81no—
330C8—122no—
331C8—122no—
332C8—121no—
333C2×C3—21no—
334C2×C3—21no—
335C2×C3#186911yescharacteristic
336C2×C3#190311yescharacteristic
337C2×C3#190511yescharacteristic
338D3—82no—
339C22—21no—
340C22—21no—
341C22—122no—
342C22—242no—
343C22—242no—
344C22—241no—
345C22—241no—
346C22#67⋊C811yescharacteristic
347C22#473011yescharacteristic
348C22#495811yescharacteristic
349C4—21no—
350C4—21no—
351C4—24no—
352C4—21no—
353C4—21no—
354C4—42no—
355C4—241no—
356C4#67⋊C811yescharacteristic
357C4#473011yescharacteristic
358C4C2×(C2×D3)⋊C811yescharacteristic
359C4#512911yescharacteristic
360C3#909011yescharacteristic, complemented, Sylow
361C2—21no—
362C2—21no—
363C2—242no—
364C2#6995211yescharacteristic
365C2#7033411yescharacteristic
366C2#7040011yescharacteristic
subgroupG/H
centerC22#4958
commutatorC2×C3×C4C22×C8
FrattiniC2×C42C22×D3
FittingC3×#1479C2
socleC22×C3#386
prime radicalC22×D12C2×C4
cosocleC2×C3×C42C23
hypercenter#1479D3
hypocenterC3#9090
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1479

Complements

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

NH ≅ G/Ntype
2. #82435C2semidirect
3. C3×#1479C2semidirect
6. #67741C2semidirect
7. #67740C2semidirect
30. #1479D3semidirect
48. C2×#215C8semidirect
360. C3#9090semidirect

Maximal subgroups

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

subgroupindex
C2×#50262
#824352
C3×#14792
#707832
#707562
#677412
#677402
#90903