Finite Group Databasebeta

← #10950913

Subgroups — #10950913

Order 768.

694 conjugacy classes of subgroups.

The subgroup lattice has 408 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#466C211yesmaximal, characteristic, complemented
2C2×#5115C211yesmaximal, characteristic, complemented
3#72035C212yesmaximal, complemented
4C2×#4549C211yesmaximal, characteristic, complemented
5#71075C212yesmaximal, complemented
6#71071C214yesmaximal, complemented
7#71068C212yesmaximal, complemented
8#70599C212yesmaximal, complemented
9#33752—31nomaximal, Sylow
10C22×C3×D8C2211yescharacteristic, complemented
11C23×C3×C8C2211yescharacteristic
12C2×C3×#121—21no—
13C2×#930—21no—
14C22×D24C2211yescharacteristic, complemented
15C3×#466—21no—
16C3×#466—21no—
17C3×#466C2212yes—
18C3×#466C2212yescomplemented
19C2×C3×#107C2211yescharacteristic, complemented
20C2×C3×#102—21no—
21C2×#826C2214yescomplemented
22#5115—21no—
23#5115—21no—
24#5115C2212yes—
25#5115C2212yescomplemented
26C2×#821—21no—
27C2×#818C2211yescharacteristic, complemented
28C22×C3⋊C16C2212yescomplemented
29#4562C2214yescomplemented
30#4549C2212yescomplemented
31#4549C2212yescomplemented
32#4491C2214yescomplemented
33#4489C2214yescomplemented
34C2×#466—31no—
35C2×#466D311yescharacteristic, complemented
36#2232—32no—
37#2232—32no—
38#2215—34no—
39#2184—32no—
40#2184—32no—
41C2×C22⋊C16—31no—
42C22×C3×D4—21no—
43C22×C3×D4—41no—
44C23×C3×C4D411yescharacteristic
45C22×#67—41no—
46C22×D12—21no—
47C2×C3×D8—21no—
48C2×C3×D8—42no—
49C2×C3×D8C2312yes—
50C22×C3×C8—42no—
51C22×C3×C8D411yescharacteristic
52C22×C3×C8D411yescharacteristic
53C22×C3×C8D411yescharacteristic
54C22×C3×C8C2311yescharacteristic
55C22×C3×C8C2311yescharacteristic
56C22×C3×C8C2311yescharacteristic
57C3×#121—22no—
58C3×#121—22no—
59C3×#121—41no—
60C3×#121—41no—
61C2×C3×C4⋊C4—21no—
62C2×C3×C22⋊C4—41no—
63#930—22no—
64#930—22no—
65#930—41no—
66#930—41no—
67C2×(C2×D3)⋊C4—41no—
68C2×C4⋊Dic3—21no—
69C2×D24—21no—
70C2×D24—42no—
71C2×D24C2312yes—
72C3×#107—21no—
73C3×#107C2312yes—
74C3×#102—22no—
75C3×#102—22no—
76#826—44no—
77#821—22no—
78#821—22no—
79#818—21no—
80#818C2312yes—
81C2×C3⋊C16—42no—
82C2×C3⋊C16C2314yes—
83C22×D8—31no—
84C22×D8C2×D311yescharacteristic, complemented
85C23×C8C2×D311yescharacteristic
86C2×#121—21no—
87C2×#121—61no—
88C2×D16—34no—
89C22×C16—32no—
90#466—21no—
91#466—21no—
92#466—32no—
93#466—32no—
94#466—61no—
95#466—61no—
96#466C2×D312yes—
97#466C2×D312yescomplemented
98C2×#107—31no—
99C2×#107C2×D311yescharacteristic, complemented
100C2×#102—21no—
101C2×#102—61no—
102#366—34no—
103#357—34no—
104#357—34no—
105C22⋊C16—32no—
106C22⋊C16—32no—
107C24×C3—41no—
108C24×C3D811yescharacteristic
109C23×D3—41no—
110C2×C3×D4—22no—
111C2×C3×D4—42no—
112C2×C3×D4—42no—
113C2×C3×D4—42no—
114C2×C3×D4—42no—
115C2×C3×D4—41no—
116C2×C3×D4—84no—
117C22×C3×C4—41no—
118C22×C3×C4—41no—
119C22×C3×C4—41no—
120C22×C3×C4D811yescharacteristic
121C22×C3×C4D811yescharacteristic
122C22×C3×C4D811yescharacteristic
123C22×C3×C4C2×D411yescharacteristic
124C22×C3×C4C2×D411yescharacteristic
125C22×C3×C4C2×D411yescharacteristic
126C2×#67—42no—
127C2×#67—42no—
128C2×#67—84no—
129C22×Dic3—41no—
130C2×D12—22no—
131C2×D12—42no—
132C2×D12—42no—
133C2×D12—41no—
134C3×D8—24no—
135C3×D8—42no—
136C2×C3×C8—22no—
137C2×C3×C8—22no—
138C2×C3×C8—22no—
139C2×C3×C8—41no—
140C2×C3×C8—41no—
141C2×C3×C8—41no—
142C2×C3×C8C2×D411yescharacteristic
143C2×C3×C8C2×D411yescharacteristic
144C2×C3×C8C2×D411yescharacteristic
145C2×C3×C8C2411yescharacteristic
146C3×C4⋊C4—22no—
147C3×C4⋊C4—41no—
148C3×C22⋊C4—42no—
149C3×C22⋊C4—42no—
150(C2×D3)⋊C4—42no—
151(C2×D3)⋊C4—42no—
152C4⋊Dic3—22no—
153C4⋊Dic3—41no—
154D24—24no—
155D24—42no—
156C3⋊C16—24no—
157C22×D4—21no—
158C22×D4—41no—
159C22×D4—61no—
160C22×D4—121no—
161C23×C4#6711yescharacteristic
162C2×D8—21no—
163C2×D8—32no—
164C2×D8—42no—
165C2×D8—61no—
166C2×D8—122no—
167C2×D8C22×D312yes—
168C22×C8—42no—
169C22×C8#6711yescharacteristic
170C22×C8#6711yescharacteristic
171C22×C8#6711yescharacteristic
172C22×C8C22×D311yescharacteristic
173C22×C8C22×D311yescharacteristic
174C22×C8C22×D311yescharacteristic
175#121—22no—
176#121—22no—
177#121—41no—
178#121—41no—
179#121—62no—
180#121—62no—
181#121—121no—
182#121—121no—
183C2×C4⋊C4—21no—
184C2×C4⋊C4—61no—
185C2×C22⋊C4—41no—
186C2×C22⋊C4—121no—
187D16—124no—
188C2×C16—34no—
189C2×C16—122no—
190#107—21no—
191#107—32no—
192#107—61no—
193#107C22×D312yes—
194#102—22no—
195#102—22no—
196#102—62no—
197#102—62no—
198C23×C3—41no—
199C23×C3—41no—
200C23×C3—42no—
201C23×C3—82no—
202C23×C3—82no—
203C23×C3—82no—
204C23×C3D1614yescomplemented
205C23×C3C2×D811yescharacteristic
206C23×C3C2×D811yescharacteristic
207C23×C3C2×D811yescharacteristic
208C22×D3—42no—
209C22×D3—82no—
210C22×D3—82no—
211C22×D3—82no—
212C3×D4—42no—
213C3×D4—42no—
214C3×D4—82no—
215C3×D4—84no—
216C3×D4—84no—
217C2×C3×C4—21no—
218C2×C3×C4—21no—
219C2×C3×C4—21no—
220C2×C3×C4—21no—
221C2×C3×C4—21no—
222C2×C3×C4—21no—
223C2×C3×C4—41no—
224C2×C3×C4—41no—
225C2×C3×C4—41no—
226C2×C3×C4—42no—
227C2×C3×C4—82no—
228C2×C3×C4C2×D811yescharacteristic
229C2×C3×C4C2×D811yescharacteristic
230C2×C3×C4C2×D811yescharacteristic
231C2×C3×C4C22×D411yescharacteristic
232#67—84no—
233#67—84no—
234C2×Dic3—42no—
235C2×Dic3—82no—
236D12—42no—
237D12—42no—
238D12—82no—
239C3×C8—21no—
240C3×C8—21no—
241C3×C8—21no—
242C3×C8H(5,𝔽2)12yes—
243C24—41no—
244C24—121no—
245C24#21311yescharacteristic
246C2×D4—22no—
247C2×D4—42no—
248C2×D4—42no—
249C2×D4—42no—
250C2×D4—42no—
251C2×D4—41no—
252C2×D4—62no—
253C2×D4—84no—
254C2×D4—122no—
255C2×D4—122no—
256C2×D4—122no—
257C2×D4—122no—
258C2×D4—121no—
259C2×D4—244no—
260C22×C4—41no—
261C22×C4—41no—
262C22×C4—41no—
263C22×C4—121no—
264C22×C4#21311yescharacteristic
265C22×C4#21311yescharacteristic
266C22×C4#21311yescharacteristic
267C22×C4C2×#6711yescharacteristic
268C22×C4C2×#6711yescharacteristic
269C22×C4C2×#6711yescharacteristic
270D8—24no—
271D8—42no—
272D8—64no—
273D8—122no—
274C2×C8—22no—
275C2×C8—22no—
276C2×C8—22no—
277C2×C8—41no—
278C2×C8—41no—
279C2×C8—41no—
280C2×C8C2×#6711yescharacteristic
281C2×C8C2×#6711yescharacteristic
282C2×C8C2×#6711yescharacteristic
283C2×C8C23×D311yescharacteristic
284C4⋊C4—22no—
285C4⋊C4—41no—
286C4⋊C4—62no—
287C4⋊C4—121no—
288C22⋊C4—42no—
289C22⋊C4—42no—
290C22⋊C4—122no—
291C22⋊C4—122no—
292C16—64no—
293C22×C3—21no—
294C22×C3—21no—
295C22×C3—21no—
296C22×C3—21no—
297C22×C3—21no—
298C22×C3—21no—
299C22×C3—44no—
300C22×C3—82no—
301C22×C3—82no—
302C22×C3—82no—
303C22×C3—164no—
304C22×C3C2×D1612yes—
305C22×C3C2×D1612yes—
306C22×C3C2×D1612yes—
307C22×C3C22×D811yescharacteristic
308C2×D3—82no—
309C2×D3—82no—
310C2×D3—82no—
311C2×D3—164no—
312C3×C4—21no—
313C3×C4—21no—
314C3×C4—21no—
315C3×C4—82no—
316C3×C4#53411yescharacteristic
317C3×C4#57511yescharacteristic
318Dic3—82no—
319C23—41no—
320C23—41no—
321C23—42no—
322C23—82no—
323C23—82no—
324C23—82no—
325C23—122no—
326C23—242no—
327C23—242no—
328C23—242no—
329C23#82614yescomplemented
330C23C2×#21311yescharacteristic
331C23C2×#21311yescharacteristic
332C23C2×#21311yescharacteristic
333D4—42no—
334D4—42no—
335D4—82no—
336D4—84no—
337D4—84no—
338D4—122no—
339D4—122no—
340D4—242no—
341D4—244no—
342D4—244no—
343C2×C4—21no—
344C2×C4—21no—
345C2×C4—21no—
346C2×C4—21no—
347C2×C4—21no—
348C2×C4—21no—
349C2×C4—41no—
350C2×C4—41no—
351C2×C4—41no—
352C2×C4—42no—
353C2×C4—82no—
354C2×C4—122no—
355C2×C4—242no—
356C2×C4C2×#21311yescharacteristic
357C2×C4C2×#21311yescharacteristic
358C2×C4C2×#21311yescharacteristic
359C2×C4C22×#6711yescharacteristic
360C8—21no—
361C8—21no—
362C8—21no—
363C8#100412yes—
364C2×C3—22no—
365C2×C3—22no—
366C2×C3—22no—
367C2×C3—162no—
368C2×C3#318611yescharacteristic
369C2×C3C22×D1611yescharacteristic
370C2×C3#341411yescharacteristic
371D3—162no—
372C22—21no—
373C22—21no—
374C22—21no—
375C22—21no—
376C22—21no—
377C22—21no—
378C22—44no—
379C22—82no—
380C22—82no—
381C22—82no—
382C22—164no—
383C22—242no—
384C22—242no—
385C22—242no—
386C22—484no—
387C22C2×#82612yes—
388C22C2×#82612yes—
389C22C2×#82612yes—
390C22C22×#21311yescharacteristic
391C4—21no—
392C4—21no—
393C4—21no—
394C4—82no—
395C4—242no—
396C4#580411yescharacteristic
397C4#582211yescharacteristic
398C3#3375211yescharacteristic, complemented, Sylow
399C2—22no—
400C2—22no—
401C2—22no—
402C2—162no—
403C2—482no—
404C2#8262511yescharacteristic
405C2C22×#82611yescharacteristic
406C2#8389411yescharacteristic
subgroupG/H
centerC22C22×#213
commutator, cosocleC2×C3×C8C24
FrattiniC2×C8C23×D3
FittingC2×C3×#466C2
socleC22×C3C22×D8
hypercenterC2×#466D3
hypocenterC3#33752
LCSG ▷ C2×C3×C8 ▷ C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23×C4 ◁ C23×C8 ◁ C2×#466

Complements

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

NH ≅ G/Ntype
1. C2×C3×#466C2semidirect
2. C2×#5115C2semidirect
3. #72035C2semidirect
4. C2×#4549C2semidirect
5. #71075C2semidirect
6. #71071C2semidirect
7. #71068C2semidirect
8. #70599C2semidirect
10. C22×C3×D8C22semidirect
14. C22×D24C22semidirect
18. C3×#466C22semidirect
19. C2×C3×#107C22semidirect
21. C2×#826C22semidirect
25. #5115C22semidirect
27. C2×#818C22semidirect
28. C22×C3⋊C16C22semidirect
29. #4562C22semidirect
30. #4549C22semidirect
31. #4549C22semidirect
32. #4491C22semidirect
33. #4489C22semidirect
35. C2×#466D3semidirect
84. C22×D8C2×D3semidirect
97. #466C2×D3semidirect
99. C2×#107C2×D3semidirect
204. C23×C3D16semidirect
329. C23#826semidirect
398. C3#33752semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#4662
C2×#51152
#720352
C2×#45492
#710752
#710712
#710682
#705992
#337523