Finite Group Databasebeta

← #10691437

Subgroups — #10691437

Order 768.

686 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
1#79298C211yesmaximal, characteristic
2C42×D12C211yesmaximal, characteristic
3#70792C211yesmaximal, characteristic, complemented
4#70762C211yesmaximal, characteristic
5#69396C211yesmaximal, characteristic, complemented
6C3×#1276C211yesmaximal, characteristic, complemented
7#67746C211yesmaximal, characteristic, complemented
8#9231—31nomaximal, Sylow
9C2×C4×D12—21no—
10C2×C4×D12C2211yescharacteristic
11C2×C42×D3—21no—
12#5470C2211yescharacteristic
13C3×#431C2211yescharacteristic
14C3×#431C2211yescharacteristic
15C3×C43C2211yescharacteristic
16C4×(C2×D3)⋊C4—21no—
17C4×(C2×D3)⋊C4—22no—
18C4×C4⋊Dic3—21no—
19C4×C4⋊Dic3C2211yescharacteristic
20#4924C2211yescharacteristic
21#4842C412yescomplemented
22#4841C412yescomplemented
23#4731—21no—
24#4721—21no—
25C4×#204—22no—
26#4523—21no—
27#4523—21no—
28#4467—22no—
29#4459—22no—
30#3101—31no—
31C42×D4—31no—
32#1994—31no—
33#1503—31no—
34#1486—31no—
35#1303—31no—
36#1276D311yescharacteristic, complemented
37C22×D12D411yescharacteristic
38C22×C4×D3—22no—
39C22×C4×D3—21no—
40C2×C3×C42—21no—
41C2×C3×C42—21no—
42C2×C3×C42D411yescharacteristic
43C2×C3×C42D411yescharacteristic
44C2×C3×C42C2311yescharacteristic
45C2×(C2×D3)⋊C4—22no—
46C2×(C2×D3)⋊C4—21no—
47#926—21no—
48C2×C4⋊Dic3—21no—
49C2×C4⋊Dic3D411yescharacteristic
50#924—21no—
51#924—21no—
52C2×C4×Dic3—22no—
53C2×C4×Dic3—21no—
54C2×#208—21no—
55C2×#204—22no—
56C4×D12—28no—
57C4×D12—21no—
58C4×D12—21no—
59C4×D12C2×C412yes—
60C4×D12D412yes—
61C42×D3—42no—
62C42×D3—42no—
63#870—21no—
64#869C2×C411yescharacteristic, complemented
65#869C2×C411yescharacteristic, complemented
66C4×Dic6—21no—
67C4×Dic6—21no—
68C3×C4⋊C8—21no—
69C3×C4⋊C8C2×C412yescomplemented
70C3×C22⋊C8—21no—
71C3×C22⋊C8—21no—
72C3×C4×C8—21no—
73C3×C4×C8—21no—
74C3⋊#95—22no—
75C3⋊#95D412yes—
76#821—22no—
77#817—22no—
78#816—22no—
79C4⋊(C3⋊C8)—21no—
80C4⋊(C3⋊C8)—21no—
81#533—31no—
82C2×C4×D4—31no—
83C2×C4×D4—61no—
84C22×C42—61no—
85#483—61no—
86#476—32no—
87#461—32no—
88#450—61no—
89C4×SD16—62no—
90#431C2×D311yescharacteristic
91#431C2×D311yescharacteristic
92#423—31no—
93C4×C4⋊C4—31no—
94C4×C4⋊C4—61no—
95C4×C22⋊C4—62no—
96C4×C22⋊C4—61no—
97C43C2×D311yescharacteristic
98#339—61no—
99#339—61no—
100#328—62no—
101D4⋊C8—62no—
102C23×D3—21no—
103C22×C3×C4—21no—
104C22×C3×C4—21no—
105C22×C3×C4C2×D411yescharacteristic
106C22×C3×C4C2×D411yescharacteristic
107C22×C3×C4C2×D411yescharacteristic
108C22×Dic3—21no—
109C22×Dic3—22no—
110C2×D12—21no—
111C2×D12—24no—
112C2×D12C22⋊C412yes—
113C2×C4×D3—28no—
114C2×C4×D3—22no—
115C2×C4×D3—22no—
116C2×C4×D3—44no—
117C2×C4×D3—44no—
118C2×C4×D3—41no—
119C2×C4×D3—41no—
120C2×C4×D3—41no—
121C2×C4×D3—41no—
122C2×Dic6—21no—
123C2×Dic6—21no—
124C2×C3×C8—22no—
125C2×C3×C8—41no—
126C3×C42—24no—
127C3×C42—21no—
128C3×C42—21no—
129C3×C42—21no—
130C3×C42—42no—
131C3×C42SD1612yescomplemented
132C3×C42C22×C411yescharacteristic
133C3×C42C2×D411yescharacteristic
134C3×C42#4111yescharacteristic
135C3×C42#4111yescharacteristic
136#217—41no—
137(C2×D3)⋊C4—28no—
138(C2×D3)⋊C4—22no—
139(C2×D3)⋊C4—22no—
140C4⋊Dic3—24no—
141C4⋊Dic3—21no—
142C4⋊Dic3—21no—
143C4⋊Dic3—21no—
144C4⋊Dic3—21no—
145C4⋊Dic3—21no—
146C4⋊Dic3C22⋊C412yes—
147Dic3⋊C4—41no—
148Dic3⋊C4—41no—
149C4×Dic3—44no—
150C4×Dic3—41no—
151C4×Dic3—41no—
152C4×Dic3—41no—
153#208—41no—
154#208—41no—
155C2×C3⋊C8—41no—
156#204—44no—
157C22×D4—31no—
158C23×C4—62no—
159C23×C4—61no—
160C2×SD16—62no—
161C2×M16—61no—
162C4⋊Q8—31no—
163C4⋊Q8—31no—
164#125—61no—
165C22⋊Q8—61no—
166C22⋊Q8—61no—
167C4×Q8—61no—
168C4×Q8—61no—
169C4×D4—32no—
170C4×D4—32no—
171C4×D4—68no—
172C4×D4—61no—
173C4×D4—61no—
174#117—61no—
175C2×C4⋊C4—31no—
176C2×C4⋊C4—61no—
177C2×C22⋊C4—62no—
178C2×C22⋊C4—61no—
179C2×C42—21no—
180C2×C42—21no—
181C2×C42—62no—
182C2×C42—61no—
183C2×C42—122no—
184C2×C42—122no—
185C2×C42D1211yescharacteristic
186C2×C42D1211yescharacteristic
187C2×C42C22×D311yescharacteristic
188#106—62no—
189C4⋊C8—21no—
190C4⋊C8—61no—
191C4⋊C8—61no—
192C4⋊C8C4×D312yescomplemented
193#103—62no—
194#102—62no—
195C22⋊C8—21no—
196C22⋊C8—21no—
197C4×C8—21no—
198C4×C8—21no—
199#95—32no—
200#95—62no—
201C23×C3#12011yescharacteristic
202C22×D3—24no—
203C22×D3—22no—
204C22×D3—42no—
205C22×D3—41no—
206C22×D3—41no—
207C2×C3×C4—22no—
208C2×C3×C4—22no—
209C2×C3×C4—22no—
210C2×C3×C4—22no—
211C2×C3×C4—22no—
212C2×C3×C4—41no—
213C2×C3×C4—41no—
214C2×C3×C4—41no—
215C2×C3×C4C2×C22⋊C411yescharacteristic
216C2×C3×C4C4×D411yescharacteristic
217C2×C3×C4C4×D411yescharacteristic
218C2×C3×C4#12111yescharacteristic
219C2×C3×C4#12111yescharacteristic
220C2×C3×C4#12311yescharacteristic
221C2×C3×C4C2×SD1611yescharacteristic
222C2×C3×C4#13511yescharacteristic
223C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
224C2×C3×C4#13711yescharacteristic
225C2×Dic3—24no—
226C2×Dic3—22no—
227C2×Dic3—42no—
228C2×Dic3—42no—
229C2×Dic3—41no—
230C2×Dic3—41no—
231C2×Dic3—41no—
232C2×Dic3—41no—
233D12—24no—
234D12—22no—
235D12#10414yes—
236C4×D3—416no—
237C4×D3—42no—
238C4×D3—42no—
239C4×D3—42no—
240C4×D3—42no—
241Dic6—42no—
242Dic6—81no—
243C3×C8—42no—
244C3×C8—81no—
245C3⋊C8—81no—
246C24—61no—
247C2×Q8—61no—
248C2×Q8—61no—
249C2×D4—32no—
250C2×D4—64no—
251C2×D4—61no—
252C22×C4—21no—
253C22×C4—21no—
254C22×C4—68no—
255C22×C4—62no—
256C22×C4—61no—
257C22×C4—62no—
258C22×C4—62no—
259C22×C4—124no—
260C22×C4—124no—
261C22×C4—121no—
262C22×C4—121no—
263C22×C4—121no—
264C22×C4—121no—
265C22×C4C2×D1211yescharacteristic
266C22×C4D3×D411yescharacteristic
267C22×C4D3×D411yescharacteristic
268SD16—124no—
269M16—121no—
270M16—121no—
271C2×C8—22no—
272C2×C8—41no—
273C2×C8—121no—
274C4⋊C4—32no—
275C4⋊C4—64no—
276C4⋊C4—61no—
277C4⋊C4—61no—
278C4⋊C4—61no—
279C4⋊C4—61no—
280C4⋊C4—61no—
281C4⋊C4—121no—
282C4⋊C4—121no—
283C22⋊C4—68no—
284C22⋊C4—62no—
285C22⋊C4—62no—
286C22⋊C4—121no—
287C42—24no—
288C42—21no—
289C42—21no—
290C42—21no—
291C42—42no—
292C42—124no—
293C42—121no—
294C42—121no—
295C42—121no—
296C42#20412yescomplemented
297C42C2×C4×D311yescharacteristic
298C42#23511yescharacteristic
299C42D3×D411yescharacteristic
300C42#23911yescharacteristic
301C22×C3—21no—
302C22×C3—21no—
303C22×C3#38611yescharacteristic
304C22×C3#45011yescharacteristic
305C22×C3#45211yescharacteristic
306C2×D3—24no—
307C2×D3—44no—
308C2×D3—44no—
309C2×D3—42no—
310C2×D3—42no—
311C3×C4—24no—
312C3×C4—21no—
313C3×C4—21no—
314C3×C4—21no—
315C3×C4—42no—
316C3×C4C2×#10412yes—
317C3×C4C4×SD1611yescharacteristic
318C3×C4#44011yescharacteristic
319C3×C4#46111yescharacteristic
320C3×C4#46311yescharacteristic
321Dic3—44no—
322Dic3—42no—
323Dic3—81no—
324Dic3—81no—
325C23—64no—
326C23—62no—
327C23—122no—
328C23—121no—
329C23—121no—
330C23#88211yescharacteristic
331Q8—122no—
332Q8—241no—
333D4—34no—
334D4—64no—
335D4—62no—
336C2×C4—22no—
337C2×C4—22no—
338C2×C4—22no—
339C2×C4—22no—
340C2×C4—22no—
341C2×C4—41no—
342C2×C4—41no—
343C2×C4—41no—
344C2×C4—64no—
345C2×C4—62no—
346C2×C4—1216no—
347C2×C4—122no—
348C2×C4—122no—
349C2×C4—121no—
350C2×C4—121no—
351C2×C4—122no—
352C2×C4—122no—
353C2×C4—122no—
354C2×C4—122no—
355C2×C4—121no—
356C2×C4—121no—
357C2×C4C4×D1211yescharacteristic
358C2×C4D3×C22⋊C411yescharacteristic
359C2×C4#88411yescharacteristic
360C2×C4#89311yescharacteristic
361C2×C4#89411yescharacteristic
362C2×C4#89511yescharacteristic
363C2×C4C2×#20411yescharacteristic
364C2×C4#90411yescharacteristic
365C2×C4#90811yescharacteristic
366C2×C4#90911yescharacteristic
367C8—42no—
368C8—81no—
369C8—241no—
370C2×C3—21no—
371C2×C3—21no—
372C2×C3#187711yescharacteristic
373C2×C3#189611yescharacteristic
374C2×C3#190611yescharacteristic
375D3—44no—
376C22—21no—
377C22—21no—
378C22—64no—
379C22—124no—
380C22—124no—
381C22—122no—
382C22—122no—
383C22#466511yescharacteristic
384C22#473111yescharacteristic
385C22#473311yescharacteristic
386C4—24no—
387C4—21no—
388C4—21no—
389C4—21no—
390C4—42no—
391C4—124no—
392C4—122no—
393C4—241no—
394C4—241no—
395C4C4×#20411yescharacteristic
396C4#471011yescharacteristic
397C4D3×#10412yes—
398C4#484111yescharacteristic
399C4#484411yescharacteristic
400C3#923111yescharacteristic, complemented, Sylow
401C2—21no—
402C2—21no—
403C2—124no—
404C2#6867211yescharacteristic
405C2#6875611yescharacteristic
406C2#6877811yescharacteristic
subgroupG/H
centerC2×C4#895
commutatorC3×C42C22×C4
FrattiniC2×C42C22×D3
FittingC3×#1276C2
socleC22×C3#386
prime radicalC22×D12D4
cosocleC2×C3×C42C23
hypercenter#1276D3
hypocenterC3#9231
LCSG ▷ C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #1276

Complements

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

NH ≅ G/Ntype
3. #70792C2semidirect
5. #69396C2semidirect
6. C3×#1276C2semidirect
7. #67746C2semidirect
21. #4842C4semidirect
22. #4841C4semidirect
36. #1276D3semidirect
64. #869C2×C4semidirect
65. #869C2×C4semidirect
69. C3×C4⋊C8C2×C4semidirect
131. C3×C42SD16semidirect
192. C4⋊C8C4×D3semidirect
296. C42#204semidirect
400. C3#9231semidirect

Maximal subgroups

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

subgroupindex
#792982
C42×D122
#707922
#707622
#693962
C3×#12762
#677462
#92313