Finite Group Databasebeta

← #10691435

Subgroups — #10691435

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#83351C211yesmaximal, characteristic
2C4×#893C211yesmaximal, characteristic
3C3×#1503C211yesmaximal, characteristic, complemented
4#71839C211yesmaximal, characteristic, complemented
5#70762C211yesmaximal, characteristic
6#67736C211yesmaximal, characteristic, complemented
7#67620C211yesmaximal, characteristic, complemented
8#9231—31nomaximal, Sylow
9C3×#533C2211yescharacteristic
10C2×#893—21no—
11C2×C4×D12C2211yescharacteristic
12C2×C42×D3—21no—
13C3×#431C2211yescharacteristic
14C3×C4×C4⋊C4C2211yescharacteristic
15#5084C412yescomplemented
16#5083C412yescomplemented
17#5046—21no—
18#5043—21no—
19C4×#215—22no—
20Dic3×C4⋊C4—21no—
21C4×(C2×D3)⋊C4—21no—
22C42×Dic3C2211yescharacteristic
23#4925C2211yescharacteristic
24#4924C2211yescharacteristic
25#4666—22no—
26C3×#328—22no—
27#4532—21no—
28#4532—21no—
29#4459—22no—
30#3101—31no—
31C42×D4—31no—
32#1994—31no—
33#1503D311yescharacteristic, complemented
34#1486—31no—
35#1303—31no—
36#1276—31no—
37C22×D12D411yescharacteristic
38C22×C4×D3—22no—
39C22×C4×D3—21no—
40C3×C4⋊Q8C2×C411yescharacteristic, complemented
41C3×C4⋊Q8C2×C411yescharacteristic, complemented
42C3×#125—21no—
43C3×C22⋊Q8—21no—
44C3×C22⋊Q8—21no—
45C3×C4×Q8—21no—
46C3×C4×Q8—21no—
47C3×#117—21no—
48C2×C3×C4⋊C4—21no—
49C2×C3×C42—21no—
50C2×C3×C42C2311yescharacteristic
51C2×#215—22no—
52C2×(C2×D3)⋊C4—22no—
53C2×(C2×D3)⋊C4—21no—
54C2×C4⋊Dic3D411yescharacteristic
55C2×C4×Dic3—21no—
56C2×C4×Dic3—22no—
57C2×C4×Dic3—21no—
58C2×C4×Dic3D411yescharacteristic
59C2×C4×Dic3D411yescharacteristic
60C2×#208—21no—
61#893—28no—
62C4×D12—21no—
63C4×D12—21no—
64C4×D12C2×C412yes—
65C4×D12D412yes—
66C42×D3—42no—
67C42×D3—42no—
68C3×C4⋊C8—21no—
69C3×C22⋊C8—21no—
70C3×C4×C8—21no—
71C3×#95D412yes—
72#835—22no—
73C3⋊#95—22no—
74#830—21no—
75#809—22no—
76#808—22no—
77C4⋊(C3⋊C8)—21no—
78C4⋊(C3⋊C8)—21no—
79C4⋊(C3⋊C8)C2×C412yescomplemented
80C4×C3⋊C8—21no—
81#533C2×D311yescharacteristic
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#431—31no—
91#431C2×D311yescharacteristic
92#423—31no—
93C4×C4⋊C4—61no—
94C4×C4⋊C4C2×D311yescharacteristic
95C4×C22⋊C4—61no—
96C4×C22⋊C4—62no—
97C43—31no—
98#339—61no—
99#339—61no—
100#328—22no—
101D4⋊C8—62no—
102C23×D3—21no—
103C2×C3×Q8—21no—
104C2×C3×Q8—21no—
105C22×C3×C4—22no—
106C22×C3×C4C2×D411yescharacteristic
107C22×C3×C4C2×D411yescharacteristic
108C22×C3×C4C2×D411yescharacteristic
109C22×Dic3—21no—
110C22×Dic3—21no—
111C22×Dic3—21no—
112C2×D12—21no—
113C2×D12—24no—
114C2×D12C22⋊C412yes—
115C2×C4×D3—28no—
116C2×C4×D3—22no—
117C2×C4×D3—22no—
118C2×C4×D3—44no—
119C2×C4×D3—44no—
120C2×C4×D3—41no—
121C2×C4×D3—41no—
122C2×C4×D3—41no—
123C2×C4×D3—41no—
124C2×C3×C8—41no—
125C3×C4⋊C4—24no—
126C3×C4⋊C4—21no—
127C3×C4⋊C4—21no—
128C3×C4⋊C4—21no—
129C3×C4⋊C4—21no—
130C3×C4⋊C4—41no—
131C3×C4⋊C4—41no—
132C3×C22⋊C4—41no—
133C3×C42—41no—
134C3×C42—41no—
135C3×C42—41no—
136C3×C42C22×C411yescharacteristic
137C3×C42C2×D411yescharacteristic
138C3×C42#4111yescharacteristic
139C3×C42#4111yescharacteristic
140#215—44no—
141(C2×D3)⋊C4—28no—
142(C2×D3)⋊C4—22no—
143(C2×D3)⋊C4—22no—
144C4⋊Dic3—21no—
145C4⋊Dic3C22⋊C412yes—
146C4×Dic3—24no—
147C4×Dic3—21no—
148C4×Dic3—21no—
149C4×Dic3—21no—
150C4×Dic3—44no—
151C4×Dic3—42no—
152C4×Dic3SD1612yescomplemented
153#208—41no—
154#208—41no—
155C2×C3⋊C8—22no—
156C2×C3⋊C8—41no—
157C22×D4—31no—
158C23×C4—62no—
159C23×C4—61no—
160C2×SD16—62no—
161C2×M16—61no—
162C4⋊Q8C4×D311yescharacteristic, complemented
163C4⋊Q8C4×D311yescharacteristic, complemented
164#125—21no—
165C22⋊Q8—21no—
166C22⋊Q8—21no—
167C4×Q8—21no—
168C4×Q8—21no—
169C4×D4—32no—
170C4×D4—32no—
171C4×D4—68no—
172C4×D4—61no—
173C4×D4—61no—
174#117—21no—
175C2×C4⋊C4—21no—
176C2×C4⋊C4—31no—
177C2×C22⋊C4—62no—
178C2×C22⋊C4—61no—
179C2×C42—21no—
180C2×C42—31no—
181C2×C42—31no—
182C2×C42—61no—
183C2×C42—62no—
184C2×C42—61no—
185C2×C42—122no—
186C2×C42—122no—
187C2×C42C22×D311yescharacteristic
188#106—62no—
189C4⋊C8—21no—
190C4⋊C8—32no—
191C4⋊C8—61no—
192C4⋊C8—61no—
193#103—62no—
194#102—62no—
195C22⋊C8—21no—
196C22⋊C8—61no—
197C4×C8—21no—
198C4×C8—61no—
199#95—62no—
200#95#6712yes—
201C23×C3#12011yescharacteristic
202C22×D3—24no—
203C22×D3—22no—
204C22×D3—42no—
205C22×D3—41no—
206C22×D3—41no—
207C3×Q8—42no—
208C3×Q8—81no—
209C2×C3×C4—22no—
210C2×C3×C4—24no—
211C2×C3×C4—41no—
212C2×C3×C4—42no—
213C2×C3×C4—42no—
214C2×C3×C4—41no—
215C2×C3×C4—41no—
216C2×C3×C4C2×C22⋊C411yescharacteristic
217C2×C3×C4C4×D411yescharacteristic
218C2×C3×C4C4×D411yescharacteristic
219C2×C3×C4#12111yescharacteristic
220C2×C3×C4#12111yescharacteristic
221C2×C3×C4#12311yescharacteristic
222C2×C3×C4C2×SD1611yescharacteristic
223C2×C3×C4#13511yescharacteristic
224C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
225C2×C3×C4#13711yescharacteristic
226C2×Dic3—22no—
227C2×Dic3—22no—
228C2×Dic3—22no—
229C2×Dic3—22no—
230C2×Dic3—22no—
231C2×Dic3—41no—
232C2×Dic3—41no—
233C2×Dic3—41no—
234C2×Dic3—41no—
235D12—24no—
236D12—22no—
237D12#10414yes—
238C4×D3—416no—
239C4×D3—42no—
240C4×D3—42no—
241C4×D3—42no—
242C4×D3—42no—
243C3×C8—81no—
244C3⋊C8—42no—
245C3⋊C8—81no—
246C24—61no—
247C2×Q8—21no—
248C2×Q8—21no—
249C2×D4—32no—
250C2×D4—64no—
251C2×D4—61no—
252C22×C4—22no—
253C22×C4—68no—
254C22×C4—61no—
255C22×C4—61no—
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×C4D3×D411yescharacteristic
266C22×C4D3×D411yescharacteristic
267C22×C4C2×#6711yescharacteristic
268SD16—124no—
269M16—121no—
270M16—121no—
271C2×C8—41no—
272C2×C8—62no—
273C2×C8—121no—
274C4⋊C4—24no—
275C4⋊C4—21no—
276C4⋊C4—21no—
277C4⋊C4—21no—
278C4⋊C4—21no—
279C4⋊C4—32no—
280C4⋊C4—41no—
281C4⋊C4—41no—
282C4⋊C4—61no—
283C22⋊C4—41no—
284C22⋊C4—68no—
285C22⋊C4—62no—
286C22⋊C4—62no—
287C42—32no—
288C42—41no—
289C42—41no—
290C42—41no—
291C42—64no—
292C42—61no—
293C42—61no—
294C42—61no—
295C42—124no—
296C42—122no—
297C42C2×C4×D311yescharacteristic
298C42#23511yescharacteristic
299C42D3×D411yescharacteristic
300C42#23711yescharacteristic
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—21no—
312C3×C4—21no—
313C3×C4—21no—
314C3×C4—44no—
315C3×C4—81no—
316C3×C4—81no—
317C3×C4C2×#10412yes—
318C3×C4C4×SD1611yescharacteristic
319C3×C4#44011yescharacteristic
320C3×C4#46111yescharacteristic
321C3×C4#46311yescharacteristic
322Dic3—24no—
323Dic3—42no—
324Dic3—42no—
325C23—64no—
326C23—62no—
327C23—122no—
328C23—121no—
329C23—121no—
330C23#93711yescharacteristic
331Q8—42no—
332Q8—81no—
333D4—34no—
334D4—64no—
335D4—62no—
336C2×C4—22no—
337C2×C4—24no—
338C2×C4—41no—
339C2×C4—42no—
340C2×C4—42no—
341C2×C4—41no—
342C2×C4—41no—
343C2×C4—62no—
344C2×C4—62no—
345C2×C4—62no—
346C2×C4—62no—
347C2×C4—62no—
348C2×C4—1216no—
349C2×C4—121no—
350C2×C4—121no—
351C2×C4—121no—
352C2×C4—121no—
353C2×C4—122no—
354C2×C4—122no—
355C2×C4—122no—
356C2×C4—122no—
357C2×C4D3×C22⋊C411yescharacteristic
358C2×C4#67⋊C411yescharacteristic
359C2×C4#88311yescharacteristic
360C2×C4#88411yescharacteristic
361C2×C4D3×SD1611yescharacteristic
362C2×C4#91411yescharacteristic
363C2×C4#91811yescharacteristic
364C2×C4#91911yescharacteristic
365C2×C4C4×#6711yescharacteristic
366C2×C4#93811yescharacteristic
367C8—81no—
368C8—122no—
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#495611yescharacteristic
384C22#516911yescharacteristic
385C22#518711yescharacteristic
386C4—21no—
387C4—21no—
388C4—21no—
389C4—44no—
390C4—64no—
391C4—81no—
392C4—81no—
393C4—122no—
394C4—122no—
395C4#478811yescharacteristic
396C4#481711yescharacteristic
397C4#481811yescharacteristic
398C4#481911yescharacteristic
399C4D3×#10412yes—
400C3#923111yescharacteristic, complemented, Sylow
401C2—21no—
402C2—21no—
403C2—124no—
404C2#7005511yescharacteristic
405C2#7027611yescharacteristic
406C2#7038211yescharacteristic
subgroupG/H
centerC2×C4#884
commutatorC3×C42C22×C4
FrattiniC2×C42C22×D3
FittingC3×#1503C2
socleC22×C3#386
prime radicalC22×D12D4
cosocleC2×C3×C42C23
hypercenter#1503D3
hypocenterC3#9231
LCSG ▷ C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #1503

Complements

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

NH ≅ G/Ntype
3. C3×#1503C2semidirect
4. #71839C2semidirect
6. #67736C2semidirect
7. #67620C2semidirect
15. #5084C4semidirect
16. #5083C4semidirect
33. #1503D3semidirect
40. C3×C4⋊Q8C2×C4semidirect
41. C3×C4⋊Q8C2×C4semidirect
79. C4⋊(C3⋊C8)C2×C4semidirect
152. C4×Dic3SD16semidirect
162. C4⋊Q8C4×D3semidirect
163. C4⋊Q8C4×D3semidirect
400. C3#9231semidirect

Maximal subgroups

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

subgroupindex
#833512
C4×#8932
C3×#15032
#718392
#707622
#677362
#676202
#92313