Finite Group Databasebeta

← #10691026

Subgroups — #10691026

Order 768.

486 conjugacy classes of subgroups.

The subgroup lattice has 422 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#3110C211yesmaximal, characteristic
2#82303C211yesmaximal, characteristic
3#71835C211yesmaximal, characteristic, complemented
4#71822C211yesmaximal, characteristic, complemented
5#71609C211yesmaximal, characteristic
6#67751C211yesmaximal, characteristic, complemented
7#67750C211yesmaximal, characteristic, complemented
8#9165—31nomaximal, Sylow
9C3×#533C2211yescharacteristic
10C2×C3×C4×D4C2211yescharacteristic
11C3×#484—21no—
12C3×#474C411yescharacteristic, complemented
13C3×#474C411yescharacteristic, complemented
14C3×#461C411yescharacteristic, complemented
15C3×#461C411yescharacteristic, complemented
16C3×#450—21no—
17C3×#440—21no—
18C3×#440—21no—
19C3×#432C2211yescharacteristic
20C2×#832—21no—
21#5207—21no—
22C2×#830—21no—
23#5012—21no—
24#5011—22no—
25D4×C3⋊C8—21no—
26C4×C4⋊Dic3C2211yescharacteristic
27#4926C2211yescharacteristic
28#4925C2211yescharacteristic
29C2×#803C2211yescharacteristic
30#4543—22no—
31#4538—21no—
32#4538—21no—
33#4529—21no—
34#3110D311yescharacteristic
35#2974—31no—
36#1762—31no—
37#1531—31no—
38#1503—31no—
39#1302—31no—
40#1285—31no—
41C22×C3×D4D411yescharacteristic
42C23×C3×C4—21no—
43C2×C3×SD16—21no—
44C2×C3×SD16—21no—
45C3×C4⋊Q8C2×C411yescharacteristic
46C3×C4⋊Q8C2×C411yescharacteristic
47C3×#125—21no—
48C3×C22⋊Q8—21no—
49C3×C22⋊Q8—21no—
50C3×C4×Q8—21no—
51C3×C4×Q8—21no—
52C3×C4×D4—21no—
53C3×C4×D4—21no—
54C3×C4×D4C2×C411yescharacteristic
55C3×C4×D4C2×C411yescharacteristic
56C3×C4×D4D411yescharacteristic
57C3×C4×D4D411yescharacteristic
58C3×#117—21no—
59C2×C3×C4⋊C4D411yescharacteristic
60C2×C3×C22⋊C4—21no—
61C2×C3×C42C2311yescharacteristic
62C2×C4⋊Dic3—21no—
63C2×C4×Dic3—21no—
64C2×#208—22no—
65C22×C3⋊C8—21no—
66C22×C3⋊C8—22no—
67C3×#107—21no—
68C3×#107—21no—
69C3×C4⋊C8C2×C411yescharacteristic
70C3×C4⋊C8C2×C411yescharacteristic
71C3×#103—21no—
72C3×#103—21no—
73C3×#102—21no—
74C3×#102—21no—
75C3×C22⋊C8—21no—
76C3×#97—21no—
77#832—24no—
78C3⋊#95—21no—
79#830—22no—
80#830—22no—
81#830—21no—
82#830D411yescharacteristic
83#830D411yescharacteristic
84C4⋊(C3⋊C8)—21no—
85C4⋊(C3⋊C8)—21no—
86#803—21no—
87#803—21no—
88#803D412yes—
89C4×C3⋊C8—21no—
90#533C2×D311yescharacteristic
91C2×C4×D4C2×D311yescharacteristic
92#484—21no—
93#474Dic311yescharacteristic, complemented
94#474Dic311yescharacteristic, complemented
95#461Dic311yescharacteristic, complemented
96#461Dic311yescharacteristic, complemented
97#450—21no—
98#440—21no—
99#440—21no—
100#436—61no—
101#435—62no—
102C8×D4—61no—
103#432—31no—
104#432C2×D311yescharacteristic
105#431—31no—
106C2×#102—61no—
107#407—61no—
108C2×C22⋊C8—61no—
109C2×#97—31no—
110C4×C4⋊C4—31no—
111#340—61no—
112#330—62no—
113#328—61no—
114#328—61no—
115C24×C3—21no—
116C2×C3×Q8—21no—
117C2×C3×Q8—21no—
118C2×C3×D4—22no—
119C2×C3×D4—21no—
120C2×C3×D4—41no—
121C2×C3×D4C22⋊C411yescharacteristic
122C2×C3×D4C22⋊C411yescharacteristic
123C22×C3×C4—21no—
124C22×C3×C4—21no—
125C22×C3×C4—21no—
126C22×C3×C4—21no—
127C22×C3×C4—21no—
128C22×C3×C4—82no—
129C22×C3×C4C2×D411yescharacteristic
130C22×C3×C4C2×D411yescharacteristic
131C22×C3×C4#4111yescharacteristic
132C22×Dic3—41no—
133C3×SD16—42no—
134C3×SD16—42no—
135C2×C3×C8—21no—
136C2×C3×C8—21no—
137C3×C4⋊C4—21no—
138C3×C4⋊C4—21no—
139C3×C4⋊C4—21no—
140C3×C4⋊C4—21no—
141C3×C4⋊C4—21no—
142C3×C4⋊C4—41no—
143C3×C4⋊C4—41no—
144C3×C4⋊C4C22⋊C411yescharacteristic
145C3×C4⋊C4C22⋊C411yescharacteristic
146C3×C22⋊C4—21no—
147C3×C22⋊C4—21no—
148C3×C22⋊C4—21no—
149C3×C22⋊C4—21no—
150C3×C22⋊C4—41no—
151C3×C42—41no—
152C3×C42C22×C411yescharacteristic
153C3×C42C2×D411yescharacteristic
154C3×C42C2×D411yescharacteristic
155C3×C42#4111yescharacteristic
156C4⋊Dic3—22no—
157C4⋊Dic3—41no—
158C4×Dic3—81no—
159#208—82no—
160C2×C3⋊C8—22no—
161C2×C3⋊C8—21no—
162C2×C3⋊C8—21no—
163C2×C3⋊C8—22no—
164C2×C3⋊C8—41no—
165C2×C3⋊C8—41no—
166C2×C3⋊C8—82no—
167C22×D4#6711yescharacteristic
168C23×C4—21no—
169C2×SD16—21no—
170C2×SD16—21no—
171C2×M16—62no—
172C22×C8—61no—
173C22×C8—62no—
174C4⋊Q8C2×Dic311yescharacteristic
175C4⋊Q8C2×Dic311yescharacteristic
176#125—21no—
177C22⋊Q8—21no—
178C22⋊Q8—21no—
179C4×Q8—21no—
180C4×Q8—21no—
181C4×D4—21no—
182C4×D4—21no—
183C4×D4C2×Dic311yescharacteristic
184C4×D4C2×Dic311yescharacteristic
185C4×D4#6711yescharacteristic
186C4×D4#6711yescharacteristic
187#117—21no—
188C2×C4⋊C4—61no—
189C2×C4⋊C4#6711yescharacteristic
190C2×C22⋊C4—21no—
191C2×C42—61no—
192C2×C42C22×D311yescharacteristic
193#107—21no—
194#107—21no—
195C4⋊C8—61no—
196C4⋊C8—61no—
197C4⋊C8C2×Dic311yescharacteristic
198C4⋊C8C2×Dic311yescharacteristic
199#103—21no—
200#103—21no—
201#102—21no—
202#102—21no—
203#102—64no—
204C22⋊C8—21no—
205C22⋊C8—31no—
206C22⋊C8—31no—
207C22⋊C8—62no—
208C22⋊C8—62no—
209C22⋊C8—61no—
210#97—21no—
211#97—32no—
212#97—61no—
213#97—61no—
214C4×C8—61no—
215#95—61no—
216C23×C3—21no—
217C23×C3—21no—
218C23×C3—42no—
219C23×C3—41no—
220C23×C3—41no—
221C23×C3—81no—
222C23×C3#12111yescharacteristic
223C3×Q8—41no—
224C3×Q8—41no—
225C3×Q8—81no—
226C3×D4—21no—
227C3×D4—21no—
228C3×D4—21no—
229C3×D4—42no—
230C3×D4#10412yes—
231C2×C3×C4—21no—
232C2×C3×C4—21no—
233C2×C3×C4—21no—
234C2×C3×C4—21no—
235C2×C3×C4—21no—
236C2×C3×C4—21no—
237C2×C3×C4—41no—
238C2×C3×C4—42no—
239C2×C3×C4—42no—
240C2×C3×C4—41no—
241C2×C3×C4—41no—
242C2×C3×C4—81no—
243C2×C3×C4—81no—
244C2×C3×C4—81no—
245C2×C3×C4C2×C22⋊C411yescharacteristic
246C2×C3×C4C4×D411yescharacteristic
247C2×C3×C4C4×D411yescharacteristic
248C2×C3×C4#12011yescharacteristic
249C2×C3×C4#12111yescharacteristic
250C2×C3×C4#12311yescharacteristic
251C2×Dic3—42no—
252C2×Dic3—81no—
253C2×Dic3—81no—
254C3×C8—41no—
255C3×C8—41no—
256C3⋊C8—42no—
257C3⋊C8—41no—
258C3⋊C8—42no—
259C3⋊C8—41no—
260C24—21no—
261C2×Q8—21no—
262C2×Q8—21no—
263C2×D4—22no—
264C2×D4—21no—
265C2×D4—41no—
266C2×D4#21711yescharacteristic
267C2×D4#21711yescharacteristic
268C22×C4—21no—
269C22×C4—21no—
270C22×C4—21no—
271C22×C4—21no—
272C22×C4—21no—
273C22×C4—82no—
274C22×C4—121no—
275C22×C4D3×D411yescharacteristic
276C22×C4#23711yescharacteristic
277C22×C4C2×#6711yescharacteristic
278SD16—42no—
279SD16—42no—
280M16—242no—
281C2×C8—21no—
282C2×C8—21no—
283C2×C8—62no—
284C2×C8—61no—
285C2×C8—61no—
286C2×C8—62no—
287C2×C8—121no—
288C2×C8—121no—
289C2×C8—242no—
290C4⋊C4—21no—
291C4⋊C4—21no—
292C4⋊C4—21no—
293C4⋊C4—21no—
294C4⋊C4—21no—
295C4⋊C4—41no—
296C4⋊C4—41no—
297C4⋊C4—62no—
298C4⋊C4—121no—
299C4⋊C4#21711yescharacteristic
300C4⋊C4#21711yescharacteristic
301C22⋊C4—21no—
302C22⋊C4—21no—
303C22⋊C4—21no—
304C22⋊C4—21no—
305C22⋊C4—41no—
306C42—41no—
307C42—241no—
308C42D3×D411yescharacteristic
309C42#23711yescharacteristic
310C42C22×Dic311yescharacteristic
311C42C2×#6711yescharacteristic
312C22×C3—21no—
313C22×C3—21no—
314C22×C3—22no—
315C22×C3—41no—
316C22×C3—41no—
317C22×C3—41no—
318C22×C3—41no—
319C22×C3—41no—
320C22×C3—82no—
321C22×C3—82no—
322C22×C3#38611yescharacteristic
323C22×C3#47111yescharacteristic
324C22×C3#47311yescharacteristic
325C3×C4—21no—
326C3×C4—21no—
327C3×C4—21no—
328C3×C4—21no—
329C3×C4—21no—
330C3×C4—41no—
331C3×C4—41no—
332C3×C4—81no—
333C3×C4—81no—
334C3×C4C2×#10411yescharacteristic
335C3×C4#42111yescharacteristic
336Dic3—82no—
337C23—21no—
338C23—21no—
339C23—42no—
340C23—41no—
341C23—41no—
342C23—81no—
343C23#93911yescharacteristic
344Q8—41no—
345Q8—41no—
346Q8—81no—
347D4—21no—
348D4—21no—
349D4—21no—
350D4—42no—
351D4#83712yes—
352C2×C4—21no—
353C2×C4—21no—
354C2×C4—21no—
355C2×C4—21no—
356C2×C4—21no—
357C2×C4—21no—
358C2×C4—41no—
359C2×C4—42no—
360C2×C4—42no—
361C2×C4—41no—
362C2×C4—41no—
363C2×C4—81no—
364C2×C4—81no—
365C2×C4—81no—
366C2×C4—122no—
367C2×C4—241no—
368C2×C4—241no—
369C2×C4D4×Dic311yescharacteristic
370C2×C4D4×Dic311yescharacteristic
371C2×C4#93511yescharacteristic
372C2×C4#93711yescharacteristic
373C2×C4#93911yescharacteristic
374C2×C4C2×#21711yescharacteristic
375C8—41no—
376C8—41no—
377C8—122no—
378C8—121no—
379C8—122no—
380C8—121no—
381C2×C3—21no—
382C2×C3—21no—
383C2×C3—42no—
384C2×C3—81no—
385C2×C3#187111yescharacteristic
386C2×C3#190411yescharacteristic
387C2×C3#190811yescharacteristic
388C22—21no—
389C22—21no—
390C22—22no—
391C22—41no—
392C22—41no—
393C22—41no—
394C22—41no—
395C22—41no—
396C22—82no—
397C22—82no—
398C22#520311yescharacteristic
399C22#520411yescharacteristic
400C22#522011yescharacteristic
401C4—21no—
402C4—21no—
403C4—21no—
404C4—21no—
405C4—21no—
406C4—41no—
407C4—41no—
408C4—81no—
409C4—81no—
410C4—242no—
411C4C2×#83711yescharacteristic
412C4#523611yescharacteristic
413C3#916511yescharacteristic, complemented, Sylow
414C2—21no—
415C2—21no—
416C2—42no—
417C2—81no—
418C2#7171311yescharacteristic
419C2#7208911yescharacteristic
420C2#7210311yescharacteristic
subgroupG/H
centerC22#5220
commutatorC3×C42C22×C4
FrattiniC2×C42C22×D3
FittingC3×#3110C2
socleC22×C3#386
prime radicalC22×C3×D4D4
cosocleC2×C3×C42C23
hypercenter#3110D3
hypocenterC3#9165
LCSG ▷ C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3110

Complements

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

NH ≅ G/Ntype
3. #71835C2semidirect
4. #71822C2semidirect
6. #67751C2semidirect
7. #67750C2semidirect
12. C3×#474C4semidirect
13. C3×#474C4semidirect
14. C3×#461C4semidirect
15. C3×#461C4semidirect
93. #474Dic3semidirect
94. #474Dic3semidirect
95. #461Dic3semidirect
96. #461Dic3semidirect
413. C3#9165semidirect

Maximal subgroups

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

subgroupindex
C3×#31102
#823032
#718352
#718222
#716092
#677512
#677502
#91653