Finite Group Databasebeta

← #10691678

Subgroups — #10691678

Order 768.

438 conjugacy classes of subgroups.

The subgroup lattice has 426 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#3211C211yesmaximal, characteristic
2#83128C211yesmaximal, characteristic, complemented
3#71748C211yesmaximal, characteristic, complemented
4#71720C211yesmaximal, characteristic, complemented
5C2×#4540C211yesmaximal, characteristic, complemented
6#67742C211yesmaximal, characteristic, complemented
7#67741C211yesmaximal, characteristic, complemented
8#9265—31nomaximal, Sylow
9C3×#529C2211yescharacteristic
10C2×C3×#124C2211yescharacteristic
11C3×#495—21no—
12C3×#488—21no—
13C3×#464—21no—
14C3×#463—21no—
15C3×#451C411yescharacteristic
16C3×#450C411yescharacteristic
17C3×#448C411yescharacteristic
18C3×#447C411yescharacteristic
19C3×#431C2211yescharacteristic
20#5024—21no—
21#5011—21no—
22#5011—21no—
23D4×C3⋊C8—21no—
24#5000—21no—
25#4997—21no—
26#4926C2211yescharacteristic
27#4925C2211yescharacteristic
28C2×#803C2211yescharacteristic
29C2×#803C2211yescharacteristic
30#4543—21no—
31#4540—21no—
32#4540—21no—
33#4540—21no—
34#4525—21no—
35#4525—21no—
36#3211D311yescharacteristic
37#2968—31no—
38#1532—31no—
39C2×#329—31no—
40#1469—31no—
41#1302—31no—
42#1281—31no—
43C22×C3×Q8C2×C411yescharacteristic
44C22×C3×D4C2×C411yescharacteristic
45C2×C3×Q16—21no—
46C2×C3×SD16—21no—
47C2×C3×SD16—21no—
48C2×C3×D8—21no—
49C3×#126—21no—
50C3×#125D411yescharacteristic
51C3×#124—21no—
52C3×#124—21no—
53C3×#124—21no—
54C3×#124—21no—
55C3×#124D411yescharacteristic
56C3×#123—21no—
57C3×C22⋊Q8C2×C411yescharacteristic
58C3×#121C2×C411yescharacteristic
59C3×C4×Q8D411yescharacteristic
60C3×C4×D4—21no—
61C3×C4×D4D411yescharacteristic
62C3×#117—21no—
63C2×C3×C22⋊C4—21no—
64C2×C3×C22⋊C4—21no—
65C2×C3×C42C2311yescharacteristic
66C2×#208—21no—
67C22×C3⋊C8—21no—
68C22×C3⋊C8—21no—
69C22×C3⋊C8—21no—
70C3×C4⋊C8—21no—
71C3×#103—21no—
72C3×#103—21no—
73C3×#102—21no—
74C3×#102—21no—
75C3×C22⋊C8C2×C411yescharacteristic
76C3×C22⋊C8C2×C411yescharacteristic
77C3×C4×C8—21no—
78#830—21no—
79#830—21no—
80#830—21no—
81#830D411yescharacteristic
82#830D411yescharacteristic
83C4⋊(C3⋊C8)—21no—
84C4⋊(C3⋊C8)—21no—
85C4⋊(C3⋊C8)—21no—
86C4⋊(C3⋊C8)—21no—
87#803—21no—
88#803—21no—
89#803—21no—
90#803—21no—
91#803—21no—
92#803D411yescharacteristic
93#803D411yescharacteristic
94C4×C3⋊C8—21no—
95C4×C3⋊C8—21no—
96#529C2×D311yescharacteristic
97C2×#124C2×D311yescharacteristic
98#495—21no—
99#488—21no—
100#464—21no—
101#463—21no—
102#451Dic311yescharacteristic
103#450Dic311yescharacteristic
104#448Dic311yescharacteristic
105#447Dic311yescharacteristic
106#446—61no—
107#435—61no—
108#435—61no—
109C8×D4—61no—
110#433—61no—
111#432—31no—
112#431—31no—
113#431C2×D311yescharacteristic
114#405—61no—
115C2×#97—31no—
116C2×#97—31no—
117#330—61no—
118#329—61no—
119#329—61no—
120#329—61no—
121#323—61no—
122#323—61no—
123C24×C3—21no—
124C2×C3×Q8—21no—
125C2×C3×Q8—21no—
126C2×C3×Q8—81no—
127C2×C3×Q8C22⋊C411yescharacteristic
128C2×C3×D4—21no—
129C2×C3×D4—21no—
130C2×C3×D4—21no—
131C2×C3×D4—81no—
132C2×C3×D4C22⋊C411yescharacteristic
133C22×C3×C4—21no—
134C22×C3×C4—21no—
135C22×C3×C4C22×C411yescharacteristic
136C22×C3×C4C2×D411yescharacteristic
137C22×C3×C4#4111yescharacteristic
138C3×Q16—81no—
139C3×SD16—81no—
140C3×SD16—81no—
141C3×D8—81no—
142C2×C3×C8—21no—
143C2×C3×C8—21no—
144C3×C4⋊C4—21no—
145C3×C4⋊C4—21no—
146C3×C4⋊C4—21no—
147C3×C4⋊C4—21no—
148C3×C4⋊C4C22⋊C411yescharacteristic
149C3×C4⋊C4C22⋊C411yescharacteristic
150C3×C22⋊C4—21no—
151C3×C22⋊C4—21no—
152C3×C22⋊C4—21no—
153C3×C22⋊C4—21no—
154C3×C22⋊C4—21no—
155C3×C22⋊C4—41no—
156C3×C22⋊C4—41no—
157C3×C22⋊C4—41no—
158C3×C22⋊C4—41no—
159C3×C42—21no—
160C3×C42C2×D411yescharacteristic
161C3×C42C2×D411yescharacteristic
162C3×C42C2×D411yescharacteristic
163C3×C42#4111yescharacteristic
164#208—42no—
165C2×C3⋊C8—21no—
166C2×C3⋊C8—22no—
167C2×C3⋊C8—21no—
168C2×C3⋊C8—21no—
169C2×C3⋊C8—21no—
170C2×C3⋊C8—42no—
171C2×C3⋊C8—42no—
172C2×C3⋊C8—41no—
173C2×C3⋊C8—41no—
174C2×C3⋊C8—41no—
175C22×Q8C2×Dic311yescharacteristic
176C22×D4C2×Dic311yescharacteristic
177C2×Q16—21no—
178C2×SD16—21no—
179C2×SD16—21no—
180C2×D8—21no—
181C2×M16—61no—
182C22×C8—61no—
183C22×C8—61no—
184C22×C8—61no—
185#126—21no—
186#125#6711yescharacteristic
187#124—21no—
188#124—21no—
189#124—21no—
190#124—21no—
191#124#6711yescharacteristic
192#123—21no—
193C22⋊Q8C2×Dic311yescharacteristic
194#121C2×Dic311yescharacteristic
195C4×Q8#6711yescharacteristic
196C4×D4—21no—
197C4×D4#6711yescharacteristic
198#117—21no—
199C2×C22⋊C4—21no—
200C2×C22⋊C4—21no—
201C2×C42C22×D311yescharacteristic
202C4⋊C8—21no—
203C4⋊C8—61no—
204C4⋊C8—61no—
205C4⋊C8—61no—
206C4⋊C8—61no—
207#103—21no—
208#103—21no—
209#102—21no—
210#102—21no—
211C22⋊C8—31no—
212C22⋊C8—31no—
213C22⋊C8—61no—
214C22⋊C8—61no—
215C22⋊C8—61no—
216C22⋊C8C2×Dic311yescharacteristic
217C22⋊C8C2×Dic311yescharacteristic
218#97—31no—
219#97—31no—
220#97—61no—
221#97—61no—
222#97—61no—
223#97—61no—
224#97—61no—
225C4×C8—21no—
226C4×C8—61no—
227C4×C8—61no—
228C23×C3—21no—
229C23×C3—41no—
230C23×C3—41no—
231C23×C3—41no—
232C23×C3—81no—
233C23×C3—81no—
234C23×C3C4×D411yescharacteristic
235C3×Q8—41no—
236C3×Q8—41no—
237C3×Q8—41no—
238C3×Q8—81no—
239C3×D4—41no—
240C3×D4—41no—
241C3×D4—41no—
242C3×D4—41no—
243C3×D4—41no—
244C3×D4—81no—
245C2×C3×C4—21no—
246C2×C3×C4—21no—
247C2×C3×C4—21no—
248C2×C3×C4—21no—
249C2×C3×C4—21no—
250C2×C3×C4—41no—
251C2×C3×C4—41no—
252C2×C3×C4—41no—
253C2×C3×C4—41no—
254C2×C3×C4—41no—
255C2×C3×C4—81no—
256C2×C3×C4C2×C22⋊C411yescharacteristic
257C2×C3×C4C4×D411yescharacteristic
258C2×C3×C4#12011yescharacteristic
259C2×C3×C4#12111yescharacteristic
260C2×C3×C4#12111yescharacteristic
261C2×C3×C4#12311yescharacteristic
262C3×C8—41no—
263C3×C8—41no—
264C3⋊C8—22no—
265C3⋊C8—42no—
266C3⋊C8—41no—
267C3⋊C8—41no—
268C3⋊C8—41no—
269C3⋊C8—81no—
270C24—21no—
271C2×Q8—21no—
272C2×Q8—21no—
273C2×Q8—81no—
274C2×Q8#21711yescharacteristic
275C2×D4—21no—
276C2×D4—21no—
277C2×D4—21no—
278C2×D4—81no—
279C2×D4#21711yescharacteristic
280C22×C4—21no—
281C22×C4—21no—
282C22×C4D3×D411yescharacteristic
283C22×C4#23711yescharacteristic
284C22×C4C22×Dic311yescharacteristic
285Q16—81no—
286SD16—81no—
287SD16—81no—
288D8—81no—
289M16—122no—
290C2×C8—21no—
291C2×C8—21no—
292C2×C8—61no—
293C2×C8—62no—
294C2×C8—61no—
295C2×C8—61no—
296C2×C8—61no—
297C2×C8—122no—
298C2×C8—122no—
299C2×C8—121no—
300C2×C8—121no—
301C2×C8—121no—
302C4⋊C4—21no—
303C4⋊C4—21no—
304C4⋊C4—21no—
305C4⋊C4—21no—
306C4⋊C4#21711yescharacteristic
307C4⋊C4#21711yescharacteristic
308C22⋊C4—21no—
309C22⋊C4—21no—
310C22⋊C4—21no—
311C22⋊C4—21no—
312C22⋊C4—21no—
313C22⋊C4—41no—
314C22⋊C4—41no—
315C22⋊C4—41no—
316C22⋊C4—41no—
317C42—21no—
318C42D3×D411yescharacteristic
319C42#23711yescharacteristic
320C42C2×#6711yescharacteristic
321C42C2×#6711yescharacteristic
322C22×C3—21no—
323C22×C3—21no—
324C22×C3—41no—
325C22×C3—41no—
326C22×C3—41no—
327C22×C3—81no—
328C22×C3—81no—
329C22×C3—81no—
330C22×C3—161no—
331C22×C3—161no—
332C22×C3#38611yescharacteristic
333C22×C3#44311yescharacteristic
334C22×C3#44411yescharacteristic
335C3×C4—21no—
336C3×C4—21no—
337C3×C4—21no—
338C3×C4—21no—
339C3×C4—21no—
340C3×C4—21no—
341C3×C4—41no—
342C3×C4—41no—
343C3×C4—41no—
344C3×C4—81no—
345C23—21no—
346C23—41no—
347C23—41no—
348C23—41no—
349C23—81no—
350C23—81no—
351C23D4×Dic311yescharacteristic
352Q8—41no—
353Q8—41no—
354Q8—41no—
355Q8—81no—
356D4—41no—
357D4—41no—
358D4—41no—
359D4—41no—
360D4—41no—
361D4—81no—
362C2×C4—21no—
363C2×C4—21no—
364C2×C4—21no—
365C2×C4—21no—
366C2×C4—21no—
367C2×C4—41no—
368C2×C4—41no—
369C2×C4—41no—
370C2×C4—41no—
371C2×C4—41no—
372C2×C4—81no—
373C2×C4D4×Dic311yescharacteristic
374C2×C4#93511yescharacteristic
375C2×C4#93711yescharacteristic
376C2×C4#93911yescharacteristic
377C2×C4#93911yescharacteristic
378C2×C4C2×#21711yescharacteristic
379C8—41no—
380C8—41no—
381C8—62no—
382C8—122no—
383C8—121no—
384C8—121no—
385C8—121no—
386C8—241no—
387C2×C3—21no—
388C2×C3—21no—
389C2×C3—81no—
390C2×C3—161no—
391C2×C3#187611yescharacteristic
392C2×C3#190611yescharacteristic
393C2×C3#191411yescharacteristic
394C22—21no—
395C22—21no—
396C22—41no—
397C22—41no—
398C22—41no—
399C22—81no—
400C22—81no—
401C22—81no—
402C22—161no—
403C22—161no—
404C22#519611yescharacteristic
405C22#519711yescharacteristic
406C22#522011yescharacteristic
407C4—21no—
408C4—21no—
409C4—21no—
410C4—21no—
411C4—21no—
412C4—21no—
413C4—41no—
414C4—41no—
415C4—41no—
416C4—81no—
417C3#926511yescharacteristic, complemented, Sylow
418C2—21no—
419C2—21no—
420C2—81no—
421C2—161no—
422C2#7185511yescharacteristic
423C2#7208711yescharacteristic
424C2#7211411yescharacteristic
subgroupG/H
centerC22#5220
commutatorC22×C3×C4C22×C4
FrattiniC2×C42C22×D3
FittingC3×#3211C2
socleC22×C3#386
prime radicalC3×#447C4
cosocleC2×C3×C42C23
hypercenter#3211D3
hypocenterC3#9265
LCSG ▷ C22×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3211

Complements

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

NH ≅ G/Ntype
2. #83128C2semidirect
3. #71748C2semidirect
4. #71720C2semidirect
5. C2×#4540C2semidirect
6. #67742C2semidirect
7. #67741C2semidirect
417. C3#9265semidirect

Maximal subgroups

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

subgroupindex
C3×#32112
#831282
#717482
#717202
C2×#45402
#677422
#677412
#92653