Finite Group Databasebeta

← #10692266

Subgroups — #10692266

Order 768.

474 conjugacy classes of subgroups.

The subgroup lattice has 416 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#3040C211yesmaximal, characteristic
2#82518C211yesmaximal, characteristic, complemented
3#71720C211yesmaximal, characteristic, complemented
4#71718C211yesmaximal, characteristic, complemented
5#71640C211yesmaximal, characteristic, complemented
6#67716C211yesmaximal, characteristic, complemented
7#67710C211yesmaximal, characteristic, complemented
8#9349—31nomaximal, Sylow
9C2×C3×#137C411yescharacteristic
10C2×C3×AGL(1,ℤ/8)C411yescharacteristic
11C2×C3×#124C2211yescharacteristic
12C3×C4×#41C2211yescharacteristic
13C3×#464—21no—
14C3×#464—21no—
15C3×#463—21no—
16C3×#463—21no—
17C3×#423C2211yescharacteristic
18C3×#419C411yescharacteristic
19C3×#418C411yescharacteristic
20#5024—21no—
21#5011—22no—
22D4×C3⋊C8—21no—
23#4926—21no—
24#4926C2211yescharacteristic
25#4926C2211yescharacteristic
26#4925C2211yescharacteristic
27C4×#208C2211yescharacteristic
28C2×#803—21no—
29#4540—21no—
30#4540—21no—
31#4535—21no—
32D4⋊(C3⋊C8)—21no—
33#4525—22no—
34#3040D311yescharacteristic
35#2970—31no—
36#1527—31no—
37#1488—31no—
38#1469—31no—
39#1307—31no—
40#1299—31no—
41C2×C3×#41C2×C411yescharacteristic
42C22×C3×Q8C2×C411yescharacteristic
43C22×C3×D4C2×C411yescharacteristic
44C3×#137—42no—
45C3×AGL(1,ℤ/8)—42no—
46C2×C3×Q16—21no—
47C2×C3×SD16—21no—
48C2×C3×SD16—21no—
49C2×C3×D8—21no—
50C2×C3×M16C2×C411yescharacteristic
51C2×C3×M16C2×C411yescharacteristic
52C3×#124—21no—
53C3×#124—21no—
54C3×#124—21no—
55C3×#124—21no—
56C3×C4×Q8D411yescharacteristic
57C3×C4×Q8D411yescharacteristic
58C3×C4×D4—21no—
59C3×C4×D4—21no—
60C3×C4×D4D411yescharacteristic
61C3×C4×D4D411yescharacteristic
62C3×#117—21no—
63C3×#117C2×C411yescharacteristic
64C2×C3×C22⋊C4—21no—
65C2×C3×C22⋊C4—21no—
66C2×C3×C42—21no—
67C2×C3×C42C2311yescharacteristic
68C2×#208—21no—
69C22×C3⋊C8—22no—
70C3×C4⋊C8—21no—
71C3×C4⋊C8—21no—
72C3×#103—21no—
73C3×#103—21no—
74C3×#102—21no—
75C3×#102—21no—
76#830—22no—
77#830—21no—
78#830—21no—
79#830D412yes—
80C4⋊(C3⋊C8)—22no—
81C4⋊(C3⋊C8)—21no—
82C4⋊(C3⋊C8)—21no—
83C4⋊(C3⋊C8)—21no—
84#803—22no—
85#803—21no—
86#803—21no—
87#803D411yescharacteristic
88#803D411yescharacteristic
89C4×C3⋊C8—21no—
90C4×C3⋊C8—21no—
91C2×#137Dic311yescharacteristic
92C2×AGL(1,ℤ/8)Dic311yescharacteristic
93C2×#124C2×D311yescharacteristic
94C4×#41C2×D311yescharacteristic
95#464—21no—
96#464—21no—
97#463—21no—
98#463—21no—
99#446—61no—
100#435—62no—
101C8×D4—61no—
102#432—31no—
103#432—31no—
104#432—61no—
105#431—31no—
106#423C2×D311yescharacteristic
107#419Dic311yescharacteristic
108#418Dic311yescharacteristic
109C4×M16—31no—
110C2×#97—61no—
111#329—61no—
112#329—61no—
113Q8⋊C8—61no—
114D4⋊C8—61no—
115#323—62no—
116C24×C3—21no—
117C3×#41—22no—
118C3×#41—41no—
119C2×C3×Q8—21no—
120C2×C3×Q8—21no—
121C2×C3×Q8—41no—
122C2×C3×Q8—41no—
123C2×C3×Q8C22⋊C411yescharacteristic
124C2×C3×D4—21no—
125C2×C3×D4—21no—
126C2×C3×D4—21no—
127C2×C3×D4—41no—
128C2×C3×D4—41no—
129C2×C3×D4C22⋊C411yescharacteristic
130C22×C3×C4—21no—
131C22×C3×C4—21no—
132C22×C3×C4—21no—
133C22×C3×C4—21no—
134C22×C3×C4C22×C411yescharacteristic
135C22×C3×C4C2×D411yescharacteristic
136C22×C3×C4#4111yescharacteristic
137C3×Q16—42no—
138C3×SD16—42no—
139C3×SD16—42no—
140C3×D8—42no—
141C3×M16—41no—
142C3×M16—41no—
143C2×C3×C8—21no—
144C2×C3×C8—21no—
145C3×C4⋊C4—21no—
146C3×C4⋊C4—21no—
147C3×C4⋊C4C22⋊C411yescharacteristic
148C3×C4⋊C4C22⋊C411yescharacteristic
149C3×C22⋊C4—21no—
150C3×C22⋊C4—21no—
151C3×C22⋊C4—21no—
152C3×C22⋊C4—41no—
153C3×C22⋊C4—41no—
154C3×C22⋊C4—41no—
155C3×C22⋊C4—41no—
156C3×C42—21no—
157C3×C42—21no—
158C3×C42—21no—
159C3×C42C2×D411yescharacteristic
160C3×C42C2×D411yescharacteristic
161C3×C42C2×D411yescharacteristic
162C3×C42#4111yescharacteristic
163#208—42no—
164C2×C3⋊C8—21no—
165C2×C3⋊C8—21no—
166C2×C3⋊C8—22no—
167C2×C3⋊C8—44no—
168C2×C3⋊C8—41no—
169C2×C3⋊C8—41no—
170C2×#41C2×Dic311yescharacteristic
171C22×Q8C2×Dic311yescharacteristic
172C22×D4C2×Dic311yescharacteristic
173#137—42no—
174AGL(1,ℤ/8)—42no—
175C2×Q16—21no—
176C2×SD16—21no—
177C2×SD16—21no—
178C2×D8—21no—
179C2×M16—61no—
180C2×M16C2×Dic311yescharacteristic
181C2×M16C2×Dic311yescharacteristic
182C22×C8—62no—
183#124—21no—
184#124—21no—
185#124—21no—
186#124—21no—
187C4×Q8#6711yescharacteristic
188C4×Q8#6711yescharacteristic
189C4×D4—21no—
190C4×D4—21no—
191C4×D4#6711yescharacteristic
192C4×D4#6711yescharacteristic
193#117—21no—
194#117C2×Dic311yescharacteristic
195C2×C22⋊C4—21no—
196C2×C22⋊C4—21no—
197C2×C42—21no—
198C2×C42C22×D311yescharacteristic
199C4⋊C8—21no—
200C4⋊C8—21no—
201C4⋊C8—62no—
202C4⋊C8—61no—
203C4⋊C8—61no—
204C4⋊C8—61no—
205#103—21no—
206#103—21no—
207#102—21no—
208#102—21no—
209C22⋊C8—32no—
210C22⋊C8—62no—
211C22⋊C8—61no—
212C22⋊C8—61no—
213#97—31no—
214#97—31no—
215#97—62no—
216#97—61no—
217#97—61no—
218C4×C8—61no—
219C4×C8—61no—
220C23×C3—21no—
221C23×C3—42no—
222C23×C3—41no—
223C23×C3—81no—
224C23×C3—81no—
225C23×C3C4×D411yescharacteristic
226C3×Q8—21no—
227C3×Q8—21no—
228C3×Q8—41no—
229C3×Q8—41no—
230C3×Q8—81no—
231C3×D4—21no—
232C3×D4—42no—
233C3×D4—41no—
234C3×D4—41no—
235C3×D4—81no—
236C3×D4#10012yes—
237C2×C3×C4—21no—
238C2×C3×C4—21no—
239C2×C3×C4—21no—
240C2×C3×C4—21no—
241C2×C3×C4—21no—
242C2×C3×C4—41no—
243C2×C3×C4—41no—
244C2×C3×C4—42no—
245C2×C3×C4—42no—
246C2×C3×C4—41no—
247C2×C3×C4—81no—
248C2×C3×C4—81no—
249C2×C3×C4C2×C22⋊C411yescharacteristic
250C2×C3×C4C4×D411yescharacteristic
251C2×C3×C4#12011yescharacteristic
252C2×C3×C4#12111yescharacteristic
253C2×C3×C4#12111yescharacteristic
254C2×C3×C4#12311yescharacteristic
255C3×C8—41no—
256C3×C8—41no—
257C3⋊C8—22no—
258C3⋊C8—42no—
259C3⋊C8—41no—
260C3⋊C8—81no—
261C3⋊C8—81no—
262C24—21no—
263#41—22no—
264#41—41no—
265C2×Q8—21no—
266C2×Q8—21no—
267C2×Q8—41no—
268C2×Q8—41no—
269C2×Q8#21711yescharacteristic
270C2×D4—21no—
271C2×D4—21no—
272C2×D4—21no—
273C2×D4—41no—
274C2×D4—41no—
275C2×D4#21711yescharacteristic
276C22×C4—21no—
277C22×C4—21no—
278C22×C4—21no—
279C22×C4—21no—
280C22×C4D3×D411yescharacteristic
281C22×C4#23711yescharacteristic
282C22×C4C22×Dic311yescharacteristic
283Q16—42no—
284SD16—42no—
285SD16—42no—
286D8—42no—
287M16—41no—
288M16—41no—
289M16—122no—
290C2×C8—21no—
291C2×C8—21no—
292C2×C8—61no—
293C2×C8—61no—
294C2×C8—62no—
295C2×C8—124no—
296C2×C8—121no—
297C2×C8—121no—
298C4⋊C4—21no—
299C4⋊C4—21no—
300C4⋊C4#21711yescharacteristic
301C4⋊C4#21711yescharacteristic
302C22⋊C4—21no—
303C22⋊C4—21no—
304C22⋊C4—21no—
305C22⋊C4—41no—
306C22⋊C4—41no—
307C22⋊C4—41no—
308C22⋊C4—41no—
309C42—21no—
310C42—21no—
311C42—21no—
312C42D3×D411yescharacteristic
313C42#23711yescharacteristic
314C42C2×#6711yescharacteristic
315C42C2×#6711yescharacteristic
316C22×C3—21no—
317C22×C3—21no—
318C22×C3—22no—
319C22×C3—21no—
320C22×C3—41no—
321C22×C3—41no—
322C22×C3—81no—
323C22×C3—81no—
324C22×C3—81no—
325C22×C3—162no—
326C22×C3#38611yescharacteristic
327C3×C4—21no—
328C3×C4—21no—
329C3×C4—21no—
330C3×C4—21no—
331C3×C4—21no—
332C3×C4—41no—
333C3×C4—41no—
334C3×C4—41no—
335C3×C4—81no—
336C3×C4C2×#10011yescharacteristic
337C3×C4#41311yescharacteristic
338C23—21no—
339C23—42no—
340C23—41no—
341C23—81no—
342C23—81no—
343C23D4×Dic311yescharacteristic
344Q8—21no—
345Q8—21no—
346Q8—41no—
347Q8—41no—
348Q8—81no—
349D4—21no—
350D4—42no—
351D4—41no—
352D4—41no—
353D4—81no—
354D4#83312yes—
355C2×C4—21no—
356C2×C4—21no—
357C2×C4—21no—
358C2×C4—21no—
359C2×C4—21no—
360C2×C4—41no—
361C2×C4—41no—
362C2×C4—42no—
363C2×C4—42no—
364C2×C4—41no—
365C2×C4—81no—
366C2×C4—81no—
367C2×C4D4×Dic311yescharacteristic
368C2×C4#93511yescharacteristic
369C2×C4#93711yescharacteristic
370C2×C4#93911yescharacteristic
371C2×C4#93911yescharacteristic
372C2×C4C2×#21711yescharacteristic
373C8—41no—
374C8—41no—
375C8—62no—
376C8—122no—
377C8—121no—
378C8—241no—
379C8—241no—
380C2×C3—41no—
381C2×C3—42no—
382C2×C3—161no—
383C2×C3#188111yescharacteristic
384C2×C3#190011yescharacteristic
385C2×C3#190611yescharacteristic
386C22—21no—
387C22—22no—
388C22—21no—
389C22—21no—
390C22—41no—
391C22—41no—
392C22—81no—
393C22—81no—
394C22—81no—
395C22—162no—
396C22#522011yescharacteristic
397C4—21no—
398C4—21no—
399C4—21no—
400C4—21no—
401C4—21no—
402C4—41no—
403C4—41no—
404C4—41no—
405C4—81no—
406C4C2×#83311yescharacteristic
407C4#523711yescharacteristic
408C3#934911yescharacteristic, complemented, Sylow
409C2—41no—
410C2—42no—
411C2—161no—
412C2#7168311yescharacteristic
413C2#7211411yescharacteristic
414C2#7211611yescharacteristic
subgroupG/H
centerC22#5220
commutatorC22×C3×C4C22×C4
FrattiniC2×C42C22×D3
FittingC3×#3040C2
socleC22×C3#386
prime radicalC2×C3×AGL(1,ℤ/8)C4
cosocleC2×C3×C42C23
hypercenter#3040D3
hypocenterC3#9349
LCSG ▷ C22×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3040

Complements

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

NH ≅ G/Ntype
2. #82518C2semidirect
3. #71720C2semidirect
4. #71718C2semidirect
5. #71640C2semidirect
6. #67716C2semidirect
7. #67710C2semidirect
408. C3#9349semidirect

Maximal subgroups

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

subgroupindex
C3×#30402
#825182
#717202
#717182
#716402
#677162
#677102
#93493