Finite Group Databasebeta

← #10692037

Subgroups — #10692037

Order 768.

496 conjugacy classes of subgroups.

The subgroup lattice has 396 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#83212C211yesmaximal, characteristic
2#79291C211yesmaximal, characteristic
3C3×#1549C211yesmaximal, characteristic, complemented
4#71778C211yesmaximal, characteristic, complemented
5#70529C211yesmaximal, characteristic
6#67746C211yesmaximal, characteristic, complemented
7#67711C211yesmaximal, characteristic, complemented
8#9317—31nomaximal, Sylow
9C2×C3×C4⋊Q8C2211yescharacteristic
10C4×#235C2211yescharacteristic
11C3×#431C2211yescharacteristic
12C3×#423C2211yescharacteristic
13C2×#835—22no—
14C2×#825—21no—
15#5094—21no—
16#5091C412yescomplemented
17#5087—21no—
18#5084C412yescomplemented
19#4966—21no—
20#4964—21no—
21C4×C4⋊Dic3C2211yescharacteristic
22#4924C2211yescharacteristic
23C2×C4⋊(C3⋊C8)C2211yescharacteristic
24#4833—21no—
25D3×C4⋊C8—21no—
26#67⋊C8—22no—
27C8×D12—21no—
28C8×Dic6—21no—
29C3×#332—21no—
30C3×#332—21no—
31#4555—21no—
32#3079—31no—
33#2988—31no—
34#1981—31no—
35#1806—31no—
36#1549D311yescharacteristic, complemented
37#1303—31no—
38#1292—31no—
39C22×C3×Q8—21no—
40C2×#235D411yescharacteristic
41C2×C3×M16—21no—
42C3×C4⋊Q8—21no—
43C3×C4⋊Q8—21no—
44C3×C4⋊Q8—21no—
45C3×C4⋊Q8C2×C412yes—
46C2×C3×C4⋊C4—21no—
47C2×C3×C4⋊C4—21no—
48C2×C3×C42C2311yescharacteristic
49C4×#67—21no—
50C4×#67—21no—
51#926D411yescharacteristic
52C2×C4⋊Dic3—21no—
53C2×C4×Dic3—21no—
54C2×#208—21no—
55C22×C3⋊C8D412yes—
56C2×D3×C8—22no—
57C4×D12C2×C411yescharacteristic
58C4×D12D411yescharacteristic
59#872—21no—
60C42×D3—21no—
61C4×Dic6C2×C411yescharacteristic
62C4×Dic6D411yescharacteristic
63C3×C4⋊C8—21no—
64C3×C4⋊C8—21no—
65C3×C4⋊C8D411yescharacteristic
66C3×C4⋊C8D411yescharacteristic
67C3×C22⋊C8—21no—
68C3×C4×C8—21no—
69#835—44no—
70C3⋊#95—21no—
71#825—42no—
72(C2×D3)⋊C8—22no—
73Dic3⋊C8—22no—
74C8×Dic3—22no—
75#810—22no—
76#809—22no—
77#808—22no—
78#807—22no—
79C4⋊(C3⋊C8)—21no—
80C4⋊(C3⋊C8)—21no—
81C4⋊(C3⋊C8)—21no—
82C4⋊(C3⋊C8)C2×C412yes—
83C2×C4⋊Q8C2×D311yescharacteristic
84C4×#41—31no—
85#477—32no—
86#476—32no—
87#475—61no—
88#474—61no—
89C8×Q8—61no—
90C8×D4—61no—
91C8×D4—62no—
92#431—61no—
93#431C2×D311yescharacteristic
94#428—61no—
95#423—31no—
96#423C2×D311yescharacteristic
97C2×C4⋊C8—31no—
98C2×C4⋊C8—61no—
99C2×#104—61no—
100#417—61no—
101C2×#103—62no—
102C4×C4⋊C4—31no—
103#339—61no—
104#332—21no—
105#332—21no—
106C2×C3×Q8—22no—
107C2×C3×Q8—42no—
108C2×C3×Q8—42no—
109C2×C3×Q8—41no—
110C22×C3×C4—41no—
111C22×C3×C4C2×D411yescharacteristic
112C22×C3×C4C2×D411yescharacteristic
113C22×C3×C4#4111yescharacteristic
114C2×#67—21no—
115C22×Dic3—41no—
116#235—21no—
117#235—21no—
118#235SD1612yes—
119#235Q1612yes—
120C2×D12C22⋊C411yescharacteristic
121C2×C4×D3—21no—
122C2×C4×D3—21no—
123C2×C4×D3—21no—
124C2×Dic6C22⋊C411yescharacteristic
125C3×M16—41no—
126C3×M16—41no—
127C2×C3×C8—21no—
128C2×C3×C8—21no—
129C2×C3×C8—41no—
130C3×C4⋊C4—22no—
131C3×C4⋊C4—22no—
132C3×C4⋊C4—41no—
133C3×C4⋊C4—41no—
134C3×C42C22×C411yescharacteristic
135C3×C42C2×D411yescharacteristic
136C3×C42C2×D411yescharacteristic
137C3×C42#4111yescharacteristic
138#217—21no—
139(C2×D3)⋊C4—21no—
140(C2×D3)⋊C4—21no—
141C4⋊Dic3—22no—
142C4⋊Dic3—41no—
143C4⋊Dic3C22⋊C411yescharacteristic
144C4⋊Dic3C22⋊C411yescharacteristic
145Dic3⋊C4—21no—
146Dic3⋊C4—21no—
147C4×Dic3—21no—
148C4×Dic3—21no—
149C4×Dic3—21no—
150C4×Dic3—41no—
151C4×Dic3—41no—
152#208—41no—
153#208—41no—
154C2×C3⋊C8—22no—
155C2×C3⋊C8—22no—
156C2×C3⋊C8—42no—
157D3×C8—82no—
158C2×#41—31no—
159C22×Q8—21no—
160C2×M16—21no—
161C2×M16—61no—
162C22×C8—32no—
163C22×C8—62no—
164C4⋊Q8—21no—
165C4⋊Q8—21no—
166C4⋊Q8—21no—
167C4⋊Q8C4×D312yes—
168C4×Q8—31no—
169C4×Q8—31no—
170C4×D4—31no—
171C4×D4—31no—
172C4×D4—61no—
173C4×D4—61no—
174#117—31no—
175#117—61no—
176C2×C4⋊C4—21no—
177C2×C4⋊C4—21no—
178C2×C4⋊C4—61no—
179C2×C42—61no—
180C2×C42—61no—
181C2×C42C22×D311yescharacteristic
182#107—62no—
183#106—62no—
184C4⋊C8—21no—
185C4⋊C8—21no—
186C4⋊C8—32no—
187C4⋊C8—62no—
188C4⋊C8—61no—
189C4⋊C8—61no—
190C4⋊C8—61no—
191C4⋊C8#6711yescharacteristic
192C4⋊C8#6711yescharacteristic
193#104—122no—
194#103—62no—
195#103—124no—
196#102—62no—
197C22⋊C8—21no—
198C22⋊C8—62no—
199C4×C8—21no—
200C4×C8—62no—
201#95—61no—
202C23×C3#12111yescharacteristic
203C22×D3—21no—
204C3×Q8—44no—
205C3×Q8—81no—
206C3×Q8—81no—
207C2×C3×C4—21no—
208C2×C3×C4—21no—
209C2×C3×C4—41no—
210C2×C3×C4—42no—
211C2×C3×C4—82no—
212C2×C3×C4C2×C22⋊C411yescharacteristic
213C2×C3×C4C4×D411yescharacteristic
214C2×C3×C4C4×D411yescharacteristic
215C2×C3×C4#12011yescharacteristic
216C2×C3×C4#12111yescharacteristic
217C2×C3×C4#12311yescharacteristic
218C2×C3×C4C2×SD1611yescharacteristic
219C2×C3×C4C2×Q1611yescharacteristic
220C2×C3×C4#13511yescharacteristic
221C2×C3×C4#13511yescharacteristic
222#67—22no—
223#67—22no—
224C2×Dic3—21no—
225C2×Dic3—21no—
226C2×Dic3—21no—
227C2×Dic3—42no—
228C2×Dic3—81no—
229C2×Dic3—81no—
230D12—21no—
231D12#10312yescomplemented
232C4×D3—22no—
233C4×D3—22no—
234C4×D3—41no—
235C4×D3—41no—
236C4×D3—81no—
237Dic6—21no—
238Dic6#10312yescomplemented
239C3×C8—41no—
240C3×C8—41no—
241C3×C8—81no—
242C3⋊C8—42no—
243C3⋊C8—42no—
244#41—32no—
245#41—32no—
246#41—61no—
247#41—61no—
248C2×Q8—22no—
249C2×Q8—31no—
250C2×Q8—42no—
251C2×Q8—42no—
252C2×Q8—41no—
253C2×D4—31no—
254C2×D4—61no—
255C22×C4—41no—
256C22×C4—61no—
257C22×C4—61no—
258C22×C4—61no—
259C22×C4—121no—
260C22×C4#23511yescharacteristic
261C22×C4D3×D411yescharacteristic
262C22×C4D3×D411yescharacteristic
263M16—41no—
264M16—41no—
265M16—121no—
266M16—121no—
267C2×C8—21no—
268C2×C8—21no—
269C2×C8—41no—
270C2×C8—62no—
271C2×C8—62no—
272C2×C8—122no—
273C2×C8—242no—
274C4⋊C4—22no—
275C4⋊C4—22no—
276C4⋊C4—31no—
277C4⋊C4—31no—
278C4⋊C4—41no—
279C4⋊C4—41no—
280C4⋊C4—62no—
281C4⋊C4—61no—
282C4⋊C4—61no—
283C4⋊C4—121no—
284C22⋊C4—61no—
285C22⋊C4—61no—
286C22⋊C4—61no—
287C42—61no—
288C42—61no—
289C42—61no—
290C42—121no—
291C42—121no—
292C42C2×C4×D311yescharacteristic
293C42D3×D411yescharacteristic
294C42#23711yescharacteristic
295C42C2×#6711yescharacteristic
296C22×C3—21no—
297C22×C3—21no—
298C22×C3#38611yescharacteristic
299C22×C3#46511yescharacteristic
300C22×C3#46711yescharacteristic
301C2×D3—22no—
302C2×D3—41no—
303C2×D3—41no—
304C3×C4—21no—
305C3×C4—21no—
306C3×C4—21no—
307C3×C4—82no—
308C3×C4C2×#10311yescharacteristic
309C3×C4#41611yescharacteristic
310C3×C4#45311yescharacteristic
311C3×C4#45411yescharacteristic
312C3×C4#46511yescharacteristic
313C3×C4#46711yescharacteristic
314Dic3—22no—
315Dic3—41no—
316Dic3—41no—
317Dic3—82no—
318C23—61no—
319C23#88411yescharacteristic
320Q8—32no—
321Q8—44no—
322Q8—61no—
323Q8—81no—
324Q8—81no—
325D4—32no—
326D4—62no—
327D4—62no—
328D4—61no—
329C2×C4—21no—
330C2×C4—21no—
331C2×C4—41no—
332C2×C4—42no—
333C2×C4—61no—
334C2×C4—62no—
335C2×C4—62no—
336C2×C4—61no—
337C2×C4—61no—
338C2×C4—82no—
339C2×C4—121no—
340C2×C4—121no—
341C2×C4—122no—
342C2×C4—241no—
343C2×C4—241no—
344C2×C4—241no—
345C2×C4D3×C22⋊C411yescharacteristic
346C2×C4#67⋊C411yescharacteristic
347C2×C4#88311yescharacteristic
348C2×C4D3×SD1611yescharacteristic
349C2×C4#91611yescharacteristic
350C2×C4D3×Q1611yescharacteristic
351C2×C4#91911yescharacteristic
352C2×C4C4×#6711yescharacteristic
353C2×C4#93711yescharacteristic
354C2×C4#93811yescharacteristic
355C8—41no—
356C8—41no—
357C8—81no—
358C8—122no—
359C8—122no—
360C2×C3—21no—
361C2×C3—21no—
362C2×C3#188311yescharacteristic
363C2×C3#189211yescharacteristic
364C2×C3#190211yescharacteristic
365D3—42no—
366C22—21no—
367C22—21no—
368C22—62no—
369C22—121no—
370C22—121no—
371C22#480711yescharacteristic
372C22#481311yescharacteristic
373C22#495611yescharacteristic
374C4—21no—
375C4—21no—
376C4—21no—
377C4—62no—
378C4—82no—
379C4—121no—
380C4—121no—
381C4—242no—
382C4D3×#10311yescharacteristic
383C4#480411yescharacteristic
384C4#517111yescharacteristic
385C4#518811yescharacteristic
386C4#519811yescharacteristic
387C4#520111yescharacteristic
388C3#931711yescharacteristic, complemented, Sylow
389C2—21no—
390C2—21no—
391C2—122no—
392C2#7012211yescharacteristic
393C2#7027011yescharacteristic
394C2#7032811yescharacteristic
subgroupG/H
centerC22#4956
commutatorC3×C42C22×C4
FrattiniC2×C42C22×D3
FittingC3×#1549C2
socleC22×C3#386
prime radicalC2×#235D4
cosocleC2×C3×C42C23
hypercenter#1549D3
hypocenterC3#9317
LCSG ▷ C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1549

Complements

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

NH ≅ G/Ntype
3. C3×#1549C2semidirect
4. #71778C2semidirect
6. #67746C2semidirect
7. #67711C2semidirect
16. #5091C4semidirect
18. #5084C4semidirect
36. #1549D3semidirect
231. D12#103semidirect
238. Dic6#103semidirect
388. C3#9317semidirect

Maximal subgroups

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

subgroupindex
#832122
#792912
C3×#15492
#717782
#705292
#677462
#677112
#93173