Finite Group Databasebeta

← #10692025

Subgroups — #10692025

Order 768.

480 conjugacy classes of subgroups.

The subgroup lattice has 390 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#79060C211yesmaximal, characteristic
2#79013C211yesmaximal, characteristic
3#70529C211yesmaximal, characteristic
4#70507C211yesmaximal, characteristic, complemented
5#70459C211yesmaximal, characteristic, complemented
6C3×#1292C211yesmaximal, characteristic, complemented
7#67715C211yesmaximal, characteristic, complemented
8#9315—31nomaximal, Sylow
9C4×#235C2211yescharacteristic
10C2×#869C2211yescharacteristic
11C3×#423C2211yescharacteristic
12C3×#423C2211yescharacteristic
13C2×C3×C4⋊C8C2211yescharacteristic
14C2×#825—21no—
15#5133—21no—
16#5121—21no—
17C8×#67—22no—
18C4×C4⋊Dic3C2211yescharacteristic
19#4925C2211yescharacteristic
20#4851C412yescomplemented
21#4850—21no—
22#4846C412yescomplemented
23#4842—21no—
24#4834—21no—
25#4833—21no—
26D3×C4⋊C8—21no—
27#4830—21no—
28#4555—21no—
29#4551—22no—
30#4481—21no—
31#4481—21no—
32#3079—31no—
33#2988—31no—
34#1987—31no—
35#1806—31no—
36#1548—31no—
37#1303—31no—
38#1292D311yescharacteristic, complemented
39C2×#235D411yescharacteristic
40C22×Dic6—21no—
41C2×C3×M16—21no—
42C2×C3×M16—21no—
43C22×C3×C8D412yes—
44C2×C3×C42C2311yescharacteristic
45C4×#67—21no—
46C4×#67—21no—
47#926D411yescharacteristic
48C2×C4⋊Dic3—21no—
49C2×C4⋊Dic3—21no—
50C2×C4⋊Dic3—21no—
51C2×C4×Dic3—21no—
52C2×D3×C8—22no—
53C4×D12C2×C411yescharacteristic
54C4×D12D411yescharacteristic
55#872—21no—
56C42×D3—21no—
57#869—21no—
58#869—21no—
59#869—21no—
60#869C2×C412yes—
61C4×Dic6C2×C411yescharacteristic
62C4×Dic6D411yescharacteristic
63C3×C4⋊C8—21no—
64C3×C4⋊C8—21no—
65C3×C4⋊C8—21no—
66C3×C4⋊C8—21no—
67C3×C4⋊C8—21no—
68C3×C4⋊C8C2×C412yes—
69C3⋊#95—21no—
70#830—21no—
71#825—42no—
72#821—22no—
73(C2×D3)⋊C8—22no—
74#818—22no—
75#817—22no—
76#816—22no—
77Dic3⋊C8—22no—
78C8×Dic3—22no—
79C4⋊(C3⋊C8)D411yescharacteristic
80C4⋊(C3⋊C8)D411yescharacteristic
81C4×C3⋊C8—21no—
82C2×C4⋊Q8—31no—
83C4×#41—31no—
84#477—61no—
85#476—61no—
86#475—32no—
87#474—32no—
88C8×Q8—61no—
89C8×D4—61no—
90C8×D4—62no—
91#431—31no—
92#431—61no—
93#428—61no—
94#423C2×D311yescharacteristic
95#423C2×D311yescharacteristic
96C2×C4⋊C8—61no—
97C2×C4⋊C8C2×D311yescharacteristic
98C2×#104—61no—
99#417—61no—
100C4×C4⋊C4—31no—
101#340—62no—
102#339—61no—
103#333—61no—
104#333—61no—
105C22×C3×C4C2×D411yescharacteristic
106C22×C3×C4C2×D411yescharacteristic
107C22×C3×C4#4111yescharacteristic
108C2×#67—21no—
109C22×Dic3—41no—
110C22×Dic3—41no—
111#235—21no—
112#235—21no—
113#235SD1612yes—
114#235Q1612yes—
115C2×D12C22⋊C411yescharacteristic
116C2×C4×D3—21no—
117C2×C4×D3—21no—
118C2×C4×D3—21no—
119C2×Dic6—22no—
120C2×Dic6—41no—
121C2×Dic6—82no—
122C2×Dic6C22⋊C411yescharacteristic
123C3×M16—41no—
124C3×M16—41no—
125C3×M16—41no—
126C3×M16—41no—
127C2×C3×C8—22no—
128C2×C3×C8—22no—
129C2×C3×C8—42no—
130C2×C3×C8—41no—
131C3×C42C22×C411yescharacteristic
132C3×C42C2×D411yescharacteristic
133C3×C42C2×D411yescharacteristic
134C3×C42#4111yescharacteristic
135#217—21no—
136(C2×D3)⋊C4—21no—
137(C2×D3)⋊C4—21no—
138C4⋊Dic3—22no—
139C4⋊Dic3—22no—
140C4⋊Dic3—22no—
141C4⋊Dic3—41no—
142C4⋊Dic3—41no—
143C4⋊Dic3—41no—
144C4⋊Dic3C22⋊C411yescharacteristic
145C4⋊Dic3C22⋊C411yescharacteristic
146Dic3⋊C4—21no—
147Dic3⋊C4—21no—
148C4×Dic3—21no—
149C4×Dic3—21no—
150C4×Dic3—21no—
151C4×Dic3—41no—
152C4×Dic3—41no—
153C2×C3⋊C8—21no—
154C2×C3⋊C8—21no—
155D3×C8—82no—
156C2×#41—31no—
157C22×Q8—61no—
158C2×M16—21no—
159C2×M16—21no—
160C22×C8—62no—
161C22×C8D1212yes—
162C4⋊Q8—32no—
163C4⋊Q8—61no—
164C4⋊Q8—61no—
165C4⋊Q8—61no—
166C4×Q8—31no—
167C4×Q8—31no—
168C4×D4—31no—
169C4×D4—31no—
170C4×D4—61no—
171C4×D4—61no—
172#117—31no—
173#117—61no—
174C2×C4⋊C4—61no—
175C2×C4⋊C4—61no—
176C2×C4⋊C4—61no—
177C2×C42—61no—
178C2×C42—61no—
179C2×C42C22×D311yescharacteristic
180#107—62no—
181#106—62no—
182C4⋊C8—21no—
183C4⋊C8—21no—
184C4⋊C8—21no—
185C4⋊C8—21no—
186C4⋊C8—21no—
187C4⋊C8—31no—
188C4⋊C8—31no—
189C4⋊C8—62no—
190C4⋊C8C4×D312yes—
191#104—122no—
192#103—62no—
193#102—62no—
194C22⋊C8—62no—
195C22⋊C8—61no—
196C4×C8—62no—
197C4×C8—61no—
198#95—61no—
199C23×C3#12111yescharacteristic
200C22×D3—21no—
201C2×C3×C4—21no—
202C2×C3×C4—21no—
203C2×C3×C4—41no—
204C2×C3×C4C2×C22⋊C411yescharacteristic
205C2×C3×C4C4×D411yescharacteristic
206C2×C3×C4C4×D411yescharacteristic
207C2×C3×C4#12011yescharacteristic
208C2×C3×C4#12111yescharacteristic
209C2×C3×C4#12311yescharacteristic
210C2×C3×C4C2×SD1611yescharacteristic
211C2×C3×C4C2×Q1611yescharacteristic
212C2×C3×C4#13511yescharacteristic
213C2×C3×C4#13511yescharacteristic
214#67—22no—
215#67—22no—
216C2×Dic3—21no—
217C2×Dic3—21no—
218C2×Dic3—21no—
219C2×Dic3—42no—
220C2×Dic3—42no—
221C2×Dic3—82no—
222C2×Dic3—81no—
223C2×Dic3—81no—
224D12—21no—
225D12#10312yes—
226C4×D3—22no—
227C4×D3—22no—
228C4×D3—41no—
229C4×D3—41no—
230C4×D3—81no—
231Dic6—21no—
232Dic6—82no—
233Dic6—82no—
234Dic6#10312yes—
235C3×C8—42no—
236C3×C8—42no—
237C3×C8—81no—
238C3⋊C8—41no—
239C3⋊C8—41no—
240#41—32no—
241#41—32no—
242#41—61no—
243#41—61no—
244C2×Q8—31no—
245C2×Q8—62no—
246C2×Q8—121no—
247C2×Q8—242no—
248C2×D4—31no—
249C2×D4—61no—
250C22×C4—61no—
251C22×C4—61no—
252C22×C4—61no—
253C22×C4—121no—
254C22×C4—121no—
255C22×C4C2×D1211yescharacteristic
256C22×C4D3×D411yescharacteristic
257C22×C4#23911yescharacteristic
258M16—41no—
259M16—41no—
260M16—41no—
261M16—41no—
262C2×C8—22no—
263C2×C8—22no—
264C2×C8—42no—
265C2×C8—41no—
266C2×C8—61no—
267C2×C8—61no—
268C2×C8—242no—
269C4⋊C4—31no—
270C4⋊C4—31no—
271C4⋊C4—62no—
272C4⋊C4—61no—
273C4⋊C4—61no—
274C4⋊C4—62no—
275C4⋊C4—62no—
276C4⋊C4—121no—
277C4⋊C4—121no—
278C4⋊C4—121no—
279C22⋊C4—61no—
280C22⋊C4—61no—
281C22⋊C4—61no—
282C42—61no—
283C42—61no—
284C42—61no—
285C42—121no—
286C42—121no—
287C42C2×C4×D311yescharacteristic
288C42#23511yescharacteristic
289C42D3×D411yescharacteristic
290C42D3×D411yescharacteristic
291C22×C3—21no—
292C22×C3—21no—
293C22×C3#38611yescharacteristic
294C22×C3#46511yescharacteristic
295C22×C3#46711yescharacteristic
296C2×D3—22no—
297C2×D3—41no—
298C2×D3—41no—
299C3×C4—21no—
300C3×C4—21no—
301C3×C4—21no—
302C3×C4C2×#10311yescharacteristic
303C3×C4#41611yescharacteristic
304C3×C4#45511yescharacteristic
305C3×C4#45611yescharacteristic
306C3×C4#46511yescharacteristic
307C3×C4#46711yescharacteristic
308Dic3—22no—
309Dic3—41no—
310Dic3—41no—
311Dic3—82no—
312Dic3—82no—
313C23—61no—
314C23#89511yescharacteristic
315Q8—32no—
316Q8—61no—
317Q8—242no—
318Q8—242no—
319D4—32no—
320D4—62no—
321D4—62no—
322D4—61no—
323C2×C4—21no—
324C2×C4—21no—
325C2×C4—41no—
326C2×C4—61no—
327C2×C4—62no—
328C2×C4—62no—
329C2×C4—61no—
330C2×C4—61no—
331C2×C4—121no—
332C2×C4—121no—
333C2×C4—122no—
334C2×C4—122no—
335C2×C4—241no—
336C2×C4—242no—
337C2×C4—241no—
338C2×C4—241no—
339C2×C4C4×D1211yescharacteristic
340C2×C4D3×C22⋊C411yescharacteristic
341C2×C4#88211yescharacteristic
342C2×C4#88411yescharacteristic
343C2×C4#89311yescharacteristic
344C2×C4#89411yescharacteristic
345C2×C4#91211yescharacteristic
346C2×C4D3×SD1611yescharacteristic
347C2×C4#91611yescharacteristic
348C2×C4D3×Q1611yescharacteristic
349C8—42no—
350C8—42no—
351C8—81no—
352C8—121no—
353C8—121no—
354C2×C3—21no—
355C2×C3—21no—
356C2×C3#188211yescharacteristic
357C2×C3#189211yescharacteristic
358C2×C3#190311yescharacteristic
359D3—42no—
360C22—21no—
361C22—21no—
362C22—62no—
363C22—121no—
364C22—121no—
365C22#466511yescharacteristic
366C22#486411yescharacteristic
367C22#488711yescharacteristic
368C4—21no—
369C4—21no—
370C4—21no—
371C4—62no—
372C4—121no—
373C4—121no—
374C4—242no—
375C4—242no—
376C4#474611yescharacteristic
377C4#475211yescharacteristic
378C4#477211yescharacteristic
379C4#477811yescharacteristic
380C4D3×#10311yescharacteristic
381C4#481311yescharacteristic
382C3#931511yescharacteristic, complemented, Sylow
383C2—21no—
384C2—21no—
385C2—122no—
386C2#6872611yescharacteristic
387C2#6888211yescharacteristic
388C2#6900411yescharacteristic
subgroupG/H
centerC22#4665
commutatorC3×C42C22×C4
FrattiniC2×C42C22×D3
FittingC3×#1292C2
socleC22×C3#386
prime radicalC2×#235D4
cosocleC2×C3×C42C23
hypercenter#1292D3
hypocenterC3#9315
LCSG ▷ C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1292

Complements

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

NH ≅ G/Ntype
4. #70507C2semidirect
5. #70459C2semidirect
6. C3×#1292C2semidirect
7. #67715C2semidirect
20. #4851C4semidirect
22. #4846C4semidirect
38. #1292D3semidirect
382. C3#9315semidirect

Maximal subgroups

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

subgroupindex
#790602
#790132
#705292
#705072
#704592
C3×#12922
#677152
#93153