Finite Group Databasebeta

← #10849657

Subgroups — #10849657

Order 768.

1072 conjugacy classes of subgroups.

The subgroup lattice has 400 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#85848C211yesmaximal, characteristic
2C3×#3080C211yesmaximal, characteristic, complemented
3#83067C211yesmaximal, characteristic, complemented
4#83015C211yesmaximal, characteristic, complemented
5#71643C211yesmaximal, characteristic, complemented
6#70674C214yesmaximal, complemented
7#70521C211yesmaximal, characteristic, complemented
8#70515C211yesmaximal, characteristic, complemented
9#70306C214yesmaximal, complemented
10#25690—31nomaximal, Sylow
11C2×C3×#125C2211yescharacteristic
12C3×C4×#41C2211yescharacteristic
13#5830C2212yes—
14#5585—22no—
15#5553—22no—
16#5550—24no—
17#5535—22no—
18#5521C2212yes—
19#5477—22no—
20C2×#874C2211yescharacteristic
21#5470C2211yescharacteristic
22C3×#479—22no—
23C3×#478—22no—
24C3×#428C2212yes—
25C3×#423C2211yescharacteristic
26C3×#417C2212yes—
27C2×#837C2212yes—
28C2×#823C2214yes—
29#5065—22no—
30#5064—22no—
31#4991—24no—
32#4988C2214yes—
33#4964C2214yes—
34#4932C2211yescharacteristic
35C4×#208C2211yescharacteristic
36#4555C2214yes—
37#4532C2212yes—
38#3508—31no—
39#3166—31no—
40#3080D311yescharacteristic, complemented
41#2641—31no—
42#2037—34no—
43#1995—31no—
44#1965—31no—
45#1856—34no—
46#1759—31no—
47C2×C3×#41C2311yescharacteristic
48C2×#235—24no—
49C22×D12D414yes—
50C22×D12C2312yes—
51C2×C3×M16C2312yes—
52C3×#125—21no—
53C3×#125—21no—
54C3×#125—22no—
55C3×C4×Q8D412yes—
56C3×C4×D4—22no—
57C3×C4×D4D412yes—
58C3×#117—21no—
59C3×#117C2311yescharacteristic
60C2×C3×C4⋊C4—22no—
61C2×C3×C4⋊C4C2312yes—
62C2×C3×C42—21no—
63C2×C3×C42C2311yescharacteristic
64#947—22no—
65#940—22no—
66#937—24no—
67#930—24no—
68#930—24no—
69C2×(C2×D3)⋊C4—28no—
70#926C2312yes—
71#924—22no—
72C2×C4×Dic3C2312yes—
73C2×#208C2312yes—
74#894—28no—
75#893—24no—
76#884—28no—
77D3×C22⋊C4—24no—
78#875—24no—
79#874—21no—
80#874—21no—
81#874—22no—
82C4×D12—24no—
83C4×D12—24no—
84#870D412yes—
85#869D412yes—
86C4×Dic6—22no—
87C3×#107—22no—
88C3×#106—22no—
89C3×C4⋊C8—22no—
90C3×#103—22no—
91C3×#102—22no—
92#837—24no—
93C3⋊#95—22no—
94#823—28no—
95#808—24no—
96#807—24no—
97#803—21no—
98C4×C3⋊C8—21no—
99#559—62no—
100#550—62no—
101#546—64no—
102#540—62no—
103#535—32no—
104#533—31no—
105#532—62no—
106C2×#127—31no—
107C2×#125C2×D311yescharacteristic
108#518—32no—
109C4×#41C2×D311yescharacteristic
110#501—62no—
111#499—62no—
112#479—22no—
113#478—22no—
114#428—34no—
115#428C2×D312yes—
116#423C2×D311yescharacteristic
117C2×#104—32no—
118#417C2×D312yes—
119C2×#100—34no—
120C4×M16—31no—
121#394—64no—
122#390—34no—
123#376—31no—
124#339—34no—
125#339—32no—
126C23×D3C2×D414yes—
127C3×#41—22no—
128C3×#41—22no—
129C2×C3×Q8C2×D411yescharacteristic
130C2×C3×D4—21no—
131C2×C3×D4C2×D411yescharacteristic
132C22×C3×C4—22no—
133C22×C3×C4—21no—
134C22×C3×C4—22no—
135C22×C3×C4C2×D412yes—
136C22×C3×C4C2411yescharacteristic
137C2×#67—28no—
138C22×Dic3—22no—
139#235—88no—
140C2×D12—24no—
141C2×D12—22no—
142C2×D12—22no—
143C2×D12—24no—
144C2×D12—24no—
145C2×D12—24no—
146C2×D12—84no—
147C2×C4×D3—28no—
148C2×Dic6—22no—
149C3×M16—24no—
150C2×C3×C8—22no—
151C3×C4⋊C4—22no—
152C3×C4⋊C4—22no—
153C3×C4⋊C4—22no—
154C3×C4⋊C4—42no—
155C3×C4⋊C4—42no—
156C3×C4⋊C4C2×D412yes—
157C3×C22⋊C4—21no—
158C3×C22⋊C4—22no—
159C3×C42—22no—
160C3×C42—21no—
161C3×C42C2×D411yescharacteristic
162C3×C42C2×D411yescharacteristic
163C3×C42C2×D412yes—
164#217—22no—
165(C2×D3)⋊C4—28no—
166(C2×D3)⋊C4—28no—
167(C2×D3)⋊C4—88no—
168C4⋊Dic3—22no—
169C4⋊Dic3C2×D414yes—
170Dic3⋊C4—24no—
171C4×Dic3—22no—
172C4×Dic3—22no—
173C4×Dic3—22no—
174#208—22no—
175#208#4114yescomplemented
176C2×C3⋊C8—22no—
177C2×#41—64no—
178C2×#41C22×D311yescharacteristic
179C22×D4—34no—
180C22×D4—32no—
181C2×M16—32no—
182C2×M16C22×D312yes—
183C4⋊Q8—32no—
184#127—62no—
185#127—61no—
186#127—61no—
187#127—62no—
188#125—21no—
189#125—21no—
190#125—22no—
191#125—32no—
192#124—64no—
193#124—62no—
194#123—68no—
195C22⋊Q8—62no—
196#121—68no—
197#121—64no—
198#121—64no—
199#120—64no—
200C4×Q8—62no—
201C4×Q8#6712yes—
202C4×D4—22no—
203C4×D4—64no—
204C4×D4—64no—
205C4×D4—64no—
206C4×D4#6712yes—
207#117—21no—
208#117—32no—
209#117C22×D311yescharacteristic
210C2×C4⋊C4—22no—
211C2×C4⋊C4C22×D312yes—
212C2×C22⋊C4—64no—
213C2×C22⋊C4—68no—
214C2×C42—21no—
215C2×C42—32no—
216C2×C42C22×D311yescharacteristic
217#107—22no—
218#107—64no—
219#106—22no—
220#106—64no—
221C4⋊C8—22no—
222#104—64no—
223#103—22no—
224#102—22no—
225#100—68no—
226#97—61no—
227C4×C8—61no—
228#95—62no—
229C23×C3—21no—
230C23×C3C22×D411yescharacteristic
231C22×D3—24no—
232C22×D3—28no—
233C22×D3—48no—
234C3×Q8—21no—
235C3×Q8—21no—
236C3×D4—21no—
237C3×D4—21no—
238C3×D4—42no—
239C2×C3×C4—21no—
240C2×C3×C4—22no—
241C2×C3×C4—42no—
242C2×C3×C4—44no—
243C2×C3×C4—42no—
244C2×C3×C4—44no—
245C2×C3×C4—42no—
246C2×C3×C4#12012yes—
247C2×C3×C4#12012yes—
248C2×C3×C4C22×D411yescharacteristic
249C2×C3×C4C22×D411yescharacteristic
250C2×C3×C4C2×#4112yes—
251C2×C3×C42−1+412yes—
252#67—88no—
253C2×Dic3—24no—
254C2×Dic3—44no—
255C2×Dic3—42no—
256D12—48no—
257D12—42no—
258D12—42no—
259D12—44no—
260D12—84no—
261C4×D3—88no—
262Dic6—44no—
263C3×C8—42no—
264C3⋊C8—24no—
265C24—34no—
266#41—22no—
267#41—22no—
268#41—248no—
269C2×Q8—62no—
270C2×Q8C2×#6711yescharacteristic
271C2×D4—21no—
272C2×D4—68no—
273C2×D4—64no—
274C2×D4—64no—
275C2×D4—62no—
276C2×D4—64no—
277C2×D4—64no—
278C2×D4—62no—
279C2×D4—244no—
280C2×D4C2×#6711yescharacteristic
281C22×C4—22no—
282C22×C4—21no—
283C22×C4—22no—
284C22×C4—68no—
285C22×C4—62no—
286C22×C4D3×D412yes—
287C22×C4C23×D311yescharacteristic
288M16—24no—
289M16—34no—
290M16—62no—
291C2×C8—22no—
292C2×C8—62no—
293C4⋊C4—22no—
294C4⋊C4—22no—
295C4⋊C4—22no—
296C4⋊C4—34no—
297C4⋊C4—42no—
298C4⋊C4—42no—
299C4⋊C4—62no—
300C4⋊C4—64no—
301C4⋊C4C2×#6712yes—
302C22⋊C4—21no—
303C22⋊C4—22no—
304C22⋊C4—68no—
305C22⋊C4—68no—
306C22⋊C4—62no—
307C22⋊C4—248no—
308C42—22no—
309C42—21no—
310C42—62no—
311C42—62no—
312C42—62no—
313C42D3×D411yescharacteristic
314C42D3×D411yescharacteristic
315C42C2×#6712yes—
316C22×C3—22no—
317C22×C3—41no—
318C22×C3—42no—
319C22×C3C2×#12011yescharacteristic
320C22×C3#54712yes—
321C2×D3—44no—
322C2×D3—48no—
323C2×D3—88no—
324C3×C4—22no—
325C3×C4—22no—
326C3×C4—41no—
327C3×C4—42no—
328C3×C4—82no—
329C3×C4#53211yescharacteristic
330C3×C4#53611yescharacteristic
331C3×C4#54812yes—
332Dic3—44no—
333Dic3—82no—
334C23—21no—
335C23—64no—
336C23—68no—
337C23—128no—
338C23C2×D3×D411yescharacteristic
339Q8—21no—
340Q8—21no—
341Q8—124no—
342D4—21no—
343D4—21no—
344D4—42no—
345D4—128no—
346D4—124no—
347D4—122no—
348D4—122no—
349D4—244no—
350D4—248no—
351C2×C4—22no—
352C2×C4—21no—
353C2×C4—42no—
354C2×C4—44no—
355C2×C4—42no—
356C2×C4—44no—
357C2×C4—42no—
358C2×C4—64no—
359C2×C4—124no—
360C2×C4—122no—
361C2×C4—248no—
362C2×C4#93712yes—
363C2×C4#93712yes—
364C2×C4C2×D3×D411yescharacteristic
365C2×C4#100712yes—
366C2×C4D3×#4112yes—
367C2×C4C22×#6711yescharacteristic
368C8—42no—
369C8—64no—
370C2×C3—22no—
371C2×C3—81no—
372C2×C3#243211yescharacteristic
373C2×C3#301912yes—
374D3—84no—
375C22—22no—
376C22—42no—
377C22—41no—
378C22—128no—
379C22—124no—
380C22—248no—
381C22#563112yes—
382C22C2×#93711yescharacteristic
383C4—22no—
384C4—22no—
385C4—42no—
386C4—41no—
387C4—82no—
388C4—124no—
389C4—242no—
390C4#563312yes—
391C4#581611yescharacteristic
392C4#583211yescharacteristic
393C3#2569011yescharacteristic, complemented, Sylow
394C2—22no—
395C2—81no—
396C2—244no—
397C2#8275911yescharacteristic
398C2#8400512yes—
subgroupG/H
centerC22C2×#937
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#3080C2
socleC22×C3C2×#120
prime radical#85848C2
hypercenter#3080D3
hypocenterC3#25690
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3080

Complements

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

NH ≅ G/Ntype
2. C3×#3080C2semidirect
3. #83067C2semidirect
4. #83015C2semidirect
5. #71643C2semidirect
6. #70674C2semidirect
7. #70521C2semidirect
8. #70515C2semidirect
9. #70306C2semidirect
40. #3080D3semidirect
175. #208#41semidirect
393. C3#25690semidirect

Maximal subgroups

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

subgroupindex
#858482
C3×#30802
#830672
#830152
#716432
#706742
#705212
#705152
#703062
#256903