Finite Group Databasebeta

← #10849656

Subgroups — #10849656

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
2#82509C211yesmaximal, characteristic, complemented
3#78507C211yesmaximal, characteristic, complemented
4#73429C211yesmaximal, characteristic, complemented
5C3×#1759C211yesmaximal, characteristic, complemented
6#70359C214yesmaximal, complemented
7#69876C211yesmaximal, characteristic, complemented
8#69441C211yesmaximal, characteristic, complemented
9#68758C214yesmaximal, complemented
10#25690—31nomaximal, Sylow
11C3×C4×#41C2211yescharacteristic
12#5830C2212yes—
13#5585—22no—
14#5553—22no—
15#5550—24no—
16#5535—22no—
17#5521C2212yes—
18#5477—22no—
19C2×#874C2211yescharacteristic
20#5470C2211yescharacteristic
21C2×#870C2211yescharacteristic
22C2×C3×#104C2212yes—
23C3×C4×M16C2211yescharacteristic
24C3×#376C2211yescharacteristic
25#5232C2212yes—
26C2×#823C2214yes—
27#5121C2214yes—
28#5022—22no—
29#5009—22no—
30#4995C2212yes—
31#4939C2214yes—
32#4924C2211yescharacteristic
33#4702—22no—
34#4683—22no—
35#4672—24no—
36C3×#339C2212yes—
37#4523C2214yes—
38#3508—31no—
39#3166—31no—
40#3080—31no—
41#2641—31no—
42#2037—34no—
43#1995—31no—
44#1965—31no—
45#1856—34no—
46#1759D311yescharacteristic, complemented
47C2×C3×#41C2311yescharacteristic
48C2×#235—24no—
49C22×D12D414yes—
50C22×D12C2312yes—
51C2×C3×M16C2312yes—
52C3×C4×Q8D412yes—
53C3×C4×D4—22no—
54C3×C4×D4D412yes—
55C3×#117—21no—
56C3×#117C2311yescharacteristic
57C2×C3×C42—21no—
58C2×C3×C42C2312yes—
59C2×C3×C42C2311yescharacteristic
60#947—22no—
61#940—22no—
62#937—24no—
63#930—24no—
64#930—24no—
65C2×(C2×D3)⋊C4—28no—
66#926C2312yes—
67C2×C4⋊Dic3C2312yes—
68#924—22no—
69C2×Dic3⋊C4—22no—
70C2×#208C2312yes—
71#894—28no—
72#893—24no—
73#884—28no—
74D3×C22⋊C4—24no—
75#875—24no—
76#874—21no—
77#874—21no—
78#874—22no—
79C4×D12—24no—
80C4×D12—24no—
81#870—21no—
82#870—21no—
83#870—22no—
84#870D412yes—
85#869D412yes—
86C4×Dic6—22no—
87C3×#104—24no—
88C3×#97—21no—
89C3×C4×C8—21no—
90C3×#95—22no—
91#835—22no—
92#832—22no—
93#823—28no—
94#818—24no—
95#817—24no—
96#808—22no—
97#807—22no—
98C4⋊(C3⋊C8)—22no—
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×#125—31no—
108#518—32no—
109C4×#41C2×D311yescharacteristic
110#501—62no—
111#499—62no—
112#479—62no—
113#478—62no—
114#428—34no—
115#428—32no—
116#423—31no—
117C2×#104C2×D312yes—
118#417—32no—
119C2×#100—34no—
120C4×M16C2×D311yescharacteristic
121#394—64no—
122#390—34no—
123#376C2×D311yescharacteristic
124#339—34no—
125#339C2×D312yes—
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—22no—
150C3×M16#4114yescomplemented
151C2×C3×C8—22no—
152C3×C4⋊C4—22no—
153C3×C4⋊C4C2×D412yes—
154C3×C22⋊C4—21no—
155C3×C22⋊C4—22no—
156C3×C42—22no—
157C3×C42—21no—
158C3×C42—22no—
159C3×C42—22no—
160C3×C42C2×D412yes—
161C3×C42C2×D411yescharacteristic
162C3×C42C2×D411yescharacteristic
163#217—22no—
164(C2×D3)⋊C4—28no—
165(C2×D3)⋊C4—28no—
166(C2×D3)⋊C4—88no—
167C4⋊Dic3—22no—
168C4⋊Dic3—22no—
169C4⋊Dic3—22no—
170C4⋊Dic3C2×D414yes—
171Dic3⋊C4—24no—
172Dic3⋊C4—42no—
173Dic3⋊C4—42no—
174C4×Dic3—22no—
175#208—24no—
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—62no—
186#127—61no—
187#127—61no—
188#125—32no—
189#125—62no—
190#125—61no—
191#125—61no—
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×Q8D1212yes—
202C4×D4—22no—
203C4×D4—64no—
204C4×D4—64no—
205C4×D4—64no—
206C4×D4D1212yes—
207#117—21no—
208#117—32no—
209#117C22×D311yescharacteristic
210C2×C4⋊C4—32no—
211C2×C4⋊C4—62no—
212C2×C22⋊C4—64no—
213C2×C22⋊C4—68no—
214C2×C42—21no—
215C2×C42C22×D312yes—
216C2×C42C22×D311yescharacteristic
217#107—64no—
218#107—62no—
219#106—64no—
220#106—62no—
221C4⋊C8—62no—
222#104—24no—
223#103—62no—
224#102—62no—
225#100—68no—
226#97—21no—
227C4×C8—21no—
228#95—22no—
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—24no—
264C3⋊C8—42no—
265C24—34no—
266#41—22no—
267#41—22no—
268#41—248no—
269C2×Q8—62no—
270C2×Q8C2×D1211yescharacteristic
271C2×D4—21no—
272C2×D4—68no—
273C2×D4—64no—
274C2×D4—64no—
275C2×D4—62no—
276C2×D4—64no—
277C2×D4—62no—
278C2×D4—64no—
279C2×D4—244no—
280C2×D4C2×D1211yescharacteristic
281C22×C4—22no—
282C22×C4—21no—
283C22×C4—22no—
284C22×C4—68no—
285C22×C4—62no—
286C22×C4D3×D412yes—
287C22×C4C23×D311yescharacteristic
288M16—22no—
289M16—64no—
290M16#23514yescomplemented
291C2×C8—22no—
292C2×C8—62no—
293C4⋊C4—22no—
294C4⋊C4—34no—
295C4⋊C4—62no—
296C4⋊C4—64no—
297C4⋊C4—62no—
298C4⋊C4—62no—
299C4⋊C4—122no—
300C4⋊C4—122no—
301C4⋊C4C2×D1212yes—
302C22⋊C4—21no—
303C22⋊C4—22no—
304C22⋊C4—68no—
305C22⋊C4—68no—
306C22⋊C4—62no—
307C22⋊C4—248no—
308C42—22no—
309C42—21no—
310C42—22no—
311C42—22no—
312C42—62no—
313C42C2×D1212yes—
314C42D3×D411yescharacteristic
315C42D3×D411yescharacteristic
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—122no—
347D4—122no—
348D4—124no—
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#88212yes—
363C2×C4#88212yes—
364C2×C4C22×D1211yescharacteristic
365C2×C4C2×#23512yes—
366C2×C4C2×D3×D411yescharacteristic
367C2×C4#101012yes—
368C8—24no—
369C8—122no—
370C2×C3—22no—
371C2×C3—81no—
372C2×C3#243211yescharacteristic
373C2×C3#301912yes—
374D3—84no—
375C22—22no—
376C22—41no—
377C22—42no—
378C22—124no—
379C22—128no—
380C22—248no—
381C22C2×#88211yescharacteristic
382C22#555112yes—
383C4—22no—
384C4—22no—
385C4—41no—
386C4—42no—
387C4—82no—
388C4—124no—
389C4—242no—
390C4#547711yescharacteristic
391C4#552611yescharacteristic
392C4#555212yes—
393C3#2569011yescharacteristic, complemented, Sylow
394C2—22no—
395C2—81no—
396C2—244no—
397C2#7348911yescharacteristic
398C2#7899312yes—
subgroupG/H
centerC22C2×#882
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#1759C2
socleC22×C3C2×#120
prime radical#85848C2
hypercenter#1759D3
hypocenterC3#25690
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1759

Complements

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

NH ≅ G/Ntype
2. #82509C2semidirect
3. #78507C2semidirect
4. #73429C2semidirect
5. C3×#1759C2semidirect
6. #70359C2semidirect
7. #69876C2semidirect
8. #69441C2semidirect
9. #68758C2semidirect
46. #1759D3semidirect
150. C3×M16#41semidirect
290. M16#235semidirect
393. C3#25690semidirect

Maximal subgroups

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

subgroupindex
#858482
#825092
#785072
#734292
C3×#17592
#703592
#698762
#694412
#687582
#256903