Finite Group Databasebeta

← #10848613

Subgroups — #10848613

Order 768.

1120 conjugacy classes of subgroups.

The subgroup lattice has 412 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#3509C211yesmaximal, characteristic, complemented
2#83336C211yesmaximal, characteristic, complemented
3#83251C211yesmaximal, characteristic, complemented
4#82526C211yesmaximal, characteristic
5#71867C214yesmaximal, complemented
6#71839C211yesmaximal, characteristic, complemented
7#71789C211yesmaximal, characteristic, complemented
8#71661C214yesmaximal, complemented
9C4×#837C211yesmaximal, characteristic, complemented
10#25591—31nomaximal, Sylow
11C3×#557—22no—
12C3×#556—22no—
13C3×#552—24no—
14C3×#541—22no—
15C3×#536C2212yes—
16C3×#533—22no—
17C3×#533C2211yescharacteristic
18C2×C3×C4⋊Q8C2211yescharacteristic
19C3×#519C2212yes—
20C3×C4×#41C2211yescharacteristic
21C2×#949C2212yes—
22C2×#947C2214yes—
23C2×#874C2211yescharacteristic
24C2×#837C2212yes—
25C2×#836C2214yes—
26#5088—22no—
27#5067—22no—
28#5027—22no—
29#5015—22no—
30#5001C2212yes—
31#5001C2214yes—
32#4988—24no—
33C42×Dic3C2211yescharacteristic
34#4924C2211yescharacteristic
35C4×#208C2211yescharacteristic
36#4532C2214yes—
37#4532C2212yes—
38#3509D311yescharacteristic, complemented
39#3138—31no—
40#3038—31no—
41#2858—31no—
42#2003—34no—
43#1994—31no—
44#1966—31no—
45#1888—34no—
46C4×#104—31no—
47C2×C3×#41C2311yescharacteristic
48C22×C3×Q8C2312yes—
49C22×D12—22no—
50C22×D12C2312yes—
51C3×C4⋊Q8—22no—
52C3×C4⋊Q8—21no—
53C3×C4⋊Q8—21no—
54C3×C4⋊Q8—22no—
55C3×C4⋊Q8D412yes—
56C3×#126—28no—
57C3×#125—24no—
58C3×#125—28no—
59C3×#125D412yes—
60C3×#124—22no—
61C3×#123—24no—
62C3×C22⋊Q8—24no—
63C3×C22⋊Q8—22no—
64C3×C22⋊Q8—24no—
65C3×C4×Q8—24no—
66C3×C4×Q8—24no—
67C3×C4×Q8—24no—
68C3×C4×Q8—22no—
69C3×C4×Q8D412yes—
70C3×C4×D4—22no—
71C3×C4×D4D412yes—
72C3×#117—24no—
73C3×#117—24no—
74C3×#117—21no—
75C3×#117C2311yescharacteristic
76C3×#117C2312yes—
77C2×C3×C4⋊C4D414yes—
78C2×C3×C42—21no—
79C2×C3×C42C2311yescharacteristic
80#949—44no—
81#947—28no—
82#947—44no—
83C2×#215—22no—
84C2×#213—22no—
85C2×(C2×D3)⋊C4—28no—
86C2×C4×Dic3—22no—
87C2×C4×Dic3C2312yes—
88C2×#208C2312yes—
89C2×#208C2312yes—
90#874—22no—
91#874—21no—
92#874—21no—
93#837—24no—
94#836—44no—
95#809—24no—
96#809—28no—
97C4⋊(C3⋊C8)—22no—
98#803—21no—
99C4×C3⋊C8—21no—
100C2×AGL(1,ℤ/8)—32no—
101#557—22no—
102#556—22no—
103#552—24no—
104#541—22no—
105#536C2×D312yes—
106#533—22no—
107#533C2×D311yescharacteristic
108C2×C4⋊Q8C2×D311yescharacteristic
109C2×#127—31no—
110C2×#124—34no—
111#519C2×D312yes—
112C4×#41C2×D311yescharacteristic
113#490—62no—
114#486—62no—
115#460—62no—
116#459—62no—
117#423—31no—
118C2×#104—32no—
119#418—34no—
120#418—32no—
121C2×#101—34no—
122C4×M16—31no—
123#390—64no—
124C43—31no—
125#339—32no—
126#339—34no—
127C23×D3—22no—
128C3×#41—22no—
129C3×#41—22no—
130C2×C3×Q8—22no—
131C2×C3×Q8—22no—
132C2×C3×Q8—22no—
133C2×C3×Q8—24no—
134C2×C3×Q8—24no—
135C2×C3×Q8C2×D411yescharacteristic
136C2×C3×D4—21no—
137C2×C3×D4C2×D411yescharacteristic
138C22×C3×C4—22no—
139C22×C3×C4—21no—
140C22×C3×C4C2×D414yes—
141C22×C3×C4C2×D412yes—
142C22×C3×C4C2411yescharacteristic
143C22×Dic3—22no—
144C2×D12—22no—
145C2×D12—22no—
146C2×D12—24no—
147C2×D12—24no—
148C2×D12—48no—
149C2×D12—42no—
150C2×D12—42no—
151C3×C4⋊C4—28no—
152C3×C4⋊C4—28no—
153C3×C4⋊C4—24no—
154C3×C4⋊C4—24no—
155C3×C4⋊C4—22no—
156C3×C4⋊C4—22no—
157C3×C4⋊C4—24no—
158C3×C4⋊C4C2×D412yes—
159C3×C4⋊C4C2×D414yes—
160C3×C22⋊C4—28no—
161C3×C22⋊C4—21no—
162C3×C22⋊C4—22no—
163C3×C22⋊C4—22no—
164C3×C42—28no—
165C3×C42—22no—
166C3×C42—21no—
167C3×C42—22no—
168C3×C42C2×D411yescharacteristic
169C3×C42C2×D411yescharacteristic
170C3×C42C2×D412yes—
171#215—44no—
172#213—44no—
173(C2×D3)⋊C4—416no—
174C4×Dic3—22no—
175C4×Dic3—44no—
176C4×Dic3#4114yes—
177#208—22no—
178#208—22no—
179#208—24no—
180C2×C3⋊C8—22no—
181C2×C3⋊C8—22no—
182C2×#41C22×D311yescharacteristic
183C22×Q8C22×D312yes—
184C22×D4—32no—
185C22×D4—62no—
186AGL(1,ℤ/8)—124no—
187C2×SD16—62no—
188C2×D8—62no—
189C2×M16—32no—
190C2×M16—32no—
191C4⋊Q8—22no—
192C4⋊Q8—21no—
193C4⋊Q8—21no—
194C4⋊Q8—22no—
195C4⋊Q8#6712yes—
196#127—62no—
197#127—61no—
198#127—61no—
199#126—28no—
200#125—24no—
201#125—28no—
202#125#6712yes—
203#124—22no—
204#124—68no—
205#124—124no—
206#123—24no—
207C22⋊Q8—24no—
208C22⋊Q8—22no—
209C22⋊Q8—24no—
210C4×Q8—24no—
211C4×Q8—24no—
212C4×Q8—24no—
213C4×Q8—22no—
214C4×Q8#6712yes—
215C4×D4—22no—
216C4×D4#6712yes—
217#117—24no—
218#117—24no—
219#117—21no—
220#117C22×D311yescharacteristic
221#117C22×D312yes—
222C2×C4⋊C4#6714yes—
223C2×C22⋊C4—68no—
224C2×C42—21no—
225C2×C42—32no—
226C2×C42—62no—
227C2×C42C22×D311yescharacteristic
228C4⋊C8—62no—
229#104—64no—
230#102—64no—
231#102—68no—
232#101—124no—
233#97—61no—
234C4×C8—61no—
235C23×C3—21no—
236C23×C3C22×D411yescharacteristic
237C22×D3—48no—
238C22×D3—42no—
239C22×D3—44no—
240C3×Q8—21no—
241C3×Q8—21no—
242C3×Q8—42no—
243C3×Q8—42no—
244C3×Q8—44no—
245C3×Q8—44no—
246C3×D4—21no—
247C3×D4—21no—
248C3×D4—42no—
249C2×C3×C4—24no—
250C2×C3×C4—22no—
251C2×C3×C4—24no—
252C2×C3×C4—24no—
253C2×C3×C4—21no—
254C2×C3×C4—28no—
255C2×C3×C4—42no—
256C2×C3×C4—44no—
257C2×C3×C4#12012yes—
258C2×C3×C4#12012yes—
259C2×C3×C4C22×D411yescharacteristic
260C2×C3×C4C22×D411yescharacteristic
261C2×C3×C4C2×#4112yes—
262C2×C3×C4H(5,𝔽2)12yes—
263C2×Dic3—24no—
264C2×Dic3—44no—
265D12—42no—
266D12—42no—
267D12—44no—
268D12—84no—
269D12—84no—
270C3⋊C8—42no—
271C3⋊C8—42no—
272C24—62no—
273#41—22no—
274#41—22no—
275C2×Q8—22no—
276C2×Q8—24no—
277C2×Q8—24no—
278C2×Q8—22no—
279C2×Q8—22no—
280C2×Q8C2×#6711yescharacteristic
281C2×D4—21no—
282C2×D4—64no—
283C2×D4—64no—
284C2×D4—62no—
285C2×D4—62no—
286C2×D4—128no—
287C2×D4—122no—
288C2×D4—122no—
289C2×D4C2×#6711yescharacteristic
290C22×C4—22no—
291C22×C4—21no—
292C22×C4—62no—
293C22×C4C2×#6714yes—
294C22×C4C2×#6712yes—
295C22×C4C23×D311yescharacteristic
296SD16—124no—
297D8—124no—
298M16—62no—
299M16—62no—
300M16—64no—
301C2×C8—62no—
302C2×C8—62no—
303C4⋊C4—28no—
304C4⋊C4—28no—
305C4⋊C4—24no—
306C4⋊C4—24no—
307C4⋊C4—22no—
308C4⋊C4—22no—
309C4⋊C4—24no—
310C4⋊C4C2×#6712yes—
311C4⋊C4C2×#6714yes—
312C22⋊C4—28no—
313C22⋊C4—21no—
314C22⋊C4—22no—
315C22⋊C4—22no—
316C22⋊C4—1216no—
317C42—28no—
318C42—22no—
319C42—21no—
320C42—22no—
321C42—34no—
322C42—62no—
323C42—124no—
324C42C2×#6711yescharacteristic
325C42C2×#6711yescharacteristic
326C42C2×#6712yes—
327C22×C3—22no—
328C22×C3—41no—
329C22×C3—42no—
330C22×C3C2×#12011yescharacteristic
331C22×C3#54612yes—
332C2×D3—88no—
333C2×D3—82no—
334C2×D3—84no—
335C3×C4—22no—
336C3×C4—44no—
337C3×C4—41no—
338C3×C4—42no—
339C3×C4—44no—
340C3×C4#45414yes—
341C3×C4#53211yescharacteristic
342C3×C4#53511yescharacteristic
343C3×C4#55012yes—
344Dic3—44no—
345C23—21no—
346C23—128no—
347C23—124no—
348C23—122no—
349C23C22×#6711yescharacteristic
350Q8—21no—
351Q8—21no—
352Q8—44no—
353Q8—42no—
354Q8—42no—
355Q8—44no—
356D4—21no—
357D4—21no—
358D4—42no—
359D4—122no—
360D4—122no—
361D4—124no—
362D4—244no—
363D4—244no—
364C2×C4—24no—
365C2×C4—28no—
366C2×C4—22no—
367C2×C4—24no—
368C2×C4—24no—
369C2×C4—21no—
370C2×C4—44no—
371C2×C4—42no—
372C2×C4—64no—
373C2×C4—124no—
374C2×C4#95312yes—
375C2×C4#95312yes—
376C2×C4D3×#4112yes—
377C2×C4#100912yes—
378C2×C4C22×#6711yescharacteristic
379C2×C4C22×#6711yescharacteristic
380C8—122no—
381C8—122no—
382C2×C3—22no—
383C2×C3—81no—
384C2×C3#243111yescharacteristic
385C2×C3C2×#45412yes—
386D3—162no—
387C22—22no—
388C22—41no—
389C22—42no—
390C22—248no—
391C22—242no—
392C22—244no—
393C22#582912yes—
394C22C2×#95311yescharacteristic
395C4—22no—
396C4—44no—
397C4—41no—
398C4—42no—
399C4—44no—
400C4—124no—
401C4#524514yes—
402C4#583011yescharacteristic
403C4#583211yescharacteristic
404C4#583412yes—
405C3#2559111yescharacteristic, complemented, Sylow
406C2—22no—
407C2—81no—
408C2—482no—
409C2#8409211yescharacteristic
410C2C2×#524512yes—
subgroupG/H
centerC22C2×#953
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#3509C2
socleC22×C3C2×#120
prime radical#82526C2
hypercenter#3509D3
hypocenterC3#25591
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3509

Complements

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

NH ≅ G/Ntype
1. C3×#3509C2semidirect
2. #83336C2semidirect
3. #83251C2semidirect
5. #71867C2semidirect
6. #71839C2semidirect
7. #71789C2semidirect
8. #71661C2semidirect
9. C4×#837C2semidirect
38. #3509D3semidirect
405. C3#25591semidirect

Maximal subgroups

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

subgroupindex
C3×#35092
#833362
#832512
#825262
#718672
#718392
#717892
#716612
C4×#8372
#255913