Finite Group Databasebeta

← #10691735

Subgroups — #10691735

Order 768.

560 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
1#83166C211yesmaximal, characteristic
2#78522C211yesmaximal, characteristic, complemented
3C3×#1965C211yesmaximal, characteristic, complemented
4C2×#4544C211yesmaximal, characteristic, complemented
5#70503C211yesmaximal, characteristic, complemented
6#67714C211yesmaximal, characteristic, complemented
7#67668C211yesmaximal, characteristic, complemented
8#9273—31nomaximal, Sylow
9C2×C3×#127C2211yescharacteristic
10C2×#907—21no—
11#5477C2211yescharacteristic
12C3×#423C2211yescharacteristic
13C2×C3×#100—21no—
14C2×C3×#100—21no—
15C3×C4×M16C2211yescharacteristic
16#5217C414yes—
17#5129—21no—
18#5073—21no—
19#5072—22no—
20#5070—21no—
21#4925C2211yescharacteristic
22C2×C4⋊(C3⋊C8)C2211yescharacteristic
23C2×C4⋊(C3⋊C8)C2211yescharacteristic
24#4706—22no—
25#4688—21no—
26C8×D12—21no—
27#4544—21no—
28#4544—21no—
29#4485—21no—
30#4482—21no—
31#3205—31no—
32#2935—31no—
33#1965D311yescharacteristic, complemented
34#1547—31no—
35C2×#331—31no—
36#1301—31no—
37#1283—31no—
38C22×C3×D4—21no—
39C22×C3×D4—21no—
40C22×C3×D4C2×C412yes—
41C2×#235—21no—
42C2×C3×M16—21no—
43C2×C3×M16D411yescharacteristic
44C2×C3×M16D411yescharacteristic
45C3×#127—21no—
46C3×#127—21no—
47C3×#127—21no—
48C3×#127—21no—
49C2×C3×C42C2311yescharacteristic
50C2×#213—24no—
51#930—21no—
52#930C2×C412yes—
53C22×C3⋊C8—21no—
54C22×C3⋊C8—21no—
55#907—42no—
56#907—42no—
57C2×ΓL(1,𝔽25)—22no—
58C2×D3×C8—22no—
59#875—21no—
60#874D411yescharacteristic
61C4×D12—21no—
62C4×D12D412yes—
63#869D411yescharacteristic
64C3×C4⋊C8—21no—
65C3×C4⋊C8—21no—
66C3×#100—42no—
67C3×#100—42no—
68C3×#97—21no—
69C3×C4×C8—21no—
70#832—24no—
71#830C2×C412yes—
72(C2×D3)⋊C8—22no—
73(C2×D3)⋊C8—22no—
74C4⋊(C3⋊C8)—21no—
75C4⋊(C3⋊C8)—21no—
76C4⋊(C3⋊C8)—21no—
77C4⋊(C3⋊C8)—21no—
78C4⋊(C3⋊C8)D411yescharacteristic
79C4⋊(C3⋊C8)D411yescharacteristic
80C4×C3⋊C8—21no—
81C2×#131—61no—
82#532—31no—
83C2×#127C2×D311yescharacteristic
84#493—61no—
85#486—61no—
86#459—62no—
87#447—34no—
88#436—61no—
89#435—62no—
90C8×D4—61no—
91#431—31no—
92#423C2×D311yescharacteristic
93C2×C4⋊C8—31no—
94C2×C4⋊C8—31no—
95C2×#100—21no—
96C2×#100—21no—
97#408—61no—
98C4×M16C2×D311yescharacteristic
99#332—61no—
100#331—61no—
101#331—61no—
102#331—61no—
103C24×C3—22no—
104C2×C3×D4—22no—
105C2×C3×D4—22no—
106C2×C3×D4—41no—
107C2×C3×D4—41no—
108C2×C3×D4—41no—
109C2×C3×D4—41no—
110C2×C3×D4—41no—
111C2×C3×D4—41no—
112C2×C3×D4—41no—
113C2×C3×D4—41no—
114C2×C3×D4—81no—
115C2×C3×D4—81no—
116C2×C3×D4—82no—
117C22×C3×C4C22×C411yescharacteristic
118C22×C3×C4C2×D411yescharacteristic
119C22×C3×C4#4111yescharacteristic
120C2×#67—22no—
121#235—41no—
122#235—41no—
123#235—41no—
124#235—41no—
125C2×D12—21no—
126C2×D12—21no—
127C2×D12C22⋊C412yes—
128C2×C4×D3—22no—
129C2×Dic6—21no—
130C3×M16—21no—
131C3×M16—21no—
132C3×M16—21no—
133C3×M16—41no—
134C3×M16—41no—
135C3×M16D812yescomplemented
136C2×C3×C8—21no—
137C2×C3×C8—21no—
138C2×C3×C8—41no—
139C3×C42C2×D411yescharacteristic
140C3×C42C2×D411yescharacteristic
141C3×C42C2×D411yescharacteristic
142C3×C42#4111yescharacteristic
143#213—84no—
144(C2×D3)⋊C4—22no—
145(C2×D3)⋊C4—22no—
146C4⋊Dic3—21no—
147C4⋊Dic3C22⋊C412yes—
148C2×C3⋊C8—22no—
149C2×C3⋊C8—21no—
150C2×C3⋊C8—21no—
151C2×C3⋊C8—41no—
152C2×C3⋊C8—41no—
153C2×C3⋊C8—41no—
154C2×C3⋊C8—81no—
155ΓL(1,𝔽25)—44no—
156D3×C8—44no—
157C2×#41—61no—
158C22×D4—21no—
159C22×D4—21no—
160C22×D4C4×D312yes—
161C2×D8—64no—
162#131—122no—
163#131—122no—
164C2×M16—21no—
165C2×M16—62no—
166C2×M16#6711yescharacteristic
167C2×M16#6711yescharacteristic
168C22×C8—62no—
169C22×C8—61no—
170C22×C8—61no—
171C4⋊Q8—31no—
172#127—21no—
173#127—21no—
174#127—21no—
175#127—21no—
176#127—31no—
177#124—61no—
178#121—32no—
179#121—61no—
180C4×D4—32no—
181C4×D4—61no—
182C2×C42C22×D311yescharacteristic
183C4⋊C8—21no—
184C4⋊C8—21no—
185C4⋊C8—31no—
186C4⋊C8—31no—
187C4⋊C8—61no—
188C4⋊C8—61no—
189C4⋊C8—61no—
190C4⋊C8—61no—
191#102—64no—
192#100—42no—
193#100—42no—
194C22⋊C8—32no—
195C22⋊C8—62no—
196C22⋊C8—62no—
197#97—21no—
198C4×C8—21no—
199C4×C8—61no—
200C23×C3—42no—
201C23×C3—42no—
202C23×C3—42no—
203C23×C3—82no—
204C23×C3—82no—
205C23×C3C4×D411yescharacteristic
206C22×D3—22no—
207C3×D4—44no—
208C3×D4—82no—
209C3×D4—82no—
210C3×D4—81no—
211C3×D4—81no—
212C3×D4—81no—
213C3×D4—81no—
214C3×D4—81no—
215C3×D4—81no—
216C2×C3×C4—21no—
217C2×C3×C4—21no—
218C2×C3×C4—41no—
219C2×C3×C4C2×C22⋊C411yescharacteristic
220C2×C3×C4C4×D411yescharacteristic
221C2×C3×C4#12011yescharacteristic
222C2×C3×C4#12111yescharacteristic
223C2×C3×C4#12111yescharacteristic
224C2×C3×C4#12311yescharacteristic
225C2×C3×C4C2×D811yescharacteristic
226C2×C3×C4#13511yescharacteristic
227C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
228C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
229#67—42no—
230#67—42no—
231C2×Dic3—22no—
232D12—42no—
233D12—41no—
234D12—41no—
235D12—41no—
236D12—41no—
237C4×D3—42no—
238C4×D3—42no—
239Dic6—41no—
240Dic6—41no—
241C3×C8—22no—
242C3×C8—41no—
243C3×C8—81no—
244C3⋊C8—42no—
245C3⋊C8—41no—
246C3⋊C8—41no—
247C3⋊C8—81no—
248C24—22no—
249#41—121no—
250#41—121no—
251#41—121no—
252#41—121no—
253C2×Q8—61no—
254C2×D4—22no—
255C2×D4—22no—
256C2×D4—32no—
257C2×D4—41no—
258C2×D4—41no—
259C2×D4—41no—
260C2×D4—41no—
261C2×D4—41no—
262C2×D4—41no—
263C2×D4—41no—
264C2×D4—41no—
265C2×D4—62no—
266C2×D4—61no—
267C2×D4—61no—
268C2×D4—81no—
269C2×D4—81no—
270C2×D4—82no—
271C22×C4—62no—
272C22×C4C2×C4×D311yescharacteristic
273C22×C4#23511yescharacteristic
274C22×C4C2×#6711yescharacteristic
275D8—244no—
276M16—21no—
277M16—21no—
278M16—21no—
279M16—41no—
280M16—41no—
281M16—124no—
282M16#21312yescomplemented
283C2×C8—21no—
284C2×C8—21no—
285C2×C8—41no—
286C2×C8—62no—
287C2×C8—61no—
288C2×C8—61no—
289C2×C8—124no—
290C2×C8—121no—
291C2×C8—121no—
292C2×C8—121no—
293C2×C8—241no—
294C4⋊C4—32no—
295C4⋊C4—61no—
296C22⋊C4—62no—
297C22⋊C4—62no—
298C42D3×D411yescharacteristic
299C42D3×D411yescharacteristic
300C42D3×D411yescharacteristic
301C42#23711yescharacteristic
302C22×C3—21no—
303C22×C3—21no—
304C22×C3—82no—
305C22×C3—82no—
306C22×C3—82no—
307C22×C3—162no—
308C22×C3—162no—
309C22×C3#38611yescharacteristic
310C22×C3C4×D811yescharacteristic
311C22×C3#44211yescharacteristic
312C2×D3—42no—
313C2×D3—42no—
314C2×D3—42no—
315C3×C4—21no—
316C3×C4—21no—
317C3×C4—21no—
318C3×C4#44711yescharacteristic
319C3×C4#44911yescharacteristic
320C3×C4#45911yescharacteristic
321C3×C4#46311yescharacteristic
322C3×C4#47111yescharacteristic
323C3×C4#47211yescharacteristic
324Dic3—42no—
325C23—42no—
326C23—42no—
327C23—42no—
328C23—62no—
329C23—82no—
330C23—82no—
331C23C4×#6711yescharacteristic
332Q8—121no—
333Q8—121no—
334D4—44no—
335D4—82no—
336D4—82no—
337D4—81no—
338D4—81no—
339D4—81no—
340D4—81no—
341D4—81no—
342D4—81no—
343D4—122no—
344D4—122no—
345D4—122no—
346D4—121no—
347D4—121no—
348D4—121no—
349D4—121no—
350C2×C4—21no—
351C2×C4—21no—
352C2×C4—41no—
353C2×C4—62no—
354C2×C4—122no—
355C2×C4—122no—
356C2×C4D3×C22⋊C411yescharacteristic
357C2×C4#67⋊C411yescharacteristic
358C2×C4#88311yescharacteristic
359C2×C4#88411yescharacteristic
360C2×C4C2×#21311yescharacteristic
361C2×C4#93211yescharacteristic
362C2×C4#93711yescharacteristic
363C2×C4#93811yescharacteristic
364C2×C4#94911yescharacteristic
365C2×C4#95011yescharacteristic
366C8—22no—
367C8—41no—
368C8—81no—
369C8—122no—
370C8—121no—
371C8—121no—
372C8—241no—
373C2×C3—21no—
374C2×C3—21no—
375C2×C3—162no—
376C2×C3#187911yescharacteristic
377C2×C3#190711yescharacteristic
378C2×C3#191411yescharacteristic
379D3—82no—
380C22—21no—
381C22—21no—
382C22—82no—
383C22—82no—
384C22—82no—
385C22—122no—
386C22—122no—
387C22—122no—
388C22—162no—
389C22—162no—
390C22#495611yescharacteristic
391C22C4×#21311yescharacteristic
392C22#501411yescharacteristic
393C4—21no—
394C4—21no—
395C4—21no—
396C4—122no—
397C4#474811yescharacteristic
398C4#474911yescharacteristic
399C4#503611yescharacteristic
400C4#503711yescharacteristic
401C4#505711yescharacteristic
402C4#507211yescharacteristic
403C3#927311yescharacteristic, complemented, Sylow
404C2—21no—
405C2—21no—
406C2—162no—
407C2—242no—
408C2#6965711yescharacteristic
409C2#7008811yescharacteristic
410C2#7008911yescharacteristic
subgroupG/H
centerC22#4956
commutatorC22×C3×C4C22×C4
FrattiniC2×C42C22×D3
FittingC3×#1965C2
socleC22×C3#386
prime radical#83166C2
cosocleC2×C3×C42C23
hypercenter#1965D3
hypocenterC3#9273
LCSG ▷ C22×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1965

Complements

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

NH ≅ G/Ntype
2. #78522C2semidirect
3. C3×#1965C2semidirect
4. C2×#4544C2semidirect
5. #70503C2semidirect
6. #67714C2semidirect
7. #67668C2semidirect
33. #1965D3semidirect
135. C3×M16D8semidirect
282. M16#213semidirect
403. C3#9273semidirect

Maximal subgroups

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

subgroupindex
#831662
#785222
C3×#19652
C2×#45442
#705032
#677142
#676682
#92733