Finite Group Databasebeta

← #10691731

Subgroups — #10691731

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#79303C211yesmaximal, characteristic
2#78522C211yesmaximal, characteristic, complemented
3#70432C211yesmaximal, characteristic, complemented
4C2×#4485C211yesmaximal, characteristic, complemented
5#69876C211yesmaximal, characteristic, complemented
6C3×#1283C211yesmaximal, characteristic, complemented
7#67745C211yesmaximal, characteristic, complemented
8#9273—31nomaximal, Sylow
9C2×#907—21no—
10#5477C2211yescharacteristic
11C2×#874C2211yescharacteristic
12C3×#431C2211yescharacteristic
13C2×C3×C4⋊C8C2211yescharacteristic
14C3×C4×M16C2211yescharacteristic
15C2×#823—21no—
16C2×#823—21no—
17#5129—21no—
18#4924C2211yescharacteristic
19C2×C4⋊(C3⋊C8)C2211yescharacteristic
20#4843—22no—
21#4732C414yes—
22#4706—22no—
23#4695—21no—
24#4694—21no—
25#4688—21no—
26C8×D12—21no—
27#4486—21no—
28#4485—21no—
29#4485—21no—
30#4485—21no—
31#3205—31no—
32#2935—31no—
33#1965—31no—
34#1547—31no—
35C2×#331—31no—
36#1301—31no—
37#1283D311yescharacteristic, complemented
38C2×#235—21no—
39C22×D12—21no—
40C22×D12—21no—
41C22×D12C2×C412yes—
42C2×C3×M16D411yescharacteristic
43C2×C3×M16D411yescharacteristic
44C22×C3×C8—21no—
45C2×C3×C42C2311yescharacteristic
46#930—21no—
47#930C2×C412yes—
48C2×#208—21no—
49C22×C3⋊C8—21no—
50#907—42no—
51#907—42no—
52C2×D24—24no—
53C2×ΓL(1,𝔽25)—22no—
54C2×D3×C8—22no—
55#875—21no—
56#874—21no—
57#874—21no—
58#874—21no—
59#874—21no—
60#874D411yescharacteristic
61C4×D12—21no—
62C4×D12D412yes—
63#869D411yescharacteristic
64C3×C4⋊C8—21no—
65C3×C4⋊C8—21no—
66C3×C4⋊C8—21no—
67C3×C22⋊C8C2×C412yes—
68C3×#97—21no—
69C3×C4×C8—21no—
70C3×C4×C8—21no—
71#823—42no—
72#823—42no—
73#821—24no—
74(C2×D3)⋊C8—22no—
75(C2×D3)⋊C8—22no—
76C4⋊(C3⋊C8)—21no—
77C4⋊(C3⋊C8)—21no—
78C4⋊(C3⋊C8)—21no—
79C4⋊(C3⋊C8)D411yescharacteristic
80C4⋊(C3⋊C8)D411yescharacteristic
81C2×#131—61no—
82#532—31no—
83C2×#127—31no—
84#493—61no—
85#486—61no—
86#459—62no—
87#447—34no—
88#436—61no—
89#435—62no—
90C8×D4—61no—
91#431C2×D311yescharacteristic
92#423—31no—
93C2×C4⋊C8—31no—
94C2×C4⋊C8C2×D311yescharacteristic
95C2×#100—61no—
96C2×#100—61no—
97#408—61no—
98C4×M16C2×D311yescharacteristic
99#332—61no—
100#331—61no—
101#331—61no—
102#331—61no—
103C23×D3—22no—
104C22×C3×C4C22×C411yescharacteristic
105C22×C3×C4C2×D411yescharacteristic
106C22×C3×C4#4111yescharacteristic
107C2×#67—22no—
108#235—41no—
109#235—41no—
110#235—41no—
111#235—41no—
112C2×D12—22no—
113C2×D12—21no—
114C2×D12—21no—
115C2×D12—22no—
116C2×D12—41no—
117C2×D12—41no—
118C2×D12—41no—
119C2×D12—41no—
120C2×D12—41no—
121C2×D12—41no—
122C2×D12—41no—
123C2×D12—41no—
124C2×D12—82no—
125C2×D12—81no—
126C2×D12—81no—
127C2×D12C22⋊C412yes—
128C2×C4×D3—22no—
129C2×Dic6—21no—
130C3×M16—21no—
131C3×M16—21no—
132C3×M16—21no—
133C3×M16D812yescomplemented
134C2×C3×C8—21no—
135C2×C3×C8—22no—
136C2×C3×C8—21no—
137C2×C3×C8—41no—
138C2×C3×C8—81no—
139C3×C42C2×D411yescharacteristic
140C3×C42C2×D411yescharacteristic
141C3×C42C2×D411yescharacteristic
142C3×C42#4111yescharacteristic
143(C2×D3)⋊C4—22no—
144(C2×D3)⋊C4—22no—
145C4⋊Dic3—21no—
146C4⋊Dic3C22⋊C412yes—
147#208—41no—
148#208—41no—
149C2×C3⋊C8—21no—
150C2×C3⋊C8—21no—
151C2×C3⋊C8—41no—
152C2×C3⋊C8—41no—
153C2×C3⋊C8—41no—
154D24—84no—
155ΓL(1,𝔽25)—44no—
156D3×C8—44no—
157C2×#41—61no—
158C22×D4—32no—
159C22×D4—61no—
160C22×D4—61no—
161C2×D8—64no—
162#131—122no—
163#131—122no—
164C2×M16—62no—
165C2×M16—61no—
166C2×M16D1211yescharacteristic
167C2×M16D1211yescharacteristic
168C22×C8—21no—
169C22×C8—62no—
170C22×C8—61no—
171C4⋊Q8—31no—
172#127—31no—
173#127—61no—
174#127—61no—
175#127—61no—
176#127—61no—
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—21no—
186C4⋊C8—31no—
187C4⋊C8—31no—
188C4⋊C8—61no—
189C4⋊C8—61no—
190C4⋊C8—61no—
191#102—64no—
192#100—122no—
193#100—122no—
194C22⋊C8—62no—
195C22⋊C8—62no—
196C22⋊C8C4×D312yes—
197#97—21no—
198C4×C8—21no—
199C4×C8—21no—
200C23×C3C4×D411yescharacteristic
201C22×D3—22no—
202C22×D3—42no—
203C22×D3—42no—
204C22×D3—42no—
205C22×D3—82no—
206C22×D3—82no—
207C2×C3×C4—21no—
208C2×C3×C4—21no—
209C2×C3×C4—41no—
210C2×C3×C4C2×C22⋊C411yescharacteristic
211C2×C3×C4C4×D411yescharacteristic
212C2×C3×C4#12011yescharacteristic
213C2×C3×C4#12111yescharacteristic
214C2×C3×C4#12111yescharacteristic
215C2×C3×C4#12311yescharacteristic
216C2×C3×C4C2×D811yescharacteristic
217C2×C3×C4#13511yescharacteristic
218C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
219C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
220#67—42no—
221#67—42no—
222C2×Dic3—22no—
223D12—42no—
224D12—44no—
225D12—41no—
226D12—41no—
227D12—41no—
228D12—41no—
229D12—82no—
230D12—82no—
231D12—81no—
232D12—81no—
233D12—81no—
234D12—81no—
235D12—81no—
236D12—81no—
237C4×D3—42no—
238C4×D3—42no—
239Dic6—41no—
240Dic6—41no—
241C3×C8—22no—
242C3×C8—42no—
243C3×C8—41no—
244C3×C8—81no—
245C3⋊C8—41no—
246C3⋊C8—41no—
247C3⋊C8—81no—
248C24—62no—
249#41—121no—
250#41—121no—
251#41—121no—
252#41—121no—
253C2×Q8—61no—
254C2×D4—32no—
255C2×D4—62no—
256C2×D4—62no—
257C2×D4—61no—
258C2×D4—62no—
259C2×D4—61no—
260C2×D4—121no—
261C2×D4—121no—
262C2×D4—121no—
263C2×D4—121no—
264C2×D4—121no—
265C2×D4—121no—
266C2×D4—121no—
267C2×D4—121no—
268C2×D4—241no—
269C2×D4—241no—
270C2×D4—242no—
271C22×C4—62no—
272C22×C4C2×C4×D311yescharacteristic
273C22×C4C2×D1211yescharacteristic
274C22×C4#23511yescharacteristic
275D8—244no—
276M16—21no—
277M16—21no—
278M16—21no—
279M16—124no—
280M16—121no—
281M16—121no—
282M16D2412yescomplemented
283C2×C8—21no—
284C2×C8—22no—
285C2×C8—21no—
286C2×C8—41no—
287C2×C8—61no—
288C2×C8—61no—
289C2×C8—81no—
290C2×C8—124no—
291C2×C8—121no—
292C2×C8—121no—
293C2×C8—121no—
294C4⋊C4—32no—
295C4⋊C4—61no—
296C22⋊C4—62no—
297C22⋊C4—62no—
298C42D3×D411yescharacteristic
299C42D3×D411yescharacteristic
300C42D3×D411yescharacteristic
301C42#23911yescharacteristic
302C22×C3—21no—
303C22×C3—21no—
304C22×C3#38611yescharacteristic
305C22×C3C4×D811yescharacteristic
306C22×C3#44211yescharacteristic
307C2×D3—42no—
308C2×D3—42no—
309C2×D3—42no—
310C2×D3—82no—
311C2×D3—82no—
312C2×D3—82no—
313C2×D3—162no—
314C2×D3—162no—
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—62no—
326C23—122no—
327C23—122no—
328C23—122no—
329C23—242no—
330C23—242no—
331C23C4×D1211yescharacteristic
332Q8—121no—
333Q8—121no—
334D4—122no—
335D4—124no—
336D4—122no—
337D4—122no—
338D4—121no—
339D4—121no—
340D4—121no—
341D4—121no—
342D4—242no—
343D4—242no—
344D4—241no—
345D4—241no—
346D4—241no—
347D4—241no—
348D4—241no—
349D4—241no—
350C2×C4—21no—
351C2×C4—21no—
352C2×C4—41no—
353C2×C4—62no—
354C2×C4—122no—
355C2×C4—122no—
356C2×C4D3×C22⋊C411yescharacteristic
357C2×C4#88211yescharacteristic
358C2×C4#88411yescharacteristic
359C2×C4#89311yescharacteristic
360C2×C4#89411yescharacteristic
361C2×C4#89511yescharacteristic
362C2×C4C2×D2411yescharacteristic
363C2×C4#90411yescharacteristic
364C2×C4#90811yescharacteristic
365C2×C4#90811yescharacteristic
366C8—22no—
367C8—42no—
368C8—41no—
369C8—81no—
370C8—121no—
371C8—121no—
372C8—241no—
373C2×C3—21no—
374C2×C3—21no—
375C2×C3#187911yescharacteristic
376C2×C3#190711yescharacteristic
377C2×C3#191411yescharacteristic
378D3—82no—
379D3—162no—
380C22—21no—
381C22—21no—
382C22—122no—
383C22—122no—
384C22—122no—
385C22—242no—
386C22—242no—
387C22—242no—
388C22—482no—
389C22—482no—
390C22#466511yescharacteristic
391C22C4×D2411yescharacteristic
392C22#471111yescharacteristic
393C4—21no—
394C4—21no—
395C4—21no—
396C4—122no—
397C4#473211yescharacteristic
398C4#473411yescharacteristic
399C4#474911yescharacteristic
400C4#475411yescharacteristic
401C4#484311yescharacteristic
402C4#484411yescharacteristic
403C3#927311yescharacteristic, complemented, Sylow
404C2—21no—
405C2—21no—
406C2—242no—
407C2—482no—
408C2#6867311yescharacteristic
409C2#6900611yescharacteristic
410C2#6904011yescharacteristic
subgroupG/H
centerC22#4665
commutatorC22×C3×C4C22×C4
FrattiniC2×C42C22×D3
FittingC3×#1283C2
socleC22×C3#386
prime radical#79303C2
cosocleC2×C3×C42C23
hypercenter#1283D3
hypocenterC3#9273
LCSG ▷ C22×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1283

Complements

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

NH ≅ G/Ntype
2. #78522C2semidirect
3. #70432C2semidirect
4. C2×#4485C2semidirect
5. #69876C2semidirect
6. C3×#1283C2semidirect
7. #67745C2semidirect
37. #1283D3semidirect
133. C3×M16D8semidirect
282. M16D24semidirect
403. C3#9273semidirect

Maximal subgroups

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

subgroupindex
#793032
#785222
#704322
C2×#44852
#698762
C3×#12832
#677452
#92733