Finite Group Databasebeta

← #10692076

Subgroups — #10692076

Order 768.

558 conjugacy classes of subgroups.

The subgroup lattice has 442 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#79292C211yesmaximal, characteristic, complemented
2#79056C211yesmaximal, characteristic
3#70774C211yesmaximal, characteristic, complemented
4#69494C211yesmaximal, characteristic, complemented
5#69487C211yesmaximal, characteristic, complemented
6C3×#1303C211yesmaximal, characteristic, complemented
7#67711C211yesmaximal, characteristic, complemented
8#9322—31nomaximal, Sylow
9C2×#908—21no—
10C2×#874C2211yescharacteristic
11C4×#235C2211yescharacteristic
12C3×#431C2211yescharacteristic
13C3×#423C2211yescharacteristic
14C3×C4×C4⋊C4C2211yescharacteristic
15#5131—21no—
16C2×#809—22no—
17C2×#805—21no—
18#4924C2211yescharacteristic
19C2×C4⋊(C3⋊C8)C2211yescharacteristic
20#4848—21no—
21#4848C412yescomplemented
22#4843—21no—
23#4843C412yescomplemented
24#4837—21no—
25#4835—21no—
26#4730—22no—
27#4688—21no—
28#4679—21no—
29#4523—21no—
30#4485—21no—
31#4485—21no—
32#3038—31no—
33#2989—31no—
34#1982—31no—
35#1806—31no—
36#1545—31no—
37#1303D311yescharacteristic, complemented
38#1292—31no—
39C2×#235D411yescharacteristic
40C22×D12—21no—
41C22×D12—21no—
42C22×D12—21no—
43C2×C3×M16—21no—
44C2×C3×C4⋊C4—21no—
45C2×C3×C42—21no—
46C2×C3×C42C2311yescharacteristic
47C4×#67—21no—
48C4×#67—21no—
49#926D411yescharacteristic
50C2×#208—21no—
51C22×C3⋊C8D412yes—
52#908—42no—
53#908—42no—
54C2×D24—22no—
55C2×#204—22no—
56C2×ΓL(1,𝔽25)—22no—
57#874—21no—
58#874—21no—
59#874—21no—
60#874C2×C412yes—
61C4×D12C2×C411yescharacteristic
62C4×D12D411yescharacteristic
63#872—21no—
64C42×D3—21no—
65C4×Dic6C2×C411yescharacteristic
66C4×Dic6D411yescharacteristic
67C3×C4⋊C8—21no—
68C3×C4⋊C8C2×C412yes—
69C3×C4⋊C8D411yescharacteristic
70C3×C4⋊C8D411yescharacteristic
71C3×C22⋊C8—21no—
72C3×C4×C8—21no—
73C3×#95—21no—
74#821—22no—
75#821—22no—
76(C2×D3)⋊C8—22no—
77#815—22no—
78Dic3⋊C8—22no—
79#809—44no—
80#805—42no—
81C4⋊(C3⋊C8)—21no—
82C4⋊(C3⋊C8)—21no—
83C4⋊(C3⋊C8)—21no—
84C4⋊(C3⋊C8)—21no—
85C2×AGL(1,ℤ/8)—61no—
86C2×#127—31no—
87C4×#41—31no—
88#460—32no—
89#460—61no—
90#459—32no—
91#459—61no—
92#446—61no—
93#436—61no—
94#435—62no—
95#432—61no—
96#431C2×D311yescharacteristic
97#423—31no—
98#423—61no—
99#423C2×D311yescharacteristic
100C2×C4⋊C8—31no—
101C2×#104—61no—
102#418—61no—
103C2×#102—62no—
104C4×C4⋊C4C2×D311yescharacteristic
105#339—61no—
106#331—61no—
107#331—61no—
108C23×D3—41no—
109C22×C3×C4—41no—
110C22×C3×C4C2×D411yescharacteristic
111C22×C3×C4C2×D411yescharacteristic
112C22×C3×C4#4111yescharacteristic
113C2×#67—21no—
114#235—21no—
115#235—21no—
116#235—21no—
117#235—21no—
118C2×D12—22no—
119C2×D12—22no—
120C2×D12—22no—
121C2×D12—42no—
122C2×D12—42no—
123C2×D12—41no—
124C2×D12—41no—
125C2×D12—41no—
126C2×D12—41no—
127C2×D12—41no—
128C2×D12—41no—
129C2×D12—41no—
130C2×D12—82no—
131C2×D12C22⋊C411yescharacteristic
132C2×C4×D3—21no—
133C2×C4×D3—21no—
134C2×C4×D3—21no—
135C2×Dic6C22⋊C411yescharacteristic
136C3×M16—41no—
137C3×M16—41no—
138C2×C3×C8—22no—
139C2×C3×C8—21no—
140C2×C3×C8—21no—
141C3×C4⋊C4—22no—
142C3×C4⋊C4—41no—
143C3×C42—41no—
144C3×C42—41no—
145C3×C42C22×C411yescharacteristic
146C3×C42C2×D411yescharacteristic
147C3×C42C2×D411yescharacteristic
148C3×C42#4111yescharacteristic
149#217—21no—
150(C2×D3)⋊C4—21no—
151(C2×D3)⋊C4—21no—
152C4⋊Dic3C22⋊C411yescharacteristic
153C4⋊Dic3C22⋊C411yescharacteristic
154Dic3⋊C4—21no—
155Dic3⋊C4—21no—
156C4×Dic3—21no—
157C4×Dic3—21no—
158C4×Dic3—21no—
159#208—41no—
160#208—41no—
161C2×C3⋊C8—22no—
162C2×C3⋊C8—42no—
163C2×C3⋊C8—41no—
164D24—44no—
165#204—44no—
166ΓL(1,𝔽25)—82no—
167C2×#41—31no—
168C22×D4—61no—
169C22×D4—61no—
170C22×D4—61no—
171AGL(1,ℤ/8)—122no—
172AGL(1,ℤ/8)—122no—
173C2×SD16—62no—
174C2×D8—62no—
175C2×M16—21no—
176C2×M16—62no—
177C2×M16—61no—
178C22×C8—32no—
179#127—32no—
180#127—61no—
181#127—61no—
182#127—61no—
183C4×Q8—31no—
184C4×Q8—31no—
185C4×D4—31no—
186C4×D4—31no—
187C4×D4—61no—
188C4×D4—61no—
189#117—31no—
190#117—61no—
191C2×C4⋊C4—21no—
192C2×C42—21no—
193C2×C42—61no—
194C2×C42C22×D311yescharacteristic
195C4⋊C8—21no—
196C4⋊C8—62no—
197C4⋊C8—61no—
198C4⋊C8—61no—
199C4⋊C8—61no—
200C4⋊C8—61no—
201C4⋊C8C4×D312yes—
202C4⋊C8D1211yescharacteristic
203C4⋊C8D1211yescharacteristic
204#104—122no—
205#102—62no—
206#102—62no—
207#102—124no—
208C22⋊C8—21no—
209C22⋊C8—62no—
210#97—62no—
211C4×C8—21no—
212#95—21no—
213C23×C3#12111yescharacteristic
214C22×D3—21no—
215C22×D3—42no—
216C22×D3—82no—
217C22×D3—82no—
218C22×D3—81no—
219C22×D3—81no—
220C2×C3×C4—21no—
221C2×C3×C4—21no—
222C2×C3×C4—41no—
223C2×C3×C4—42no—
224C2×C3×C4—81no—
225C2×C3×C4—81no—
226C2×C3×C4C2×C22⋊C411yescharacteristic
227C2×C3×C4C4×D411yescharacteristic
228C2×C3×C4C4×D411yescharacteristic
229C2×C3×C4#12011yescharacteristic
230C2×C3×C4#12111yescharacteristic
231C2×C3×C4#12311yescharacteristic
232C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
233C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
234C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
235C2×C3×C4#13711yescharacteristic
236#67—41no—
237#67—41no—
238C2×Dic3—21no—
239C2×Dic3—21no—
240C2×Dic3—21no—
241D12—21no—
242D12—21no—
243D12—44no—
244D12—42no—
245D12—42no—
246D12—82no—
247D12—82no—
248D12—81no—
249D12—81no—
250D12—81no—
251D12—81no—
252C4×D3—41no—
253C4×D3—41no—
254C4×D3—41no—
255C4×D3—41no—
256C4×D3—81no—
257Dic6—21no—
258Dic6—21no—
259C3×C8—42no—
260C3×C8—41no—
261C3×C8—41no—
262C3⋊C8—42no—
263C3⋊C8—81no—
264C24—121no—
265#41—61no—
266#41—61no—
267#41—61no—
268#41—61no—
269C2×Q8—31no—
270C2×D4—31no—
271C2×D4—61no—
272C2×D4—62no—
273C2×D4—62no—
274C2×D4—62no—
275C2×D4—122no—
276C2×D4—122no—
277C2×D4—121no—
278C2×D4—121no—
279C2×D4—121no—
280C2×D4—121no—
281C2×D4—121no—
282C2×D4—121no—
283C2×D4—121no—
284C2×D4—242no—
285C22×C4—41no—
286C22×C4—61no—
287C22×C4—61no—
288C22×C4—61no—
289C22×C4#23511yescharacteristic
290C22×C4D3×D411yescharacteristic
291C22×C4D3×D411yescharacteristic
292SD16—124no—
293D8—124no—
294M16—41no—
295M16—41no—
296M16—121no—
297M16—121no—
298M16—242no—
299C2×C8—22no—
300C2×C8—21no—
301C2×C8—21no—
302C2×C8—62no—
303C2×C8—122no—
304C2×C8—121no—
305C4⋊C4—22no—
306C4⋊C4—31no—
307C4⋊C4—31no—
308C4⋊C4—41no—
309C4⋊C4—61no—
310C4⋊C4—61no—
311C22⋊C4—61no—
312C22⋊C4—61no—
313C22⋊C4—61no—
314C42—41no—
315C42—41no—
316C42—61no—
317C42—61no—
318C42—61no—
319C42C2×C4×D311yescharacteristic
320C42C2×D1211yescharacteristic
321C42D3×D411yescharacteristic
322C42#23911yescharacteristic
323C22×C3—21no—
324C22×C3—21no—
325C22×C3#38611yescharacteristic
326C22×C3#46811yescharacteristic
327C22×C3#46911yescharacteristic
328C2×D3—41no—
329C2×D3—41no—
330C2×D3—41no—
331C2×D3—82no—
332C2×D3—82no—
333C2×D3—82no—
334C2×D3—162no—
335C2×D3—162no—
336C3×C4—21no—
337C3×C4—21no—
338C3×C4—21no—
339C3×C4—82no—
340C3×C4#41711yescharacteristic
341C3×C4#41811yescharacteristic
342C3×C4#45311yescharacteristic
343C3×C4#45411yescharacteristic
344C3×C4#46811yescharacteristic
345C3×C4#46911yescharacteristic
346Dic3—41no—
347Dic3—41no—
348Dic3—41no—
349C23—61no—
350C23—122no—
351C23—242no—
352C23—242no—
353C23—241no—
354C23—241no—
355C23#88411yescharacteristic
356Q8—61no—
357Q8—61no—
358D4—61no—
359D4—61no—
360D4—124no—
361D4—121no—
362D4—121no—
363D4—122no—
364D4—122no—
365D4—242no—
366D4—242no—
367D4—241no—
368D4—241no—
369D4—241no—
370D4—241no—
371C2×C4—21no—
372C2×C4—21no—
373C2×C4—41no—
374C2×C4—42no—
375C2×C4—61no—
376C2×C4—61no—
377C2×C4—61no—
378C2×C4—81no—
379C2×C4—81no—
380C2×C4—121no—
381C2×C4—121no—
382C2×C4—121no—
383C2×C4—121no—
384C2×C4—241no—
385C2×C4C4×D1211yescharacteristic
386C2×C4D3×C22⋊C411yescharacteristic
387C2×C4#88211yescharacteristic
388C2×C4#89311yescharacteristic
389C2×C4#89411yescharacteristic
390C2×C4#89511yescharacteristic
391C2×C4#91111yescharacteristic
392C2×C4#91411yescharacteristic
393C2×C4#91411yescharacteristic
394C2×C4#91811yescharacteristic
395C8—42no—
396C8—41no—
397C8—41no—
398C8—122no—
399C8—241no—
400C2×C3—21no—
401C2×C3—21no—
402C2×C3#188311yescharacteristic
403C2×C3#189511yescharacteristic
404C2×C3#190311yescharacteristic
405D3—81no—
406D3—162no—
407C22—21no—
408C22—21no—
409C22—121no—
410C22—121no—
411C22—121no—
412C22—242no—
413C22—242no—
414C22—242no—
415C22—482no—
416C22—482no—
417C22#466511yescharacteristic
418C22#478211yescharacteristic
419C22#481211yescharacteristic
420C4—21no—
421C4—21no—
422C4—21no—
423C4—82no—
424C4—121no—
425C4—121no—
426C4—121no—
427C4#477011yescharacteristic
428C4#480311yescharacteristic
429C4#482211yescharacteristic
430C4#482711yescharacteristic
431C4#486511yescharacteristic
432C4#488611yescharacteristic
433C3#932211yescharacteristic, complemented, Sylow
434C2—21no—
435C2—21no—
436C2—241no—
437C2—482no—
438C2#6877611yescharacteristic
439C2#6888511yescharacteristic
440C2#6903811yescharacteristic
subgroupG/H
centerC22#4665
commutatorC3×C42C22×C4
FrattiniC2×C42C22×D3
FittingC3×#1303C2
socleC22×C3#386
prime radical#79056C2
cosocleC2×C3×C42C23
hypercenter#1303D3
hypocenterC3#9322
LCSG ▷ C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1303

Complements

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

NH ≅ G/Ntype
1. #79292C2semidirect
3. #70774C2semidirect
4. #69494C2semidirect
5. #69487C2semidirect
6. C3×#1303C2semidirect
7. #67711C2semidirect
21. #4848C4semidirect
23. #4843C4semidirect
37. #1303D3semidirect
433. C3#9322semidirect

Maximal subgroups

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

subgroupindex
#792922
#790562
#707742
#694942
#694872
C3×#13032
#677112
#93223