Finite Group Databasebeta

← #10690407

Subgroups — #10690407

Order 768.

538 conjugacy classes of subgroups.

The subgroup lattice has 432 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#79004C211yesmaximal, characteristic, complemented
2#78470C211yesmaximal, characteristic
3#70426C211yesmaximal, characteristic, complemented
4#69492C211yesmaximal, characteristic, complemented
5#69469C211yesmaximal, characteristic, complemented
6C3×#1279C211yesmaximal, characteristic, complemented
7#67656C211yesmaximal, characteristic, complemented
8#9053—31nomaximal, Sylow
9C2×#901—21no—
10#5477C2211yescharacteristic
11#5477C2211yescharacteristic
12C2×C3×C4⋊C8C2211yescharacteristic
13C2×C3×C4×C8C2211yescharacteristic
14C3×C4×C4⋊C4C2211yescharacteristic
15#5115—22no—
16#5110—21no—
17C2×#805—21no—
18C2×#805—21no—
19#4925C2211yescharacteristic
20C4×#208C2211yescharacteristic
21#4838—21no—
22#4836—22no—
23#4834—21no—
24#4695C411yescharacteristic, complemented
25#4694C411yescharacteristic, complemented
26#4694C411yescharacteristic, complemented
27C4×D24—22no—
28#4682C411yescharacteristic, complemented
29#4483—21no—
30#4483—21no—
31#4451—22no—
32#3169—31no—
33#2935—31no—
34#1980—31no—
35#1502—31no—
36#1480—31no—
37#1279D311yescharacteristic, complemented
38#1278—31no—
39C2×#235—21no—
40C2×#235—21no—
41C22×C3×C8—21no—
42C22×C3×C8—21no—
43C2×C3×C4⋊C4—21no—
44C2×C3×C42—21no—
45C2×C3×C42C2311yescharacteristic
46#930—21no—
47#930—21no—
48#930—21no—
49#930D412yes—
50C2×#208D411yescharacteristic
51C2×#208D411yescharacteristic
52C2×D24—22no—
53#901—42no—
54#901—42no—
55C2×ΓL(1,𝔽25)—22no—
56C2×D3×C8—22no—
57#875—21no—
58#875—21no—
59#874C2×C411yescharacteristic
60#874C2×C411yescharacteristic
61C4×D12—21no—
62C4×D12—21no—
63C4×D12—21no—
64C4×D12D412yes—
65#869C2×C411yescharacteristic
66#869C2×C411yescharacteristic
67C3×C4⋊C8—21no—
68C3×C4⋊C8D411yescharacteristic
69C3×C4⋊C8D411yescharacteristic
70C3×C4×C8—21no—
71C3×C4×C8C2×C411yescharacteristic
72C3×C4×C8C2×C411yescharacteristic
73C3×#95—21no—
74#830—21no—
75#821—22no—
76#821—22no—
77(C2×D3)⋊C8—22no—
78(C2×D3)⋊C8—22no—
79#818—22no—
80#805—42no—
81#805—42no—
82C4⋊(C3⋊C8)—21no—
83#803—21no—
84C4×C3⋊C8—21no—
85C4×C3⋊C8—21no—
86C2×#131—61no—
87#532—31no—
88#532—31no—
89#500—31no—
90#493—31no—
91#486—31no—
92#486—31no—
93#466—62no—
94C4×D8—62no—
95#436—61no—
96#435—62no—
97C8×D4—61no—
98#431—31no—
99C2×C4⋊C8C2×D311yescharacteristic
100C2×#104—61no—
101C2×#104—61no—
102#408—61no—
103C4×M16—31no—
104C2×C4×C8C2×D311yescharacteristic
105C4×C4⋊C4C2×D311yescharacteristic
106#327—62no—
107D4⋊C8—61no—
108D4⋊C8—61no—
109C22×C3×C4—41no—
110C22×C3×C4C2×D411yescharacteristic
111C22×C3×C4C2×D411yescharacteristic
112C22×C3×C4#4111yescharacteristic
113C2×#67—22no—
114C2×#67—41no—
115#235—41no—
116#235—41no—
117#235—41no—
118#235—41no—
119#235—41no—
120#235—41no—
121#235—41no—
122#235—41no—
123C2×D12—21no—
124C2×D12—21no—
125C2×D12—21no—
126C2×D12—21no—
127C2×D12—21no—
128C2×D12C22⋊C412yes—
129C2×C4×D3—22no—
130C2×C4×D3—41no—
131C2×Dic6—21no—
132C2×Dic6—21no—
133C2×C3×C8—21no—
134C2×C3×C8—21no—
135C2×C3×C8—21no—
136C2×C3×C8—21no—
137C2×C3×C8—42no—
138C2×C3×C8—41no—
139C2×C3×C8—41no—
140C3×C4⋊C4—22no—
141C3×C4⋊C4—41no—
142C3×C42—41no—
143C3×C42—41no—
144C3×C42C22×C411yescharacteristic
145C3×C42C2×D411yescharacteristic
146C3×C42C2×D411yescharacteristic
147C3×C42#4111yescharacteristic
148(C2×D3)⋊C4—22no—
149(C2×D3)⋊C4—22no—
150(C2×D3)⋊C4—41no—
151(C2×D3)⋊C4—41no—
152C4⋊Dic3—21no—
153C4⋊Dic3—21no—
154C4⋊Dic3—21no—
155C4⋊Dic3C22⋊C412yes—
156#208—21no—
157#208—21no—
158#208—21no—
159#208D812yescomplemented
160C2×C3⋊C8—21no—
161C2×C3⋊C8—21no—
162C2×C3⋊C8—41no—
163D24—44no—
164ΓL(1,𝔽25)—44no—
165D3×C8—44no—
166C2×#41—61no—
167C2×#41—61no—
168C2×D8—62no—
169#131—122no—
170#131—122no—
171C2×M16—31no—
172C2×M16—31no—
173C2×M16—62no—
174C22×C8—21no—
175C22×C8—21no—
176C22×C8—62no—
177C4⋊Q8—31no—
178C4⋊Q8—31no—
179#127—31no—
180#127—31no—
181#124—61no—
182#124—61no—
183#121—32no—
184#121—61no—
185#121—61no—
186#121—61no—
187C4×D4—32no—
188C4×D4—61no—
189C4×D4—61no—
190C4×D4—61no—
191C2×C4⋊C4—21no—
192C2×C42—21no—
193C2×C42C22×D311yescharacteristic
194#107—62no—
195C4⋊C8—21no—
196C4⋊C8—61no—
197C4⋊C8D1211yescharacteristic
198C4⋊C8D1211yescharacteristic
199#104—122no—
200#104—122no—
201#102—62no—
202#102—62no—
203C22⋊C8—62no—
204C22⋊C8—62no—
205C22⋊C8—61no—
206#97—61no—
207C4×C8—21no—
208C4×C8—61no—
209C4×C8—61no—
210C4×C8C4×D311yescharacteristic
211C4×C8C4×D311yescharacteristic
212#95—21no—
213C23×C3#12111yescharacteristic
214C22×D3—22no—
215C22×D3—41no—
216C2×C3×C4—21no—
217C2×C3×C4—21no—
218C2×C3×C4—41no—
219C2×C3×C4—42no—
220C2×C3×C4—81no—
221C2×C3×C4—81no—
222C2×C3×C4C2×C22⋊C411yescharacteristic
223C2×C3×C4C4×D411yescharacteristic
224C2×C3×C4C4×D411yescharacteristic
225C2×C3×C4#12011yescharacteristic
226C2×C3×C4#12111yescharacteristic
227C2×C3×C4#12311yescharacteristic
228C2×C3×C4C2×D811yescharacteristic
229C2×C3×C4#13511yescharacteristic
230C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
231C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
232#67—42no—
233#67—42no—
234#67—81no—
235#67—81no—
236C2×Dic3—22no—
237C2×Dic3—41no—
238D12—42no—
239D12—42no—
240D12—41no—
241D12—41no—
242D12—41no—
243D12—41no—
244D12—41no—
245D12—41no—
246D12—41no—
247D12—41no—
248C4×D3—42no—
249C4×D3—42no—
250C4×D3—81no—
251C4×D3—81no—
252Dic6—41no—
253Dic6—41no—
254Dic6—41no—
255Dic6—41no—
256C3×C8—22no—
257C3×C8—41no—
258C3×C8—41no—
259C3×C8—41no—
260C3⋊C8—22no—
261C3⋊C8—41no—
262C3⋊C8—81no—
263#41—121no—
264#41—121no—
265#41—121no—
266#41—121no—
267#41—121no—
268#41—121no—
269#41—121no—
270#41—121no—
271C2×Q8—61no—
272C2×Q8—61no—
273C2×D4—32no—
274C2×D4—62no—
275C2×D4—61no—
276C2×D4—61no—
277C2×D4—61no—
278C2×D4—61no—
279C2×D4—61no—
280C2×D4—121no—
281C22×C4—41no—
282C22×C4—62no—
283C22×C4—121no—
284C22×C4#23511yescharacteristic
285C22×C4D3×D411yescharacteristic
286C22×C4D3×D411yescharacteristic
287D8—124no—
288M16—32no—
289M16—61no—
290M16—61no—
291M16—61no—
292M16—124no—
293C2×C8—21no—
294C2×C8—21no—
295C2×C8—21no—
296C2×C8—21no—
297C2×C8—42no—
298C2×C8—41no—
299C2×C8—41no—
300C2×C8—61no—
301C2×C8—61no—
302C2×C8—124no—
303C2×C8—121no—
304C4⋊C4—22no—
305C4⋊C4—32no—
306C4⋊C4—41no—
307C4⋊C4—61no—
308C4⋊C4—61no—
309C4⋊C4—61no—
310C22⋊C4—62no—
311C22⋊C4—62no—
312C22⋊C4—121no—
313C22⋊C4—121no—
314C42—41no—
315C42—41no—
316C42C2×C4×D311yescharacteristic
317C42C2×D1211yescharacteristic
318C42D3×D411yescharacteristic
319C42#23911yescharacteristic
320C22×C3—21no—
321C22×C3—21no—
322C22×C3#38611yescharacteristic
323C22×C3#45911yescharacteristic
324C22×C3#46311yescharacteristic
325C2×D3—42no—
326C2×D3—42no—
327C2×D3—42no—
328C2×D3—81no—
329C2×D3—81no—
330C2×D3—81no—
331C3×C4—21no—
332C3×C4—21no—
333C3×C4—21no—
334C3×C4—82no—
335C3×C4C4×D811yescharacteristic
336C3×C4#44211yescharacteristic
337C3×C4#44311yescharacteristic
338C3×C4#44411yescharacteristic
339C3×C4#44711yescharacteristic
340C3×C4#44911yescharacteristic
341Dic3—42no—
342Dic3—81no—
343C23—62no—
344C23—121no—
345C23#88411yescharacteristic
346Q8—121no—
347Q8—121no—
348Q8—121no—
349Q8—121no—
350D4—122no—
351D4—122no—
352D4—122no—
353D4—122no—
354D4—121no—
355D4—121no—
356D4—121no—
357D4—121no—
358D4—121no—
359D4—121no—
360D4—121no—
361D4—121no—
362D4—241no—
363D4—241no—
364C2×C4—21no—
365C2×C4—21no—
366C2×C4—41no—
367C2×C4—42no—
368C2×C4—62no—
369C2×C4—81no—
370C2×C4—81no—
371C2×C4—122no—
372C2×C4—122no—
373C2×C4—121no—
374C2×C4—241no—
375C2×C4—241no—
376C2×C4C4×D1211yescharacteristic
377C2×C4D3×C22⋊C411yescharacteristic
378C2×C4#88211yescharacteristic
379C2×C4#89311yescharacteristic
380C2×C4#89411yescharacteristic
381C2×C4#89511yescharacteristic
382C2×C4D3×D811yescharacteristic
383C2×C4#91411yescharacteristic
384C2×C4#91411yescharacteristic
385C2×C4#91911yescharacteristic
386C8—22no—
387C8—41no—
388C8—41no—
389C8—41no—
390C8—62no—
391C8—121no—
392C8—241no—
393C2×C3—21no—
394C2×C3—21no—
395C2×C3#187611yescharacteristic
396C2×C3#187911yescharacteristic
397C2×C3#188411yescharacteristic
398D3—82no—
399D3—161no—
400C22—21no—
401C22—21no—
402C22—122no—
403C22—122no—
404C22—122no—
405C22—241no—
406C22—241no—
407C22—241no—
408C22#466511yescharacteristic
409C22#478611yescharacteristic
410C22#481911yescharacteristic
411C4—21no—
412C4—21no—
413C4—21no—
414C4—82no—
415C4—122no—
416C4—241no—
417C4#470011yescharacteristic
418C4#471711yescharacteristic
419C4#477311yescharacteristic
420C4#481011yescharacteristic
421C4#486911yescharacteristic
422C4#487211yescharacteristic
423C3#905311yescharacteristic, complemented, Sylow
424C2—21no—
425C2—21no—
426C2—242no—
427C2—481no—
428C2#6869511yescharacteristic
429C2#6877711yescharacteristic
430C2#6889711yescharacteristic
subgroupG/H
centerC2×C4#882
commutatorC3×C42C22×C4
FrattiniC2×C42C22×D3
FittingC3×#1279C2
socleC22×C3#386
prime radical#78470C2
cosocleC2×C3×C42C23
hypercenter#1279D3
hypocenterC3#9053
LCSG ▷ C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #1279

Complements

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

NH ≅ G/Ntype
1. #79004C2semidirect
3. #70426C2semidirect
4. #69492C2semidirect
5. #69469C2semidirect
6. C3×#1279C2semidirect
7. #67656C2semidirect
24. #4695C4semidirect
25. #4694C4semidirect
26. #4694C4semidirect
28. #4682C4semidirect
37. #1279D3semidirect
159. #208D8semidirect
423. C3#9053semidirect

Maximal subgroups

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

subgroupindex
#790042
#784702
#704262
#694922
#694692
C3×#12792
#676562
#90533