Finite Group Databasebeta

← #10691219

Subgroups — #10691219

Order 768.

624 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#79048C211yesmaximal, characteristic, complemented
2#74244C211yesmaximal, characteristic
3#70782C211yesmaximal, characteristic, complemented
4#70774C211yesmaximal, characteristic, complemented
5#70502C211yesmaximal, characteristic, complemented
6C3×#1286C211yesmaximal, characteristic, complemented
7#67631C211yesmaximal, characteristic, complemented
8#9197—31nomaximal, Sylow
9C2×D3×M16—21no—
10C2×#895—21no—
11C2×#874C2211yescharacteristic
12C2×#872C2211yescharacteristic
13C3×#431C2211yescharacteristic
14C3×#423C2211yescharacteristic
15C3×#382C2211yescharacteristic
16#5118—21no—
17#4988—21no—
18#4925C2211yescharacteristic
19#4924C2211yescharacteristic
20#4839—22no—
21#4839—22no—
22#4672—21no—
23#4672C412yes—
24#4671—21no—
25#4671C412yes—
26#4485—22no—
27#4485—22no—
28#4469—21no—
29#4468—21no—
30#2905—31no—
31#2619—31no—
32#1545—31no—
33#1545—31no—
34#1458—31no—
35#1286—31no—
36#1286D311yescharacteristic, complemented
37C22×D12—21no—
38C22×D12—22no—
39C22×D12—21no—
40C22×D12C2×C412yes—
41C22×C4×D3D411yescharacteristic
42C2×C3×M16—21no—
43C2×C3×C4⋊C4—21no—
44C2×C3×C42C2311yescharacteristic
45C2×(C2×D3)⋊C4—22no—
46C2×(C2×D3)⋊C4—22no—
47C2×(C2×D3)⋊C4—22no—
48C2×(C2×D3)⋊C4C2×C411yescharacteristic
49C2×(C2×D3)⋊C4D411yescharacteristic
50C2×Dic3⋊C4C2×C411yescharacteristic
51C2×Dic3⋊C4D411yescharacteristic
52C2×C4×Dic3D411yescharacteristic
53C2×#208—21no—
54D3×M16—42no—
55D3×M16—42no—
56C2×ΓL(1,𝔽25)—22no—
57C2×D3×C8—22no—
58#895—42no—
59#895—42no—
60#874—21no—
61#874—21no—
62#874—21no—
63#874—21no—
64#872—21no—
65#872—21no—
66#872—21no—
67#872—21no—
68C3×C4⋊C8—21no—
69C3×C4⋊C8—21no—
70C3×C4⋊C8D412yescomplemented
71C3×C22⋊C8—21no—
72C3×C4×C8—21no—
73C3×#95—21no—
74C3×#95C2×C412yes—
75#830—21no—
76Dic3⋊C8—24no—
77C4⋊(C3⋊C8)—21no—
78C4⋊(C3⋊C8)—21no—
79C4⋊(C3⋊C8)D412yescomplemented
80C4×C3⋊C8—21no—
81C22×M16—61no—
82C2×#127—31no—
83C2×#121—61no—
84C2×#117—31no—
85#431—31no—
86#431C2×D311yescharacteristic
87#424—62no—
88#424—62no—
89#423—31no—
90#423—61no—
91#423C2×D311yescharacteristic
92#394—32no—
93#394—61no—
94#392—32no—
95#392—61no—
96#390—61no—
97#382C2×D311yescharacteristic
98#331—62no—
99#331—62no—
100#324—61no—
101#323—61no—
102C23×D3—22no—
103C23×D3C22⋊C411yescharacteristic
104C22×C3×C4—22no—
105C22×C3×C4C22×C411yescharacteristic
106C22×C3×C4C2×D411yescharacteristic
107C22×C3×C4C2×D411yescharacteristic
108C22×Dic3C22⋊C411yescharacteristic
109C22×Dic3C22⋊C411yescharacteristic
110C22×Dic3C22⋊C411yescharacteristic
111C2×D12—22no—
112C2×D12—22no—
113C2×D12—44no—
114C2×D12—41no—
115C2×D12—41no—
116C2×D12—41no—
117C2×D12—41no—
118C2×D12—41no—
119C2×D12—41no—
120C2×D12—41no—
121C2×D12—41no—
122C2×D12—82no—
123C2×D12—82no—
124C2×D12—82no—
125C2×D12—82no—
126C2×C4×D3—21no—
127C2×C4×D3—21no—
128C2×C4×D3—21no—
129C2×C4×D3—21no—
130C2×C4×D3D812yes—
131C2×C4×D3SD1612yes—
132C3×M16—41no—
133C3×M16—41no—
134C2×C3×C8—22no—
135C2×C3×C8—41no—
136C3×C4⋊C4—41no—
137C3×C4⋊C4—41no—
138C3×C42C2×D411yescharacteristic
139C3×C42C2×D411yescharacteristic
140C3×C42#4111yescharacteristic
141C3×C42#4111yescharacteristic
142(C2×D3)⋊C4—21no—
143(C2×D3)⋊C4—21no—
144(C2×D3)⋊C4—21no—
145(C2×D3)⋊C4—21no—
146(C2×D3)⋊C4—44no—
147(C2×D3)⋊C4—82no—
148(C2×D3)⋊C4—82no—
149Dic3⋊C4—21no—
150Dic3⋊C4—21no—
151Dic3⋊C4—21no—
152Dic3⋊C4—21no—
153C4×Dic3—21no—
154C4×Dic3—21no—
155#208—41no—
156#208—41no—
157C2×C3⋊C8—22no—
158C2×C3⋊C8—41no—
159ΓL(1,𝔽25)—44no—
160D3×C8—44no—
161C22×D4—32no—
162C22×D4—62no—
163C22×D4—61no—
164C22×D4—61no—
165C23×C4—31no—
166C2×M16—21no—
167C2×M16—62no—
168C2×M16—61no—
169C2×M16—122no—
170C2×M16—122no—
171C22×C8—62no—
172#127—61no—
173#127—61no—
174#127—61no—
175#127—61no—
176#121—122no—
177#121—122no—
178#117—61no—
179#117—61no—
180#117—61no—
181#117—61no—
182C2×C4⋊C4—21no—
183C2×C4⋊C4—31no—
184C2×C4⋊C4—31no—
185C2×C22⋊C4—31no—
186C2×C22⋊C4—31no—
187C2×C22⋊C4—62no—
188C2×C22⋊C4—62no—
189C2×C22⋊C4—62no—
190C2×C42—31no—
191C2×C42C22×D311yescharacteristic
192C4⋊C8—21no—
193C4⋊C8—21no—
194C4⋊C8—32no—
195C4⋊C8—64no—
196C4⋊C8—61no—
197C4⋊C8—61no—
198C4⋊C8D1212yescomplemented
199C22⋊C8—21no—
200C22⋊C8—61no—
201C4×C8—21no—
202C4×C8—61no—
203#95—21no—
204#95C4×D312yes—
205C23×C3C2×C22⋊C411yescharacteristic
206C22×D3—21no—
207C22×D3—21no—
208C22×D3—41no—
209C22×D3—41no—
210C22×D3—42no—
211C22×D3—42no—
212C22×D3—42no—
213C22×D3—42no—
214C22×D3—42no—
215C22×D3—42no—
216C22×D3—42no—
217C22×D3#10212yes—
218C2×C3×C4—41no—
219C2×C3×C4—41no—
220C2×C3×C4—42no—
221C2×C3×C4—42no—
222C2×C3×C4—42no—
223C2×C3×C4C4×D411yescharacteristic
224C2×C3×C4C4×D411yescharacteristic
225C2×C3×C4#12011yescharacteristic
226C2×C3×C4#12111yescharacteristic
227C2×C3×C4#12111yescharacteristic
228C2×C3×C4#12311yescharacteristic
229C2×C3×C4C2×D811yescharacteristic
230C2×C3×C4C2×SD1611yescharacteristic
231C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
232C2×C3×C4#13711yescharacteristic
233C2×Dic3—21no—
234C2×Dic3—21no—
235C2×Dic3—21no—
236C2×Dic3—21no—
237C2×Dic3—41no—
238C2×Dic3—41no—
239C2×Dic3#10212yes—
240D12—82no—
241D12—82no—
242D12—82no—
243D12—82no—
244D12—88no—
245D12—81no—
246D12—81no—
247D12—81no—
248D12—81no—
249C4×D3—22no—
250C4×D3—22no—
251C4×D3—22no—
252C4×D3—22no—
253C3×C8—42no—
254C3×C8—81no—
255C3⋊C8—42no—
256C3⋊C8—81no—
257C24—31no—
258C24—62no—
259C2×D4—62no—
260C2×D4—62no—
261C2×D4—124no—
262C2×D4—121no—
263C2×D4—121no—
264C2×D4—121no—
265C2×D4—121no—
266C2×D4—121no—
267C2×D4—121no—
268C2×D4—121no—
269C2×D4—121no—
270C2×D4—242no—
271C2×D4—242no—
272C2×D4—242no—
273C2×D4—242no—
274C22×C4—22no—
275C22×C4—31no—
276C22×C4—32no—
277C22×C4—32no—
278C22×C4—31no—
279C22×C4—31no—
280C22×C4—61no—
281C22×C4—61no—
282C22×C4—61no—
283C22×C4—61no—
284C22×C4C2×C4×D311yescharacteristic
285C22×C4D3×D411yescharacteristic
286C22×C4D3×D411yescharacteristic
287M16—41no—
288M16—41no—
289M16—124no—
290M16—121no—
291M16—121no—
292C2×C8—22no—
293C2×C8—41no—
294C2×C8—62no—
295C2×C8—124no—
296C2×C8—121no—
297C4⋊C4—41no—
298C4⋊C4—41no—
299C4⋊C4—61no—
300C4⋊C4—61no—
301C4⋊C4—61no—
302C4⋊C4—61no—
303C22⋊C4—61no—
304C22⋊C4—61no—
305C22⋊C4—61no—
306C22⋊C4—61no—
307C22⋊C4—124no—
308C22⋊C4—242no—
309C22⋊C4—242no—
310C42—61no—
311C42—61no—
312C42C2×D1211yescharacteristic
313C42#23511yescharacteristic
314C42D3×D411yescharacteristic
315C42#23911yescharacteristic
316C22×C3—21no—
317C22×C3—21no—
318C22×C3#38611yescharacteristic
319C22×C3C2×#10211yescharacteristic
320C22×C3#41711yescharacteristic
321C2×D3—22no—
322C2×D3—41no—
323C2×D3—41no—
324C2×D3—42no—
325C2×D3—81no—
326C2×D3—82no—
327C2×D3—82no—
328C2×D3—82no—
329C2×D3—82no—
330C2×D3—82no—
331C2×D3—82no—
332C2×D3—82no—
333C3×C4—21no—
334C3×C4—21no—
335C3×C4—21no—
336C3×C4—21no—
337C3×C4—82no—
338C3×C4#44711yescharacteristic
339C3×C4#44811yescharacteristic
340C3×C4#48011yescharacteristic
341C3×C4#48111yescharacteristic
342Dic3—22no—
343Dic3—41no—
344Dic3—41no—
345C23—32no—
346C23—61no—
347C23—61no—
348C23—121no—
349C23—121no—
350C23—122no—
351C23—122no—
352C23—122no—
353C23—122no—
354C23—122no—
355C23—122no—
356C23—122no—
357C23D3×C22⋊C411yescharacteristic
358D4—242no—
359D4—242no—
360D4—242no—
361D4—242no—
362D4—248no—
363D4—241no—
364D4—241no—
365D4—241no—
366D4—241no—
367C2×C4—32no—
368C2×C4—41no—
369C2×C4—41no—
370C2×C4—42no—
371C2×C4—42no—
372C2×C4—42no—
373C2×C4—61no—
374C2×C4—61no—
375C2×C4—62no—
376C2×C4—62no—
377C2×C4—62no—
378C2×C4—62no—
379C2×C4—61no—
380C2×C4—61no—
381C2×C4—121no—
382C2×C4—121no—
383C2×C4C4×D1211yescharacteristic
384C2×C4#88211yescharacteristic
385C2×C4#88411yescharacteristic
386C2×C4#89311yescharacteristic
387C2×C4#89411yescharacteristic
388C2×C4#89511yescharacteristic
389C2×C4D3×D811yescharacteristic
390C2×C4D3×SD1611yescharacteristic
391C2×C4#91411yescharacteristic
392C2×C4#91811yescharacteristic
393C8—42no—
394C8—81no—
395C8—122no—
396C8—241no—
397C2×C3—21no—
398C2×C3—21no—
399C2×C3#187411yescharacteristic
400C2×C3#191011yescharacteristic
401C2×C3#191411yescharacteristic
402D3—42no—
403D3—162no—
404C22—21no—
405C22—21no—
406C22—62no—
407C22—122no—
408C22—121no—
409C22—121no—
410C22—241no—
411C22—242no—
412C22—242no—
413C22—242no—
414C22—242no—
415C22—242no—
416C22—242no—
417C22—242no—
418C22#466511yescharacteristic
419C22D3×#10211yescharacteristic
420C22#480311yescharacteristic
421C4—21no—
422C4—21no—
423C4—21no—
424C4—21no—
425C4—62no—
426C4—82no—
427C4—121no—
428C4—121no—
429C4#477311yescharacteristic
430C4#480611yescharacteristic
431C4#486311yescharacteristic
432C4#488211yescharacteristic
433C3#919711yescharacteristic, complemented, Sylow
434C2—21no—
435C2—21no—
436C2—122no—
437C2—482no—
438C2#6871311yescharacteristic
439C2#6888011yescharacteristic
440C2#6904011yescharacteristic
subgroupG/H
centerC22#4665
commutatorC22×C3×C4C22×C4
FrattiniC2×C42C22×D3
FittingC3×#1286C2
socleC22×C3#386
prime radical#74244C2
cosocleC2×C3×C42C23
hypercenter#1286D3
hypocenterC3#9197
LCSG ▷ C22×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1286

Complements

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

NH ≅ G/Ntype
1. #79048C2semidirect
3. #70782C2semidirect
4. #70774C2semidirect
5. #70502C2semidirect
6. C3×#1286C2semidirect
7. #67631C2semidirect
36. #1286D3semidirect
70. C3×C4⋊C8D4semidirect
79. C4⋊(C3⋊C8)D4semidirect
198. C4⋊C8D12semidirect
433. C3#9197semidirect

Maximal subgroups

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

subgroupindex
#790482
#742442
#707822
#707742
#705022
C3×#12862
#676312
#91973