Finite Group Databasebeta

← #10691217

Subgroups — #10691217

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#83189C211yesmaximal, characteristic
2#79048C211yesmaximal, characteristic, complemented
3C3×#1545C211yesmaximal, characteristic, complemented
4#71759C211yesmaximal, characteristic, complemented
5#70782C211yesmaximal, characteristic, complemented
6#67738C211yesmaximal, characteristic, complemented
7#67713C211yesmaximal, characteristic, complemented
8#9197—31nomaximal, Sylow
9C2×C3×#127C2211yescharacteristic
10C2×#938—21no—
11C2×D3×M16—21no—
12C2×#872C2211yescharacteristic
13C3×#431C2211yescharacteristic
14C3×#423C2211yescharacteristic
15#5223—21no—
16#5223C412yes—
17#5222—21no—
18#5222C412yes—
19#5221—21no—
20#5118—21no—
21#4936C2211yescharacteristic
22#4925C2211yescharacteristic
23#4924C2211yescharacteristic
24#4839—22no—
25#4839—22no—
26C3×#331—22no—
27#4544—22no—
28#4469—21no—
29#4468—21no—
30#2905—31no—
31#2619—31no—
32#1545—31no—
33#1545D311yescharacteristic, complemented
34#1458—31no—
35#1286—31no—
36#1286—31no—
37C22×C3×D4—21no—
38C22×C3×D4—21no—
39C22×C3×D4C2×C412yes—
40C22×#67—22no—
41C22×C4×D3D411yescharacteristic
42C2×C3×M16—21no—
43C3×#127—21no—
44C3×#127—21no—
45C3×#127—21no—
46C3×#127—21no—
47C2×C3×C42C2311yescharacteristic
48C2×#217—22no—
49C2×#217—22no—
50C2×#217—22no—
51#938—42no—
52#938—42no—
53C2×(C2×D3)⋊C4C2×C411yescharacteristic
54C2×(C2×D3)⋊C4D411yescharacteristic
55C2×C4⋊Dic3—21no—
56C2×Dic3⋊C4C2×C411yescharacteristic
57C2×Dic3⋊C4D411yescharacteristic
58C2×C4×Dic3D411yescharacteristic
59C2×#208—21no—
60D3×M16—42no—
61D3×M16—42no—
62C2×ΓL(1,𝔽25)—22no—
63C2×D3×C8—22no—
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×#127C2×D311yescharacteristic
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#382—31no—
98#331—22no—
99#331—62no—
100#324—61no—
101#323—61no—
102C24×C3—22no—
103C23×D3C22⋊C411yescharacteristic
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—82no—
115C2×C3×D4—82no—
116C22×C3×C4C22×C411yescharacteristic
117C22×C3×C4C2×D411yescharacteristic
118C22×C3×C4C2×D411yescharacteristic
119C2×#67—44no—
120C2×#67—82no—
121C2×#67—82no—
122C22×Dic3—22no—
123C22×Dic3C22⋊C411yescharacteristic
124C22×Dic3C22⋊C411yescharacteristic
125C22×Dic3C22⋊C411yescharacteristic
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×C42C2×D411yescharacteristic
137C3×C42C2×D411yescharacteristic
138C3×C42#4111yescharacteristic
139C3×C42#4111yescharacteristic
140#217—44no—
141#217—82no—
142#217—82no—
143(C2×D3)⋊C4—21no—
144(C2×D3)⋊C4—21no—
145(C2×D3)⋊C4—21no—
146(C2×D3)⋊C4—21no—
147C4⋊Dic3—41no—
148C4⋊Dic3—41no—
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—21no—
162C22×D4—21no—
163C22×D4—62no—
164C22×D4C4×D312yes—
165C23×C4—31no—
166C2×M16—21no—
167C2×M16—62no—
168C2×M16—61no—
169C2×M16—122no—
170C2×M16—122no—
171C22×C8—62no—
172#127—21no—
173#127—21no—
174#127—21no—
175#127—21no—
176#121—122no—
177#121—122no—
178#117—61no—
179#117—61no—
180#117—61no—
181#117—61no—
182C2×C4⋊C4—31no—
183C2×C4⋊C4—31no—
184C2×C4⋊C4—61no—
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⋊C8#6712yescomplemented
199C22⋊C8—21no—
200C22⋊C8—61no—
201C4×C8—21no—
202C4×C8—61no—
203#95—32no—
204#95—61no—
205C23×C3—42no—
206C23×C3—42no—
207C23×C3—42no—
208C23×C3—42no—
209C23×C3—42no—
210C23×C3—42no—
211C23×C3—42no—
212C23×C3C2×C22⋊C411yescharacteristic
213C22×D3—21no—
214C22×D3—21no—
215C22×D3—41no—
216C22×D3—41no—
217C22×D3#10212yes—
218C3×D4—82no—
219C3×D4—82no—
220C3×D4—82no—
221C3×D4—82no—
222C3×D4—81no—
223C3×D4—81no—
224C3×D4—81no—
225C3×D4—81no—
226C2×C3×C4—41no—
227C2×C3×C4—41no—
228C2×C3×C4C4×D411yescharacteristic
229C2×C3×C4C4×D411yescharacteristic
230C2×C3×C4#12011yescharacteristic
231C2×C3×C4#12111yescharacteristic
232C2×C3×C4#12111yescharacteristic
233C2×C3×C4#12311yescharacteristic
234C2×C3×C4C2×D811yescharacteristic
235C2×C3×C4C2×SD1611yescharacteristic
236C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
237C2×C3×C4#13711yescharacteristic
238#67—88no—
239C2×Dic3—21no—
240C2×Dic3—21no—
241C2×Dic3—21no—
242C2×Dic3—21no—
243C2×Dic3—41no—
244C2×Dic3—41no—
245C2×Dic3—42no—
246C2×Dic3—42no—
247C2×Dic3—42no—
248C2×Dic3#10212yes—
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—22no—
258C24—31no—
259C2×D4—22no—
260C2×D4—22no—
261C2×D4—41no—
262C2×D4—41no—
263C2×D4—41no—
264C2×D4—41no—
265C2×D4—41no—
266C2×D4—41no—
267C2×D4—41no—
268C2×D4—41no—
269C2×D4—82no—
270C2×D4—82no—
271C2×D4—124no—
272C2×D4—242no—
273C2×D4—242no—
274C22×C4—31no—
275C22×C4—32no—
276C22×C4—32no—
277C22×C4—31no—
278C22×C4—31no—
279C22×C4—61no—
280C22×C4—61no—
281C22×C4—61no—
282C22×C4—61no—
283C22×C4—62no—
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—61no—
298C4⋊C4—61no—
299C4⋊C4—61no—
300C4⋊C4—61no—
301C4⋊C4—121no—
302C4⋊C4—121no—
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—
312C42#23511yescharacteristic
313C42D3×D411yescharacteristic
314C42#23711yescharacteristic
315C42C2×#6711yescharacteristic
316C22×C3—21no—
317C22×C3—21no—
318C22×C3—82no—
319C22×C3—82no—
320C22×C3—82no—
321C22×C3—82no—
322C22×C3—82no—
323C22×C3—82no—
324C22×C3—82no—
325C22×C3#38611yescharacteristic
326C22×C3C2×#10211yescharacteristic
327C22×C3#41711yescharacteristic
328C2×D3—22no—
329C2×D3—41no—
330C2×D3—41no—
331C2×D3—42no—
332C2×D3—81no—
333C3×C4—21no—
334C3×C4—21no—
335C3×C4—21no—
336C3×C4—21no—
337C3×C4#44711yescharacteristic
338C3×C4#44811yescharacteristic
339C3×C4#48011yescharacteristic
340C3×C4#48111yescharacteristic
341Dic3—22no—
342Dic3—41no—
343Dic3—41no—
344Dic3—82no—
345C23—32no—
346C23—42no—
347C23—42no—
348C23—42no—
349C23—42no—
350C23—42no—
351C23—42no—
352C23—42no—
353C23—61no—
354C23—61no—
355C23—121no—
356C23—121no—
357C23D3×C22⋊C411yescharacteristic
358D4—82no—
359D4—82no—
360D4—82no—
361D4—82no—
362D4—81no—
363D4—81no—
364D4—81no—
365D4—81no—
366D4—248no—
367C2×C4—32no—
368C2×C4—41no—
369C2×C4—41no—
370C2×C4—61no—
371C2×C4—61no—
372C2×C4—62no—
373C2×C4—62no—
374C2×C4—62no—
375C2×C4—62no—
376C2×C4—61no—
377C2×C4—61no—
378C2×C4—121no—
379C2×C4—121no—
380C2×C4—122no—
381C2×C4—122no—
382C2×C4—122no—
383C2×C4#67⋊C411yescharacteristic
384C2×C4#88311yescharacteristic
385C2×C4#88411yescharacteristic
386C2×C4D3×D811yescharacteristic
387C2×C4#91111yescharacteristic
388C2×C4D3×SD1611yescharacteristic
389C2×C4#91511yescharacteristic
390C2×C4C4×#6711yescharacteristic
391C2×C4#93711yescharacteristic
392C2×C4#93811yescharacteristic
393C8—42no—
394C8—81no—
395C8—122no—
396C8—241no—
397C2×C3—21no—
398C2×C3—21no—
399C2×C3—162no—
400C2×C3#187411yescharacteristic
401C2×C3#191011yescharacteristic
402C2×C3#191411yescharacteristic
403D3—42no—
404C22—21no—
405C22—21no—
406C22—62no—
407C22—82no—
408C22—82no—
409C22—82no—
410C22—82no—
411C22—82no—
412C22—82no—
413C22—82no—
414C22—122no—
415C22—121no—
416C22—121no—
417C22—241no—
418C22D3×#10211yescharacteristic
419C22#477111yescharacteristic
420C22#495611yescharacteristic
421C4—21no—
422C4—21no—
423C4—21no—
424C4—21no—
425C4—62no—
426C4—121no—
427C4—121no—
428C4—242no—
429C4#477411yescharacteristic
430C4#477711yescharacteristic
431C4#515611yescharacteristic
432C4#517011yescharacteristic
433C3#919711yescharacteristic, complemented, Sylow
434C2—21no—
435C2—21no—
436C2—122no—
437C2—162no—
438C2#7008911yescharacteristic
439C2#7020011yescharacteristic
440C2#7033011yescharacteristic
subgroupG/H
centerC22#4956
commutatorC22×C3×C4C22×C4
FrattiniC2×C42C22×D3
FittingC3×#1545C2
socleC22×C3#386
prime radical#83189C2
cosocleC2×C3×C42C23
hypercenter#1545D3
hypocenterC3#9197
LCSG ▷ C22×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1545

Complements

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

NH ≅ G/Ntype
2. #79048C2semidirect
3. C3×#1545C2semidirect
4. #71759C2semidirect
5. #70782C2semidirect
6. #67738C2semidirect
7. #67713C2semidirect
33. #1545D3semidirect
70. C3×C4⋊C8D4semidirect
79. C4⋊(C3⋊C8)D4semidirect
198. C4⋊C8#67semidirect
433. C3#9197semidirect

Maximal subgroups

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

subgroupindex
#831892
#790482
C3×#15452
#717592
#707822
#677382
#677132
#91973