Finite Group Databasebeta

← #10691278

Subgroups — #10691278

Order 768.

532 conjugacy classes of subgroups.

The subgroup lattice has 460 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#79276C211yesmaximal, characteristic, complemented
2#79021C211yesmaximal, characteristic
3#70789C211yesmaximal, characteristic, complemented
4#70529C211yesmaximal, characteristic, complemented
5#70498C211yesmaximal, characteristic, complemented
6C3×#1301C211yesmaximal, characteristic, complemented
7#67704C211yesmaximal, characteristic, complemented
8#9208—31nomaximal, Sylow
9C2×#904—21no—
10#5477C2211yescharacteristic
11C4×#235C2211yescharacteristic
12C3×#431C2211yescharacteristic
13C3×#423C2211yescharacteristic
14C2×C3×C4⋊C8C2211yescharacteristic
15C2×#825—21no—
16#5113—21no—
17C4×C4⋊Dic3C2211yescharacteristic
18C2×C4×C3⋊C8C2211yescharacteristic
19#4849C411yescharacteristic, complemented
20#4848C411yescharacteristic, complemented
21#4845—21no—
22#4844—21no—
23#4843C411yescharacteristic, complemented
24#4841C411yescharacteristic, complemented
25#4834—21no—
26#4830—21no—
27#67⋊C8—22no—
28#4685—21no—
29C4×D3×C8—21no—
30#4555—21no—
31#4483—21no—
32#4483—21no—
33#4470—22no—
34#3035—31no—
35C8×#41—31no—
36#1806—31no—
37#1502—31no—
38#1492—31no—
39#1301D311yescharacteristic, complemented
40#1279—31no—
41C2×#235—21no—
42C2×#235D411yescharacteristic
43C2×C3×M16—21no—
44C22×C3×C8—21no—
45C2×C3×C42C2311yescharacteristic
46#930—21no—
47#930—21no—
48C4×#67—21no—
49C4×#67—21no—
50#926D411yescharacteristic
51C2×C4⋊Dic3—21no—
52C2×C4×Dic3—21no—
53C22×C3⋊C8—21no—
54C2×Dic12—21no—
55#904—42no—
56#904—42no—
57C2×D24—21no—
58C2×#204—21no—
59C2×#204—21no—
60C2×D3×C8—22no—
61#875—21no—
62#874C2×C411yescharacteristic
63C4×D12—21no—
64C4×D12—21no—
65C4×D12C2×C411yescharacteristic
66C4×D12D411yescharacteristic
67#872—21no—
68C42×D3—21no—
69#869C2×C411yescharacteristic
70C4×Dic6C2×C411yescharacteristic
71C4×Dic6D411yescharacteristic
72C3×C4⋊C8—21no—
73C3×C4⋊C8—21no—
74C3×C4⋊C8—21no—
75C3×C4⋊C8—21no—
76C3×C4⋊C8C2×C411yescharacteristic
77C3×C4⋊C8C2×C411yescharacteristic
78C3×C22⋊C8D412yes—
79C3×C4×C8—21no—
80C3⋊#95—21no—
81#825—42no—
82#821—21no—
83#821—21no—
84(C2×D3)⋊C8—22no—
85#816—21no—
86#816—21no—
87Dic3⋊C8—22no—
88C8×Dic3—22no—
89C4×C3⋊C8—21no—
90C4×C3⋊C8D411yescharacteristic
91C4×C3⋊C8D411yescharacteristic
92C2×#135—61no—
93#532—31no—
94C4×#41—31no—
95#464—61no—
96#463—61no—
97#462—31no—
98#461—31no—
99#460—31no—
100#459—31no—
101C8×Q8—61no—
102C8×D4—61no—
103C8×D4—62no—
104#431—61no—
105#431C2×D311yescharacteristic
106#423C2×D311yescharacteristic
107C2×C4⋊C8C2×D311yescharacteristic
108C2×#104—61no—
109#416—61no—
110C2×C4×C8—31no—
111C2×C4×C8—61no—
112C4×C4⋊C4—31no—
113#339—61no—
114#327—62no—
115D4⋊C8—61no—
116D4⋊C8—61no—
117C22×C3×C4C2×D411yescharacteristic
118C22×C3×C4C2×D411yescharacteristic
119C22×C3×C4#4111yescharacteristic
120C2×#67—21no—
121C2×#67—41no—
122C22×Dic3—41no—
123#235—21no—
124#235—21no—
125#235—41no—
126#235—41no—
127#235—41no—
128#235—41no—
129#235D812yes—
130#235SD1612yes—
131C2×D12—21no—
132C2×D12—21no—
133C2×D12—21no—
134C2×D12C22⋊C411yescharacteristic
135C2×C4×D3—21no—
136C2×C4×D3—21no—
137C2×C4×D3—21no—
138C2×C4×D3—41no—
139C2×Dic6—21no—
140C2×Dic6C22⋊C411yescharacteristic
141C3×M16—41no—
142C3×M16—41no—
143C2×C3×C8—21no—
144C2×C3×C8—21no—
145C2×C3×C8—22no—
146C2×C3×C8—41no—
147C2×C3×C8—41no—
148C2×C3×C8—41no—
149C3×C42C22×C411yescharacteristic
150C3×C42C2×D411yescharacteristic
151C3×C42C2×D411yescharacteristic
152C3×C42#4111yescharacteristic
153#217—21no—
154(C2×D3)⋊C4—21no—
155(C2×D3)⋊C4—21no—
156(C2×D3)⋊C4—41no—
157(C2×D3)⋊C4—41no—
158C4⋊Dic3—22no—
159C4⋊Dic3—21no—
160C4⋊Dic3—21no—
161C4⋊Dic3—41no—
162C4⋊Dic3C22⋊C411yescharacteristic
163C4⋊Dic3C22⋊C411yescharacteristic
164Dic3⋊C4—21no—
165Dic3⋊C4—21no—
166C4×Dic3—21no—
167C4×Dic3—21no—
168C4×Dic3—21no—
169C4×Dic3—41no—
170C4×Dic3—41no—
171C2×C3⋊C8—21no—
172C2×C3⋊C8—21no—
173C2×C3⋊C8—42no—
174Dic12—42no—
175D24—42no—
176#204—42no—
177#204—42no—
178D3×C8—44no—
179C2×#41—31no—
180C2×#41—61no—
181#135—122no—
182#135—122no—
183C2×Q16—61no—
184C2×SD16—61no—
185C2×SD16—61no—
186C2×D8—61no—
187C2×M16—21no—
188C22×C8—21no—
189C22×C8—62no—
190C22×C8—61no—
191C4⋊Q8—31no—
192#127—31no—
193#124—61no—
194#121—61no—
195#121—61no—
196C4×Q8—31no—
197C4×Q8—31no—
198C4×D4—31no—
199C4×D4—31no—
200C4×D4—61no—
201C4×D4—61no—
202C4×D4—61no—
203C4×D4—61no—
204#117—31no—
205#117—61no—
206C2×C4⋊C4—61no—
207C2×C42—61no—
208C2×C42—61no—
209C2×C42C22×D311yescharacteristic
210C4⋊C8—21no—
211C4⋊C8—21no—
212C4⋊C8—21no—
213C4⋊C8—21no—
214C4⋊C8—62no—
215C4⋊C8C4×D311yescharacteristic
216C4⋊C8C4×D311yescharacteristic
217#104—122no—
218#103—61no—
219#103—61no—
220#102—61no—
221#102—61no—
222C22⋊C8—62no—
223C22⋊C8D1212yes—
224C4×C8—21no—
225C4×C8—31no—
226C4×C8—31no—
227C4×C8—62no—
228C4×C8—61no—
229#95—61no—
230C23×C3#12111yescharacteristic
231C22×D3—21no—
232C22×D3—41no—
233C2×C3×C4—21no—
234C2×C3×C4—21no—
235C2×C3×C4—41no—
236C2×C3×C4C2×C22⋊C411yescharacteristic
237C2×C3×C4C4×D411yescharacteristic
238C2×C3×C4C4×D411yescharacteristic
239C2×C3×C4#12011yescharacteristic
240C2×C3×C4#12111yescharacteristic
241C2×C3×C4#12311yescharacteristic
242C2×C3×C4C2×D811yescharacteristic
243C2×C3×C4C2×SD1611yescharacteristic
244C2×C3×C4#13511yescharacteristic
245C2×C3×C4#13511yescharacteristic
246#67—22no—
247#67—22no—
248#67—81no—
249#67—81no—
250C2×Dic3—21no—
251C2×Dic3—21no—
252C2×Dic3—21no—
253C2×Dic3—42no—
254C2×Dic3—41no—
255C2×Dic3—81no—
256C2×Dic3—81no—
257D12—21no—
258D12—41no—
259D12—41no—
260D12—41no—
261D12—41no—
262D12—81no—
263D12#10212yes—
264C4×D3—22no—
265C4×D3—22no—
266C4×D3—42no—
267C4×D3—41no—
268C4×D3—41no—
269C4×D3—81no—
270C4×D3—81no—
271Dic6—21no—
272Dic6—41no—
273Dic6—41no—
274Dic6#10212yes—
275C3×C8—42no—
276C3×C8—41no—
277C3×C8—41no—
278C3×C8—81no—
279C3⋊C8—22no—
280C3⋊C8—41no—
281#41—32no—
282#41—32no—
283#41—61no—
284#41—61no—
285#41—121no—
286#41—121no—
287#41—121no—
288#41—121no—
289C2×Q8—31no—
290C2×Q8—61no—
291C2×D4—31no—
292C2×D4—61no—
293C2×D4—61no—
294C2×D4—61no—
295C2×D4—61no—
296C2×D4—121no—
297C22×C4—61no—
298C22×C4—61no—
299C22×C4—61no—
300C22×C4—121no—
301C22×C4—121no—
302C22×C4C2×D1211yescharacteristic
303C22×C4D3×D411yescharacteristic
304C22×C4#23911yescharacteristic
305Q16—122no—
306SD16—122no—
307SD16—122no—
308D8—122no—
309M16—41no—
310M16—41no—
311C2×C8—21no—
312C2×C8—21no—
313C2×C8—22no—
314C2×C8—41no—
315C2×C8—41no—
316C2×C8—41no—
317C2×C8—61no—
318C2×C8—61no—
319C2×C8—124no—
320C2×C8—122no—
321C4⋊C4—31no—
322C4⋊C4—31no—
323C4⋊C4—62no—
324C4⋊C4—61no—
325C4⋊C4—61no—
326C4⋊C4—61no—
327C4⋊C4—61no—
328C4⋊C4—121no—
329C22⋊C4—61no—
330C22⋊C4—61no—
331C22⋊C4—61no—
332C22⋊C4—121no—
333C22⋊C4—121no—
334C42—61no—
335C42—61no—
336C42—61no—
337C42—121no—
338C42—121no—
339C42C2×C4×D311yescharacteristic
340C42#23511yescharacteristic
341C42D3×D411yescharacteristic
342C42D3×D411yescharacteristic
343C22×C3—21no—
344C22×C3—21no—
345C22×C3#38611yescharacteristic
346C22×C3#46511yescharacteristic
347C22×C3#46611yescharacteristic
348C2×D3—22no—
349C2×D3—41no—
350C2×D3—41no—
351C2×D3—81no—
352C2×D3—81no—
353C2×D3—81no—
354C3×C4—21no—
355C3×C4—21no—
356C3×C4—21no—
357C3×C4C2×#10211yescharacteristic
358C3×C4#41611yescharacteristic
359C3×C4#44311yescharacteristic
360C3×C4#44411yescharacteristic
361C3×C4#46511yescharacteristic
362C3×C4#46611yescharacteristic
363Dic3—22no—
364Dic3—41no—
365Dic3—41no—
366Dic3—82no—
367Dic3—81no—
368C23—61no—
369C23—121no—
370C23#89511yescharacteristic
371Q8—32no—
372Q8—61no—
373Q8—121no—
374Q8—121no—
375D4—32no—
376D4—62no—
377D4—62no—
378D4—61no—
379D4—121no—
380D4—121no—
381D4—121no—
382D4—121no—
383D4—241no—
384D4—241no—
385D4—241no—
386C2×C4—21no—
387C2×C4—21no—
388C2×C4—41no—
389C2×C4—61no—
390C2×C4—62no—
391C2×C4—62no—
392C2×C4—61no—
393C2×C4—61no—
394C2×C4—122no—
395C2×C4—121no—
396C2×C4—121no—
397C2×C4—122no—
398C2×C4—121no—
399C2×C4—241no—
400C2×C4—241no—
401C2×C4—241no—
402C2×C4—241no—
403C2×C4C4×D1211yescharacteristic
404C2×C4D3×C22⋊C411yescharacteristic
405C2×C4#88211yescharacteristic
406C2×C4#88411yescharacteristic
407C2×C4#89311yescharacteristic
408C2×C4#89411yescharacteristic
409C2×C4D3×D811yescharacteristic
410C2×C4D3×SD1611yescharacteristic
411C2×C4#91611yescharacteristic
412C2×C4#91911yescharacteristic
413C8—42no—
414C8—41no—
415C8—41no—
416C8—62no—
417C8—81no—
418C8—121no—
419C2×C3—21no—
420C2×C3—21no—
421C2×C3#187611yescharacteristic
422C2×C3#189311yescharacteristic
423C2×C3#190611yescharacteristic
424D3—42no—
425D3—161no—
426C22—21no—
427C22—21no—
428C22—62no—
429C22—121no—
430C22—121no—
431C22—241no—
432C22—241no—
433C22—241no—
434C22#466511yescharacteristic
435C22#486411yescharacteristic
436C22#488311yescharacteristic
437C4—21no—
438C4—21no—
439C4—21no—
440C4—62no—
441C4—121no—
442C4—121no—
443C4—242no—
444C4—241no—
445C4D3×#10211yescharacteristic
446C4#477511yescharacteristic
447C4#480411yescharacteristic
448C4#480711yescharacteristic
449C4#489411yescharacteristic
450C4#489511yescharacteristic
451C3#920811yescharacteristic, complemented, Sylow
452C2—21no—
453C2—21no—
454C2—122no—
455C2—481no—
456C2#6888311yescharacteristic
457C2#6894611yescharacteristic
458C2#6903911yescharacteristic
subgroupG/H
centerC2×C4#884
commutatorC3×C42C22×C4
FrattiniC2×C42C22×D3
FittingC3×#1301C2
socleC22×C3#386
prime radical#79021C2
cosocleC2×C3×C42C23
hypercenter#1301D3
hypocenterC3#9208
LCSG ▷ C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #1301

Complements

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

NH ≅ G/Ntype
1. #79276C2semidirect
3. #70789C2semidirect
4. #70529C2semidirect
5. #70498C2semidirect
6. C3×#1301C2semidirect
7. #67704C2semidirect
19. #4849C4semidirect
20. #4848C4semidirect
23. #4843C4semidirect
24. #4841C4semidirect
39. #1301D3semidirect
451. C3#9208semidirect

Maximal subgroups

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

subgroupindex
#792762
#790212
#707892
#705292
#704982
C3×#13012
#677042
#92083