Finite Group Databasebeta

← #10690052

Subgroups — #10690052

Order 768.

500 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#79309C211yesmaximal, characteristic
2M16×Dic6C211yesmaximal, characteristic, complemented
3#69867C211yesmaximal, characteristic
4C4×#816C211yesmaximal, characteristic, complemented
5#69735C211yesmaximal, characteristic
6C3×#1278C211yesmaximal, characteristic, complemented
7#67746C211yesmaximal, characteristic, complemented
8#8992—31nomaximal, Sylow
9#5477C2211yescharacteristic
10C2×C4×Dic6C2211yescharacteristic
11C3×#431C2211yescharacteristic
12C3×C4×M16C2211yescharacteristic
13C2×C3×C4×C8C2211yescharacteristic
14C2×#825—21no—
15C2×#825—21no—
16#5118—21no—
17Dic3×M16—21no—
18C2×#816—21no—
19C4×C4⋊Dic3C2211yescharacteristic
20#4924C2211yescharacteristic
21#4851C411yescharacteristic, complemented
22#4851C411yescharacteristic, complemented
23#4848C411yescharacteristic, complemented
24#4848C411yescharacteristic, complemented
25#4738—21no—
26#4735—21no—
27#4701—22no—
28C4×#204—21no—
29C4×#204—21no—
30#4679—21no—
31C8×Dic6—21no—
32#4551—21no—
33#4459—22no—
34#3103—31no—
35Q8×M16—31no—
36#1980—31no—
37C4×#103—31no—
38#1492—31no—
39#1303—31no—
40#1278D311yescharacteristic, complemented
41C2×#235—21no—
42C22×Dic6D411yescharacteristic
43C2×C3×M16D411yescharacteristic
44C2×C3×M16D411yescharacteristic
45C22×C3×C8—21no—
46C2×C3×C42C2311yescharacteristic
47#930—21no—
48#930—21no—
49C2×C4⋊Dic3—21no—
50C2×C4⋊Dic3D411yescharacteristic
51C2×Dic3⋊C4—21no—
52C2×C4×Dic3—21no—
53C2×C4×Dic3—21no—
54C2×#208—21no—
55C2×#204—21no—
56C2×#204—21no—
57#875—21no—
58#874C2×C411yescharacteristic
59C4×D12—21no—
60C4×D12—21no—
61#869C2×C411yescharacteristic
62C4×Dic6—21no—
63C4×Dic6—21no—
64C4×Dic6C2×C411yescharacteristic
65C4×Dic6C2×C411yescharacteristic
66C4×Dic6D411yescharacteristic
67C4×Dic6D411yescharacteristic
68C3×C4⋊C8C2×C411yescharacteristic
69C3×C4⋊C8C2×C411yescharacteristic
70C3×C22⋊C8—21no—
71C3×#97—21no—
72C3×C4×C8—21no—
73C3×C4×C8—21no—
74C3×C4×C8—21no—
75C3×C4×C8—21no—
76C3⋊#95—21no—
77#825—42no—
78#825—42no—
79#821—21no—
80#821—21no—
81#817—21no—
82#817—21no—
83#816—24no—
84#816—21no—
85#816—21no—
86#815—22no—
87Dic3⋊C8—22no—
88Dic3⋊C8—22no—
89C8×Dic3—22no—
90C4⋊(C3⋊C8)—21no—
91C4⋊(C3⋊C8)D412yes—
92#532—31no—
93C2×C4×Q8—31no—
94#481—61no—
95#475—31no—
96#475—31no—
97#460—31no—
98#460—31no—
99#448—61no—
100#446—61no—
101#446—62no—
102C8×Q8—61no—
103C4×SD16—61no—
104C4×SD16—61no—
105#431C2×D311yescharacteristic
106#423—31no—
107#423—61no—
108C2×#104—61no—
109C2×#104—61no—
110C2×#103—61no—
111C4×M16—61no—
112C4×M16C2×D311yescharacteristic
113C2×C4×C8C2×D311yescharacteristic
114C4×C4⋊C4—31no—
115#340—61no—
116D4⋊C8—62no—
117C22×C3×C4C2×D411yescharacteristic
118C22×C3×C4C2×D411yescharacteristic
119C22×C3×C4#4111yescharacteristic
120C2×#67—41no—
121C22×Dic3—21no—
122C22×Dic3—21no—
123C22×Dic3—41no—
124#235—41no—
125#235—41no—
126#235—41no—
127#235—41no—
128C2×D12—21no—
129C2×D12—21no—
130C2×D12—21no—
131C2×C4×D3—41no—
132C2×Dic6—22no—
133C2×Dic6—21no—
134C2×Dic6—21no—
135C2×Dic6—41no—
136C2×Dic6C22⋊C411yescharacteristic
137C2×Dic6C22⋊C411yescharacteristic
138C3×M16—21no—
139C3×M16—21no—
140C3×M16—21no—
141C3×M16SD1612yescomplemented
142C2×C3×C8—21no—
143C2×C3×C8—22no—
144C2×C3×C8—21no—
145C2×C3×C8—21no—
146C2×C3×C8—21no—
147C2×C3×C8—41no—
148C2×C3×C8—41no—
149C3×C42C22×C411yescharacteristic
150C3×C42C2×D411yescharacteristic
151C3×C42C2×D411yescharacteristic
152C3×C42#4111yescharacteristic
153(C2×D3)⋊C4—41no—
154(C2×D3)⋊C4—41no—
155C4⋊Dic3—21no—
156C4⋊Dic3—22no—
157C4⋊Dic3—21no—
158C4⋊Dic3—21no—
159C4⋊Dic3—41no—
160C4⋊Dic3C22⋊C411yescharacteristic
161C4⋊Dic3C22⋊C411yescharacteristic
162Dic3⋊C4—21no—
163Dic3⋊C4—21no—
164Dic3⋊C4—21no—
165Dic3⋊C4—21no—
166C4×Dic3—21no—
167C4×Dic3—21no—
168C4×Dic3—21no—
169C4×Dic3—21no—
170C4×Dic3—41no—
171C4×Dic3—41no—
172#208—41no—
173#208—41no—
174C2×C3⋊C8—22no—
175#204—42no—
176#204—42no—
177C2×#41—61no—
178C22×Q8—31no—
179C2×SD16—61no—
180C2×SD16—61no—
181C2×M16—61no—
182C2×M16D1211yescharacteristic
183C2×M16D1211yescharacteristic
184C22×C8—21no—
185C4⋊Q8—31no—
186#127—31no—
187#124—61no—
188#121—61no—
189#121—61no—
190C4×Q8—31no—
191C4×Q8—31no—
192C4×Q8—31no—
193C4×Q8—31no—
194C4×Q8—61no—
195C4×Q8—61no—
196C4×D4—61no—
197C4×D4—61no—
198C2×C4⋊C4—31no—
199C2×C4⋊C4—61no—
200C2×C4⋊C4—61no—
201C2×C42—61no—
202C2×C42—61no—
203C2×C42C22×D311yescharacteristic
204#106—61no—
205#106—61no—
206C4⋊C8—32no—
207C4⋊C8—62no—
208C4⋊C8—62no—
209C4⋊C8—61no—
210C4⋊C8C4×D311yescharacteristic
211C4⋊C8C4×D311yescharacteristic
212#104—122no—
213#104—122no—
214#103—61no—
215#103—61no—
216#103—64no—
217#102—61no—
218#102—61no—
219C22⋊C8—21no—
220#97—21no—
221#97—62no—
222C4×C8—21no—
223C4×C8—21no—
224C4×C8—21no—
225C4×C8—21no—
226C4×C8—62no—
227#95—61no—
228C23×C3#12111yescharacteristic
229C22×D3—41no—
230C2×C3×C4—21no—
231C2×C3×C4—21no—
232C2×C3×C4—41no—
233C2×C3×C4C2×C22⋊C411yescharacteristic
234C2×C3×C4C4×D411yescharacteristic
235C2×C3×C4C4×D411yescharacteristic
236C2×C3×C4#12011yescharacteristic
237C2×C3×C4#12111yescharacteristic
238C2×C3×C4#12311yescharacteristic
239C2×C3×C4C2×SD1611yescharacteristic
240C2×C3×C4#13511yescharacteristic
241C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
242C2×C3×C4#13711yescharacteristic
243#67—81no—
244#67—81no—
245C2×Dic3—21no—
246C2×Dic3—21no—
247C2×Dic3—21no—
248C2×Dic3—21no—
249C2×Dic3—42no—
250C2×Dic3—42no—
251C2×Dic3—41no—
252C2×Dic3—81no—
253C2×Dic3—81no—
254C2×Dic3—81no—
255D12—42no—
256D12—41no—
257D12—41no—
258D12—41no—
259D12—41no—
260C4×D3—81no—
261C4×D3—81no—
262Dic6—21no—
263Dic6—21no—
264Dic6—21no—
265Dic6—42no—
266Dic6—41no—
267Dic6—41no—
268Dic6#10412yescomplemented
269C3×C8—22no—
270C3×C8—42no—
271C3×C8—41no—
272C3×C8—41no—
273C3×C8—41no—
274C3⋊C8—42no—
275#41—121no—
276#41—121no—
277#41—121no—
278#41—121no—
279C2×Q8—31no—
280C2×Q8—31no—
281C2×Q8—62no—
282C2×Q8—61no—
283C2×Q8—61no—
284C2×Q8—121no—
285C2×D4—61no—
286C2×D4—61no—
287C2×D4—61no—
288C2×D4—121no—
289C22×C4—61no—
290C22×C4—61no—
291C22×C4—121no—
292C22×C4—121no—
293C22×C4C2×D1211yescharacteristic
294C22×C4D3×D411yescharacteristic
295C22×C4#23911yescharacteristic
296SD16—122no—
297SD16—122no—
298M16—21no—
299M16—21no—
300M16—21no—
301M16—121no—
302M16—121no—
303M16#20412yescomplemented
304C2×C8—21no—
305C2×C8—22no—
306C2×C8—21no—
307C2×C8—21no—
308C2×C8—21no—
309C2×C8—41no—
310C2×C8—41no—
311C2×C8—62no—
312C4⋊C4—31no—
313C4⋊C4—31no—
314C4⋊C4—62no—
315C4⋊C4—61no—
316C4⋊C4—61no—
317C4⋊C4—61no—
318C4⋊C4—61no—
319C4⋊C4—61no—
320C4⋊C4—61no—
321C4⋊C4—61no—
322C4⋊C4—121no—
323C22⋊C4—121no—
324C22⋊C4—121no—
325C42—61no—
326C42—61no—
327C42—61no—
328C42—61no—
329C42—121no—
330C42—121no—
331C42C2×C4×D311yescharacteristic
332C42#23511yescharacteristic
333C42D3×D411yescharacteristic
334C42D3×D411yescharacteristic
335C22×C3—21no—
336C22×C3—21no—
337C22×C3#38611yescharacteristic
338C22×C3#46011yescharacteristic
339C22×C3#46411yescharacteristic
340C2×D3—81no—
341C2×D3—81no—
342C2×D3—81no—
343C3×C4—21no—
344C3×C4—21no—
345C3×C4—21no—
346C3×C4C2×#10411yescharacteristic
347C3×C4#42111yescharacteristic
348C3×C4C4×SD1611yescharacteristic
349C3×C4#44011yescharacteristic
350C3×C4#44811yescharacteristic
351C3×C4#44911yescharacteristic
352Dic3—22no—
353Dic3—41no—
354Dic3—41no—
355Dic3—41no—
356Dic3—82no—
357Dic3—81no—
358C23—121no—
359C23#89511yescharacteristic
360Q8—32no—
361Q8—61no—
362Q8—61no—
363Q8—61no—
364Q8—122no—
365Q8—121no—
366Q8—121no—
367D4—122no—
368D4—121no—
369D4—121no—
370D4—121no—
371D4—121no—
372D4—241no—
373D4—241no—
374C2×C4—21no—
375C2×C4—21no—
376C2×C4—41no—
377C2×C4—61no—
378C2×C4—61no—
379C2×C4—61no—
380C2×C4—61no—
381C2×C4—122no—
382C2×C4—122no—
383C2×C4—121no—
384C2×C4—241no—
385C2×C4—241no—
386C2×C4—241no—
387C2×C4—241no—
388C2×C4—241no—
389C2×C4C4×D1211yescharacteristic
390C2×C4D3×C22⋊C411yescharacteristic
391C2×C4#88211yescharacteristic
392C2×C4#88411yescharacteristic
393C2×C4#89311yescharacteristic
394C2×C4#89411yescharacteristic
395C2×C4C2×#20411yescharacteristic
396C2×C4#90411yescharacteristic
397C2×C4#90811yescharacteristic
398C2×C4#90911yescharacteristic
399C8—22no—
400C8—42no—
401C8—41no—
402C8—41no—
403C8—41no—
404C8—122no—
405C2×C3—21no—
406C2×C3—21no—
407C2×C3#187111yescharacteristic
408C2×C3#188011yescharacteristic
409C2×C3#188111yescharacteristic
410D3—161no—
411C22—21no—
412C22—21no—
413C22—241no—
414C22—241no—
415C22—241no—
416C22#466511yescharacteristic
417C22#484511yescharacteristic
418C22#484811yescharacteristic
419C4—21no—
420C4—21no—
421C4—21no—
422C4—62no—
423C4—121no—
424C4—121no—
425C4—121no—
426C4—242no—
427C4—241no—
428C4C4×#20411yescharacteristic
429C4#471011yescharacteristic
430C4#473411yescharacteristic
431C4#473811yescharacteristic
432C4D3×#10411yescharacteristic
433C4#482111yescharacteristic
434C3#899211yescharacteristic, complemented, Sylow
435C2—21no—
436C2—21no—
437C2—481no—
438C2#6868611yescharacteristic
439C2#6875511yescharacteristic
440C2#6894711yescharacteristic
subgroupG/H
centerC2×C4#882
commutatorC3×C42C22×C4
FrattiniC2×C42C22×D3
FittingC3×#1278C2
socleC22×C3#386
prime radical#5477C22
cosocleC2×C3×C42C23
hypercenter#1278D3
hypocenterC3#8992
LCSG ▷ C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #1278

Complements

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

NH ≅ G/Ntype
2. M16×Dic6C2semidirect
4. C4×#816C2semidirect
6. C3×#1278C2semidirect
7. #67746C2semidirect
21. #4851C4semidirect
22. #4851C4semidirect
23. #4848C4semidirect
24. #4848C4semidirect
40. #1278D3semidirect
141. C3×M16SD16semidirect
268. Dic6#104semidirect
303. M16#204semidirect
434. C3#8992semidirect

Maximal subgroups

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

subgroupindex
#793092
M16×Dic62
#698672
C4×#8162
#697352
C3×#12782
#677462
#89923