Finite Group Databasebeta

← #10690053

Subgroups — #10690053

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
1C3×Q8×M16C211yesmaximal, characteristic, complemented
2#82442C211yesmaximal, characteristic
3C4×#835C211yesmaximal, characteristic, complemented
4#70498C211yesmaximal, characteristic
5#69867C211yesmaximal, characteristic
6#67715C211yesmaximal, characteristic, complemented
7#67667C211yesmaximal, characteristic, complemented
8#8992—31nomaximal, Sylow
9C2×C3×C4×Q8C2211yescharacteristic
10#5477C2211yescharacteristic
11C3×#446—21no—
12C3×#446—22no—
13C3×C8×Q8—21no—
14C3×#423—21no—
15C3×#423C2211yescharacteristic
16C3×C4×M16—21no—
17C3×C4×M16C2211yescharacteristic
18#5227—21no—
19C2×#835—21no—
20C2×#825—21no—
21C2×#825—21no—
22#5027C411yescharacteristic, complemented
23#5027C411yescharacteristic, complemented
24C4×#215—21no—
25C4×#215—21no—
26#5020C411yescharacteristic, complemented
27#5020C411yescharacteristic, complemented
28#4968—21no—
29C4×C4⋊Dic3C2211yescharacteristic
30#4925C2211yescharacteristic
31C2×C4×C3⋊C8C2211yescharacteristic
32#4529—21no—
33#4483—22no—
34#3103—31no—
35Q8×M16D311yescharacteristic, complemented
36#1980—31no—
37C4×#103—31no—
38#1492—31no—
39#1303—31no—
40#1278—31no—
41C22×C3×Q8D411yescharacteristic
42C2×#235—21no—
43C2×C3×M16—21no—
44C2×C3×M16D411yescharacteristic
45C2×C3×M16D411yescharacteristic
46C3×C4×Q8—21no—
47C3×C4×Q8—21no—
48C3×C4×Q8C2×C411yescharacteristic
49C3×C4×Q8C2×C411yescharacteristic
50C3×C4×Q8D411yescharacteristic
51C3×C4×Q8D411yescharacteristic
52C2×C3×C4⋊C4—21no—
53C2×C3×C4⋊C4D411yescharacteristic
54C2×C3×C42—21no—
55C2×C3×C42C2311yescharacteristic
56C2×#215—21no—
57C2×#215—21no—
58#930—21no—
59#930—21no—
60C2×C4⋊Dic3—21no—
61C2×C4×Dic3—21no—
62C22×C3⋊C8—21no—
63#875—21no—
64#874C2×C411yescharacteristic
65C4×D12—21no—
66C4×D12—21no—
67#869C2×C411yescharacteristic
68C3×C4⋊C8—22no—
69C3×C4⋊C8—22no—
70C3×C4⋊C8—21no—
71C3×C4⋊C8D412yes—
72C3×#97—22no—
73C3×#97—21no—
74C3×C4×C8—22no—
75C3×C4×C8—21no—
76#835—24no—
77#835—21no—
78#835—21no—
79C3⋊#95—21no—
80#830—21no—
81#825—42no—
82#825—42no—
83#809—21no—
84#809—21no—
85#808—21no—
86#808—21no—
87C4⋊(C3⋊C8)C2×C411yescharacteristic
88C4⋊(C3⋊C8)C2×C411yescharacteristic
89C4×C3⋊C8—21no—
90C4×C3⋊C8—21no—
91C4×C3⋊C8—21no—
92#532—31no—
93C2×C4×Q8C2×D311yescharacteristic
94#481—61no—
95#475—31no—
96#475—31no—
97#460—31no—
98#460—31no—
99#448—61no—
100#446—21no—
101#446—22no—
102C8×Q8—21no—
103C4×SD16—61no—
104C4×SD16—61no—
105#431—31no—
106#423—21no—
107#423C2×D311yescharacteristic
108C2×#104—61no—
109C2×#104—61no—
110C2×#103—61no—
111C4×M16—21no—
112C4×M16C2×D311yescharacteristic
113C2×C4×C8—31no—
114C4×C4⋊C4—31no—
115#340—61no—
116D4⋊C8—62no—
117C2×C3×Q8—21no—
118C2×C3×Q8—22no—
119C2×C3×Q8—41no—
120C2×C3×Q8C22⋊C411yescharacteristic
121C2×C3×Q8C22⋊C411yescharacteristic
122C22×C3×C4—21no—
123C22×C3×C4—21no—
124C22×C3×C4C2×D411yescharacteristic
125C22×C3×C4C2×D411yescharacteristic
126C22×C3×C4#4111yescharacteristic
127C2×#67—41no—
128C22×Dic3—41no—
129#235—41no—
130#235—41no—
131#235—41no—
132#235—41no—
133C2×D12—21no—
134C2×D12—21no—
135C2×D12—21no—
136C2×C4×D3—41no—
137C2×Dic6—21no—
138C3×M16—21no—
139C3×M16—21no—
140C3×M16—21no—
141C3×M16—41no—
142C3×M16—41no—
143C3×M16SD1612yescomplemented
144C2×C3×C8—21no—
145C2×C3×C8—21no—
146C2×C3×C8—22no—
147C3×C4⋊C4—21no—
148C3×C4⋊C4—21no—
149C3×C4⋊C4—21no—
150C3×C4⋊C4—21no—
151C3×C4⋊C4—21no—
152C3×C4⋊C4C22⋊C411yescharacteristic
153C3×C4⋊C4C22⋊C411yescharacteristic
154C3×C42—21no—
155C3×C42—21no—
156C3×C42—21no—
157C3×C42—21no—
158C3×C42C22×C411yescharacteristic
159C3×C42C2×D411yescharacteristic
160C3×C42C2×D411yescharacteristic
161C3×C42#4111yescharacteristic
162#215—42no—
163#215—42no—
164(C2×D3)⋊C4—41no—
165(C2×D3)⋊C4—41no—
166C4⋊Dic3—22no—
167C4⋊Dic3—21no—
168C4⋊Dic3—21no—
169C4⋊Dic3—41no—
170C4×Dic3—41no—
171C4×Dic3—41no—
172C2×C3⋊C8—22no—
173C2×C3⋊C8—21no—
174C2×C3⋊C8—21no—
175C2×C3⋊C8—41no—
176C2×C3⋊C8—41no—
177C2×#41—61no—
178C22×Q8D1211yescharacteristic
179C2×SD16—61no—
180C2×SD16—61no—
181C2×M16—21no—
182C2×M16#6711yescharacteristic
183C2×M16#6711yescharacteristic
184C22×C8—61no—
185C4⋊Q8—31no—
186#127—31no—
187#124—61no—
188#121—61no—
189#121—61no—
190C4×Q8—21no—
191C4×Q8—21no—
192C4×Q8C4×D311yescharacteristic
193C4×Q8C4×D311yescharacteristic
194C4×Q8#6711yescharacteristic
195C4×Q8#6711yescharacteristic
196C4×D4—61no—
197C4×D4—61no—
198C2×C4⋊C4—21no—
199C2×C4⋊C4—61no—
200C2×C4⋊C4D1211yescharacteristic
201C2×C42—21no—
202C2×C42—61no—
203C2×C42C22×D311yescharacteristic
204#106—61no—
205#106—61no—
206C4⋊C8—22no—
207C4⋊C8—22no—
208C4⋊C8—21no—
209C4⋊C8—31no—
210C4⋊C8—31no—
211C4⋊C8#6712yes—
212#104—122no—
213#104—122no—
214#103—64no—
215#103—61no—
216#103—61no—
217#102—61no—
218#102—61no—
219C22⋊C8—61no—
220#97—22no—
221#97—21no—
222C4×C8—22no—
223C4×C8—21no—
224C4×C8—61no—
225C4×C8—61no—
226C4×C8—61no—
227#95—61no—
228C23×C3#12111yescharacteristic
229C22×D3—41no—
230C3×Q8—21no—
231C3×Q8—21no—
232C3×Q8—21no—
233C3×Q8—42no—
234C3×Q8#10412yescomplemented
235C2×C3×C4—21no—
236C2×C3×C4—21no—
237C2×C3×C4—21no—
238C2×C3×C4—21no—
239C2×C3×C4—21no—
240C2×C3×C4—21no—
241C2×C3×C4—41no—
242C2×C3×C4—42no—
243C2×C3×C4—81no—
244C2×C3×C4C2×C22⋊C411yescharacteristic
245C2×C3×C4C4×D411yescharacteristic
246C2×C3×C4C4×D411yescharacteristic
247C2×C3×C4#12011yescharacteristic
248C2×C3×C4#12111yescharacteristic
249C2×C3×C4#12311yescharacteristic
250C2×C3×C4C2×SD1611yescharacteristic
251C2×C3×C4#13511yescharacteristic
252C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
253C2×C3×C4#13711yescharacteristic
254#67—81no—
255#67—81no—
256C2×Dic3—42no—
257C2×Dic3—41no—
258C2×Dic3—81no—
259C2×Dic3—81no—
260D12—42no—
261D12—41no—
262D12—41no—
263D12—41no—
264D12—41no—
265C4×D3—81no—
266C4×D3—81no—
267Dic6—41no—
268Dic6—41no—
269C3×C8—22no—
270C3×C8—41no—
271C3×C8—42no—
272C3⋊C8—42no—
273C3⋊C8—41no—
274C3⋊C8—41no—
275#41—121no—
276#41—121no—
277#41—121no—
278#41—121no—
279C2×Q8—21no—
280C2×Q8—22no—
281C2×Q8—41no—
282C2×Q8—61no—
283C2×Q8(C2×D3)⋊C411yescharacteristic
284C2×Q8(C2×D3)⋊C411yescharacteristic
285C2×D4—61no—
286C2×D4—61no—
287C2×D4—61no—
288C2×D4—121no—
289C22×C4—21no—
290C22×C4—21no—
291C22×C4—121no—
292C22×C4—121no—
293C22×C4C2×D1211yescharacteristic
294C22×C4#23511yescharacteristic
295C22×C4C2×#6711yescharacteristic
296SD16—122no—
297SD16—122no—
298M16—21no—
299M16—21no—
300M16—21no—
301M16—41no—
302M16—41no—
303M16#21512yescomplemented
304C2×C8—21no—
305C2×C8—21no—
306C2×C8—22no—
307C2×C8—62no—
308C2×C8—61no—
309C2×C8—61no—
310C2×C8—121no—
311C2×C8—121no—
312C4⋊C4—21no—
313C4⋊C4—21no—
314C4⋊C4—21no—
315C4⋊C4—21no—
316C4⋊C4—21no—
317C4⋊C4—62no—
318C4⋊C4—61no—
319C4⋊C4—61no—
320C4⋊C4—121no—
321C4⋊C4(C2×D3)⋊C411yescharacteristic
322C4⋊C4(C2×D3)⋊C411yescharacteristic
323C22⋊C4—121no—
324C22⋊C4—121no—
325C42—21no—
326C42—21no—
327C42—21no—
328C42—21no—
329C42—121no—
330C42—121no—
331C42C2×C4×D311yescharacteristic
332C42#23511yescharacteristic
333C42C2×#6711yescharacteristic
334C42C2×#6711yescharacteristic
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—22no—
345C3×C4—21no—
346C3×C4—21no—
347C3×C4—41no—
348C3×C4—41no—
349C3×C4—41no—
350C3×C4C2×#10411yescharacteristic
351C3×C4#42111yescharacteristic
352C3×C4C4×SD1611yescharacteristic
353C3×C4#44011yescharacteristic
354C3×C4#44811yescharacteristic
355C3×C4#44911yescharacteristic
356Dic3—82no—
357Dic3—81no—
358C23—121no—
359C23#93011yescharacteristic
360Q8—21no—
361Q8—21no—
362Q8—21no—
363Q8—42no—
364Q8—121no—
365Q8—121no—
366Q8#82512yescomplemented
367D4—122no—
368D4—121no—
369D4—121no—
370D4—121no—
371D4—121no—
372D4—241no—
373D4—241no—
374C2×C4—21no—
375C2×C4—21no—
376C2×C4—21no—
377C2×C4—21no—
378C2×C4—21no—
379C2×C4—21no—
380C2×C4—41no—
381C2×C4—42no—
382C2×C4—81no—
383C2×C4—122no—
384C2×C4—121no—
385C2×C4—241no—
386C2×C4—241no—
387C2×C4—241no—
388C2×C4—241no—
389C2×C4C2×(C2×D3)⋊C411yescharacteristic
390C2×C4C4×#6711yescharacteristic
391C2×C4C4×#6711yescharacteristic
392C2×C4#92911yescharacteristic
393C2×C4#93011yescharacteristic
394C2×C4C2×#21511yescharacteristic
395C2×C4#94211yescharacteristic
396C2×C4#94911yescharacteristic
397C2×C4#95011yescharacteristic
398C2×C4#95311yescharacteristic
399C8—22no—
400C8—41no—
401C8—42no—
402C8—122no—
403C8—121no—
404C8—121no—
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#502711yescharacteristic
417C22#502811yescharacteristic
418C22#521311yescharacteristic
419C4—21no—
420C4—22no—
421C4—21no—
422C4—21no—
423C4—41no—
424C4—41no—
425C4—41no—
426C4—242no—
427C4—241no—
428C4C4×#21511yescharacteristic
429C4#502611yescharacteristic
430C4C2×#82511yescharacteristic
431C4#513911yescharacteristic
432C4#522711yescharacteristic
433C4#523811yescharacteristic
434C3#899211yescharacteristic, complemented, Sylow
435C2—21no—
436C2—21no—
437C2—481no—
438C2#7163311yescharacteristic
439C2#7198411yescharacteristic
440C2#7202011yescharacteristic
subgroupG/H
centerC2×C4#953
commutatorC3×C42C22×C4
FrattiniC2×C42C22×D3
FittingC3×Q8×M16C2
socleC22×C3#386
prime radical#5477C22
cosocleC2×C3×C42C23
hypercenterQ8×M16D3
hypocenterC3#8992
LCSG ▷ C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ Q8×M16

Complements

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

NH ≅ G/Ntype
1. C3×Q8×M16C2semidirect
3. C4×#835C2semidirect
6. #67715C2semidirect
7. #67667C2semidirect
22. #5027C4semidirect
23. #5027C4semidirect
26. #5020C4semidirect
27. #5020C4semidirect
35. Q8×M16D3semidirect
143. C3×M16SD16semidirect
234. C3×Q8#104semidirect
303. M16#215semidirect
366. Q8#825semidirect
434. C3#8992semidirect

Maximal subgroups

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

subgroupindex
C3×Q8×M162
#824422
C4×#8352
#704982
#698672
#677152
#676672
#89923