Finite Group Databasebeta

← #10690664

Subgroups — #10690664

Order 768.

514 conjugacy classes of subgroups.

The subgroup lattice has 428 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C4×#942C211yesmaximal, characteristic
2C8×#235C211yesmaximal, characteristic
3C2×C3×Q8⋊C8C211yesmaximal, characteristic, complemented
4#71624C211yesmaximal, characteristic, complemented
5#70425C211yesmaximal, characteristic
6#67706C211yesmaximal, characteristic, complemented
7#67656C211yesmaximal, characteristic, complemented
8#9096—31nomaximal, Sylow
9C2×C3×C4×Q8C2211yescharacteristic
10C2×#942C411yescharacteristic
11C4×#235C2211yescharacteristic
12C2×C3×C4⋊C8C2211yescharacteristic
13C2×C3×C4×C8C2211yescharacteristic
14#5225C411yescharacteristic
15C8×#67—22no—
16#5030—21no—
17C4×#216—21no—
18#5026—21no—
19C4×#215—21no—
20#4966C411yescharacteristic
21#4964C411yescharacteristic
22#4925C2211yescharacteristic
23#4925C2211yescharacteristic
24C4×#208C2211yescharacteristic
25C8×D12—21no—
26#4685—21no—
27C4×D3×C8—21no—
28C8×Dic6—21no—
29C3×Q8⋊C8—21no—
30C3×Q8⋊C8—21no—
31#4528—22no—
32#4484—21no—
33#4483—21no—
34C8×#41—31no—
35C4×#137—31no—
36#1477—31no—
37C2×Q8⋊C8D311yescharacteristic, complemented
38#1461—31no—
39#1283—31no—
40#1278—31no—
41C22×C3×Q8C2×C411yescharacteristic
42C2×#235C2×C411yescharacteristic
43C22×C3×C8—21no—
44C22×C3×C8—21no—
45C3×C4×Q8—21no—
46C3×C4×Q8—21no—
47C3×C4×Q8—21no—
48C3×C4×Q8—21no—
49C2×C3×C4⋊C4—21no—
50C2×C3×C4⋊C4C2×C411yescharacteristic
51C2×C3×C42—21no—
52C2×C3×C42C2311yescharacteristic
53C2×#216—21no—
54#942—22no—
55#942C814yescomplemented
56C2×#215—21no—
57C4×#67—21no—
58C4×#67—21no—
59#926C2×C411yescharacteristic
60C2×#208C2×C411yescharacteristic
61C2×#208C2×C411yescharacteristic
62C2×D3×C8—22no—
63C4×D12D411yescharacteristic
64C4×D12D411yescharacteristic
65#872—21no—
66C42×D3—21no—
67C4×Dic6D411yescharacteristic
68C4×Dic6D411yescharacteristic
69C3×C4⋊C8—21no—
70C3×C4⋊C8—21no—
71C3×C4×C8—21no—
72C3×C4×C8D411yescharacteristic
73C3×C4×C8D411yescharacteristic
74#835—21no—
75#835—21no—
76#830—21no—
77#830D412yes—
78(C2×D3)⋊C8—22no—
79Dic3⋊C8—22no—
80C8×Dic3—22no—
81#810—21no—
82#809—21no—
83#808—21no—
84#807—21no—
85C4⋊(C3⋊C8)—21no—
86C4⋊(C3⋊C8)—21no—
87#803—21no—
88C4×C3⋊C8—21no—
89C4×C3⋊C8—21no—
90C4×C3⋊C8—21no—
91C2×#137—31no—
92C4×#41—31no—
93C2×C4×Q8C2×D311yescharacteristic
94C8×Q8—61no—
95#441—61no—
96#440—61no—
97C4×Q16—61no—
98C4×SD16—61no—
99C8×D4—61no—
100C8×D4—62no—
101#431—31no—
102#431—31no—
103#431—61no—
104#428—31no—
105C2×C4⋊C8C2×D311yescharacteristic
106#419—31no—
107#417—31no—
108C4×M16—31no—
109C2×C4×C8—61no—
110C2×C4×C8C2×D311yescharacteristic
111Q8⋊C8—21no—
112Q8⋊C8—21no—
113Q8⋊C8—61no—
114D4⋊C8—61no—
115#324—62no—
116C2×C3×Q8—21no—
117C2×C3×Q8—21no—
118C2×C3×Q8—21no—
119C2×C3×Q8—21no—
120C2×C3×Q8—21no—
121C2×C3×Q8C2×C812yescomplemented
122C22×C3×C4—21no—
123C22×C3×C4—21no—
124C22×C3×C4C22×C411yescharacteristic
125C22×C3×C4C2×D411yescharacteristic
126C22×C3×C4#4111yescharacteristic
127C2×#67—21no—
128#235—21no—
129#235—21no—
130#235C2×C812yes—
131#235M1612yes—
132C2×D12C22⋊C411yescharacteristic
133C2×C4×D3—21no—
134C2×C4×D3—21no—
135C2×C4×D3—21no—
136C2×Dic6C22⋊C411yescharacteristic
137C2×C3×C8—21no—
138C2×C3×C8—21no—
139C2×C3×C8—42no—
140C2×C3×C8—41no—
141C2×C3×C8—81no—
142C3×C4⋊C4—21no—
143C3×C4⋊C4—21no—
144C3×C4⋊C4—41no—
145C3×C4⋊C4—41no—
146C3×C42—41no—
147C3×C42—41no—
148C3×C42C2×D411yescharacteristic
149C3×C42C2×D411yescharacteristic
150C3×C42C2×D411yescharacteristic
151C3×C42#4111yescharacteristic
152#217—21no—
153#216—24no—
154#215—24no—
155(C2×D3)⋊C4—21no—
156(C2×D3)⋊C4—21no—
157C4⋊Dic3C22⋊C411yescharacteristic
158C4⋊Dic3C22⋊C411yescharacteristic
159Dic3⋊C4—21no—
160Dic3⋊C4—21no—
161C4×Dic3—21no—
162C4×Dic3—21no—
163C4×Dic3—21no—
164#208—21no—
165#208—21no—
166#208—21no—
167#208C2×C812yescomplemented
168C2×C3⋊C8—21no—
169C2×C3⋊C8—21no—
170C2×C3⋊C8—22no—
171C2×C3⋊C8—41no—
172D3×C8—44no—
173C2×#41—31no—
174C22×Q8C4×D311yescharacteristic
175#137—34no—
176#137—62no—
177C2×Q16—61no—
178C2×SD16—61no—
179C2×M16—31no—
180C2×M16—31no—
181C22×C8—21no—
182C22×C8—21no—
183C22×C8—62no—
184C4×Q8—21no—
185C4×Q8—21no—
186C4×Q8—21no—
187C4×Q8—21no—
188C4×Q8—31no—
189C4×Q8—31no—
190C4×D4—31no—
191C4×D4—31no—
192C4×D4—61no—
193C4×D4—61no—
194#117—31no—
195#117—61no—
196C2×C4⋊C4—21no—
197C2×C4⋊C4C4×D311yescharacteristic
198C2×C42—21no—
199C2×C42—61no—
200C2×C42C22×D311yescharacteristic
201#107—61no—
202#106—61no—
203C4⋊C8—21no—
204C4⋊C8—21no—
205C4⋊C8—62no—
206C4⋊C8—61no—
207C4⋊C8—61no—
208#103—61no—
209#103—61no—
210#103—61no—
211#102—61no—
212C22⋊C8—32no—
213C22⋊C8—62no—
214C22⋊C8—61no—
215#97—61no—
216C4×C8—21no—
217C4×C8—62no—
218C4×C8—61no—
219C4×C8—61no—
220C4×C8—61no—
221C4×C8#6711yescharacteristic
222C4×C8#6711yescharacteristic
223C23×C3C4×D411yescharacteristic
224C22×D3—21no—
225C3×Q8—22no—
226C3×Q8—22no—
227C3×Q8—41no—
228C3×Q8—41no—
229C2×C3×C4—21no—
230C2×C3×C4—21no—
231C2×C3×C4—22no—
232C2×C3×C4—41no—
233C2×C3×C4—41no—
234C2×C3×C4—41no—
235C2×C3×C4—41no—
236C2×C3×C4—81no—
237C2×C3×C4C2×C22⋊C411yescharacteristic
238C2×C3×C4C4×D411yescharacteristic
239C2×C3×C4#12011yescharacteristic
240C2×C3×C4#12111yescharacteristic
241C2×C3×C4#12111yescharacteristic
242C2×C3×C4#12311yescharacteristic
243C2×C3×C4C22×C811yescharacteristic
244C2×C3×C4C2×M1611yescharacteristic
245C2×C3×C4#13111yescharacteristic
246C2×C3×C4#13111yescharacteristic
247#67—22no—
248#67—22no—
249C2×Dic3—21no—
250C2×Dic3—21no—
251C2×Dic3—21no—
252D12—21no—
253D12C22⋊C812yes—
254C4×D3—22no—
255C4×D3—22no—
256C4×D3—42no—
257C4×D3—41no—
258C4×D3—41no—
259Dic6—21no—
260Dic6C22⋊C812yes—
261C3×C8—22no—
262C3×C8—41no—
263C3×C8—81no—
264C3⋊C8—22no—
265C3⋊C8—42no—
266C3⋊C8—41no—
267C3⋊C8—81no—
268#41—32no—
269#41—32no—
270#41—61no—
271#41—61no—
272C2×Q8—21no—
273C2×Q8—21no—
274C2×Q8—21no—
275C2×Q8—21no—
276C2×Q8—21no—
277C2×Q8—31no—
278C2×Q8D3×C812yescomplemented
279C2×D4—31no—
280C2×D4—61no—
281C22×C4—21no—
282C22×C4—21no—
283C22×C4—61no—
284C22×C4—61no—
285C22×C4—61no—
286C22×C4C2×C4×D311yescharacteristic
287C22×C4D3×D411yescharacteristic
288C22×C4#23711yescharacteristic
289Q16—64no—
290SD16—64no—
291M16—32no—
292M16—61no—
293M16—61no—
294M16—61no—
295C2×C8—21no—
296C2×C8—21no—
297C2×C8—42no—
298C2×C8—41no—
299C2×C8—61no—
300C2×C8—61no—
301C2×C8—62no—
302C2×C8—81no—
303C2×C8—124no—
304C2×C8—121no—
305C4⋊C4—21no—
306C4⋊C4—21no—
307C4⋊C4—31no—
308C4⋊C4—31no—
309C4⋊C4—41no—
310C4⋊C4—41no—
311C4⋊C4—61no—
312C4⋊C4—61no—
313C22⋊C4—61no—
314C22⋊C4—61no—
315C22⋊C4—61no—
316C42—41no—
317C42—41no—
318C42—61no—
319C42—61no—
320C42—61no—
321C42#23511yescharacteristic
322C42D3×D411yescharacteristic
323C42D3×D411yescharacteristic
324C42C2×#6711yescharacteristic
325C22×C3—21no—
326C22×C3—21no—
327C22×C3#38611yescharacteristic
328C22×C3C8×D411yescharacteristic
329C22×C3#43511yescharacteristic
330C2×D3—22no—
331C2×D3—41no—
332C2×D3—41no—
333C3×C4—21no—
334C3×C4—21no—
335C3×C4—21no—
336C3×C4—42no—
337C3×C4—81no—
338C3×C4C2×C22⋊C811yescharacteristic
339C3×C4#40811yescharacteristic
340C3×C4C8×D411yescharacteristic
341C3×C4#43511yescharacteristic
342C3×C4#47111yescharacteristic
343C3×C4#47311yescharacteristic
344Dic3—22no—
345Dic3—41no—
346Dic3—41no—
347C23—61no—
348C23#67⋊C411yescharacteristic
349Q8—22no—
350Q8—22no—
351Q8—32no—
352Q8—41no—
353Q8—41no—
354Q8—61no—
355D4—32no—
356D4—62no—
357D4—62no—
358D4—61no—
359C2×C4—21no—
360C2×C4—21no—
361C2×C4—22no—
362C2×C4—41no—
363C2×C4—41no—
364C2×C4—41no—
365C2×C4—41no—
366C2×C4—61no—
367C2×C4—62no—
368C2×C4—62no—
369C2×C4—61no—
370C2×C4—61no—
371C2×C4—81no—
372C2×C4—122no—
373C2×C4—121no—
374C2×C4—121no—
375C2×C4D3×C22⋊C411yescharacteristic
376C2×C4#88311yescharacteristic
377C2×C4#88411yescharacteristic
378C2×C4C2×D3×C811yescharacteristic
379C2×C4#90111yescharacteristic
380C2×C4D3×M1611yescharacteristic
381C2×C4#90711yescharacteristic
382C2×C4C4×#6711yescharacteristic
383C2×C4#93711yescharacteristic
384C2×C4#93811yescharacteristic
385C8—22no—
386C8—41no—
387C8—62no—
388C8—81no—
389C8—122no—
390C8—121no—
391C8—241no—
392C2×C3—21no—
393C2×C3—21no—
394C2×C3#186911yescharacteristic
395C2×C3#190611yescharacteristic
396C2×C3#190811yescharacteristic
397D3—42no—
398C22—21no—
399C22—21no—
400C22—62no—
401C22—121no—
402C22—121no—
403C22#67⋊C811yescharacteristic
404C22#472811yescharacteristic
405C22#495611yescharacteristic
406C4—21no—
407C4—21no—
408C4—21no—
409C4—42no—
410C4—62no—
411C4—81no—
412C4—121no—
413C4—121no—
414C4D3×C22⋊C811yescharacteristic
415C4#472711yescharacteristic
416C4C8×#6711yescharacteristic
417C4#512711yescharacteristic
418C4#517711yescharacteristic
419C4#519211yescharacteristic
420C3#909611yescharacteristic, complemented, Sylow
421C2—21no—
422C2—21no—
423C2—122no—
424C2#6995711yescharacteristic
425C2#7028611yescharacteristic
426C2#7039111yescharacteristic
subgroupG/H
centerC2×C4#938
commutatorC2×C3×C4C22×C8
FrattiniC2×C42C22×D3
FittingC2×C3×Q8⋊C8C2
socleC22×C3#386
prime radicalC2×#235C2×C4
cosocleC2×C3×C42C23
hypercenterC2×Q8⋊C8D3
hypocenterC3#9096
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ C2×Q8⋊C8

Complements

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

NH ≅ G/Ntype
3. C2×C3×Q8⋊C8C2semidirect
4. #71624C2semidirect
6. #67706C2semidirect
7. #67656C2semidirect
37. C2×Q8⋊C8D3semidirect
55. #942C8semidirect
121. C2×C3×Q8C2×C8semidirect
167. #208C2×C8semidirect
278. C2×Q8D3×C8semidirect
420. C3#9096semidirect

Maximal subgroups

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

subgroupindex
C4×#9422
C8×#2352
C2×C3×Q8⋊C82
#716242
#704252
#677062
#676562
#90963