Finite Group Databasebeta

← #10691547

Subgroups — #10691547

Order 768.

544 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
1C2×#5083C211yesmaximal, characteristic
2#79043C211yesmaximal, characteristic
3C3×#1548C211yesmaximal, characteristic, complemented
4#71790C211yesmaximal, characteristic, complemented
5#70762C211yesmaximal, characteristic
6#67744C211yesmaximal, characteristic, complemented
7#67711C211yesmaximal, characteristic, complemented
8#9247—31nomaximal, Sylow
9C2×C3×C4⋊Q8C2211yescharacteristic
10C22×#215C412yes—
11C2×D3×M16—21no—
12C2×C4×D12C2211yescharacteristic
13C3×#431C2211yescharacteristic
14C3×#423C2211yescharacteristic
15C2×#835C412yes—
16#5126—21no—
17#5083—21no—
18#5083—21no—
19#5083—21no—
20#5083—21no—
21#4924C2211yescharacteristic
22C2×C4⋊(C3⋊C8)C2211yescharacteristic
23C4×#208C2211yescharacteristic
24#4838—21no—
25#4836—22no—
26#4834—21no—
27C3×#333—22no—
28#4530—21no—
29#4530—21no—
30#4459—21no—
31#4459—21no—
32C2×#461—31no—
33D4×M16—31no—
34#1997—31no—
35#1548D311yescharacteristic, complemented
36#1486—31no—
37#1292—31no—
38#1283—31no—
39C22×C3×Q8C2×C412yes—
40C22×D12C2×C411yescharacteristic
41C22×C4×D3—21no—
42C2×C3×M16—21no—
43C3×C4⋊Q8—21no—
44C3×C4⋊Q8—21no—
45C3×C4⋊Q8—21no—
46C3×C4⋊Q8—21no—
47C2×C3×C4⋊C4—21no—
48C2×C3×C4⋊C4—21no—
49C2×C3×C42C2311yescharacteristic
50C2×#215—22no—
51C2×#215—22no—
52C2×#215—44no—
53C2×(C2×D3)⋊C4—21no—
54C2×C4⋊Dic3C2×C411yescharacteristic
55C2×#208—21no—
56C2×#208D411yescharacteristic
57C2×#208D411yescharacteristic
58C22×C3⋊C8C2×C412yes—
59D3×M16—42no—
60D3×M16—42no—
61C2×ΓL(1,𝔽25)—22no—
62C2×D3×C8—22no—
63C4×D12—21no—
64C4×D12—21no—
65C4×D12D411yescharacteristic
66C4×D12D411yescharacteristic
67C4×D12D411yescharacteristic
68C4×D12D411yescharacteristic
69C3×C4⋊C8—21no—
70C3×C4⋊C8—21no—
71C3×C4⋊C8D412yes—
72C3×C22⋊C8—21no—
73C3×C4×C8—21no—
74#835—22no—
75#835—22no—
76(C2×D3)⋊C8—22no—
77(C2×D3)⋊C8—22no—
78C4⋊(C3⋊C8)—21no—
79C4⋊(C3⋊C8)—21no—
80C4⋊(C3⋊C8)—21no—
81C4⋊(C3⋊C8)—21no—
82#803—21no—
83C4×C3⋊C8—21no—
84C22×SD16—32no—
85C22×M16—61no—
86C2×C4⋊Q8C2×D311yescharacteristic
87C2×C4×D4—31no—
88#461—61no—
89#461—61no—
90#461—61no—
91#461—61no—
92#436—61no—
93#435—62no—
94C8×D4—61no—
95#431C2×D311yescharacteristic
96#423—31no—
97#423C2×D311yescharacteristic
98C2×C4⋊C8—31no—
99C2×#103—32no—
100#407—61no—
101C4×M16—31no—
102#343—61no—
103#343—61no—
104#333—22no—
105D4⋊C8—61no—
106D4⋊C8—61no—
107C23×D3—21no—
108C2×C3×Q8—22no—
109C2×C3×Q8—22no—
110C2×C3×Q8—42no—
111C2×C3×Q8—42no—
112C22×C3×C4—22no—
113C22×C3×C4C22×C411yescharacteristic
114C22×C3×C4C2×D411yescharacteristic
115C22×C3×C4#4111yescharacteristic
116C22×Dic3—21no—
117C2×D12—22no—
118C2×D12—21no—
119C2×D12—41no—
120C2×D12C22⋊C411yescharacteristic
121C2×D12C22⋊C411yescharacteristic
122C2×C4×D3—21no—
123C2×C4×D3—21no—
124C2×C4×D3—21no—
125C2×C4×D3—21no—
126C2×C4×D3—41no—
127C2×C4×D3—41no—
128C2×C4×D3—41no—
129C2×C4×D3—41no—
130C3×M16—41no—
131C3×M16—41no—
132C2×C3×C8—22no—
133C2×C3×C8—41no—
134C3×C4⋊C4—41no—
135C3×C4⋊C4—41no—
136C3×C4⋊C4—41no—
137C3×C4⋊C4—41no—
138C3×C42C2×D411yescharacteristic
139C3×C42C2×D411yescharacteristic
140C3×C42C2×D411yescharacteristic
141C3×C42#4111yescharacteristic
142#215—44no—
143#215—44no—
144(C2×D3)⋊C4—21no—
145(C2×D3)⋊C4—21no—
146(C2×D3)⋊C4—21no—
147(C2×D3)⋊C4—21no—
148C4⋊Dic3—21no—
149C4⋊Dic3C22⋊C411yescharacteristic
150C4⋊Dic3C22⋊C411yescharacteristic
151#208—21no—
152#208—21no—
153#208—21no—
154#208—41no—
155#208—41no—
156#208SD1612yescomplemented
157C2×C3⋊C8—21no—
158C2×C3⋊C8—22no—
159C2×C3⋊C8—21no—
160C2×C3⋊C8—42no—
161C2×C3⋊C8—41no—
162ΓL(1,𝔽25)—44no—
163D3×C8—44no—
164C22×Q8C4×D312yes—
165C22×D4—31no—
166C23×C4—61no—
167C2×SD16—62no—
168C2×SD16—62no—
169C2×SD16—124no—
170C2×M16—21no—
171C2×M16—31no—
172C2×M16—31no—
173C2×M16—62no—
174C2×M16—61no—
175C2×M16—122no—
176C2×M16—122no—
177C22×C8—32no—
178C22×C8—62no—
179C4⋊Q8—21no—
180C4⋊Q8—21no—
181C4⋊Q8—21no—
182C4⋊Q8—21no—
183C4×D4—31no—
184C4×D4—31no—
185C4×D4—31no—
186C4×D4—31no—
187C4×D4—61no—
188C4×D4—61no—
189C2×C4⋊C4—21no—
190C2×C4⋊C4—21no—
191C2×C4⋊C4—31no—
192C2×C22⋊C4—61no—
193C2×C42C22×D311yescharacteristic
194C4⋊C8—21no—
195C4⋊C8—21no—
196C4⋊C8—61no—
197C4⋊C8—61no—
198C4⋊C8—61no—
199C4⋊C8—61no—
200C4⋊C8#6712yes—
201#103—62no—
202#103—62no—
203C22⋊C8—21no—
204C22⋊C8—62no—
205C22⋊C8—62no—
206#97—61no—
207C4×C8—21no—
208C4×C8—61no—
209C23×C3C4×D411yescharacteristic
210C22×D3—21no—
211C22×D3—21no—
212C22×D3—42no—
213C22×D3—41no—
214C22×D3—41no—
215C22×D3—81no—
216C3×Q8—42no—
217C3×Q8—42no—
218C3×Q8—82no—
219C2×C3×C4—21no—
220C2×C3×C4—21no—
221C2×C3×C4—41no—
222C2×C3×C4—42no—
223C2×C3×C4—82no—
224C2×C3×C4C2×C22⋊C411yescharacteristic
225C2×C3×C4C4×D411yescharacteristic
226C2×C3×C4#12011yescharacteristic
227C2×C3×C4#12111yescharacteristic
228C2×C3×C4#12111yescharacteristic
229C2×C3×C4#12311yescharacteristic
230C2×C3×C4C2×SD1611yescharacteristic
231C2×C3×C4#13511yescharacteristic
232C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
233C2×C3×C4#13711yescharacteristic
234C2×Dic3—21no—
235C2×Dic3—21no—
236C2×Dic3—41no—
237C2×Dic3—41no—
238D12—21no—
239D12—21no—
240D12—21no—
241D12—42no—
242D12#10012yes—
243C4×D3—41no—
244C4×D3—41no—
245C4×D3—41no—
246C4×D3—41no—
247C4×D3—41no—
248C4×D3—41no—
249C4×D3—41no—
250C4×D3—41no—
251C3×C8—42no—
252C3×C8—81no—
253C3⋊C8—22no—
254C3⋊C8—42no—
255C3⋊C8—41no—
256C3⋊C8—81no—
257C24—61no—
258C2×Q8—22no—
259C2×Q8—22no—
260C2×Q8—42no—
261C2×Q8—42no—
262C2×D4—31no—
263C2×D4—31no—
264C2×D4—62no—
265C2×D4—61no—
266C2×D4—121no—
267C22×C4—22no—
268C22×C4—61no—
269C22×C4—61no—
270C22×C4—61no—
271C22×C4—61no—
272C22×C4—61no—
273C22×C4—121no—
274C22×C4—121no—
275C22×C4—121no—
276C22×C4—121no—
277C22×C4C2×C4×D311yescharacteristic
278C22×C4D3×D411yescharacteristic
279C22×C4#23711yescharacteristic
280SD16—124no—
281SD16—124no—
282M16—32no—
283M16—41no—
284M16—41no—
285M16—61no—
286M16—61no—
287M16—61no—
288M16—124no—
289M16—121no—
290M16—121no—
291C2×C8—22no—
292C2×C8—41no—
293C2×C8—61no—
294C2×C8—62no—
295C2×C8—61no—
296C2×C8—124no—
297C2×C8—122no—
298C2×C8—121no—
299C4⋊C4—31no—
300C4⋊C4—31no—
301C4⋊C4—41no—
302C4⋊C4—41no—
303C4⋊C4—41no—
304C4⋊C4—41no—
305C4⋊C4—61no—
306C22⋊C4—61no—
307C22⋊C4—61no—
308C22⋊C4—61no—
309C22⋊C4—61no—
310C42#23511yescharacteristic
311C42D3×D411yescharacteristic
312C42D3×D411yescharacteristic
313C42C2×#6711yescharacteristic
314C22×C3—21no—
315C22×C3—21no—
316C22×C3#38611yescharacteristic
317C22×C3C4×SD1611yescharacteristic
318C22×C3#44011yescharacteristic
319C2×D3—22no—
320C2×D3—41no—
321C2×D3—41no—
322C2×D3—41no—
323C2×D3—41no—
324C2×D3—41no—
325C2×D3—82no—
326C2×D3—82no—
327C3×C4—21no—
328C3×C4—21no—
329C3×C4—21no—
330C3×C4—82no—
331C3×C4C2×#10011yescharacteristic
332C3×C4#41311yescharacteristic
333C3×C4#45011yescharacteristic
334C3×C4#45211yescharacteristic
335C3×C4#46111yescharacteristic
336C3×C4#46311yescharacteristic
337Dic3—41no—
338Dic3—41no—
339C23—61no—
340C23—61no—
341C23—122no—
342C23—121no—
343C23—121no—
344C23—241no—
345C23#67⋊C411yescharacteristic
346Q8—42no—
347Q8—42no—
348Q8—82no—
349D4—32no—
350D4—61no—
351D4—61no—
352D4—61no—
353D4—122no—
354C2×C4—21no—
355C2×C4—21no—
356C2×C4—41no—
357C2×C4—42no—
358C2×C4—61no—
359C2×C4—61no—
360C2×C4—82no—
361C2×C4—121no—
362C2×C4—121no—
363C2×C4—121no—
364C2×C4—121no—
365C2×C4—121no—
366C2×C4—121no—
367C2×C4—121no—
368C2×C4—121no—
369C2×C4—121no—
370C2×C4—121no—
371C2×C4D3×C22⋊C411yescharacteristic
372C2×C4#88311yescharacteristic
373C2×C4#88411yescharacteristic
374C2×C4D3×SD1611yescharacteristic
375C2×C4#91411yescharacteristic
376C2×C4#91811yescharacteristic
377C2×C4#91911yescharacteristic
378C2×C4C4×#6711yescharacteristic
379C2×C4#93711yescharacteristic
380C2×C4#93811yescharacteristic
381C8—42no—
382C8—62no—
383C8—81no—
384C8—121no—
385C8—122no—
386C8—241no—
387C2×C3—21no—
388C2×C3—21no—
389C2×C3#187711yescharacteristic
390C2×C3#190011yescharacteristic
391C2×C3#190211yescharacteristic
392D3—42no—
393D3—81no—
394C22—21no—
395C22—21no—
396C22—62no—
397C22—121no—
398C22—121no—
399C22—121no—
400C22—121no—
401C22—121no—
402C22—242no—
403C22—242no—
404C22#478811yescharacteristic
405C22#481711yescharacteristic
406C22#495611yescharacteristic
407C4—21no—
408C4—21no—
409C4—21no—
410C4—82no—
411C4—121no—
412C4—121no—
413C4D3×#10011yescharacteristic
414C4#474511yescharacteristic
415C4#481811yescharacteristic
416C4#481911yescharacteristic
417C4#516911yescharacteristic
418C4#518711yescharacteristic
419C3#924711yescharacteristic, complemented, Sylow
420C2—21no—
421C2—21no—
422C2—122no—
423C2—241no—
424C2#7008111yescharacteristic
425C2#7027611yescharacteristic
426C2#7034911yescharacteristic
subgroupG/H
centerC22#4956
commutatorC22×C3×C4C22×C4
FrattiniC2×C42C22×D3
FittingC3×#1548C2
socleC22×C3#386
prime radicalC22×D12C2×C4
cosocleC2×C3×C42C23
hypercenter#1548D3
hypocenterC3#9247
LCSG ▷ C22×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1548

Complements

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

NH ≅ G/Ntype
3. C3×#1548C2semidirect
4. #71790C2semidirect
6. #67744C2semidirect
7. #67711C2semidirect
35. #1548D3semidirect
156. #208SD16semidirect
419. C3#9247semidirect

Maximal subgroups

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

subgroupindex
C2×#50832
#790432
C3×#15482
#717902
#707622
#677442
#677112
#92473