Finite Group Databasebeta

← #10691599

Subgroups — #10691599

Order 768.

534 conjugacy classes of subgroups.

The subgroup lattice has 414 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×Q8×M16C211yesmaximal, characteristic, complemented
2C2×#5027C211yesmaximal, characteristic
3#71622C211yesmaximal, characteristic, complemented
4#70502C211yesmaximal, characteristic
5#69876C211yesmaximal, characteristic
6#67711C211yesmaximal, characteristic, complemented
7#67668C211yesmaximal, characteristic, complemented
8#9254—31nomaximal, Sylow
9C2×C3×C4×Q8C2211yescharacteristic
10C22×#215C412yes—
11C2×#874C2211yescharacteristic
12C3×#446—21no—
13C3×#446—22no—
14C3×C8×Q8—21no—
15C3×#423—21no—
16C3×#423C2211yescharacteristic
17C3×C4×M16—21no—
18C3×C4×M16C2211yescharacteristic
19C2×#823—21no—
20C2×#823—21no—
21#5027—21no—
22#5027—21no—
23#5027—21no—
24#5027—21no—
25C2×#809C412yes—
26#4925C2211yescharacteristic
27#4924C2211yescharacteristic
28C2×C4⋊(C3⋊C8)C2211yescharacteristic
29#4535—21no—
30#4535—21no—
31#4485—22no—
32C2×#460—31no—
33Q8×M16D311yescharacteristic, complemented
34#1965—31no—
35#1545—31no—
36#1487—31no—
37#1292—31no—
38#1283—31no—
39C22×C3×Q8C2×C411yescharacteristic
40C22×D12—21no—
41C22×D12—21no—
42C22×D12C2×C412yes—
43C2×C3×M16—21no—
44C2×C3×M16D411yescharacteristic
45C2×C3×M16D411yescharacteristic
46C3×C4×Q8—21no—
47C3×C4×Q8—21no—
48C3×C4×Q8D411yescharacteristic
49C3×C4×Q8D411yescharacteristic
50C3×C4×Q8D411yescharacteristic
51C3×C4×Q8D411yescharacteristic
52C2×C3×C4⋊C4—21no—
53C2×C3×C4⋊C4C2×C411yescharacteristic
54C2×C3×C42—21no—
55C2×C3×C42C2311yescharacteristic
56C2×#215—22no—
57C2×#215—22no—
58C2×#215—44no—
59C2×#208—21no—
60C22×C3⋊C8C2×C412yes—
61#874—21no—
62#874—21no—
63#874—21no—
64#874—21no—
65C3×C4⋊C8—22no—
66C3×C4⋊C8—22no—
67C3×C4⋊C8—21no—
68C3×C4⋊C8D412yes—
69C3×#97—22no—
70C3×#97—21no—
71C3×C4×C8—22no—
72C3×C4×C8—21no—
73#830—21no—
74#823—42no—
75#823—42no—
76#809—22no—
77#809—22no—
78C4⋊(C3⋊C8)—21no—
79C4⋊(C3⋊C8)—21no—
80C4⋊(C3⋊C8)—21no—
81C4⋊(C3⋊C8)—21no—
82C4⋊(C3⋊C8)—21no—
83C4×C3⋊C8—21no—
84C22×SD16—32no—
85C2×#127—31no—
86C2×C4×Q8C2×D311yescharacteristic
87#460—61no—
88#460—61no—
89#460—61no—
90#460—61no—
91#446—21no—
92#446—22no—
93C8×Q8—21no—
94#431—31no—
95#423—21no—
96#423—31no—
97#423C2×D311yescharacteristic
98C2×C4⋊C8—31no—
99C2×#102—32no—
100C2×#100—61no—
101C2×#100—61no—
102C4×M16—21no—
103C4×M16C2×D311yescharacteristic
104#331—62no—
105Q8⋊C8—61no—
106Q8⋊C8—61no—
107C23×D3—22no—
108C2×C3×Q8—21no—
109C2×C3×Q8—22no—
110C2×C3×Q8—41no—
111C2×C3×Q8C22⋊C411yescharacteristic
112C2×C3×Q8C22⋊C411yescharacteristic
113C22×C3×C4—21no—
114C22×C3×C4—21no—
115C22×C3×C4C22×C411yescharacteristic
116C22×C3×C4C2×D411yescharacteristic
117C22×C3×C4#4111yescharacteristic
118C2×D12—22no—
119C2×D12—22no—
120C2×D12—42no—
121C2×D12—42no—
122C2×D12—41no—
123C2×D12—41no—
124C2×D12—41no—
125C2×D12—41no—
126C2×D12—41no—
127C2×D12—41no—
128C2×D12—41no—
129C2×D12—41no—
130C2×D12—82no—
131C3×M16—21no—
132C3×M16—21no—
133C3×M16—21no—
134C3×M16—41no—
135C3×M16—41no—
136C3×M16SD1612yescomplemented
137C2×C3×C8—21no—
138C2×C3×C8—21no—
139C2×C3×C8—22no—
140C3×C4⋊C4—21no—
141C3×C4⋊C4—21no—
142C3×C4⋊C4—21no—
143C3×C4⋊C4—21no—
144C3×C4⋊C4—21no—
145C3×C4⋊C4C22⋊C411yescharacteristic
146C3×C4⋊C4C22⋊C411yescharacteristic
147C3×C42—21no—
148C3×C42—21no—
149C3×C42—21no—
150C3×C42—21no—
151C3×C42C2×D411yescharacteristic
152C3×C42C2×D411yescharacteristic
153C3×C42C2×D411yescharacteristic
154C3×C42#4111yescharacteristic
155#215—44no—
156#215—44no—
157#208—41no—
158#208—41no—
159C2×C3⋊C8—22no—
160C2×C3⋊C8—42no—
161C2×C3⋊C8—41no—
162C2×C3⋊C8—41no—
163C22×Q8C4×D311yescharacteristic
164C22×D4—32no—
165C22×D4—61no—
166C22×D4—61no—
167C2×SD16—62no—
168C2×SD16—62no—
169C2×SD16—124no—
170C2×M16—21no—
171C2×M16—61no—
172C2×M16#6711yescharacteristic
173C2×M16#6711yescharacteristic
174C22×C8—32no—
175#127—61no—
176#127—61no—
177#127—61no—
178#127—61no—
179C4×Q8—21no—
180C4×Q8—21no—
181C4×Q8D1211yescharacteristic
182C4×Q8D1211yescharacteristic
183C4×Q8#6711yescharacteristic
184C4×Q8#6711yescharacteristic
185C2×C4⋊C4—21no—
186C2×C4⋊C4C4×D311yescharacteristic
187C2×C42—21no—
188C2×C42C22×D311yescharacteristic
189C4⋊C8—22no—
190C4⋊C8—22no—
191C4⋊C8—21no—
192C4⋊C8—61no—
193C4⋊C8—61no—
194C4⋊C8—61no—
195C4⋊C8—61no—
196C4⋊C8—61no—
197C4⋊C8#6712yes—
198#102—62no—
199#102—62no—
200#100—122no—
201#100—122no—
202C22⋊C8—61no—
203#97—22no—
204#97—21no—
205C4×C8—22no—
206C4×C8—21no—
207C4×C8—61no—
208C23×C3C4×D411yescharacteristic
209C22×D3—42no—
210C22×D3—42no—
211C22×D3—42no—
212C22×D3—82no—
213C22×D3—82no—
214C3×Q8—21no—
215C3×Q8—21no—
216C3×Q8—21no—
217C3×Q8—42no—
218C3×Q8#10012yescomplemented
219C2×C3×C4—21no—
220C2×C3×C4—21no—
221C2×C3×C4—21no—
222C2×C3×C4—21no—
223C2×C3×C4—21no—
224C2×C3×C4—21no—
225C2×C3×C4—41no—
226C2×C3×C4—42no—
227C2×C3×C4—81no—
228C2×C3×C4C2×C22⋊C411yescharacteristic
229C2×C3×C4C4×D411yescharacteristic
230C2×C3×C4#12011yescharacteristic
231C2×C3×C4#12111yescharacteristic
232C2×C3×C4#12111yescharacteristic
233C2×C3×C4#12311yescharacteristic
234C2×C3×C4C2×SD1611yescharacteristic
235C2×C3×C4#13511yescharacteristic
236C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
237C2×C3×C4#13711yescharacteristic
238D12—42no—
239D12—42no—
240D12—82no—
241D12—82no—
242D12—82no—
243D12—81no—
244D12—81no—
245D12—81no—
246D12—81no—
247C3×C8—22no—
248C3×C8—41no—
249C3×C8—42no—
250C3⋊C8—42no—
251C3⋊C8—81no—
252C3⋊C8—81no—
253C24—62no—
254C2×Q8—21no—
255C2×Q8—22no—
256C2×Q8—41no—
257C2×Q8(C2×D3)⋊C411yescharacteristic
258C2×Q8(C2×D3)⋊C411yescharacteristic
259C2×D4—62no—
260C2×D4—62no—
261C2×D4—122no—
262C2×D4—122no—
263C2×D4—121no—
264C2×D4—121no—
265C2×D4—121no—
266C2×D4—121no—
267C2×D4—121no—
268C2×D4—121no—
269C2×D4—121no—
270C2×D4—121no—
271C2×D4—242no—
272C22×C4—21no—
273C22×C4—21no—
274C22×C4C2×C4×D311yescharacteristic
275C22×C4#23511yescharacteristic
276C22×C4C2×#6711yescharacteristic
277SD16—124no—
278SD16—124no—
279M16—21no—
280M16—21no—
281M16—21no—
282M16—41no—
283M16—41no—
284M16—121no—
285M16—121no—
286M16#21512yescomplemented
287C2×C8—21no—
288C2×C8—21no—
289C2×C8—22no—
290C2×C8—62no—
291C2×C8—122no—
292C2×C8—121no—
293C2×C8—121no—
294C4⋊C4—21no—
295C4⋊C4—21no—
296C4⋊C4—21no—
297C4⋊C4—21no—
298C4⋊C4—21no—
299C4⋊C4(C2×D3)⋊C411yescharacteristic
300C4⋊C4(C2×D3)⋊C411yescharacteristic
301C42—21no—
302C42—21no—
303C42—21no—
304C42—21no—
305C42C2×D1211yescharacteristic
306C42#23511yescharacteristic
307C42C2×#6711yescharacteristic
308C42C2×#6711yescharacteristic
309C22×C3—21no—
310C22×C3—21no—
311C22×C3#38611yescharacteristic
312C22×C3C4×SD1611yescharacteristic
313C22×C3#44011yescharacteristic
314C2×D3—82no—
315C2×D3—82no—
316C2×D3—82no—
317C2×D3—162no—
318C2×D3—162no—
319C3×C4—21no—
320C3×C4—22no—
321C3×C4—21no—
322C3×C4—21no—
323C3×C4—41no—
324C3×C4—41no—
325C3×C4—41no—
326C3×C4C2×#10011yescharacteristic
327C3×C4#41311yescharacteristic
328C3×C4#44811yescharacteristic
329C3×C4#44911yescharacteristic
330C3×C4#46011yescharacteristic
331C3×C4#46411yescharacteristic
332C23—122no—
333C23—122no—
334C23—122no—
335C23—242no—
336C23—242no—
337C23C4×#6711yescharacteristic
338Q8—21no—
339Q8—21no—
340Q8—21no—
341Q8—42no—
342Q8#82312yescomplemented
343D4—122no—
344D4—122no—
345D4—242no—
346D4—242no—
347D4—242no—
348D4—241no—
349D4—241no—
350D4—241no—
351D4—241no—
352C2×C4—21no—
353C2×C4—21no—
354C2×C4—21no—
355C2×C4—21no—
356C2×C4—21no—
357C2×C4—21no—
358C2×C4—41no—
359C2×C4—42no—
360C2×C4—81no—
361C2×C4C2×(C2×D3)⋊C411yescharacteristic
362C2×C4C4×#6711yescharacteristic
363C2×C4#92911yescharacteristic
364C2×C4#93011yescharacteristic
365C2×C4#93011yescharacteristic
366C2×C4C2×#21511yescharacteristic
367C2×C4#94211yescharacteristic
368C2×C4#94911yescharacteristic
369C2×C4#95011yescharacteristic
370C2×C4#95311yescharacteristic
371C8—22no—
372C8—41no—
373C8—42no—
374C8—122no—
375C8—241no—
376C8—241no—
377C2×C3—21no—
378C2×C3—21no—
379C2×C3#188011yescharacteristic
380C2×C3#190011yescharacteristic
381C2×C3#190311yescharacteristic
382D3—162no—
383C22—21no—
384C22—21no—
385C22—242no—
386C22—242no—
387C22—242no—
388C22—482no—
389C22—482no—
390C22C4×#21511yescharacteristic
391C22#502611yescharacteristic
392C22#521311yescharacteristic
393C4—21no—
394C4—22no—
395C4—21no—
396C4—21no—
397C4—41no—
398C4—41no—
399C4—41no—
400C4#502711yescharacteristic
401C4#502811yescharacteristic
402C4C2×#82311yescharacteristic
403C4#513211yescharacteristic
404C4#522711yescharacteristic
405C4#523811yescharacteristic
406C3#925411yescharacteristic, complemented, Sylow
407C2—21no—
408C2—21no—
409C2—482no—
410C2#7163311yescharacteristic
411C2#7200211yescharacteristic
412C2#7202311yescharacteristic
subgroupG/H
centerC22#5213
commutatorC22×C3×C4C22×C4
FrattiniC2×C42C22×D3
FittingC3×Q8×M16C2
socleC22×C3#386
prime radicalC2×#874C22
cosocleC2×C3×C42C23
hypercenterQ8×M16D3
hypocenterC3#9254
LCSG ▷ C22×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ Q8×M16

Complements

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

NH ≅ G/Ntype
1. C3×Q8×M16C2semidirect
3. #71622C2semidirect
6. #67711C2semidirect
7. #67668C2semidirect
33. Q8×M16D3semidirect
136. C3×M16SD16semidirect
218. C3×Q8#100semidirect
286. M16#215semidirect
342. Q8#823semidirect
406. C3#9254semidirect

Maximal subgroups

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

subgroupindex
C3×Q8×M162
C2×#50272
#716222
#705022
#698762
#677112
#676682
#92543