Finite Group Databasebeta

← #10691598

Subgroups — #10691598

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
1C2×#4848C211yesmaximal, characteristic
2M16×Dic6C211yesmaximal, characteristic, complemented
3#70774C211yesmaximal, characteristic
4#70770C211yesmaximal, characteristic, complemented
5#69876C211yesmaximal, characteristic
6C3×#1283C211yesmaximal, characteristic, complemented
7#67705C211yesmaximal, characteristic, complemented
8#9254—31nomaximal, Sylow
9C22×#204C412yes—
10C2×#874C2211yescharacteristic
11C2×C4×Dic6C2211yescharacteristic
12C3×#431C2211yescharacteristic
13C2×C3×C4⋊C8C2211yescharacteristic
14C3×C4×M16C2211yescharacteristic
15C2×#823—21no—
16C2×#823—21no—
17#5118—21no—
18Dic3×M16—21no—
19C2×#821C412yes—
20#4924C2211yescharacteristic
21#4924C2211yescharacteristic
22#4848—21no—
23#4848—21no—
24#4848—21no—
25#4848—21no—
26#4701—22no—
27#4679—21no—
28C8×Dic6—21no—
29#4485—22no—
30#4456—21no—
31#4456—21no—
32C2×#460—31no—
33Q8×M16—31no—
34#1965—31no—
35#1545—31no—
36#1487—31no—
37#1292—31no—
38#1283D311yescharacteristic, complemented
39C22×D12—21no—
40C22×D12—21no—
41C22×D12C2×C412yes—
42C22×Dic6C2×C411yescharacteristic
43C2×C3×M16D411yescharacteristic
44C2×C3×M16D411yescharacteristic
45C22×C3×C8C2×C412yes—
46C2×C3×C42C2311yescharacteristic
47C2×C4⋊Dic3C2×C411yescharacteristic
48C2×Dic3⋊C4—21no—
49C2×C4×Dic3—21no—
50C2×#208—21no—
51C2×#208—21no—
52C2×#204—22no—
53C2×#204—22no—
54C2×#204—44no—
55#874—21no—
56#874—21no—
57#874—21no—
58#874—21no—
59C4×Dic6—21no—
60C4×Dic6—21no—
61C4×Dic6D411yescharacteristic
62C4×Dic6D411yescharacteristic
63C4×Dic6D411yescharacteristic
64C4×Dic6D411yescharacteristic
65C3×C4⋊C8—21no—
66C3×C4⋊C8—21no—
67C3×C4⋊C8—21no—
68C3×C22⋊C8—21no—
69C3×#97—21no—
70C3×C4×C8—21no—
71C3×C4×C8—21no—
72#823—42no—
73#823—42no—
74#821—22no—
75#821—22no—
76#815—22no—
77Dic3⋊C8—22no—
78Dic3⋊C8—22no—
79C8×Dic3—22no—
80C4⋊(C3⋊C8)—21no—
81C4⋊(C3⋊C8)—21no—
82C4⋊(C3⋊C8)—21no—
83C4⋊(C3⋊C8)D412yes—
84C22×SD16—32no—
85C2×#127—31no—
86C2×C4×Q8—31no—
87#460—61no—
88#460—61no—
89#460—61no—
90#460—61no—
91#446—61no—
92#446—62no—
93C8×Q8—61no—
94#431C2×D311yescharacteristic
95#423—31no—
96#423—31no—
97#423—61no—
98C2×C4⋊C8C2×D311yescharacteristic
99C2×#102—32no—
100C2×#100—61no—
101C2×#100—61no—
102C4×M16—61no—
103C4×M16C2×D311yescharacteristic
104#331—62no—
105Q8⋊C8—61no—
106Q8⋊C8—61no—
107C23×D3—22no—
108C22×C3×C4C22×C411yescharacteristic
109C22×C3×C4C2×D411yescharacteristic
110C22×C3×C4#4111yescharacteristic
111C22×Dic3—21no—
112C22×Dic3—21no—
113C2×D12—22no—
114C2×D12—22no—
115C2×D12—42no—
116C2×D12—42no—
117C2×D12—41no—
118C2×D12—41no—
119C2×D12—41no—
120C2×D12—41no—
121C2×D12—41no—
122C2×D12—41no—
123C2×D12—41no—
124C2×D12—41no—
125C2×D12—82no—
126C2×Dic6—22no—
127C2×Dic6—21no—
128C2×Dic6—41no—
129C2×Dic6C22⋊C411yescharacteristic
130C2×Dic6C22⋊C411yescharacteristic
131C3×M16—21no—
132C3×M16—21no—
133C3×M16—21no—
134C3×M16SD1612yescomplemented
135C2×C3×C8—21no—
136C2×C3×C8—22no—
137C2×C3×C8—21no—
138C2×C3×C8—42no—
139C2×C3×C8—41no—
140C3×C42C2×D411yescharacteristic
141C3×C42C2×D411yescharacteristic
142C3×C42C2×D411yescharacteristic
143C3×C42#4111yescharacteristic
144C4⋊Dic3—21no—
145C4⋊Dic3C22⋊C411yescharacteristic
146C4⋊Dic3C22⋊C411yescharacteristic
147Dic3⋊C4—21no—
148Dic3⋊C4—21no—
149Dic3⋊C4—21no—
150Dic3⋊C4—21no—
151C4×Dic3—21no—
152C4×Dic3—21no—
153C4×Dic3—21no—
154C4×Dic3—21no—
155#208—41no—
156#208—41no—
157#208—41no—
158#208—41no—
159C2×C3⋊C8—22no—
160C2×C3⋊C8—41no—
161#204—44no—
162#204—44no—
163C22×Q8—31no—
164C22×D4—32no—
165C22×D4—61no—
166C22×D4—61no—
167C2×SD16—62no—
168C2×SD16—62no—
169C2×SD16—124no—
170C2×M16—61no—
171C2×M16—61no—
172C2×M16D1211yescharacteristic
173C2×M16D1211yescharacteristic
174C22×C8C4×D312yes—
175#127—61no—
176#127—61no—
177#127—61no—
178#127—61no—
179C4×Q8—31no—
180C4×Q8—31no—
181C4×Q8—31no—
182C4×Q8—31no—
183C4×Q8—61no—
184C4×Q8—61no—
185C2×C4⋊C4—31no—
186C2×C4⋊C4—61no—
187C2×C42—61no—
188C2×C42C22×D311yescharacteristic
189C4⋊C8—21no—
190C4⋊C8—21no—
191C4⋊C8—21no—
192C4⋊C8—32no—
193C4⋊C8—62no—
194C4⋊C8—62no—
195C4⋊C8—61no—
196C4⋊C8—61no—
197C4⋊C8—61no—
198#102—62no—
199#102—62no—
200#100—122no—
201#100—122no—
202C22⋊C8—21no—
203#97—21no—
204#97—62no—
205C4×C8—21no—
206C4×C8—21no—
207C4×C8—62no—
208C23×C3C4×D411yescharacteristic
209C22×D3—42no—
210C22×D3—42no—
211C22×D3—42no—
212C22×D3—82no—
213C22×D3—82no—
214C2×C3×C4—21no—
215C2×C3×C4—21no—
216C2×C3×C4—41no—
217C2×C3×C4C2×C22⋊C411yescharacteristic
218C2×C3×C4C4×D411yescharacteristic
219C2×C3×C4#12011yescharacteristic
220C2×C3×C4#12111yescharacteristic
221C2×C3×C4#12111yescharacteristic
222C2×C3×C4#12311yescharacteristic
223C2×C3×C4C2×SD1611yescharacteristic
224C2×C3×C4#13511yescharacteristic
225C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
226C2×C3×C4#13711yescharacteristic
227C2×Dic3—21no—
228C2×Dic3—21no—
229C2×Dic3—21no—
230C2×Dic3—21no—
231C2×Dic3—42no—
232C2×Dic3—81no—
233D12—42no—
234D12—42no—
235D12—82no—
236D12—82no—
237D12—82no—
238D12—81no—
239D12—81no—
240D12—81no—
241D12—81no—
242Dic6—21no—
243Dic6—21no—
244Dic6—21no—
245Dic6—42no—
246Dic6#10012yescomplemented
247C3×C8—22no—
248C3×C8—42no—
249C3×C8—41no—
250C3×C8—81no—
251C3⋊C8—42no—
252C3⋊C8—81no—
253C24—62no—
254C2×Q8—31no—
255C2×Q8—31no—
256C2×Q8—62no—
257C2×Q8—61no—
258C2×Q8—121no—
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—61no—
273C22×C4—61no—
274C22×C4C2×C4×D311yescharacteristic
275C22×C4C2×D1211yescharacteristic
276C22×C4#23511yescharacteristic
277SD16—124no—
278SD16—124no—
279M16—21no—
280M16—21no—
281M16—21no—
282M16—121no—
283M16—121no—
284M16—121no—
285M16—121no—
286M16#20412yescomplemented
287C2×C8—21no—
288C2×C8—22no—
289C2×C8—21no—
290C2×C8—42no—
291C2×C8—41no—
292C2×C8—62no—
293C2×C8—121no—
294C4⋊C4—31no—
295C4⋊C4—31no—
296C4⋊C4—61no—
297C4⋊C4—61no—
298C4⋊C4—61no—
299C4⋊C4—61no—
300C4⋊C4—61no—
301C42—61no—
302C42—61no—
303C42—61no—
304C42—61no—
305C42D3×D411yescharacteristic
306C42D3×D411yescharacteristic
307C42D3×D411yescharacteristic
308C42#23911yescharacteristic
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—21no—
321C3×C4—21no—
322C3×C4C2×#10011yescharacteristic
323C3×C4#41311yescharacteristic
324C3×C4#44811yescharacteristic
325C3×C4#44911yescharacteristic
326C3×C4#46011yescharacteristic
327C3×C4#46411yescharacteristic
328Dic3—22no—
329Dic3—41no—
330Dic3—41no—
331Dic3—41no—
332C23—122no—
333C23—122no—
334C23—122no—
335C23—242no—
336C23—242no—
337C23C4×D1211yescharacteristic
338Q8—32no—
339Q8—61no—
340Q8—61no—
341Q8—61no—
342Q8—122no—
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—41no—
355C2×C4—61no—
356C2×C4—61no—
357C2×C4—61no—
358C2×C4—61no—
359C2×C4—122no—
360C2×C4—241no—
361C2×C4D3×C22⋊C411yescharacteristic
362C2×C4#88211yescharacteristic
363C2×C4#88411yescharacteristic
364C2×C4#89311yescharacteristic
365C2×C4#89411yescharacteristic
366C2×C4#89511yescharacteristic
367C2×C4C2×#20411yescharacteristic
368C2×C4#90411yescharacteristic
369C2×C4#90811yescharacteristic
370C2×C4#90911yescharacteristic
371C8—22no—
372C8—41no—
373C8—42no—
374C8—81no—
375C8—122no—
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—
390C22#466511yescharacteristic
391C22C4×#20411yescharacteristic
392C22#471011yescharacteristic
393C4—21no—
394C4—21no—
395C4—21no—
396C4—62no—
397C4—121no—
398C4—121no—
399C4—121no—
400C4#473411yescharacteristic
401C4#473811yescharacteristic
402C4D3×#10011yescharacteristic
403C4#475111yescharacteristic
404C4#484511yescharacteristic
405C4#484811yescharacteristic
406C3#925411yescharacteristic, complemented, Sylow
407C2—21no—
408C2—21no—
409C2—482no—
410C2#6868611yescharacteristic
411C2#6899211yescharacteristic
412C2#6903811yescharacteristic
subgroupG/H
centerC22#4665
commutatorC22×C3×C4C22×C4
FrattiniC2×C42C22×D3
FittingC3×#1283C2
socleC22×C3#386
prime radicalC2×#874C22
cosocleC2×C3×C42C23
hypercenter#1283D3
hypocenterC3#9254
LCSG ▷ C22×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1283

Complements

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

NH ≅ G/Ntype
2. M16×Dic6C2semidirect
4. #70770C2semidirect
6. C3×#1283C2semidirect
7. #67705C2semidirect
38. #1283D3semidirect
134. C3×M16SD16semidirect
246. Dic6#100semidirect
286. M16#204semidirect
406. C3#9254semidirect

Maximal subgroups

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

subgroupindex
C2×#48482
M16×Dic62
#707742
#707702
#698762
C3×#12832
#677052
#92543