Finite Group Databasebeta

← #10691563

Subgroups — #10691563

Order 768.

512 conjugacy classes of subgroups.

The subgroup lattice has 418 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#4842C211yesmaximal, characteristic
2M16×D12C211yesmaximal, characteristic
3#70780C211yesmaximal, characteristic, complemented
4#70762C211yesmaximal, characteristic
5#69879C211yesmaximal, characteristic, complemented
6C3×#1283C211yesmaximal, characteristic, complemented
7#67705C211yesmaximal, characteristic, complemented
8#9249—31nomaximal, Sylow
9C2×D3×M16—21no—
10C2×C4×D12C2211yescharacteristic
11C2×#869C2211yescharacteristic
12C3×#431C2211yescharacteristic
13C2×C3×C4⋊C8C2211yescharacteristic
14C3×C4×M16C2211yescharacteristic
15#5126—21no—
16C2×#821C412yes—
17C2×#817C412yes—
18#4924C2211yescharacteristic
19#4924C2211yescharacteristic
20#4842—21no—
21#4842—21no—
22#4842—21no—
23#4842—21no—
24#4706—22no—
25#4688—21no—
26C8×D12—21no—
27#4554—21no—
28#4554—21no—
29#4482—22no—
30#4459—21no—
31#4459—21no—
32C2×#476—31no—
33D4×M16—31no—
34#1997—31no—
35#1549—31no—
36#1486—31no—
37#1292—31no—
38#1283D311yescharacteristic, complemented
39C22×D12C2×C411yescharacteristic
40C22×C4×D3—21no—
41C22×Dic6—21no—
42C2×C3×M16D411yescharacteristic
43C2×C3×M16D411yescharacteristic
44C22×C3×C8C2×C412yes—
45C2×C3×C42C2311yescharacteristic
46C2×(C2×D3)⋊C4—21no—
47C2×C4⋊Dic3—21no—
48C2×C4⋊Dic3C2×C412yes—
49C2×C4⋊Dic3C2×C411yescharacteristic
50C2×#208—21no—
51C2×#208—21no—
52D3×M16—42no—
53D3×M16—42no—
54C2×ΓL(1,𝔽25)—22no—
55C2×D3×C8—22no—
56C4×D12—21no—
57C4×D12—21no—
58C4×D12D411yescharacteristic
59C4×D12D411yescharacteristic
60C4×D12D411yescharacteristic
61C4×D12D411yescharacteristic
62#869—21no—
63#869—21no—
64#869—21no—
65#869—21no—
66C3×C4⋊C8—21no—
67C3×C4⋊C8—21no—
68C3×C4⋊C8—21no—
69C3×C22⋊C8—21no—
70C3×#97—21no—
71C3×C4×C8—21no—
72C3×C4×C8—21no—
73#821—22no—
74#821—22no—
75(C2×D3)⋊C8—22no—
76(C2×D3)⋊C8—22no—
77#817—22no—
78#817—22no—
79C4⋊(C3⋊C8)—21no—
80C4⋊(C3⋊C8)—21no—
81C4⋊(C3⋊C8)—21no—
82C4⋊(C3⋊C8)D412yes—
83C22×M16—61no—
84C2×C4⋊Q8—31no—
85C2×C4×D4—31no—
86#476—61no—
87#476—61no—
88#476—61no—
89#476—61no—
90#436—61no—
91#435—62no—
92C8×D4—61no—
93#431C2×D311yescharacteristic
94C2×#106—32no—
95#423—31no—
96#423—31no—
97C2×C4⋊C8C2×D311yescharacteristic
98C2×#102—32no—
99#407—61no—
100C4×M16C2×D311yescharacteristic
101#343—61no—
102#343—61no—
103#332—62no—
104D4⋊C8—61no—
105D4⋊C8—61no—
106C23×D3—21no—
107C22×C3×C4C22×C411yescharacteristic
108C22×C3×C4C2×D411yescharacteristic
109C22×C3×C4#4111yescharacteristic
110C22×Dic3—21no—
111C22×Dic3—22no—
112C2×D12—22no—
113C2×D12—21no—
114C2×D12—41no—
115C2×D12C22⋊C411yescharacteristic
116C2×D12C22⋊C411yescharacteristic
117C2×C4×D3—21no—
118C2×C4×D3—21no—
119C2×C4×D3—21no—
120C2×C4×D3—21no—
121C2×C4×D3—41no—
122C2×C4×D3—41no—
123C2×C4×D3—41no—
124C2×C4×D3—41no—
125C2×Dic6—41no—
126C2×Dic6—41no—
127C2×Dic6—82no—
128C3×M16—21no—
129C3×M16—21no—
130C3×M16—21no—
131C3×M16SD1612yes—
132C2×C3×C8—21no—
133C2×C3×C8—22no—
134C2×C3×C8—21no—
135C2×C3×C8—42no—
136C2×C3×C8—41no—
137C3×C42C2×D411yescharacteristic
138C3×C42C2×D411yescharacteristic
139C3×C42C2×D411yescharacteristic
140C3×C42#4111yescharacteristic
141(C2×D3)⋊C4—21no—
142(C2×D3)⋊C4—21no—
143(C2×D3)⋊C4—21no—
144(C2×D3)⋊C4—21no—
145C4⋊Dic3—21no—
146C4⋊Dic3—22no—
147C4⋊Dic3—22no—
148C4⋊Dic3—41no—
149C4⋊Dic3—41no—
150C4⋊Dic3C22⋊C411yescharacteristic
151C4⋊Dic3C22⋊C411yescharacteristic
152#208—41no—
153#208—41no—
154#208—41no—
155#208—41no—
156C2×C3⋊C8—22no—
157C2×C3⋊C8—41no—
158ΓL(1,𝔽25)—44no—
159D3×C8—44no—
160C22×Q8—61no—
161C22×D4—31no—
162C23×C4—61no—
163C2×M16—62no—
164C2×M16—61no—
165C2×M16—61no—
166C2×M16—122no—
167C2×M16—122no—
168C2×M16D1211yescharacteristic
169C2×M16D1211yescharacteristic
170C22×C8—62no—
171C22×C8C4×D312yes—
172C4⋊Q8—61no—
173C4⋊Q8—61no—
174C4⋊Q8—61no—
175C4⋊Q8—61no—
176C4×D4—31no—
177C4×D4—31no—
178C4×D4—31no—
179C4×D4—31no—
180C4×D4—61no—
181C4×D4—61no—
182C2×C4⋊C4—31no—
183C2×C4⋊C4—32no—
184C2×C4⋊C4—61no—
185C2×C22⋊C4—61no—
186C2×C42C22×D311yescharacteristic
187#106—62no—
188#106—62no—
189C4⋊C8—21no—
190C4⋊C8—21no—
191C4⋊C8—21no—
192C4⋊C8—32no—
193C4⋊C8—61no—
194C4⋊C8—61no—
195C4⋊C8—61no—
196#102—62no—
197#102—62no—
198C22⋊C8—21no—
199C22⋊C8—62no—
200C22⋊C8—62no—
201#97—21no—
202C4×C8—21no—
203C4×C8—21no—
204C23×C3C4×D411yescharacteristic
205C22×D3—21no—
206C22×D3—21no—
207C22×D3—42no—
208C22×D3—41no—
209C22×D3—41no—
210C22×D3—81no—
211C2×C3×C4—21no—
212C2×C3×C4—21no—
213C2×C3×C4—41no—
214C2×C3×C4C2×C22⋊C411yescharacteristic
215C2×C3×C4C4×D411yescharacteristic
216C2×C3×C4#12011yescharacteristic
217C2×C3×C4#12111yescharacteristic
218C2×C3×C4#12111yescharacteristic
219C2×C3×C4#12311yescharacteristic
220C2×C3×C4C2×SD1611yescharacteristic
221C2×C3×C4#13511yescharacteristic
222C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
223C2×C3×C4#13711yescharacteristic
224C2×Dic3—21no—
225C2×Dic3—21no—
226C2×Dic3—41no—
227C2×Dic3—41no—
228C2×Dic3—42no—
229C2×Dic3—82no—
230D12—21no—
231D12—21no—
232D12—21no—
233D12—42no—
234D12#10112yes—
235C4×D3—41no—
236C4×D3—41no—
237C4×D3—41no—
238C4×D3—41no—
239C4×D3—41no—
240C4×D3—41no—
241C4×D3—41no—
242C4×D3—41no—
243Dic6—82no—
244Dic6—82no—
245C3×C8—22no—
246C3×C8—42no—
247C3×C8—41no—
248C3×C8—81no—
249C3⋊C8—42no—
250C3⋊C8—81no—
251C24—61no—
252C2×Q8—121no—
253C2×Q8—121no—
254C2×Q8—242no—
255C2×D4—31no—
256C2×D4—31no—
257C2×D4—62no—
258C2×D4—61no—
259C2×D4—121no—
260C22×C4—61no—
261C22×C4—61no—
262C22×C4—61no—
263C22×C4—61no—
264C22×C4—61no—
265C22×C4—62no—
266C22×C4—121no—
267C22×C4—121no—
268C22×C4—121no—
269C22×C4—121no—
270C22×C4C2×C4×D311yescharacteristic
271C22×C4C2×D1211yescharacteristic
272C22×C4#23511yescharacteristic
273M16—21no—
274M16—21no—
275M16—21no—
276M16—124no—
277M16—121no—
278M16—121no—
279M16—121no—
280M16—121no—
281M16#20412yes—
282C2×C8—21no—
283C2×C8—22no—
284C2×C8—21no—
285C2×C8—42no—
286C2×C8—41no—
287C2×C8—62no—
288C2×C8—124no—
289C2×C8—121no—
290C4⋊C4—31no—
291C4⋊C4—31no—
292C4⋊C4—61no—
293C4⋊C4—62no—
294C4⋊C4—62no—
295C4⋊C4—121no—
296C4⋊C4—121no—
297C22⋊C4—61no—
298C22⋊C4—61no—
299C22⋊C4—61no—
300C22⋊C4—61no—
301C42D3×D411yescharacteristic
302C42D3×D411yescharacteristic
303C42D3×D411yescharacteristic
304C42#23911yescharacteristic
305C22×C3—21no—
306C22×C3—21no—
307C22×C3#38611yescharacteristic
308C22×C3C4×SD1611yescharacteristic
309C22×C3#44011yescharacteristic
310C2×D3—22no—
311C2×D3—41no—
312C2×D3—41no—
313C2×D3—41no—
314C2×D3—41no—
315C2×D3—41no—
316C2×D3—82no—
317C2×D3—82no—
318C3×C4—21no—
319C3×C4—21no—
320C3×C4—21no—
321C3×C4C2×#10111yescharacteristic
322C3×C4#41311yescharacteristic
323C3×C4#45011yescharacteristic
324C3×C4#45211yescharacteristic
325C3×C4#46111yescharacteristic
326C3×C4#46311yescharacteristic
327Dic3—41no—
328Dic3—41no—
329Dic3—82no—
330C23—61no—
331C23—61no—
332C23—122no—
333C23—121no—
334C23—121no—
335C23—241no—
336C23C4×D1211yescharacteristic
337Q8—242no—
338Q8—242no—
339D4—32no—
340D4—61no—
341D4—61no—
342D4—61no—
343D4—122no—
344C2×C4—21no—
345C2×C4—21no—
346C2×C4—41no—
347C2×C4—61no—
348C2×C4—61no—
349C2×C4—121no—
350C2×C4—121no—
351C2×C4—121no—
352C2×C4—121no—
353C2×C4—121no—
354C2×C4—121no—
355C2×C4—121no—
356C2×C4—121no—
357C2×C4—121no—
358C2×C4—121no—
359C2×C4—122no—
360C2×C4—242no—
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#187711yescharacteristic
380C2×C3#190111yescharacteristic
381C2×C3#190311yescharacteristic
382D3—42no—
383D3—81no—
384C22—21no—
385C22—21no—
386C22—62no—
387C22—121no—
388C22—121no—
389C22—121no—
390C22—121no—
391C22—121no—
392C22—242no—
393C22—242no—
394C22#466511yescharacteristic
395C22C4×#20411yescharacteristic
396C22#471011yescharacteristic
397C4—21no—
398C4—21no—
399C4—21no—
400C4—121no—
401C4—121no—
402C4—242no—
403C4#473111yescharacteristic
404C4#473311yescharacteristic
405C4#474511yescharacteristic
406C4D3×#10111yescharacteristic
407C4#484111yescharacteristic
408C4#484411yescharacteristic
409C3#924911yescharacteristic, complemented, Sylow
410C2—21no—
411C2—21no—
412C2—122no—
413C2—241no—
414C2#6867211yescharacteristic
415C2#6899311yescharacteristic
416C2#6903811yescharacteristic
subgroupG/H
centerC22#4665
commutatorC22×C3×C4C22×C4
FrattiniC2×C42C22×D3
FittingC3×#1283C2
socleC22×C3#386
prime radicalC22×D12C2×C4
cosocleC2×C3×C42C23
hypercenter#1283D3
hypocenterC3#9249
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
3. #70780C2semidirect
5. #69879C2semidirect
6. C3×#1283C2semidirect
7. #67705C2semidirect
38. #1283D3semidirect
409. C3#9249semidirect

Maximal subgroups

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

subgroupindex
C2×#48422
M16×D122
#707802
#707622
#698792
C3×#12832
#677052
#92493