Finite Group Databasebeta

← #10691567

Subgroups — #10691567

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×#5084C211yesmaximal, characteristic
2M16×D12C211yesmaximal, characteristic
3C3×#1997C211yesmaximal, characteristic, complemented
4#71775C211yesmaximal, characteristic, complemented
5#70496C211yesmaximal, characteristic
6#67711C211yesmaximal, characteristic, complemented
7#67668C211yesmaximal, characteristic, complemented
8#9249—31nomaximal, Sylow
9C2×C3×C4⋊Q8C2211yescharacteristic
10C2×D3×M16—21no—
11C2×C4×D12C2211yescharacteristic
12C3×#423C2211yescharacteristic
13C3×C4×M16C2211yescharacteristic
14#5126—21no—
15#5084—21no—
16#5084—21no—
17#5084—21no—
18#5084—21no—
19C2×#809C412yes—
20C2×#808C412yes—
21#4925C2211yescharacteristic
22#4924C2211yescharacteristic
23C2×C4⋊(C3⋊C8)C2211yescharacteristic
24#4706—22no—
25#4688—21no—
26C8×D12—21no—
27C3×#343—21no—
28C3×#343—21no—
29#4547—22no—
30#4483—21no—
31#4483—21no—
32C2×#476—31no—
33D4×M16—31no—
34#1997D311yescharacteristic, complemented
35#1549—31no—
36#1486—31no—
37#1292—31no—
38#1283—31no—
39C22×C3×Q8—21no—
40C22×D12C2×C411yescharacteristic
41C22×C4×D3—21no—
42C2×C3×M16—21no—
43C2×C3×M16D411yescharacteristic
44C2×C3×M16D411yescharacteristic
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×C412yes—
51C2×C3×C42C2311yescharacteristic
52C2×(C2×D3)⋊C4—21no—
53C2×C4⋊Dic3C2×C411yescharacteristic
54C2×#208—21no—
55C22×C3⋊C8C2×C412yes—
56D3×M16—42no—
57D3×M16—42no—
58C2×ΓL(1,𝔽25)—22no—
59C2×D3×C8—22no—
60C4×D12—21no—
61C4×D12—21no—
62C4×D12D411yescharacteristic
63C4×D12D411yescharacteristic
64C4×D12D411yescharacteristic
65C4×D12D411yescharacteristic
66C3×C4⋊C8—21no—
67C3×C4⋊C8—21no—
68C3×#97—21no—
69C3×C4×C8—21no—
70#830—21no—
71(C2×D3)⋊C8—22no—
72(C2×D3)⋊C8—22no—
73#809—22no—
74#809—22no—
75#808—22no—
76#808—22no—
77C4⋊(C3⋊C8)—21no—
78C4⋊(C3⋊C8)—21no—
79C4⋊(C3⋊C8)—21no—
80C4⋊(C3⋊C8)—21no—
81C4⋊(C3⋊C8)D412yes—
82C4×C3⋊C8—21no—
83C22×M16—61no—
84C2×C4⋊Q8C2×D311yescharacteristic
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#431—31no—
94C2×#106—32no—
95#423—31no—
96#423C2×D311yescharacteristic
97C2×C4⋊C8—31no—
98C2×#102—32no—
99#407—61no—
100C4×M16C2×D311yescharacteristic
101#343—21no—
102#343—21no—
103#332—62no—
104D4⋊C8—61no—
105D4⋊C8—61no—
106C23×D3—21no—
107C2×C3×Q8—41no—
108C2×C3×Q8—41no—
109C2×C3×Q8—82no—
110C22×C3×C4—22no—
111C22×C3×C4C22×C411yescharacteristic
112C22×C3×C4C2×D411yescharacteristic
113C22×C3×C4#4111yescharacteristic
114C22×Dic3—21no—
115C2×D12—22no—
116C2×D12—21no—
117C2×D12—41no—
118C2×D12C22⋊C411yescharacteristic
119C2×D12C22⋊C411yescharacteristic
120C2×C4×D3—21no—
121C2×C4×D3—21no—
122C2×C4×D3—21no—
123C2×C4×D3—21no—
124C2×C4×D3—41no—
125C2×C4×D3—41no—
126C2×C4×D3—41no—
127C2×C4×D3—41no—
128C3×M16—21no—
129C3×M16—21no—
130C3×M16—21no—
131C3×M16—41no—
132C3×M16—41no—
133C3×M16SD1612yes—
134C2×C3×C8—21no—
135C2×C3×C8—21no—
136C2×C3×C8—41no—
137C3×C4⋊C4—22no—
138C3×C4⋊C4—22no—
139C3×C4⋊C4—41no—
140C3×C4⋊C4—41no—
141C3×C42C2×D411yescharacteristic
142C3×C42C2×D411yescharacteristic
143C3×C42C2×D411yescharacteristic
144C3×C42#4111yescharacteristic
145(C2×D3)⋊C4—21no—
146(C2×D3)⋊C4—21no—
147(C2×D3)⋊C4—21no—
148(C2×D3)⋊C4—21no—
149C4⋊Dic3—21no—
150C4⋊Dic3C22⋊C411yescharacteristic
151C4⋊Dic3C22⋊C411yescharacteristic
152#208—41no—
153#208—41no—
154C2×C3⋊C8—22no—
155C2×C3⋊C8—22no—
156C2×C3⋊C8—42no—
157C2×C3⋊C8—41no—
158ΓL(1,𝔽25)—44no—
159D3×C8—44no—
160C22×Q8—21no—
161C22×D4—31no—
162C23×C4—61no—
163C2×M16—21no—
164C2×M16—62no—
165C2×M16—61no—
166C2×M16—122no—
167C2×M16—122no—
168C2×M16#6711yescharacteristic
169C2×M16#6711yescharacteristic
170C22×C8—32no—
171C22×C8—62no—
172C4⋊Q8—21no—
173C4⋊Q8—21no—
174C4⋊Q8—21no—
175C4⋊Q8—21no—
176C4×D4—31no—
177C4×D4—31no—
178C4×D4—31no—
179C4×D4—31no—
180C4×D4—61no—
181C4×D4—61no—
182C2×C4⋊C4—21no—
183C2×C4⋊C4—31no—
184C2×C4⋊C4C4×D312yes—
185C2×C22⋊C4—61no—
186C2×C42C22×D311yescharacteristic
187#106—62no—
188#106—62no—
189C4⋊C8—21no—
190C4⋊C8—21no—
191C4⋊C8—32no—
192C4⋊C8—61no—
193C4⋊C8—61no—
194C4⋊C8—61no—
195C4⋊C8—61no—
196#102—62no—
197#102—62no—
198C22⋊C8—62no—
199C22⋊C8—62no—
200C22⋊C8—61no—
201#97—21no—
202C4×C8—21no—
203C4×C8—61no—
204C23×C3C4×D411yescharacteristic
205C22×D3—21no—
206C22×D3—21no—
207C22×D3—42no—
208C22×D3—41no—
209C22×D3—41no—
210C22×D3—81no—
211C3×Q8—82no—
212C3×Q8—82no—
213C2×C3×C4—21no—
214C2×C3×C4—21no—
215C2×C3×C4—41no—
216C2×C3×C4—42no—
217C2×C3×C4—82no—
218C2×C3×C4C2×C22⋊C411yescharacteristic
219C2×C3×C4C4×D411yescharacteristic
220C2×C3×C4#12011yescharacteristic
221C2×C3×C4#12111yescharacteristic
222C2×C3×C4#12111yescharacteristic
223C2×C3×C4#12311yescharacteristic
224C2×C3×C4C2×SD1611yescharacteristic
225C2×C3×C4#13511yescharacteristic
226C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
227C2×C3×C4#13711yescharacteristic
228C2×Dic3—21no—
229C2×Dic3—21no—
230C2×Dic3—41no—
231C2×Dic3—41no—
232D12—21no—
233D12—21no—
234D12—21no—
235D12—42no—
236D12#10112yes—
237C4×D3—41no—
238C4×D3—41no—
239C4×D3—41no—
240C4×D3—41no—
241C4×D3—41no—
242C4×D3—41no—
243C4×D3—41no—
244C4×D3—41no—
245C3×C8—22no—
246C3×C8—41no—
247C3×C8—81no—
248C3⋊C8—42no—
249C3⋊C8—42no—
250C3⋊C8—81no—
251C24—61no—
252C2×Q8—41no—
253C2×Q8—41no—
254C2×Q8—82no—
255C2×D4—31no—
256C2×D4—31no—
257C2×D4—62no—
258C2×D4—61no—
259C2×D4—121no—
260C22×C4—22no—
261C22×C4—61no—
262C22×C4—61no—
263C22×C4—61no—
264C22×C4—61no—
265C22×C4—61no—
266C22×C4—121no—
267C22×C4—121no—
268C22×C4—121no—
269C22×C4—121no—
270C22×C4C2×C4×D311yescharacteristic
271C22×C4#23511yescharacteristic
272C22×C4C2×#6711yescharacteristic
273M16—21no—
274M16—21no—
275M16—21no—
276M16—41no—
277M16—41no—
278M16—124no—
279M16—121no—
280M16—121no—
281M16#21512yes—
282C2×C8—21no—
283C2×C8—21no—
284C2×C8—41no—
285C2×C8—62no—
286C2×C8—62no—
287C2×C8—124no—
288C2×C8—122no—
289C2×C8—121no—
290C4⋊C4—22no—
291C4⋊C4—22no—
292C4⋊C4—31no—
293C4⋊C4—31no—
294C4⋊C4—41no—
295C4⋊C4—41no—
296C4⋊C4—61no—
297C22⋊C4—61no—
298C22⋊C4—61no—
299C22⋊C4—61no—
300C22⋊C4—61no—
301C42D3×D411yescharacteristic
302C42D3×D411yescharacteristic
303C42D3×D411yescharacteristic
304C42#23711yescharacteristic
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×C4—82no—
322C3×C4C2×#10111yescharacteristic
323C3×C4#41311yescharacteristic
324C3×C4#45011yescharacteristic
325C3×C4#45211yescharacteristic
326C3×C4#46111yescharacteristic
327C3×C4#46311yescharacteristic
328Dic3—41no—
329Dic3—41no—
330C23—61no—
331C23—61no—
332C23—122no—
333C23—121no—
334C23—121no—
335C23—241no—
336C23C4×#6711yescharacteristic
337Q8—82no—
338Q8—82no—
339D4—32no—
340D4—61no—
341D4—61no—
342D4—61no—
343D4—122no—
344C2×C4—21no—
345C2×C4—21no—
346C2×C4—41no—
347C2×C4—42no—
348C2×C4—61no—
349C2×C4—61no—
350C2×C4—82no—
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—121no—
360C2×C4—121no—
361C2×C4D3×C22⋊C411yescharacteristic
362C2×C4#67⋊C411yescharacteristic
363C2×C4#88311yescharacteristic
364C2×C4#88411yescharacteristic
365C2×C4#93711yescharacteristic
366C2×C4#93811yescharacteristic
367C2×C4C2×#21511yescharacteristic
368C2×C4#94211yescharacteristic
369C2×C4#94911yescharacteristic
370C2×C4#95011yescharacteristic
371C8—22no—
372C8—41no—
373C8—81no—
374C8—122no—
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#495611yescharacteristic
395C22C4×#21511yescharacteristic
396C22#502611yescharacteristic
397C4—21no—
398C4—21no—
399C4—21no—
400C4—82no—
401C4—121no—
402C4—121no—
403C4D3×#10111yescharacteristic
404C4#475111yescharacteristic
405C4#504611yescharacteristic
406C4#504711yescharacteristic
407C4#505711yescharacteristic
408C4#508311yescharacteristic
409C3#924911yescharacteristic, complemented, Sylow
410C2—21no—
411C2—21no—
412C2—122no—
413C2—241no—
414C2#6968711yescharacteristic
415C2#7008711yescharacteristic
416C2#7011511yescharacteristic
subgroupG/H
centerC22#4956
commutatorC22×C3×C4C22×C4
FrattiniC2×C42C22×D3
FittingC3×#1997C2
socleC22×C3#386
prime radicalC22×D12C2×C4
cosocleC2×C3×C42C23
hypercenter#1997D3
hypocenterC3#9249
LCSG ▷ C22×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1997

Complements

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

NH ≅ G/Ntype
3. C3×#1997C2semidirect
4. #71775C2semidirect
6. #67711C2semidirect
7. #67668C2semidirect
34. #1997D3semidirect
409. C3#9249semidirect

Maximal subgroups

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

subgroupindex
C2×#50842
M16×D122
C3×#19972
#717752
#704962
#677112
#676682
#92493