Finite Group Databasebeta

← #10777928

Subgroups — #10777928

Order 768.

642 conjugacy classes of subgroups.

The subgroup lattice has 442 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#81775C211yesmaximal, characteristic, complemented
2#80361C212yesmaximal, complemented
3#79938C211yesmaximal, characteristic, complemented
4C3×#1688C211yesmaximal, characteristic, complemented
5#71283C211yesmaximal, characteristic, complemented
6#71026C212yesmaximal, complemented
7#70965C211yesmaximal, characteristic, complemented
8#68543C211yesmaximal, characteristic, complemented
9#68527C211yesmaximal, characteristic, complemented
10#68400C212yesmaximal, complemented
11#68382C212yesmaximal, complemented
12#19160—31nomaximal, Sylow
13#5716C2212yes—
14C4×#239C2211yescharacteristic
15C3×#493C2211yescharacteristic
16C3×#460C2212yes—
17C3×#446C2211yescharacteristic
18#5172—22no—
19Dic3×SD16—22no—
20#5157—21no—
21#5157—21no—
22Dic3×D8—21no—
23Dic3×D8—21no—
24#5072C2212yes—
25#5070C2212yes—
26#5024C2211yescharacteristic
27#5021C2212yes—
28#4842C2212yes—
29#4838C2211yescharacteristic
30#4836—21no—
31#4833—21no—
32#4833C2211yescharacteristic
33#4833C2212yes—
34#4779—22no—
35#4756—22no—
36#4707—21no—
37#4706—21no—
38#4688C2211yescharacteristic
39C4×D3×C8C2211yescharacteristic
40#4682C2211yescharacteristic
41C3×#334C2211yescharacteristic
42C3×#331C2212yes—
43#4544C2212yes—
44#4481C2212yes—
45#4481C2212yes—
46#4479C2212yes—
47#4457C2211yescharacteristic
48#3169—31no—
49#3106—32no—
50#2965—31no—
51#1698—32no—
52#1697—32no—
53#1696—31no—
54#1695—31no—
55#1688D311yescharacteristic, complemented
56#1687—31no—
57#1564—31no—
58#1543—32no—
59C2×#239D411yescharacteristic
60C2×#237—22no—
61C2×C3×SD16—22no—
62C2×C3×D8—21no—
63C2×C3×D8—21no—
64C3×#127C2312yes—
65C3×C4×Q8C2311yescharacteristic
66Q8×Dic3D411yescharacteristic
67#938—22no—
68#938—22no—
69#936—22no—
70D4×Dic3—22no—
71D4×Dic3—22no—
72C2×#213—22no—
73C2×ΓL(1,𝔽25)—21no—
74C2×ΓL(1,𝔽25)—21no—
75C2×D3×C8D411yescharacteristic
76C2×D3×C8D411yescharacteristic
77#893—21no—
78#893D411yescharacteristic
79#892—21no—
80#892D411yescharacteristic
81C4×D12—21no—
82C4×D12C2311yescharacteristic
83C42×D3—21no—
84C42×D3C2311yescharacteristic
85#869C2312yes—
86C3×C4⋊C8—21no—
87C3×C4⋊C8C2312yes—
88C3×C4⋊C8C2311yescharacteristic
89C3×#102—22no—
90C3×#97—21no—
91C3×C4×C8C2311yescharacteristic
92#835—22no—
93#832—22no—
94#832—22no—
95#832—22no—
96#821—22no—
97(C2×D3)⋊C8—22no—
98(C2×D3)⋊C8—21no—
99(C2×D3)⋊C8—21no—
100(C2×D3)⋊C8—21no—
101#818—21no—
102#818—21no—
103#817—22no—
104C8×Dic3—22no—
105C8×Dic3—21no—
106C8×Dic3D411yescharacteristic
107C8×Dic3D411yescharacteristic
108#807—22no—
109C4⋊(C3⋊C8)—21no—
110C4⋊(C3⋊C8)C2311yescharacteristic
111C4⋊(C3⋊C8)C2312yes—
112#803—21no—
113C4×C3⋊C8C2311yescharacteristic
114#532—32no—
115C4×#41—31no—
116#500—31no—
117#493C2×D311yescharacteristic
118#486—32no—
119#482—62no—
120#477—32no—
121#476—32no—
122#466—61no—
123#466—61no—
124#460C2×D312yes—
125#459—32no—
126#449—62no—
127#446—31no—
128#446C2×D311yescharacteristic
129C4×SD16—62no—
130C4×D8—61no—
131C4×D8—61no—
132C4×D8—62no—
133#436—31no—
134#436—31no—
135#435—61no—
136#435—61no—
137#431—31no—
138#431—32no—
139#431—61no—
140C4×M16—61no—
141C2×C4×C8—31no—
142#335—32no—
143#334—31no—
144#334C2×D311yescharacteristic
145#333—32no—
146#333—32no—
147#331—32no—
148#331C2×D312yes—
149C2×C3×Q8C2×D411yescharacteristic
150C2×C3×D4—22no—
151C2×C3×D4—22no—
152C2×C3×D4—22no—
153C2×#67—42no—
154C22×Dic3—42no—
155#239—42no—
156#237—84no—
157C2×D12—21no—
158C2×D12C2×D411yescharacteristic
159C2×C4×D3—21no—
160C2×C4×D3—21no—
161C2×C4×D3—21no—
162C2×C4×D3C2×D411yescharacteristic
163C2×C4×D3C2×D411yescharacteristic
164C2×C4×D3#4111yescharacteristic
165C2×Dic6—22no—
166C3×SD16—44no—
167C3×D8—42no—
168C3×D8—42no—
169C2×C3×C8—22no—
170C2×C3×C8—21no—
171C2×C3×C8C2×D411yescharacteristic
172C2×C3×C8C2×D411yescharacteristic
173C3×C4⋊C4—21no—
174C3×C4⋊C4C2×D411yescharacteristic
175C3×C42—21no—
176C3×C42C2411yescharacteristic
177#217—42no—
178#217—42no—
179#213—44no—
180(C2×D3)⋊C4—21no—
181(C2×D3)⋊C4—21no—
182(C2×D3)⋊C4—21no—
183C4⋊Dic3—21no—
184C4⋊Dic3—22no—
185C4⋊Dic3—22no—
186C4⋊Dic3C2×D411yescharacteristic
187C4×Dic3—21no—
188C4×Dic3—21no—
189C4×Dic3C2×D411yescharacteristic
190C4×Dic3C2×D411yescharacteristic
191C4×Dic3#4111yescharacteristic
192C2×C3⋊C8—21no—
193C2×C3⋊C8—22no—
194C2×C3⋊C8C2×D411yescharacteristic
195C2×C3⋊C8C2×D411yescharacteristic
196ΓL(1,𝔽25)—42no—
197ΓL(1,𝔽25)—42no—
198D3×C8—22no—
199D3×C8—41no—
200C2×#41—31no—
201C2×#41—62no—
202C2×SD16—22no—
203C2×D8—21no—
204C2×D8—21no—
205C2×D8—62no—
206C2×M16—61no—
207C2×M16—61no—
208C22×C8—31no—
209C22×C8—31no—
210C4⋊Q8—32no—
211#127C22×D312yes—
212#124—62no—
213#121—62no—
214#121—62no—
215C4×Q8—31no—
216C4×Q8C22×D311yescharacteristic
217C4×D4—31no—
218C4×D4—31no—
219C4×D4—61no—
220C4×D4—62no—
221C4×D4—62no—
222C4×D4—61no—
223#117—31no—
224#117—61no—
225C2×C42—31no—
226C2×C42—61no—
227#107—61no—
228#107—61no—
229#107—62no—
230#106—62no—
231C4⋊C8—21no—
232C4⋊C8—31no—
233C4⋊C8—32no—
234C4⋊C8—61no—
235C4⋊C8C22×D311yescharacteristic
236C4⋊C8C22×D312yes—
237#103—62no—
238#102—22no—
239#102—62no—
240#102—62no—
241#102—62no—
242#102—62no—
243C22⋊C8—61no—
244C22⋊C8—61no—
245C22⋊C8—62no—
246C22⋊C8—61no—
247#97—21no—
248#97—61no—
249C4×C8—31no—
250C4×C8—31no—
251C4×C8—31no—
252C4×C8—61no—
253C4×C8—62no—
254C4×C8C22×D311yescharacteristic
255C23×C3—42no—
256C22×D3—21no—
257C22×D3#12111yescharacteristic
258C3×Q8—22no—
259C3×D4—42no—
260C3×D4—42no—
261C3×D4—42no—
262C3×D4—42no—
263C3×D4—82no—
264C2×C3×C4—21no—
265C2×C3×C4—21no—
266C2×C3×C4C22×D411yescharacteristic
267C2×C3×C4C22×D411yescharacteristic
268C2×C3×C4C2×#4111yescharacteristic
269#67—84no—
270C2×Dic3—21no—
271C2×Dic3—42no—
272C2×Dic3—84no—
273C2×Dic3#12111yescharacteristic
274C2×Dic3#12111yescharacteristic
275C2×Dic3#12111yescharacteristic
276D12—22no—
277D12—42no—
278C4×D3—21no—
279C4×D3—21no—
280C4×D3—22no—
281C4×D3—22no—
282C4×D3—42no—
283C4×D3—41no—
284C4×D3—41no—
285C4×D3—42no—
286Dic6—82no—
287C3×C8—21no—
288C3×C8—41no—
289C3×C8—42no—
290C3×C8#13512yes—
291C3⋊C8—21no—
292C3⋊C8—21no—
293C3⋊C8—41no—
294C3⋊C8—42no—
295#41—122no—
296#41—244no—
297C2×Q8—62no—
298C2×Q8D3×D411yescharacteristic
299C2×D4—22no—
300C2×D4—22no—
301C2×D4—22no—
302C2×D4—31no—
303C2×D4—61no—
304C2×D4—122no—
305C22×C4—31no—
306C22×C4—31no—
307C22×C4—31no—
308C22×C4—61no—
309C22×C4—61no—
310C22×C4—61no—
311C22×C4—122no—
312SD16—44no—
313D8—42no—
314D8—42no—
315D8—124no—
316M16—122no—
317M16—122no—
318C2×C8—21no—
319C2×C8—22no—
320C2×C8—31no—
321C2×C8—31no—
322C2×C8—62no—
323C2×C8—61no—
324C2×C8—62no—
325C2×C8—121no—
326C2×C8D3×D411yescharacteristic
327C2×C8D3×D411yescharacteristic
328C4⋊C4—21no—
329C4⋊C4—31no—
330C4⋊C4—61no—
331C4⋊C4—62no—
332C4⋊C4—62no—
333C4⋊C4D3×D411yescharacteristic
334C22⋊C4—61no—
335C22⋊C4—61no—
336C22⋊C4—61no—
337C22⋊C4—122no—
338C22⋊C4—122no—
339C42—21no—
340C42—31no—
341C42—31no—
342C42—31no—
343C42—61no—
344C42—61no—
345C42C23×D311yescharacteristic
346C22×C3—82no—
347C22×C3—82no—
348C22×C3—82no—
349C22×C3C2×#12111yescharacteristic
350C2×D3—21no—
351C2×D3—21no—
352C2×D3—21no—
353C2×D3—41no—
354C2×D3—41no—
355C2×D3—41no—
356C3×C4—21no—
357C3×C4—41no—
358C3×C4—41no—
359C3×C4C2×#13511yescharacteristic
360C3×C4#57511yescharacteristic
361C3×C4#57512yes—
362Dic3—21no—
363Dic3—21no—
364Dic3—41no—
365Dic3—82no—
366Dic3#47212yes—
367C23—31no—
368C23—42no—
369C23—61no—
370Q8—22no—
371Q8—242no—
372D4—42no—
373D4—42no—
374D4—42no—
375D4—42no—
376D4—62no—
377D4—82no—
378D4—122no—
379D4—244no—
380C2×C4—21no—
381C2×C4—21no—
382C2×C4—31no—
383C2×C4—31no—
384C2×C4—31no—
385C2×C4—62no—
386C2×C4—61no—
387C2×C4—61no—
388C2×C4—62no—
389C2×C4—61no—
390C2×C4—122no—
391C2×C4—122no—
392C2×C4—121no—
393C2×C4—121no—
394C2×C4—122no—
395C2×C4—244no—
396C2×C4C2×D3×D411yescharacteristic
397C2×C4C2×D3×D411yescharacteristic
398C2×C4D3×#4111yescharacteristic
399C8—21no—
400C8—41no—
401C8—42no—
402C8—61no—
403C8—61no—
404C8—121no—
405C8—122no—
406C8#91612yes—
407C2×C3—162no—
408C2×C3#303511yescharacteristic
409C2×C3#305411yescharacteristic
410C2×C3C2×#47211yescharacteristic
411D3—41no—
412D3—81no—
413C22—61no—
414C22—61no—
415C22—61no—
416C22—82no—
417C22—82no—
418C22—82no—
419C22—121no—
420C22—121no—
421C22—121no—
422C22D3×#12111yescharacteristic
423C4—21no—
424C4—32no—
425C4—41no—
426C4—41no—
427C4—61no—
428C4—61no—
429C4—121no—
430C4—242no—
431C4C2×#91611yescharacteristic
432C4#577312yes—
433C4#577311yescharacteristic
434C3#1916011yescharacteristic, complemented, Sylow
435C2—121no—
436C2—162no—
437C2—241no—
438C2#8038211yescharacteristic
439C2#8086311yescharacteristic
440C2D3×#47211yescharacteristic
subgroupG/H
centerC22D3×#121
commutator, cosocleC3×C42C24
FrattiniC42C23×D3
FittingC3×#1688C2
socleC22×C3C2×#121
hypercenter#1688D3
hypocenterC3#19160
LCSG ▷ C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C42 ◁ #1688

Complements

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

NH ≅ G/Ntype
1. #81775C2semidirect
2. #80361C2semidirect
3. #79938C2semidirect
4. C3×#1688C2semidirect
5. #71283C2semidirect
6. #71026C2semidirect
7. #70965C2semidirect
8. #68543C2semidirect
9. #68527C2semidirect
10. #68400C2semidirect
11. #68382C2semidirect
55. #1688D3semidirect
434. C3#19160semidirect

Maximal subgroups

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

subgroupindex
#817752
#803612
#799382
C3×#16882
#712832
#710262
#709652
#685432
#685272
#684002
#683822
#191603