Finite Group Databasebeta

← #10777933

Subgroups — #10777933

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
1C3×#2965C211yesmaximal, characteristic, complemented
2#82529C212yesmaximal, complemented
3#78470C211yesmaximal, characteristic, complemented
4#70965C211yesmaximal, characteristic, complemented
5#70960C212yesmaximal, complemented
6#70959C211yesmaximal, characteristic, complemented
7#70874C211yesmaximal, characteristic, complemented
8#70869C212yesmaximal, complemented
9#70868C211yesmaximal, characteristic, complemented
10#70773C212yesmaximal, complemented
11#69729C211yesmaximal, characteristic, complemented
12#19160—31nomaximal, Sylow
13C3×C4×#41C2211yescharacteristic
14#5477C2212yes—
15C3×#446C2211yescharacteristic
16C3×#446C2211yescharacteristic
17C3×#436C2211yescharacteristic
18C3×#436C2211yescharacteristic
19C3×#435—21no—
20C3×#435—21no—
21C3×#431—21no—
22C3×#431C2211yescharacteristic
23C3×C4×M16—21no—
24C2×C3×C4×C8C2211yescharacteristic
25#5238—22no—
26#5115—21no—
27#5115—21no—
28#5027C2212yes—
29C4×#215—22no—
30#5021C2212yes—
31#5015C2212yes—
32C4×#213—22no—
33#5008C2212yes—
34#5003—22no—
35#4925C2212yes—
36#4695C2211yescharacteristic
37#4694C2212yes—
38C4×D24—21no—
39C4×D24—21no—
40#4682C2211yescharacteristic
41#4485C2212yes—
42#4485C2212yes—
43#4481C2212yes—
44#4481C2212yes—
45#4458C2212yes—
46#4457C2211yescharacteristic
47#4457C2211yescharacteristic
48#3169—31no—
49#3106—32no—
50#2965D311yescharacteristic, complemented
51#1698—32no—
52#1697—32no—
53#1696—31no—
54#1695—31no—
55#1688—31no—
56#1687—31no—
57#1564—31no—
58#1543—32no—
59C2×C3×#41D411yescharacteristic
60C2×#235—22no—
61C2×C3×M16—21no—
62C2×C3×M16—21no—
63C22×C3×C8D411yescharacteristic
64C22×C3×C8D411yescharacteristic
65C3×C4×Q8D411yescharacteristic
66C3×C4×Q8C2311yescharacteristic
67C3×C4×D4—21no—
68C3×C4×D4—21no—
69C3×C4×D4D411yescharacteristic
70C3×C4×D4C2311yescharacteristic
71C3×#117—21no—
72C3×#117D411yescharacteristic
73C2×C3×C42—21no—
74C2×C3×C42C2311yescharacteristic
75C2×#215—22no—
76C2×#213—22no—
77#930—22no—
78#930—22no—
79C2×D24—21no—
80C2×D24—21no—
81#875—22no—
82#874C2312yes—
83C4×D12—22no—
84C4×D12—22no—
85#869C2312yes—
86C3×C4⋊C8—21no—
87C3×C4⋊C8—21no—
88C3×C4⋊C8C2311yescharacteristic
89C3×C4⋊C8C2311yescharacteristic
90C3×C22⋊C8—21no—
91C3×C22⋊C8—21no—
92C3×C22⋊C8—21no—
93C3×#97—21no—
94C3×#97—21no—
95C3×C4×C8—21no—
96C3×C4×C8D411yescharacteristic
97C3×C4×C8D411yescharacteristic
98C3×C4×C8C2311yescharacteristic
99C3×C4×C8C2311yescharacteristic
100#835—22no—
101#832—22no—
102#830—22no—
103#821—22no—
104#821—22no—
105#818—21no—
106#818—21no—
107#809—22no—
108#809—22no—
109#808—22no—
110#807—22no—
111C4⋊(C3⋊C8)C2312yes—
112C4⋊(C3⋊C8)C2312yes—
113C4×C3⋊C8—22no—
114#532—32no—
115C4×#41C2×D311yescharacteristic
116#500—31no—
117#493—31no—
118#486—32no—
119#482—62no—
120#477—32no—
121#476—32no—
122#466—61no—
123#466—61no—
124#460—32no—
125#459—32no—
126#449—62no—
127#446C2×D311yescharacteristic
128#446C2×D311yescharacteristic
129C4×SD16—62no—
130C4×D8—62no—
131C4×D8—61no—
132C4×D8—61no—
133#436C2×D311yescharacteristic
134#436C2×D311yescharacteristic
135#435—21no—
136#435—21no—
137#431—21no—
138#431—32no—
139#431C2×D311yescharacteristic
140C4×M16—21no—
141C2×C4×C8C2×D311yescharacteristic
142#335—32no—
143#334—31no—
144#334—31no—
145#333—32no—
146#333—32no—
147#331—32no—
148#331—32no—
149C3×#41—42no—
150C2×C3×Q8C2×D411yescharacteristic
151C2×C3×D4—21no—
152C2×C3×D4C2×D411yescharacteristic
153C22×C3×C4—21no—
154C22×C3×C4—21no—
155C22×C3×C4—21no—
156C22×C3×C4C2×D411yescharacteristic
157C22×C3×C4C2×D411yescharacteristic
158C22×C3×C4#4111yescharacteristic
159C2×#67—42no—
160#235—84no—
161C2×D12—22no—
162C2×D12—22no—
163C2×D12—22no—
164C2×C4×D3—42no—
165C2×Dic6—22no—
166C3×M16—42no—
167C3×M16—42no—
168C2×C3×C8—22no—
169C2×C3×C8—21no—
170C2×C3×C8—21no—
171C2×C3×C8—41no—
172C2×C3×C8C2×D411yescharacteristic
173C2×C3×C8C2×D411yescharacteristic
174C2×C3×C8C2×D411yescharacteristic
175C2×C3×C8C2×D411yescharacteristic
176C3×C4⋊C4—21no—
177C3×C4⋊C4—21no—
178C3×C4⋊C4C2×D411yescharacteristic
179C3×C4⋊C4C2×D411yescharacteristic
180C3×C22⋊C4—21no—
181C3×C22⋊C4—21no—
182C3×C22⋊C4—21no—
183C3×C42—21no—
184C3×C42—21no—
185C3×C42—21no—
186C3×C42C2×D411yescharacteristic
187C3×C42C2×D411yescharacteristic
188C3×C42#4111yescharacteristic
189C3×C42C2411yescharacteristic
190#215—44no—
191#213—44no—
192(C2×D3)⋊C4—42no—
193(C2×D3)⋊C4—42no—
194C4⋊Dic3—22no—
195C4⋊Dic3—22no—
196C2×C3⋊C8—22no—
197C2×C3⋊C8—22no—
198D24—42no—
199D24—42no—
200C2×#41—62no—
201C2×#41D1211yescharacteristic
202C2×SD16—62no—
203C2×D8—62no—
204C2×D8—61no—
205C2×D8—61no—
206C2×M16—21no—
207C2×M16—21no—
208C22×C8#6711yescharacteristic
209C22×C8#6711yescharacteristic
210C4⋊Q8—32no—
211#127—32no—
212#124—62no—
213#121—62no—
214#121—62no—
215C4×Q8D1211yescharacteristic
216C4×Q8C22×D311yescharacteristic
217C4×D4—21no—
218C4×D4—21no—
219C4×D4—62no—
220C4×D4—62no—
221C4×D4D1211yescharacteristic
222C4×D4C22×D311yescharacteristic
223#117—21no—
224#117D1211yescharacteristic
225C2×C42—21no—
226C2×C42C22×D311yescharacteristic
227#107—61no—
228#107—61no—
229#107—62no—
230#106—62no—
231C4⋊C8—21no—
232C4⋊C8—21no—
233C4⋊C8—32no—
234C4⋊C8—32no—
235C4⋊C8C22×D311yescharacteristic
236C4⋊C8C22×D311yescharacteristic
237#103—62no—
238#102—62no—
239#102—62no—
240#102—62no—
241#102—62no—
242#102—62no—
243C22⋊C8—21no—
244C22⋊C8—21no—
245C22⋊C8—21no—
246C22⋊C8—62no—
247#97—21no—
248#97—21no—
249C4×C8—21no—
250C4×C8—62no—
251C4×C8#6711yescharacteristic
252C4×C8#6711yescharacteristic
253C4×C8C22×D311yescharacteristic
254C4×C8C22×D311yescharacteristic
255C23×C3—21no—
256C23×C3#12111yescharacteristic
257C22×D3—42no—
258C3×Q8—22no—
259C3×D4—22no—
260C3×D4—42no—
261C2×C3×C4—21no—
262C2×C3×C4—21no—
263C2×C3×C4—22no—
264C2×C3×C4—22no—
265C2×C3×C4—21no—
266C2×C3×C4—21no—
267C2×C3×C4—21no—
268C2×C3×C4—42no—
269C2×C3×C4—41no—
270C2×C3×C4—41no—
271C2×C3×C4—42no—
272C2×C3×C4#12111yescharacteristic
273C2×C3×C4#12111yescharacteristic
274C2×C3×C4#12111yescharacteristic
275C2×C3×C4C22×D411yescharacteristic
276C2×C3×C4C22×D411yescharacteristic
277C2×C3×C4C2×#4111yescharacteristic
278#67—84no—
279C2×Dic3—42no—
280D12—42no—
281D12—42no—
282D12—42no—
283D12—42no—
284D12—82no—
285C4×D3—84no—
286Dic6—82no—
287C3×C8—21no—
288C3×C8—21no—
289C3×C8—21no—
290C3×C8—41no—
291C3×C8—41no—
292C3×C8#13512yes—
293C3⋊C8—42no—
294C3⋊C8—42no—
295#41—42no—
296#41—244no—
297C2×Q8—62no—
298C2×Q8C2×D1211yescharacteristic
299C2×D4—21no—
300C2×D4—62no—
301C2×D4—62no—
302C2×D4—62no—
303C2×D4—122no—
304C2×D4C2×D1211yescharacteristic
305C22×C4—21no—
306C22×C4—21no—
307C22×C4—21no—
308C22×C4—122no—
309C22×C4C2×D1211yescharacteristic
310C22×C4#23511yescharacteristic
311C22×C4C2×#6711yescharacteristic
312SD16—124no—
313D8—122no—
314D8—124no—
315D8—122no—
316M16—42no—
317M16—42no—
318C2×C8—22no—
319C2×C8—21no—
320C2×C8—21no—
321C2×C8—41no—
322C2×C8—62no—
323C2×C8—62no—
324C2×C8C2×#6711yescharacteristic
325C2×C8C2×#6711yescharacteristic
326C2×C8C2×#6711yescharacteristic
327C2×C8C2×#6711yescharacteristic
328C4⋊C4—21no—
329C4⋊C4—21no—
330C4⋊C4—62no—
331C4⋊C4—62no—
332C4⋊C4C2×D1211yescharacteristic
333C4⋊C4C2×D1211yescharacteristic
334C22⋊C4—21no—
335C22⋊C4—21no—
336C22⋊C4—21no—
337C22⋊C4—122no—
338C22⋊C4—122no—
339C42—21no—
340C42—21no—
341C42—21no—
342C42C2×D1211yescharacteristic
343C42#23511yescharacteristic
344C42C2×#6711yescharacteristic
345C42C23×D311yescharacteristic
346C22×C3—21no—
347C22×C3—21no—
348C22×C3—21no—
349C22×C3—41no—
350C22×C3—41no—
351C22×C3—41no—
352C22×C3C2×#12111yescharacteristic
353C2×D3—82no—
354C2×D3—82no—
355C2×D3—82no—
356C3×C4—21no—
357C3×C4—21no—
358C3×C4—21no—
359C3×C4—41no—
360C3×C4—41no—
361C3×C4—41no—
362C3×C4#47212yes—
363C3×C4C2×#13511yescharacteristic
364C3×C4#57512yes—
365C3×C4#57511yescharacteristic
366Dic3—82no—
367C23—21no—
368C23—122no—
369C23#93011yescharacteristic
370Q8—22no—
371Q8—242no—
372D4—22no—
373D4—42no—
374D4—122no—
375D4—122no—
376D4—122no—
377D4—122no—
378D4—242no—
379D4—244no—
380C2×C4—22no—
381C2×C4—22no—
382C2×C4—21no—
383C2×C4—21no—
384C2×C4—21no—
385C2×C4—21no—
386C2×C4—21no—
387C2×C4—42no—
388C2×C4—42no—
389C2×C4—41no—
390C2×C4—41no—
391C2×C4—122no—
392C2×C4—244no—
393C2×C4#93011yescharacteristic
394C2×C4#93011yescharacteristic
395C2×C4#93011yescharacteristic
396C2×C4C22×D1211yescharacteristic
397C2×C4C2×#23511yescharacteristic
398C2×C4C22×#6711yescharacteristic
399C8—21no—
400C8—21no—
401C8—21no—
402C8—41no—
403C8—41no—
404C8—122no—
405C8—122no—
406C8#95012yes—
407C2×C3—41no—
408C2×C3—81no—
409C2×C3#303511yescharacteristic
410C2×C3#305411yescharacteristic
411C2×C3C2×#47211yescharacteristic
412D3—162no—
413C22—21no—
414C22—21no—
415C22—21no—
416C22—41no—
417C22—41no—
418C22—41no—
419C22—242no—
420C22—242no—
421C22—242no—
422C22C2×#93011yescharacteristic
423C4—21no—
424C4—21no—
425C4—21no—
426C4—41no—
427C4—41no—
428C4—41no—
429C4—242no—
430C4#514212yes—
431C4#574812yes—
432C4C2×#95011yescharacteristic
433C4#582211yescharacteristic
434C3#1916011yescharacteristic, complemented, Sylow
435C2—41no—
436C2—81no—
437C2—482no—
438C2#8252511yescharacteristic
439C2#8377111yescharacteristic
440C2C2×#514211yescharacteristic
subgroupG/H
centerC2×C4#930
commutator, cosocleC3×C42C24
FrattiniC42C23×D3
FittingC3×#2965C2
socleC22×C3C2×#121
hypercenter#2965D3
hypocenterC3#19160
LCSG ▷ C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #2965

Complements

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

NH ≅ G/Ntype
1. C3×#2965C2semidirect
2. #82529C2semidirect
3. #78470C2semidirect
4. #70965C2semidirect
5. #70960C2semidirect
6. #70959C2semidirect
7. #70874C2semidirect
8. #70869C2semidirect
9. #70868C2semidirect
10. #70773C2semidirect
11. #69729C2semidirect
50. #2965D3semidirect
434. C3#19160semidirect

Maximal subgroups

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

subgroupindex
C3×#29652
#825292
#784702
#709652
#709602
#709592
#708742
#708692
#708682
#707732
#697292
#191603