Finite Group Databasebeta

← #10686188

Subgroups — #10686188

Order 768.

522 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
1#82916C211yesmaximal, characteristic
2C3×#1501C211yesmaximal, characteristic
3#71707C211yesmaximal, characteristic
4#69947C211yesmaximal, characteristic, complemented
5#69447C211yesmaximal, characteristic, complemented
6#67680C211yesmaximal, characteristic, complemented
7#67618C211yesmaximal, characteristic, complemented
8#8240—31nomaximal, Sylow
9C3×#525C2211yescharacteristic
10C2×#948C411yescharacteristic, complemented
11C23×C3⋊C8C2211yescharacteristic
12C2×C3×C22⋊C8C2211yescharacteristic
13C3×C4×C22⋊C4C2211yescharacteristic
14#5234C411yescharacteristic, complemented
15C4×#217C2211yescharacteristic
16#5207C2211yescharacteristic
17C2×#830C2211yescharacteristic
18C2×C8×Dic3—21no—
19#5011—21no—
20#5011—21no—
21D4×C3⋊C8—21no—
22D4×C3⋊C8—21no—
23#4999C411yescharacteristic, complemented
24#4997C411yescharacteristic, complemented
25C2×#803—21no—
26C3×#328C412yescomplemented
27C3×#327C412yescomplemented
28#4550—21no—
29#4538C412yescomplemented
30#4537C412yescomplemented
31#4524—21no—
32#2929—31no—
33#1752—31no—
34C8×C22⋊C4—31no—
35#1501—31no—
36#1501D311yescharacteristic
37#1298—31no—
38#1297—31no—
39C2×C3×#41C2×C411yescharacteristic
40C23×C3×C4C2311yescharacteristic
41C22×C3×C8—21no—
42C3×#123—21no—
43C3×C22⋊Q8C2×C411yescharacteristic
44C3×C22⋊Q8C2×C411yescharacteristic
45C3×#121C2×C411yescharacteristic
46C3×#121C2×C411yescharacteristic
47C3×#120—21no—
48C3×C4×D4—21no—
49C3×C4×D4—21no—
50C3×#117C2×C411yescharacteristic
51C2×C3×C22⋊C4—21no—
52C2×C3×C42—21no—
53C2×#217—21no—
54#948—42no—
55C2×C4×Dic3—21no—
56C2×#208—21no—
57C2×#208C2×C411yescharacteristic
58C2×#208C2×C411yescharacteristic
59C22×C3⋊C8—21no—
60C22×C3⋊C8—21no—
61C22×C3⋊C8—42no—
62C22×C3⋊C8C2×C411yescharacteristic
63C22×C3⋊C8C2×C411yescharacteristic
64C22×C3⋊C8D411yescharacteristic
65C22×C3⋊C8D411yescharacteristic
66C22×C3⋊C8D411yescharacteristic
67C22×C3⋊C8D411yescharacteristic
68C3×C22⋊C8—21no—
69C3×C22⋊C8C2×C412yes—
70C3×#95C2×C412yes—
71C3⋊#95C2×C412yes—
72#830—21no—
73#830—21no—
74#830—21no—
75#830—21no—
76#830—21no—
77#830C2×C412yes—
78C8×Dic3—42no—
79C4⋊(C3⋊C8)—21no—
80C4⋊(C3⋊C8)—21no—
81#803—21no—
82#803—42no—
83C4×C3⋊C8—21no—
84C2×#131—31no—
85C23×C8—31no—
86#525C2×D311yescharacteristic
87#435—61no—
88#435—61no—
89C8×D4—61no—
90C8×D4—61no—
91#424—31no—
92#408—31no—
93#407—31no—
94C2×C22⋊C8—31no—
95C2×C22⋊C8C2×D311yescharacteristic
96#405—31no—
97C2×#97—61no—
98C2×C4×C8—61no—
99C4×C22⋊C4—31no—
100C4×C22⋊C4C2×D311yescharacteristic
101#336—61no—
102#336—61no—
103#328—32no—
104#328Dic312yescomplemented
105#327—32no—
106#327Dic312yescomplemented
107C24×C3#4111yescharacteristic
108C3×#41—41no—
109C3×#41—41no—
110C2×C3×Q8C4211yescharacteristic
111C2×C3×D4—21no—
112C2×C3×D4—21no—
113C2×C3×D4—21no—
114C2×C3×D4C4211yescharacteristic
115C22×C3×C4—21no—
116C22×C3×C4—22no—
117C22×C3×C4—41no—
118C22×C3×C4—41no—
119C22×C3×C4C22×C411yescharacteristic
120C22×C3×C4C22×C411yescharacteristic
121C22×C3×C4C22×C411yescharacteristic
122C22×C3×C4C2×D411yescharacteristic
123C22×C3×C4C2×D411yescharacteristic
124C22×C3×C4#4111yescharacteristic
125C22×Dic3—22no—
126C2×C3×C8—22no—
127C2×C3×C8—41no—
128C2×C3×C8—41no—
129C3×C4⋊C4—21no—
130C3×C4⋊C4—21no—
131C3×C4⋊C4C4211yescharacteristic
132C3×C4⋊C4C4211yescharacteristic
133C3×C22⋊C4—21no—
134C3×C22⋊C4—21no—
135C3×C22⋊C4—21no—
136C3×C22⋊C4—21no—
137C3×C22⋊C4—21no—
138C3×C22⋊C4—42no—
139C3×C42—21no—
140C3×C42—41no—
141C3×C42—41no—
142#217—24no—
143C4×Dic3—41no—
144C4×Dic3—41no—
145#208—42no—
146#208—41no—
147#208—41no—
148C2×C3⋊C8—22no—
149C2×C3⋊C8—22no—
150C2×C3⋊C8—22no—
151C2×C3⋊C8—21no—
152C2×C3⋊C8—21no—
153C2×C3⋊C8—22no—
154C2×C3⋊C8—41no—
155C2×C3⋊C8—41no—
156C2×C3⋊C8—41no—
157C2×C3⋊C8—42no—
158C2×C3⋊C8—41no—
159C2×C3⋊C8—41no—
160C2×C3⋊C8C22⋊C411yescharacteristic
161C2×C3⋊C8C22⋊C411yescharacteristic
162C2×C3⋊C8C22⋊C411yescharacteristic
163C2×C3⋊C8C22⋊C411yescharacteristic
164C2×#41C4×D311yescharacteristic
165C23×C4C22×D311yescharacteristic
166#131—122no—
167C2×M16—31no—
168C2×M16—31no—
169C2×M16—61no—
170C22×C8—21no—
171C22×C8—31no—
172C22×C8—31no—
173C22×C8—31no—
174C22×C8—31no—
175C22×C8—31no—
176C22×C8—31no—
177C22×C8—61no—
178C22×C8—61no—
179C22×C8—122no—
180#123—21no—
181C22⋊Q8C4×D311yescharacteristic
182C22⋊Q8C2×Dic311yescharacteristic
183#121C4×D311yescharacteristic
184#121C2×Dic311yescharacteristic
185#120—21no—
186C4×D4—21no—
187C4×D4—21no—
188#117C4×D311yescharacteristic
189C2×C22⋊C4—21no—
190C2×C22⋊C4—61no—
191C2×C42—21no—
192C2×C42—61no—
193C4⋊C8—61no—
194C4⋊C8—61no—
195C22⋊C8—21no—
196C22⋊C8—32no—
197C22⋊C8—61no—
198C22⋊C8—61no—
199C22⋊C8—61no—
200C22⋊C8—61no—
201C22⋊C8—61no—
202C22⋊C8C2×Dic312yes—
203#97—61no—
204#97—122no—
205C4×C8—61no—
206C4×C8—122no—
207#95—32no—
208#95C2×Dic312yes—
209C23×C3—22no—
210C23×C3—22no—
211C23×C3—21no—
212C23×C3—41no—
213C23×C3#11711yescharacteristic
214C23×C3C4×D411yescharacteristic
215C23×C3C4×D411yescharacteristic
216C3×Q8—41no—
217C3×D4—41no—
218C3×D4—42no—
219C3×D4—42no—
220C3×D4—42no—
221C2×C3×C4—21no—
222C2×C3×C4—21no—
223C2×C3×C4—21no—
224C2×C3×C4—21no—
225C2×C3×C4—21no—
226C2×C3×C4—21no—
227C2×C3×C4—21no—
228C2×C3×C4—21no—
229C2×C3×C4—21no—
230C2×C3×C4—41no—
231C2×C3×C4—41no—
232C2×C3×C4—41no—
233C2×C3×C4—41no—
234C2×C3×C4—41no—
235C2×C3×C4—42no—
236C2×C3×C4—42no—
237C2×C3×C4—42no—
238C2×C3×C4C2×C4211yescharacteristic
239C2×C3×C4C2×C22⋊C411yescharacteristic
240C2×C3×C4C4×D411yescharacteristic
241C2×C3×C4C4×D411yescharacteristic
242C2×Dic3—24no—
243C2×Dic3—42no—
244C2×Dic3—42no—
245C3×C8—42no—
246C3⋊C8—21no—
247C3⋊C8—21no—
248C3⋊C8—21no—
249C3⋊C8—42no—
250C3⋊C8—41no—
251C3⋊C8—41no—
252C3⋊C8#9912yes—
253C24#23711yescharacteristic
254#41—41no—
255#41—41no—
256C2×Q8C4×Dic311yescharacteristic
257C2×D4—21no—
258C2×D4—21no—
259C2×D4—21no—
260C2×D4C4×Dic311yescharacteristic
261C22×C4—21no—
262C22×C4—22no—
263C22×C4—41no—
264C22×C4—41no—
265C22×C4—62no—
266C22×C4C2×C4×D311yescharacteristic
267C22×C4C2×C4×D311yescharacteristic
268C22×C4D3×D411yescharacteristic
269C22×C4D3×D411yescharacteristic
270C22×C4#23711yescharacteristic
271C22×C4C22×Dic311yescharacteristic
272M16—122no—
273M16—121no—
274M16—121no—
275C2×C8—22no—
276C2×C8—31no—
277C2×C8—31no—
278C2×C8—31no—
279C2×C8—31no—
280C2×C8—41no—
281C2×C8—41no—
282C2×C8—62no—
283C2×C8—62no—
284C2×C8—62no—
285C2×C8—61no—
286C2×C8—61no—
287C2×C8—62no—
288C2×C8—121no—
289C2×C8—121no—
290C2×C8—121no—
291C2×C8—122no—
292C2×C8—121no—
293C2×C8—121no—
294C4⋊C4—21no—
295C4⋊C4—21no—
296C4⋊C4C4×Dic311yescharacteristic
297C4⋊C4C4×Dic311yescharacteristic
298C22⋊C4—21no—
299C22⋊C4—21no—
300C22⋊C4—21no—
301C22⋊C4—21no—
302C22⋊C4—21no—
303C22⋊C4—42no—
304C22⋊C4—64no—
305C42—21no—
306C42—41no—
307C42—41no—
308C42—121no—
309C42—121no—
310C22×C3—21no—
311C22×C3—21no—
312C22×C3—21no—
313C22×C3—21no—
314C22×C3—21no—
315C22×C3—21no—
316C22×C3—21no—
317C22×C3—44no—
318C22×C3—41no—
319C22×C3—41no—
320C22×C3—41no—
321C22×C3C4×C22⋊C411yescharacteristic
322C22×C3#44312yes—
323C22×C3#44412yes—
324C3×C4—21no—
325C3×C4—21no—
326C3×C4—21no—
327C3×C4—41no—
328C3×C4—41no—
329C3×C4—41no—
330C3×C4—82no—
331C3×C4C2×#9911yescharacteristic
332C3×C4#41011yescharacteristic
333Dic3—44no—
334C23—22no—
335C23—22no—
336C23—21no—
337C23—41no—
338C23#87711yescharacteristic
339C23#67⋊C411yescharacteristic
340C23D4×Dic311yescharacteristic
341Q8—41no—
342D4—41no—
343D4—42no—
344D4—42no—
345D4—42no—
346C2×C4—21no—
347C2×C4—21no—
348C2×C4—21no—
349C2×C4—21no—
350C2×C4—21no—
351C2×C4—21no—
352C2×C4—21no—
353C2×C4—21no—
354C2×C4—21no—
355C2×C4—41no—
356C2×C4—41no—
357C2×C4—41no—
358C2×C4—41no—
359C2×C4—41no—
360C2×C4—42no—
361C2×C4—42no—
362C2×C4—42no—
363C2×C4—64no—
364C2×C4—122no—
365C2×C4—122no—
366C2×C4D3×C22⋊C411yescharacteristic
367C2×C4#67⋊C411yescharacteristic
368C2×C4C2×C4×Dic311yescharacteristic
369C2×C4D4×Dic311yescharacteristic
370C8—32no—
371C8—42no—
372C8—61no—
373C8—61no—
374C8—61no—
375C8—122no—
376C8—121no—
377C8—121no—
378C2×C3—22no—
379C2×C3—22no—
380C2×C3—41no—
381C2×C3—81no—
382C2×C3C4×#9911yescharacteristic
383C2×C3#176411yescharacteristic
384C2×C3#176511yescharacteristic
385C22—21no—
386C22—21no—
387C22—21no—
388C22—21no—
389C22—21no—
390C22—21no—
391C22—21no—
392C22—44no—
393C22—41no—
394C22—41no—
395C22—41no—
396C22#482312yes—
397C22Dic3×C22⋊C411yescharacteristic
398C22#519612yes—
399C4—21no—
400C4—21no—
401C4—21no—
402C4—41no—
403C4—41no—
404C4—41no—
405C4—82no—
406C4—124no—
407C4#474011yescharacteristic
408C4D3×#9911yescharacteristic
409C3#824011yescharacteristic, complemented, Sylow
410C2—22no—
411C2—22no—
412C2—41no—
413C2—81no—
414C2Dic3×#9911yescharacteristic
415C2#7028911yescharacteristic
416C2#7037011yescharacteristic
subgroupG/H
centerC2×C4D3×C22⋊C4
commutatorC2×C3×C4C2×C42
FrattiniC23×C4C22×D3
FittingC3×#1501C2
socleC22×C3C4×C22⋊C4
prime radicalC3×#525C22
cosocleC23×C3×C4C23
hypercenter#1501D3
hypocenterC3#8240
LCSG ▷ C2×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C2×C4 ◁ C23×C4 ◁ #1501

Complements

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

NH ≅ G/Ntype
4. #69947C2semidirect
5. #69447C2semidirect
6. #67680C2semidirect
7. #67618C2semidirect
10. C2×#948C4semidirect
14. #5234C4semidirect
23. #4999C4semidirect
24. #4997C4semidirect
26. C3×#328C4semidirect
27. C3×#327C4semidirect
29. #4538C4semidirect
30. #4537C4semidirect
104. #328Dic3semidirect
106. #327Dic3semidirect
409. C3#8240semidirect

Maximal subgroups

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

subgroupindex
#829162
C3×#15012
#717072
#699472
#694472
#676802
#676182
#82403