Finite Group Databasebeta

← #10686187

Subgroups — #10686187

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#83734C211yesmaximal, characteristic
2C3×#1752C211yesmaximal, characteristic, complemented
3C8×#217C211yesmaximal, characteristic, complemented
4#69942C211yesmaximal, characteristic
5#69443C211yesmaximal, characteristic
6#67680C211yesmaximal, characteristic, complemented
7#67618C211yesmaximal, characteristic, complemented
8#8240—31nomaximal, Sylow
9C23×C3×C8C2211yescharacteristic
10#5791C2211yescharacteristic
11C2×#901C411yescharacteristic, complemented
12C2×C3×C22⋊C8C2211yescharacteristic
13C2×C3×#97—21no—
14C3×C4×C22⋊C4C2211yescharacteristic
15C4×#217C2211yescharacteristic
16#5207C2211yescharacteristic
17C2×#830C2211yescharacteristic
18#5111—21no—
19#5111—21no—
20#5110C411yescharacteristic, complemented
21C8×#67—21no—
22C8×#67—21no—
23#5102C411yescharacteristic, complemented
24#5101C411yescharacteristic, complemented
25C2×C8×Dic3—21no—
26C3×#336—21no—
27#4550—21no—
28#4470C412yescomplemented
29#4467C412yescomplemented
30#4452C412yescomplemented
31#4451C412yescomplemented
32#2929—31no—
33#1752D311yescharacteristic, complemented
34C8×C22⋊C4—31no—
35#1501—31no—
36#1501—31no—
37#1298—31no—
38#1297—31no—
39C23×C3×C4C2311yescharacteristic
40C2×#235C2×C411yescharacteristic
41C22×C3×C8—21no—
42C22×C3×C8—42no—
43C22×C3×C8C2×C411yescharacteristic
44C22×C3×C8C2×C411yescharacteristic
45C22×C3×C8D411yescharacteristic
46C22×C3×C8D411yescharacteristic
47C22×C3×C8D411yescharacteristic
48C22×C3×C8D411yescharacteristic
49C2×C3×C22⋊C4—21no—
50C2×C3×C42—21no—
51#953—21no—
52C2×#217—21no—
53#930C2×C411yescharacteristic
54#930C2×C411yescharacteristic
55#929—21no—
56C4×#67—21no—
57C4×#67—21no—
58#926C2×C411yescharacteristic
59#924C2×C411yescharacteristic
60#924C2×C411yescharacteristic
61C2×C4×Dic3—21no—
62C2×#208C2×C411yescharacteristic
63C2×#208C2×C411yescharacteristic
64C22×C3⋊C8—21no—
65#901—42no—
66C2×ΓL(1,𝔽25)—21no—
67C2×D3×C8—21no—
68C3×C22⋊C8—21no—
69C3×C22⋊C8C2×C412yes—
70C3×#97—42no—
71C3×#95C2×C412yes—
72C3⋊#95C2×C412yes—
73#830—21no—
74#830—21no—
75#830—21no—
76#830C2×C412yes—
77(C2×D3)⋊C8—21no—
78(C2×D3)⋊C8—21no—
79#815—21no—
80Dic3⋊C8—21no—
81Dic3⋊C8—21no—
82C8×Dic3—21no—
83C8×Dic3—42no—
84C2×#131—31no—
85C23×C8C2×D311yescharacteristic
86#525—31no—
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—21no—
98C2×C4×C8—61no—
99C4×C22⋊C4—31no—
100C4×C22⋊C4C2×D311yescharacteristic
101#336—21no—
102#336—61no—
103#328—32no—
104#328—32no—
105#327—32no—
106#327—32no—
107C24×C3#4111yescharacteristic
108C22×C3×C4—22no—
109C22×C3×C4—41no—
110C22×C3×C4—41no—
111C22×C3×C4C22×C411yescharacteristic
112C22×C3×C4C22×C411yescharacteristic
113C22×C3×C4C22×C411yescharacteristic
114C22×C3×C4C2×D411yescharacteristic
115C22×C3×C4C2×D411yescharacteristic
116C22×C3×C4#4111yescharacteristic
117C2×#67—21no—
118C2×#67—21no—
119C2×#67—21no—
120C22×Dic3—22no—
121#235—41no—
122#235—41no—
123C2×D12C4211yescharacteristic
124C2×C4×D3—21no—
125C2×Dic6C4211yescharacteristic
126C2×C3×C8—22no—
127C2×C3×C8—22no—
128C2×C3×C8—22no—
129C2×C3×C8—22no—
130C2×C3×C8—41no—
131C2×C3×C8—41no—
132C2×C3×C8—41no—
133C2×C3×C8—41no—
134C2×C3×C8—41no—
135C2×C3×C8C22⋊C411yescharacteristic
136C2×C3×C8C22⋊C411yescharacteristic
137C2×C3×C8C22⋊C411yescharacteristic
138C2×C3×C8C22⋊C411yescharacteristic
139C3×C22⋊C4—42no—
140C3×C42—41no—
141C3×C42—41no—
142#217—24no—
143#217—21no—
144#217—21no—
145#217—21no—
146(C2×D3)⋊C4—21no—
147(C2×D3)⋊C4—21no—
148C4⋊Dic3C4211yescharacteristic
149C4⋊Dic3C4211yescharacteristic
150Dic3⋊C4—21no—
151Dic3⋊C4—21no—
152C4×Dic3—21no—
153C4×Dic3—41no—
154C4×Dic3—41no—
155#208—41no—
156#208—41no—
157C2×C3⋊C8—21no—
158C2×C3⋊C8—21no—
159C2×C3⋊C8—22no—
160C2×C3⋊C8—41no—
161C2×C3⋊C8—41no—
162ΓL(1,𝔽25)—42no—
163D3×C8—42no—
164C2×#41—31no—
165C23×C4C22×D311yescharacteristic
166#131—122no—
167C2×M16—31no—
168C2×M16—31no—
169C2×M16—61no—
170C22×C8—21no—
171C22×C8—42no—
172C22×C8—61no—
173C22×C8—61no—
174C22×C8C4×D311yescharacteristic
175C22×C8C4×D311yescharacteristic
176C22×C8D1211yescharacteristic
177C22×C8D1211yescharacteristic
178C22×C8#6711yescharacteristic
179C22×C8#6711yescharacteristic
180#123—61no—
181C22⋊Q8—31no—
182C22⋊Q8—31no—
183#121—31no—
184#121—31no—
185#120—61no—
186C4×D4—61no—
187C4×D4—61no—
188#117—31no—
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⋊C8C4×D312yes—
203#97—42no—
204#97—61no—
205C4×C8—61no—
206C4×C8—122no—
207#95—32no—
208#95C4×D312yes—
209C23×C3—22no—
210C23×C3—22no—
211C23×C3—41no—
212C23×C3#11711yescharacteristic
213C23×C3C4×D411yescharacteristic
214C23×C3C4×D411yescharacteristic
215C22×D3—21no—
216C2×C3×C4—21no—
217C2×C3×C4—21no—
218C2×C3×C4—21no—
219C2×C3×C4—21no—
220C2×C3×C4—21no—
221C2×C3×C4—21no—
222C2×C3×C4—41no—
223C2×C3×C4—41no—
224C2×C3×C4—41no—
225C2×C3×C4—42no—
226C2×C3×C4—42no—
227C2×C3×C4—42no—
228C2×C3×C4C2×C4211yescharacteristic
229C2×C3×C4C2×C22⋊C411yescharacteristic
230C2×C3×C4C4×D411yescharacteristic
231C2×C3×C4C4×D411yescharacteristic
232#67—42no—
233#67—42no—
234#67—42no—
235C2×Dic3—24no—
236C2×Dic3—21no—
237C2×Dic3—21no—
238C2×Dic3—21no—
239C2×Dic3—42no—
240C2×Dic3—42no—
241D12—41no—
242C4×D3—41no—
243C4×D3—41no—
244Dic6—41no—
245C3×C8—21no—
246C3×C8—21no—
247C3×C8—21no—
248C3×C8—42no—
249C3×C8#9912yes—
250C3⋊C8—42no—
251C3⋊C8—41no—
252C3⋊C8—41no—
253C24#23511yescharacteristic
254#41—121no—
255#41—121no—
256C2×Q8—31no—
257C2×D4—31no—
258C2×D4—61no—
259C2×D4—61no—
260C2×D4—61no—
261C22×C4—22no—
262C22×C4—41no—
263C22×C4—41no—
264C22×C4—62no—
265C22×C4—61no—
266C22×C4C2×C4×D311yescharacteristic
267C22×C4C2×C4×D311yescharacteristic
268C22×C4C2×C4×D311yescharacteristic
269C22×C4C2×D1211yescharacteristic
270C22×C4#23511yescharacteristic
271C22×C4C2×#6711yescharacteristic
272M16—122no—
273M16—121no—
274M16—121no—
275C2×C8—22no—
276C2×C8—22no—
277C2×C8—22no—
278C2×C8—22no—
279C2×C8—41no—
280C2×C8—41no—
281C2×C8—41no—
282C2×C8—41no—
283C2×C8—41no—
284C2×C8—62no—
285C2×C8—61no—
286C2×C8—61no—
287C2×C8—122no—
288C2×C8—121no—
289C2×C8—121no—
290C2×C8(C2×D3)⋊C411yescharacteristic
291C2×C8(C2×D3)⋊C411yescharacteristic
292C2×C8(C2×D3)⋊C411yescharacteristic
293C2×C8(C2×D3)⋊C411yescharacteristic
294C4⋊C4—31no—
295C4⋊C4—31no—
296C4⋊C4—61no—
297C4⋊C4—61no—
298C22⋊C4—42no—
299C22⋊C4—64no—
300C22⋊C4—61no—
301C22⋊C4—61no—
302C22⋊C4—61no—
303C22⋊C4—61no—
304C22⋊C4—61no—
305C42—41no—
306C42—41no—
307C42—61no—
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×C3C4×C22⋊C411yescharacteristic
319C22×C3#44312yes—
320C22×C3#44412yes—
321C2×D3—41no—
322C2×D3—41no—
323C2×D3—41no—
324C3×C4—21no—
325C3×C4—21no—
326C3×C4—21no—
327C3×C4—82no—
328C3×C4C2×#9911yescharacteristic
329C3×C4#41011yescharacteristic
330Dic3—44no—
331Dic3—41no—
332Dic3—41no—
333Dic3—41no—
334C23—22no—
335C23—22no—
336C23—41no—
337C23—61no—
338C23#87211yescharacteristic
339C23C4×D1211yescharacteristic
340C23C4×#6711yescharacteristic
341Q8—121no—
342D4—121no—
343D4—122no—
344D4—122no—
345D4—122no—
346C2×C4—21no—
347C2×C4—21no—
348C2×C4—21no—
349C2×C4—21no—
350C2×C4—21no—
351C2×C4—21no—
352C2×C4—41no—
353C2×C4—41no—
354C2×C4—41no—
355C2×C4—42no—
356C2×C4—42no—
357C2×C4—42no—
358C2×C4—64no—
359C2×C4—61no—
360C2×C4—61no—
361C2×C4—61no—
362C2×C4—122no—
363C2×C4—122no—
364C2×C4—121no—
365C2×C4—121no—
366C2×C4C42×D311yescharacteristic
367C2×C4C4×D1211yescharacteristic
368C2×C4C2×(C2×D3)⋊C411yescharacteristic
369C2×C4C4×#6711yescharacteristic
370C8—21no—
371C8—21no—
372C8—21no—
373C8—42no—
374C8—122no—
375C8—121no—
376C8—121no—
377C8#80612yes—
378C2×C3—22no—
379C2×C3—22no—
380C2×C3—41no—
381C2×C3C4×#9911yescharacteristic
382C2×C3#176411yescharacteristic
383C2×C3#176511yescharacteristic
384D3—81no—
385C22—21no—
386C22—21no—
387C22—21no—
388C22—21no—
389C22—21no—
390C22—21no—
391C22—21no—
392C22—44no—
393C22—121no—
394C22—121no—
395C22—121no—
396C22#471712yes—
397C22C4×(C2×D3)⋊C411yescharacteristic
398C22#514412yes—
399C4—21no—
400C4—21no—
401C4—21no—
402C4—82no—
403C4—124no—
404C4—121no—
405C4—121no—
406C4—121no—
407C4C2×#80611yescharacteristic
408C4#500611yescharacteristic
409C3#824011yescharacteristic, complemented, Sylow
410C2—22no—
411C2—22no—
412C2—41no—
413C2—241no—
414C2C4×#80611yescharacteristic
415C2#6983911yescharacteristic
416C2#6988711yescharacteristic
subgroupG/H
centerC2×C4C2×(C2×D3)⋊C4
commutatorC2×C3×C4C2×C42
FrattiniC23×C4C22×D3
FittingC3×#1752C2
socleC22×C3C4×C22⋊C4
prime radical#5791C22
cosocleC23×C3×C4C23
hypercenter#1752D3
hypocenterC3#8240
LCSG ▷ C2×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C2×C4 ◁ C23×C8 ◁ #1752

Complements

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

NH ≅ G/Ntype
2. C3×#1752C2semidirect
3. C8×#217C2semidirect
6. #67680C2semidirect
7. #67618C2semidirect
11. C2×#901C4semidirect
20. #5110C4semidirect
23. #5102C4semidirect
24. #5101C4semidirect
28. #4470C4semidirect
29. #4467C4semidirect
30. #4452C4semidirect
31. #4451C4semidirect
33. #1752D3semidirect
409. C3#8240semidirect

Maximal subgroups

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

subgroupindex
#837342
C3×#17522
C8×#2172
#699422
#694432
#676802
#676182
#82403