Finite Group Databasebeta

← #35217057

Subgroups — #35217057

Order 20864.

688 conjugacy classes of subgroups.

The subgroup lattice has 414 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#19104720C211yesmaximal, characteristic
2C2×#19104697C211yesmaximal, characteristic
3C2×C42×D163C211yesmaximal, characteristic, complemented
4#23781557C211yesmaximal, characteristic, complemented
5#23781556C211yesmaximal, characteristic, complemented
6#23781546C211yesmaximal, characteristic, complemented
7#23781543C211yesmaximal, characteristic
8#23781539C211yesmaximal, characteristic, complemented
9#23781527C211yesmaximal, characteristic, complemented
10C4⋊C4×Dic163C211yesmaximal, characteristic, complemented
11#23781516C211yesmaximal, characteristic, complemented
12#23781512C211yesmaximal, characteristic, complemented
13#23781512C211yesmaximal, characteristic, complemented
14#23781512C211yesmaximal, characteristic, complemented
15#387×C163C211yesmaximal, characteristic, complemented
16C22×D652C2211yescharacteristic
17C22×C4×D163C2211yescharacteristic
18C22×C4×D163C2211yescharacteristic
19C22×C4×D163C2211yescharacteristic
20C22×C4×D163C2211yescharacteristic
21#19104720C418yescomplemented
22#19104697—24no—
23C2×C4⋊Dic163C2211yescharacteristic
24C2×Dic163⋊C4C2211yescharacteristic
25C2×Dic163⋊C4C2211yescharacteristic
26C2×(C2×D163)⋊C4C2211yescharacteristic
27C2×(C2×D163)⋊C4C2211yescharacteristic
28C2×(C2×D163)⋊C4C2211yescharacteristic
29C2×(C2×D163)⋊C4C2211yescharacteristic
30C2×(C2×D163)⋊C4C2211yescharacteristic
31C2×(C2×D163)⋊C4C2211yescharacteristic
32C2×(C2×D163)⋊C4C2211yescharacteristic
33C2×(C2×D163)⋊C4C2211yescharacteristic
34C2×C4×Dic163C2211yescharacteristic
35C2×C4×Dic163C2211yescharacteristic
36C2×C4×Dic163C2211yescharacteristic
37C2×C4×Dic163C2211yescharacteristic
38C2×C4×Dic163C2211yescharacteristic
39C2×C4×Dic163C2211yescharacteristic
40C42×D163—24no—
41C163⋊#95C2211yescharacteristic
42C163⋊#95C2211yescharacteristic
43C163⋊#95C2211yescharacteristic
44C163⋊#95C2211yescharacteristic
45C163⋊#95C2211yescharacteristic
46C163⋊#95C2211yescharacteristic
47C2×C4⋊C4×C163C2211yescharacteristic
48C2×C4⋊C4×C163C2211yescharacteristic
49C2×C42×C163C2211yescharacteristic
50#95×C163C2211yescharacteristic
51#95×C163C2211yescharacteristic
52#95×C163C2211yescharacteristic
53#95×C163C2211yescharacteristic
54C23×D163C2311yescharacteristic
55C23×D163C2311yescharacteristic
56C2×D652—22no—
57C2×D652—22no—
58C2×D652C2×C414yes—
59C22×Dic163C2311yescharacteristic
60C22×Dic163C2311yescharacteristic
61C22×Dic163C2311yescharacteristic
62C22×Dic163C2311yescharacteristic
63C22×Dic163C2311yescharacteristic
64C22×Dic163C2311yescharacteristic
65C2×C4×D163—22no—
66C2×C4×D163—22no—
67C2×C4×D163—22no—
68C2×C4×D163—22no—
69C2×C4×D163—22no—
70C2×C4×D163—22no—
71C2×C4×D163—22no—
72C2×C4×D163—22no—
73C2×C4×D163—22no—
74C2×C4×D163—22no—
75C2×C4×D163C2×C414yescomplemented
76C2×C4×D163D414yes—
77C4⋊Dic163—22no—
78Dic163⋊C4—22no—
79Dic163⋊C4C2×C414yescomplemented
80(C2×D163)⋊C4—22no—
81(C2×D163)⋊C4—22no—
82(C2×D163)⋊C4—22no—
83(C2×D163)⋊C4—22no—
84(C2×D163)⋊C4—22no—
85(C2×D163)⋊C4C2×C414yescomplemented
86(C2×D163)⋊C4C2×C414yescomplemented
87(C2×D163)⋊C4C2×C414yescomplemented
88C4×Dic163—22no—
89C4×Dic163—22no—
90C4×Dic163—22no—
91C4×Dic163—22no—
92C4×Dic163—22no—
93C4×Dic163—22no—
94C22×C4×C163C2311yescharacteristic
95C22×C4×C163C2311yescharacteristic
96C22×C4×C163C2311yescharacteristic
97C22×C4×C163C2311yescharacteristic
98C22×C4×C163C2311yescharacteristic
99C22×C4×C163C2311yescharacteristic
100C22×C4×C163C2311yescharacteristic
101C4⋊C4×C163—22no—
102C4⋊C4×C163C2×C414yescomplemented
103C42×C163—22no—
104C22×D163—21no—
105C22×D163—21no—
106C22×D163—21no—
107C22×D163—21no—
108C22×D163—21no—
109C22×D163—21no—
110C22×D163—21no—
111C22×D163—21no—
112C22×D163—21no—
113C22×D163—21no—
114C22×D163C22×C412yes—
115C22×D163C22×C412yes—
116C22×D163C2×D412yes—
117C22×D163#4112yes—
118D652—28no—
119C2×Dic163—21no—
120C2×Dic163—21no—
121C2×Dic163—21no—
122C2×Dic163—21no—
123C2×Dic163—21no—
124C2×Dic163—21no—
125C2×Dic163—21no—
126C2×Dic163—21no—
127C2×Dic163—21no—
128C2×Dic163—21no—
129C2×Dic163—21no—
130C2×Dic163—21no—
131C2×Dic163C22×C412yes—
132C2×Dic163C22×C412yes—
133C2×Dic163C2×D412yes—
134C2×Dic163#4112yes—
135C2×Dic163#4112yes—
136C2×Dic163#4112yes—
137C4×D163—28no—
138C4×D163—28no—
139C4×D163—28no—
140C4×D163—44no—
141C23×C163C2411yescharacteristic
142C2×C4×C163—21no—
143C2×C4×C163—21no—
144C2×C4×C163—21no—
145C2×C4×C163—21no—
146C2×C4×C163—21no—
147C2×C4×C163—21no—
148C2×C4×C163—21no—
149C2×C4×C163—21no—
150C2×C4×C163—21no—
151C2×C4×C163—21no—
152C2×C4×C163—21no—
153C2×C4×C163—21no—
154C2×C4×C163—21no—
155C2×C4×C163—21no—
156C2×C4×C163—21no—
157C2×C4×C163—21no—
158C2×C4×C163C22×C412yes—
159C2×C4×C163C22×C412yes—
160C2×C4×C163C22×C412yes—
161C2×C4×C163C2×D412yes—
162C2×C4×C163#4112yes—
163C22×C163C23×C411yescharacteristic
164C22×C163C22×D411yescharacteristic
165C22×C163C2×#4111yescharacteristic
166C22×C163C2×#4111yescharacteristic
167C22×C163C2×#4111yescharacteristic
168C22×C163C2×#4111yescharacteristic
169C22×C163C2×#4111yescharacteristic
170C2×D163—22no—
171C2×D163—22no—
172C2×D163—22no—
173C2×D163—22no—
174C2×D163—22no—
175C2×D163—22no—
176C2×D163—22no—
177C2×D163—22no—
178C2×D163—22no—
179C2×D163—41no—
180C2×D163—41no—
181C2×D163—41no—
182C2×D163—41no—
183C2×D163C4×D414yes—
184C4×C163—22no—
185C4×C163—22no—
186C4×C163—22no—
187C4×C163—22no—
188C4×C163—22no—
189C4×C163—41no—
190C4×C163—41no—
191Dic163—22no—
192Dic163—22no—
193Dic163—22no—
194Dic163—41no—
195Dic163—41no—
196Dic163#12314yescomplemented
197C2×C163C2×C4×D411yescharacteristic
198C2×C163C4×#4111yescharacteristic
199C2×C163C4×#4111yescharacteristic
200C2×C163C2×#12311yescharacteristic
201C2×C163#52511yescharacteristic
202C2×C163#52911yescharacteristic
203C2×C163#52911yescharacteristic
204D163—24no—
205D163—42no—
206C163C4×#12311yescharacteristic, complemented, Sylow
207C4×#123—1631nomaximal, Sylow
208C2×#123—1631no—
209C2×C4×D4—1631no—
210C22×C42—1631no—
211#388—1631no—
212#388—1631no—
213#387—1631no—
214#387D16311yescharacteristic, complemented
215#386—1631no—
216#385—1631no—
217#381—1631no—
218C4×C4⋊C4—1631no—
219C4×C4⋊C4—1631no—
220C4×C22⋊C4—1631no—
221C4×C22⋊C4—1631no—
222C4×C22⋊C4—1631no—
223C22×D4—1631no—
224C23×C4—1631no—
225C23×C4—1631no—
226C23×C4—1631no—
227C23×C4—1631no—
228#123—1638no—
229C4×D4—3264no—
230C2×C4⋊C4—1631no—
231C2×C4⋊C4—1631no—
232C2×C4⋊C4—1631no—
233C2×C4⋊C4C2×D16311yescharacteristic
234C2×C4⋊C4C2×D16311yescharacteristic
235C2×C22⋊C4—1631no—
236C2×C22⋊C4—1631no—
237C2×C22⋊C4—1631no—
238C2×C22⋊C4—1631no—
239C2×C22⋊C4—1631no—
240C2×C22⋊C4—1631no—
241C2×C22⋊C4—1631no—
242C2×C22⋊C4—1631no—
243C2×C42—3264no—
244C2×C42—1631no—
245C2×C42—1631no—
246C2×C42—1631no—
247C2×C42—1631no—
248C2×C42—1631no—
249C2×C42—1631no—
250C2×C42C2×D16311yescharacteristic
251#95—1631no—
252#95—1631no—
253#95—1631no—
254#95—1631no—
255#95—1631no—
256#95—1631no—
257#95C2×D16311yescharacteristic
258#95C2×D16311yescharacteristic
259#95C2×D16311yescharacteristic
260#95C2×D16311yescharacteristic
261C24—1631no—
262C24—1631no—
263C2×D4—3262no—
264C2×D4—3262no—
265C2×D4—1634no—
266C22×C4—3262no—
267C22×C4—3262no—
268C22×C4—3262no—
269C22×C4—3262no—
270C22×C4—3262no—
271C22×C4—3262no—
272C22×C4—3262no—
273C22×C4—3262no—
274C22×C4—3262no—
275C22×C4—3262no—
276C22×C4—1631no—
277C22×C4—1631no—
278C22×C4—1631no—
279C22×C4—1631no—
280C22×C4—1631no—
281C22×C4—1634no—
282C22×C4—1634no—
283C22×C4—1631no—
284C22×C4C22×D16311yescharacteristic
285C22×C4C22×D16311yescharacteristic
286C22×C4C22×D16311yescharacteristic
287C22×C4C22×D16311yescharacteristic
288C22×C4C22×D16311yescharacteristic
289C22×C4C22×D16311yescharacteristic
290C22×C4C22×D16311yescharacteristic
291C4⋊C4—22no—
292C4⋊C4—3262no—
293C4⋊C4—3262no—
294C4⋊C4—1634no—
295C4⋊C4C4×D16314yescomplemented
296C22⋊C4—3262no—
297C22⋊C4—3262no—
298C22⋊C4—3262no—
299C22⋊C4—3262no—
300C22⋊C4—3262no—
301C22⋊C4—1634no—
302C22⋊C4—1634no—
303C22⋊C4—1634no—
304C42—22no—
305C42—3262no—
306C42—3262no—
307C42—3262no—
308C42—3262no—
309C42—3262no—
310C42—3262no—
311C23—3261no—
312C23—3261no—
313C23—3261no—
314C23—3261no—
315C23—3261no—
316C23—3261no—
317C23—3261no—
318C23—3261no—
319C23—3261no—
320C23—3261no—
321C23—1632no—
322C23—1632no—
323C23—1632no—
324C23—1632no—
325C23C23×D16311yescharacteristic
326D4—3268no—
327C2×C4—21no—
328C2×C4—21no—
329C2×C4—21no—
330C2×C4—21no—
331C2×C4—21no—
332C2×C4—21no—
333C2×C4—21no—
334C2×C4—21no—
335C2×C4—21no—
336C2×C4—21no—
337C2×C4—21no—
338C2×C4—21no—
339C2×C4—21no—
340C2×C4—21no—
341C2×C4—21no—
342C2×C4—21no—
343C2×C4—3268no—
344C2×C4—3268no—
345C2×C4—3268no—
346C2×C4—3261no—
347C2×C4—3261no—
348C2×C4—3261no—
349C2×C4—3261no—
350C2×C4—3261no—
351C2×C4—3261no—
352C2×C4—3261no—
353C2×C4—3261no—
354C2×C4—3261no—
355C2×C4—3261no—
356C2×C4—3261no—
357C2×C4—3261no—
358C2×C4—6524no—
359C2×C4—1632no—
360C2×C4—1632no—
361C2×C4—1632no—
362C2×C4—1632no—
363C2×C4—1632no—
364C2×C4—1632no—
365C2×C4C2×C4×D16312yes—
366C2×C4C2×C4×D16312yes—
367C2×C4C2×C4×D16312yes—
368C2×C4D4×D16312yes—
369C2×C4#1457970012yes—
370C22—3262no—
371C22—3262no—
372C22—3262no—
373C22—3262no—
374C22—3262no—
375C22—3262no—
376C22—3262no—
377C22—3262no—
378C22—3262no—
379C22—6521no—
380C22—6521no—
381C22—6521no—
382C22—6521no—
383C22—1634no—
384C22C22×C4×D16311yescharacteristic
385C22C2×D4×D16311yescharacteristic
386C22#41×D16311yescharacteristic
387C22#41×D16311yescharacteristic
388C22#41×D16311yescharacteristic
389C22#41×D16311yescharacteristic
390C22C2×#1457970011yescharacteristic
391C4—22no—
392C4—22no—
393C4—22no—
394C4—22no—
395C4—22no—
396C4—41no—
397C4—41no—
398C4—3262no—
399C4—3262no—
400C4—3262no—
401C4—6521no—
402C4—6521no—
403C4—1634no—
404C2—3264no—
405C2—6522no—
406C2C4×D4×D16311yescharacteristic
407C2#2378213311yescharacteristic
408C2C4×#1457970011yescharacteristic
409C2#123×D16311yescharacteristic
410C2#2378217811yescharacteristic
411C2#2378219811yescharacteristic
412C2#2378220011yescharacteristic
subgroupG/H
center, FrattiniC23C23×D163
commutatorC22×C163C23×C4
Fitting#387×C163C2
socle, cosocleC23×C163C24
prime radicalC22×D652C22
hypercenter#387D163
hypocenterC163C4×#123
LCSG ▷ C22×C163 ▷ C163
UCSC1 ◁ C23 ◁ #387

Complements

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

NH ≅ G/Ntype
3. C2×C42×D163C2semidirect
4. #23781557C2semidirect
5. #23781556C2semidirect
6. #23781546C2semidirect
8. #23781539C2semidirect
9. #23781527C2semidirect
10. C4⋊C4×Dic163C2semidirect
11. #23781516C2semidirect
12. #23781512C2semidirect
13. #23781512C2semidirect
14. #23781512C2semidirect
15. #387×C163C2semidirect
21. #19104720C4semidirect
75. C2×C4×D163C2×C4semidirect
79. Dic163⋊C4C2×C4semidirect
85. (C2×D163)⋊C4C2×C4semidirect
86. (C2×D163)⋊C4C2×C4semidirect
87. (C2×D163)⋊C4C2×C4semidirect
102. C4⋊C4×C163C2×C4semidirect
196. Dic163#123semidirect
206. C163C4×#123semidirect
214. #387D163semidirect
295. C4⋊C4C4×D163semidirect

Maximal subgroups

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

subgroupindex
C2×#191047202
C2×#191046972
C2×C42×D1632
#237815572
#237815562
#237815462
#237815432
#237815392
#237815272
C4⋊C4×Dic1632
#237815162
#237815122
#387×C1632
C4×#123163