Finite Group Databasebeta

← #35222871

Subgroups — #35222871

Order 20864.

564 conjugacy classes of subgroups.

The subgroup lattice has 376 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#19104713C211yesmaximal, characteristic
2C2×C42×D163C211yesmaximal, characteristic
3#23781569C211yesmaximal, characteristic, complemented
4#23781569C211yesmaximal, characteristic, complemented
5#23781562C211yesmaximal, characteristic, complemented
6#23781560C211yesmaximal, characteristic, complemented
7#23781550C211yesmaximal, characteristic, complemented
8#23781549C211yesmaximal, characteristic, complemented
9#23781546C211yesmaximal, characteristic, complemented
10#23781539C211yesmaximal, characteristic
11#23781536C211yesmaximal, characteristic
12#23781527C211yesmaximal, characteristic
13#23781527C211yesmaximal, characteristic
14#23781524C211yesmaximal, characteristic
15#387×C163C211yesmaximal, characteristic, complemented
16C22×Dic326C2211yescharacteristic
17C22×C4×D163C2211yescharacteristic
18C22×C4×D163C2211yescharacteristic
19C22×C4×D163C2211yescharacteristic
20#19104713—24no—
21C2×C4⋊Dic163C2211yescharacteristic
22C2×Dic163⋊C4C2211yescharacteristic
23C2×Dic163⋊C4C2211yescharacteristic
24C2×Dic163⋊C4C2211yescharacteristic
25C2×Dic163⋊C4C2211yescharacteristic
26C2×Dic163⋊C4C2211yescharacteristic
27C2×Dic163⋊C4C2211yescharacteristic
28C2×(C2×D163)⋊C4C2211yescharacteristic
29C2×(C2×D163)⋊C4C2211yescharacteristic
30C2×(C2×D163)⋊C4C2211yescharacteristic
31C2×(C2×D163)⋊C4C2211yescharacteristic
32C2×C4×Dic163C2211yescharacteristic
33C2×C4×Dic163C2211yescharacteristic
34C2×C4×Dic163C2211yescharacteristic
35C42×D163—24no—
36C163⋊#95C2211yescharacteristic
37C163⋊#95C2211yescharacteristic
38C163⋊#95C2211yescharacteristic
39C163⋊#95C2211yescharacteristic
40C163⋊#95C2211yescharacteristic
41C163⋊#95C2211yescharacteristic
42C163⋊#95C2211yescharacteristic
43C163⋊#95C2211yescharacteristic
44C163⋊#95C2211yescharacteristic
45C163⋊#95C2211yescharacteristic
46C2×C4⋊C4×C163C2211yescharacteristic
47C2×C4⋊C4×C163C2211yescharacteristic
48C2×C42×C163C2211yescharacteristic
49#95×C163C2211yescharacteristic
50#95×C163C2211yescharacteristic
51#95×C163C2211yescharacteristic
52#95×C163C2211yescharacteristic
53C23×D163C2311yescharacteristic
54C2×Dic326—22no—
55C2×Dic326—22no—
56C2×Dic326—22no—
57C22×Dic163C2311yescharacteristic
58C22×Dic163C2311yescharacteristic
59C22×Dic163C2311yescharacteristic
60C22×Dic163C2311yescharacteristic
61C22×Dic163C2311yescharacteristic
62C22×Dic163C2311yescharacteristic
63C22×Dic163C2311yescharacteristic
64C2×C4×D163—22no—
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×D163D414yes—
72C2×C4×D163Q814yescomplemented
73C4⋊Dic163—22no—
74Dic163⋊C4—22no—
75Dic163⋊C4—22no—
76Dic163⋊C4—22no—
77Dic163⋊C4—22no—
78Dic163⋊C4—22no—
79Dic163⋊C4—22no—
80(C2×D163)⋊C4—22no—
81(C2×D163)⋊C4—22no—
82(C2×D163)⋊C4—22no—
83(C2×D163)⋊C4—22no—
84C4×Dic163—22no—
85C4×Dic163—22no—
86C4×Dic163—22no—
87C22×C4×C163C2311yescharacteristic
88C22×C4×C163C2311yescharacteristic
89C22×C4×C163C2311yescharacteristic
90C22×C4×C163C2311yescharacteristic
91C22×C4×C163C2311yescharacteristic
92C22×C4×C163C2311yescharacteristic
93C22×C4×C163C2311yescharacteristic
94C4⋊C4×C163—22no—
95C4⋊C4×C163—22no—
96C42×C163—22no—
97C22×D163—21no—
98C22×D163—21no—
99C22×D163—21no—
100C22×D163—21no—
101C22×D163C2×D412yes—
102C22×D163C2×Q812yes—
103C22×D163#4112yes—
104Dic326—44no—
105C2×Dic163—21no—
106C2×Dic163—21no—
107C2×Dic163—21no—
108C2×Dic163—21no—
109C2×Dic163—21no—
110C2×Dic163—21no—
111C2×Dic163—21no—
112C2×Dic163—21no—
113C2×Dic163—21no—
114C2×Dic163—21no—
115C2×Dic163—21no—
116C2×Dic163—21no—
117C2×Dic163—21no—
118C2×Dic163—21no—
119C2×Dic163—21no—
120C2×Dic163—21no—
121C2×Dic163C2×D412yes—
122C2×Dic163C2×Q812yes—
123C2×Dic163#4112yes—
124C2×Dic163#4112yes—
125C2×Dic163#4112yes—
126C4×D163—28no—
127C4×D163—28no—
128C4×D163—28no—
129C23×C163C2411yescharacteristic
130C2×C4×C163—21no—
131C2×C4×C163—21no—
132C2×C4×C163—21no—
133C2×C4×C163—21no—
134C2×C4×C163—21no—
135C2×C4×C163—21no—
136C2×C4×C163—21no—
137C2×C4×C163—21no—
138C2×C4×C163—21no—
139C2×C4×C163—21no—
140C2×C4×C163—21no—
141C2×C4×C163—21no—
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×C163C2×D412yes—
149C2×C4×C163C2×Q812yes—
150C2×C4×C163#4112yes—
151C22×C163C22×D411yescharacteristic
152C22×C163C22×Q811yescharacteristic
153C22×C163C2×#4111yescharacteristic
154C22×C163C2×#4111yescharacteristic
155C22×C163C2×#4111yescharacteristic
156C22×C163C2×#4111yescharacteristic
157C22×C163C2×#4111yescharacteristic
158C2×D163—22no—
159C2×D163—22no—
160C2×D163—22no—
161C2×D163—22no—
162C2×D163—22no—
163C2×D163—22no—
164C2×D163C22⋊Q814yes—
165C4×C163—22no—
166C4×C163—22no—
167C4×C163—22no—
168C4×C163—41no—
169C4×C163—41no—
170C4×C163—41no—
171C4×C163—41no—
172Dic163—22no—
173Dic163—22no—
174Dic163—41no—
175Dic163—41no—
176Dic163—41no—
177Dic163—41no—
178Dic163#12314yes—
179C2×C163C2×C22⋊Q811yescharacteristic
180C2×C163C2×#12311yescharacteristic
181C2×C163#52511yescharacteristic
182C2×C163#52911yescharacteristic
183C2×C163#52911yescharacteristic
184C2×C163#53311yescharacteristic
185C2×C163#53311yescharacteristic
186D163—24no—
187C163#285511yescharacteristic, complemented, Sylow
188#2855—1631nomaximal, Sylow
189C2×C22⋊Q8—1631no—
190C22×C42—1631no—
191#391—1631no—
192#391—1631no—
193#389—1631no—
194#389—1631no—
195#387—1631no—
196#387—1631no—
197#387—1631no—
198#387D16311yescharacteristic, complemented
199#385—1631no—
200#385—1631no—
201#381—1631no—
202#381—1631no—
203#380—1631no—
204C22×Q8—1631no—
205C23×C4—1631no—
206C23×C4—1631no—
207C23×C4—1631no—
208C22⋊Q8—3264no—
209C2×C4⋊C4—1631no—
210C2×C4⋊C4—1631no—
211C2×C4⋊C4—1631no—
212C2×C4⋊C4—1631no—
213C2×C4⋊C4—1631no—
214C2×C4⋊C4—1631no—
215C2×C4⋊C4—1631no—
216C2×C4⋊C4C2×D16311yescharacteristic
217C2×C4⋊C4C2×D16311yescharacteristic
218C2×C22⋊C4—1631no—
219C2×C22⋊C4—1631no—
220C2×C22⋊C4—1631no—
221C2×C22⋊C4—1631no—
222C2×C42—3264no—
223C2×C42—1631no—
224C2×C42—1631no—
225C2×C42—1631no—
226C2×C42C2×D16311yescharacteristic
227#95—1631no—
228#95—1631no—
229#95—1631no—
230#95—1631no—
231#95—1631no—
232#95—1631no—
233#95—1631no—
234#95—1631no—
235#95—1631no—
236#95—1631no—
237#95C2×D16311yescharacteristic
238#95C2×D16311yescharacteristic
239#95C2×D16311yescharacteristic
240#95C2×D16311yescharacteristic
241C24—1631no—
242C2×Q8—3262no—
243C2×Q8—3262no—
244C2×Q8—3262no—
245C22×C4—3262no—
246C22×C4—3262no—
247C22×C4—3262no—
248C22×C4—3262no—
249C22×C4—3262no—
250C22×C4—3262no—
251C22×C4—3262no—
252C22×C4—1631no—
253C22×C4—1631no—
254C22×C4—1631no—
255C22×C4—1631no—
256C22×C4—1631no—
257C22×C4—1631no—
258C22×C4—1634no—
259C22×C4—1634no—
260C22×C4—1631no—
261C22×C4C22×D16311yescharacteristic
262C22×C4C22×D16311yescharacteristic
263C22×C4C22×D16311yescharacteristic
264C22×C4C22×D16311yescharacteristic
265C22×C4C22×D16311yescharacteristic
266C22×C4C22×D16311yescharacteristic
267C22×C4C22×D16311yescharacteristic
268C4⋊C4—22no—
269C4⋊C4—22no—
270C4⋊C4—3262no—
271C4⋊C4—3262no—
272C4⋊C4—3262no—
273C4⋊C4—3262no—
274C4⋊C4—3262no—
275C4⋊C4—3262no—
276C4⋊C4—3262no—
277C22⋊C4—3262no—
278C22⋊C4—3262no—
279C22⋊C4—3262no—
280C22⋊C4—3262no—
281C42—22no—
282C42—3262no—
283C42—3262no—
284C42—3262no—
285C23—3261no—
286C23—3261no—
287C23—3261no—
288C23—3261no—
289C23—1632no—
290C23—1632no—
291C23—1632no—
292C23C23×D16311yescharacteristic
293Q8—6524no—
294C2×C4—21no—
295C2×C4—21no—
296C2×C4—21no—
297C2×C4—21no—
298C2×C4—21no—
299C2×C4—21no—
300C2×C4—21no—
301C2×C4—21no—
302C2×C4—21no—
303C2×C4—21no—
304C2×C4—21no—
305C2×C4—21no—
306C2×C4—21no—
307C2×C4—21no—
308C2×C4—21no—
309C2×C4—21no—
310C2×C4—21no—
311C2×C4—21no—
312C2×C4—3268no—
313C2×C4—3268no—
314C2×C4—3268no—
315C2×C4—3261no—
316C2×C4—3261no—
317C2×C4—3261no—
318C2×C4—3261no—
319C2×C4—3261no—
320C2×C4—3261no—
321C2×C4—3261no—
322C2×C4—3261no—
323C2×C4—3261no—
324C2×C4—3261no—
325C2×C4—3261no—
326C2×C4—3261no—
327C2×C4—3261no—
328C2×C4—3261no—
329C2×C4—3261no—
330C2×C4—3261no—
331C2×C4—1632no—
332C2×C4—1632no—
333C2×C4—1632no—
334C2×C4—1632no—
335C2×C4—1632no—
336C2×C4D4×D16312yes—
337C2×C4Q8×D16312yes—
338C2×C4#1457969912yes—
339C22—3262no—
340C22—3262no—
341C22—3262no—
342C22—3262no—
343C22—3262no—
344C22—3262no—
345C22—1634no—
346C22C2×D4×D16311yescharacteristic
347C22C2×Q8×D16311yescharacteristic
348C22#41×D16311yescharacteristic
349C22#41×D16311yescharacteristic
350C22#41×D16311yescharacteristic
351C22#41×D16311yescharacteristic
352C22C2×#1457969911yescharacteristic
353C4—22no—
354C4—22no—
355C4—22no—
356C4—41no—
357C4—41no—
358C4—41no—
359C4—41no—
360C4—3262no—
361C4—3262no—
362C4—6521no—
363C4—6521no—
364C4—6521no—
365C4—6521no—
366C4—1634no—
367C2—3264no—
368C2C22⋊Q8×D16311yescharacteristic
369C2#123×D16311yescharacteristic
370C2#2378217411yescharacteristic
371C2#2378219311yescharacteristic
372C2#2378219811yescharacteristic
373C2#2378221911yescharacteristic
374C2#2378222111yescharacteristic
subgroupG/H
center, FrattiniC23C23×D163
commutator, socle, cosocleC23×C163C24
Fitting#387×C163C2
prime radicalC23×D163C23
hypercenter#387D163
hypocenterC163#2855
LCSG ▷ C23×C163 ▷ C163
UCSC1 ◁ C23 ◁ #387

Complements

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

NH ≅ G/Ntype
3. #23781569C2semidirect
4. #23781569C2semidirect
5. #23781562C2semidirect
6. #23781560C2semidirect
7. #23781550C2semidirect
8. #23781549C2semidirect
9. #23781546C2semidirect
15. #387×C163C2semidirect
72. C2×C4×D163Q8semidirect
187. C163#2855semidirect
198. #387D163semidirect

Maximal subgroups

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

subgroupindex
C2×#191047132
C2×C42×D1632
#237815692
#237815622
#237815602
#237815502
#237815492
#237815462
#237815392
#237815362
#237815272
#237815242
#387×C1632
#2855163