Finite Group Databasebeta

← #35225219

Subgroups — #35225219

Order 20864.

374 conjugacy classes of subgroups.

The subgroup lattice has 358 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23782286C211yesmaximal, characteristic, complemented
2#23782270C211yesmaximal, characteristic, complemented
3#23782074C211yesmaximal, characteristic, complemented
4#23782054C211yesmaximal, characteristic, complemented
5#23782020C211yesmaximal, characteristic, complemented
6#23782012C211yesmaximal, characteristic, complemented
7#23781918C211yesmaximal, characteristic, complemented
8#23781911C211yesmaximal, characteristic, complemented
9#23781901C211yesmaximal, characteristic, complemented
10#23781894C211yesmaximal, characteristic, complemented
11#23781778C211yesmaximal, characteristic, complemented
12#23781767C211yesmaximal, characteristic, complemented
13C4×#14579688C211yesmaximal, characteristic, complemented
14C4×D1304C211yesmaximal, characteristic, complemented
15#405×C163C211yesmaximal, characteristic, complemented
16C2×#14579688C2211yescharacteristic
17C2×#14579688C2211yescharacteristic
18C2×D1304C2211yescharacteristic
19C2×D1304C2211yescharacteristic
20#19104735C2211yescharacteristic
21#19104732C2211yescharacteristic
22#19104730—21no—
23#19104727C2211yescharacteristic
24#19104723C2211yescharacteristic
25#19104721—21no—
26#19104720—21no—
27#19104720—21no—
28#19104717—21no—
29#19104714C2211yescharacteristic
30#19104713—21no—
31#19104709C2211yescharacteristic
32#19104709C2211yescharacteristic
33#19104709C2211yescharacteristic
34#19104708—21no—
35#19104708—21no—
36C4×Dic326C2211yescharacteristic
37C4×D652C2211yescharacteristic
38C4×D652C2211yescharacteristic
39C4×D652C2211yescharacteristic
40#19104669C2211yescharacteristic
41#19104669C2211yescharacteristic
42#19104667C2211yescharacteristic
43#19104667C2211yescharacteristic
44#19104660C2211yescharacteristic
45#19104660C2211yescharacteristic
46#19104657C2211yescharacteristic
47#19104657C2211yescharacteristic
48#19104657C2211yescharacteristic
49#19104657C2211yescharacteristic
50#19104657C2211yescharacteristic
51#19104657C2211yescharacteristic
52C2×M16×C163C2211yescharacteristic
53C22×C8×C163C2211yescharacteristic
54#117×C163C2211yescharacteristic
55#97×C163C2211yescharacteristic
56#97×C163C2211yescharacteristic
57C4×C8×C163C2211yescharacteristic
58C4×C8×C163C2211yescharacteristic
59C2×Dic326—21no—
60C2×Dic326C2311yescharacteristic
61C2×D652—21no—
62C2×D652—21no—
63C2×D652C2311yescharacteristic
64C2×D652C2311yescharacteristic
65C2×D652C2311yescharacteristic
66C2×C22⋊D163—21no—
67C2×C22⋊D163—21no—
68C2×C22⋊D163—21no—
69C2×C4×D163—21no—
70C2×C4×D163—21no—
71C2×C4×D163—21no—
72#14579688—22no—
73#14579688—22no—
74D1304—22no—
75D1304—22no—
76C4⋊Dic163—21no—
77C4⋊Dic163C2311yescharacteristic
78C4⋊Dic163C2311yescharacteristic
79C4⋊Dic163C2311yescharacteristic
80C4⋊Dic163C2311yescharacteristic
81Dic163⋊C4—21no—
82Dic163⋊C4—21no—
83Dic163⋊C4—21no—
84Dic163⋊C4—21no—
85C22⋊Dic163—21no—
86(C2×D163)⋊C4—21no—
87(C2×D163)⋊C4—21no—
88(C2×D163)⋊C4—21no—
89(C2×D163)⋊C4—21no—
90(C2×D163)⋊C4—21no—
91(C2×D163)⋊C4—21no—
92(C2×D163)⋊C4—21no—
93(C2×D163)⋊C4—21no—
94C4×Dic163—21no—
95C22×C4×C163C2311yescharacteristic
96M16×C163—22no—
97C2×C8×C163—22no—
98C2×C8×C163C2311yescharacteristic
99C2×C8×C163C2311yescharacteristic
100C2×C8×C163C2311yescharacteristic
101C2×C8×C163C2311yescharacteristic
102C4⋊C4×C163D411yescharacteristic
103C4⋊C4×C163D411yescharacteristic
104C22⋊C4×C163D411yescharacteristic
105C22⋊C4×C163D411yescharacteristic
106C42×C163C2311yescharacteristic
107C42×C163C2311yescharacteristic
108C22×D163—21no—
109C22×D163—21no—
110C22×D163—21no—
111Dic326—21no—
112Dic326—21no—
113Dic326—81no—
114D652—21no—
115D652—21no—
116D652—21no—
117D652—21no—
118D652—21no—
119D652—21no—
120D652—81no—
121D652—81no—
122C22⋊D163—81no—
123C22⋊D163—81no—
124C22⋊D163—81no—
125C2×Dic163—21no—
126C2×Dic163—21no—
127C2×Dic163—21no—
128C2×Dic163—21no—
129C2×Dic163—21no—
130C4×D163—81no—
131C4×D163—81no—
132C4×D163—81no—
133C23×C163C2×D411yescharacteristic
134C2×C4×C163—21no—
135C2×C4×C163—21no—
136C2×C4×C163C2×D411yescharacteristic
137C2×C4×C163C2×D411yescharacteristic
138C2×C4×C163C2×D411yescharacteristic
139C2×C4×C163C2×D411yescharacteristic
140C2×C4×C163C2×D411yescharacteristic
141C2×C4×C163C2411yescharacteristic
142C8×C163—21no—
143C8×C163—21no—
144C8×C163#4112yes—
145C8×C163#4112yes—
146C22×C163—21no—
147C22×C163—21no—
148C22×C163—21no—
149C22×C163C22×D411yescharacteristic
150C2×D163—41no—
151C2×D163—41no—
152C2×D163—41no—
153C2×D163—41no—
154C2×D163—41no—
155C2×D163—41no—
156C2×D163—41no—
157C2×D163—41no—
158C2×D163—41no—
159C4×C163—21no—
160C4×C163—21no—
161C4×C163—21no—
162C4×C163—21no—
163C4×C163—21no—
164C4×C163C2×#4111yescharacteristic
165C4×C163C2×#4111yescharacteristic
166Dic163—41no—
167Dic163—41no—
168Dic163—41no—
169Dic163—41no—
170Dic163—41no—
171C2×C163—41no—
172C2×C163#53211yescharacteristic
173C2×C163#57611yescharacteristic
174C2×C163#57711yescharacteristic
175D163—81no—
176D163—81no—
177D163—81no—
178C163#317911yescharacteristic, complemented, Sylow
179#3179—1631nomaximal, Sylow
180#542—1631no—
181#540—1631no—
182#501—1631no—
183#496—1631no—
184#488—1631no—
185#486—1631no—
186#469—1631no—
187#468—1631no—
188#466—1631no—
189#465—1631no—
190#442—1631no—
191#440—1631no—
192C4×SD16—1631no—
193C4×D8—1631no—
194#405D16311yescharacteristic, complemented
195C2×SD16—1631no—
196C2×SD16—1631no—
197C2×D8—1631no—
198C2×D8—1631no—
199C2×M16C2×D16311yescharacteristic
200C22×C8C2×D16311yescharacteristic
201C4⋊Q8—1631no—
202#127—1631no—
203#126—3261no—
204#125—1631no—
205#124—3261no—
206#124—1631no—
207#123—3261no—
208#123—3261no—
209#123—3261no—
210C22⋊Q8—3261no—
211C22⋊Q8—1631no—
212#121—3261no—
213#121—3261no—
214#121—1631no—
215#121—1631no—
216#121—1631no—
217C4×Q8—1631no—
218C4×D4—1631no—
219C4×D4—1631no—
220C4×D4—1631no—
221#117C2×D16311yescharacteristic
222#107—1631no—
223#107—1631no—
224#106—1631no—
225#106—1631no—
226#103—1631no—
227#103—1631no—
228#102—1631no—
229#102—1631no—
230#102—1631no—
231#102—1631no—
232#102—1631no—
233#102—1631no—
234#97C2×D16311yescharacteristic
235#97C2×D16311yescharacteristic
236C4×C8C2×D16311yescharacteristic
237C4×C8C2×D16311yescharacteristic
238C2×Q8—3261no—
239C2×Q8—1631no—
240C2×D4—3261no—
241C2×D4—3261no—
242C2×D4—3261no—
243C2×D4—3261no—
244C2×D4—3261no—
245C2×D4—1631no—
246C2×D4—1631no—
247C2×D4—1631no—
248C22×C4—3261no—
249C22×C4—3261no—
250C22×C4—3261no—
251C22×C4C22×D16311yescharacteristic
252SD16—3262no—
253SD16—3262no—
254D8—3262no—
255D8—3262no—
256M16—22no—
257C2×C8—22no—
258C2×C8C22×D16311yescharacteristic
259C2×C8C22×D16311yescharacteristic
260C2×C8C22×D16311yescharacteristic
261C2×C8C22×D16311yescharacteristic
262C4⋊C4—3261no—
263C4⋊C4—3261no—
264C4⋊C4—3261no—
265C4⋊C4—3261no—
266C4⋊C4—3261no—
267C4⋊C4—1631no—
268C4⋊C4—1631no—
269C4⋊C4—1631no—
270C4⋊C4—1631no—
271C4⋊C4D65211yescharacteristic
272C4⋊C4D65211yescharacteristic
273C22⋊C4—3261no—
274C22⋊C4—3261no—
275C22⋊C4—3261no—
276C22⋊C4—3261no—
277C22⋊C4—3261no—
278C22⋊C4—3261no—
279C22⋊C4—3261no—
280C22⋊C4—3261no—
281C22⋊C4—3261no—
282C22⋊C4D65211yescharacteristic
283C22⋊C4D65211yescharacteristic
284C42—3261no—
285C42C22×D16311yescharacteristic
286C42C22×D16311yescharacteristic
287C23—3261no—
288C23—3261no—
289C23—3261no—
290C23C2×D65211yescharacteristic
291Q8—13041no—
292Q8—3261no—
293Q8—3261no—
294D4—13041no—
295D4—13041no—
296D4—13041no—
297D4—13041no—
298D4—13041no—
299D4—3261no—
300D4—3261no—
301D4—3261no—
302D4—3261no—
303D4—3261no—
304D4—3261no—
305C2×C4—21no—
306C2×C4—21no—
307C2×C4—13041no—
308C2×C4—13041no—
309C2×C4—13041no—
310C2×C4—3261no—
311C2×C4—3261no—
312C2×C4—3261no—
313C2×C4—3261no—
314C2×C4—3261no—
315C2×C4C2×D65211yescharacteristic
316C2×C4C2×D65211yescharacteristic
317C2×C4C2×D65211yescharacteristic
318C2×C4C2×D65211yescharacteristic
319C2×C4C2×D65211yescharacteristic
320C2×C4C23×D16311yescharacteristic
321C8—21no—
322C8—21no—
323C8#1457969912yes—
324C8#1457969912yes—
325C22—21no—
326C22—21no—
327C22—21no—
328C22—6521no—
329C22—6521no—
330C22—6521no—
331C22—6521no—
332C22—6521no—
333C22—6521no—
334C22—6521no—
335C22—6521no—
336C22—6521no—
337C22C22×D65211yescharacteristic
338C4—21no—
339C4—21no—
340C4—21no—
341C4—21no—
342C4—21no—
343C4—6521no—
344C4—6521no—
345C4—6521no—
346C4—6521no—
347C4—6521no—
348C4C2×#1457969911yescharacteristic
349C4C2×#1457969911yescharacteristic
350C2—41no—
351C2—13041no—
352C2—13041no—
353C2—13041no—
354C2#2378221511yescharacteristic
355C2#2378247011yescharacteristic
356C2#2378247711yescharacteristic
subgroupG/H
centerC22C22×D652
commutator, cosocleC2×C4×C163C24
FrattiniC2×C4C23×D163
Fitting#405×C163C2
socleC22×C163C22×D4
hypercenter#405D163
hypocenterC163#3179
LCSG ▷ C2×C4×C163 ▷ C2×C163 ▷ C163
UCSC1 ◁ C22 ◁ #117 ◁ #405

Complements

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

NH ≅ G/Ntype
1. #23782286C2semidirect
2. #23782270C2semidirect
3. #23782074C2semidirect
4. #23782054C2semidirect
5. #23782020C2semidirect
6. #23782012C2semidirect
7. #23781918C2semidirect
8. #23781911C2semidirect
9. #23781901C2semidirect
10. #23781894C2semidirect
11. #23781778C2semidirect
12. #23781767C2semidirect
13. C4×#14579688C2semidirect
14. C4×D1304C2semidirect
15. #405×C163C2semidirect
178. C163#3179semidirect
194. #405D163semidirect

Maximal subgroups

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

subgroupindex
#237822862
#237822702
#237820742
#237820542
#237820202
#237820122
#237819182
#237819112
#237819012
#237818942
#237817782
#237817672
C4×#145796882
C4×D13042
#405×C1632
#3179163