Finite Group Databasebeta

← #35226589

Subgroups — #35226589

Order 20864.

376 conjugacy classes of subgroups.

The subgroup lattice has 350 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23782355C211yesmaximal, characteristic
2D4×Dic326C211yesmaximal, characteristic
3#23782034C211yesmaximal, characteristic, complemented
4#23782007C211yesmaximal, characteristic, complemented
5#23781996C211yesmaximal, characteristic, complemented
6#23781973C211yesmaximal, characteristic, complemented
7#23781923C211yesmaximal, characteristic, complemented
8#23781906C211yesmaximal, characteristic, complemented
9#23781876C211yesmaximal, characteristic, complemented
10#23781823C211yesmaximal, characteristic, complemented
11#23781823C211yesmaximal, characteristic, complemented
12#23781773C211yesmaximal, characteristic, complemented
13#23781675C211yesmaximal, characteristic, complemented
14C2×#19104660C211yesmaximal, characteristic, complemented
15#435×C163C211yesmaximal, characteristic
16C22×Dic326C2211yescharacteristic
17C22×Dic326C2211yescharacteristic
18C2×Dic652C2211yescharacteristic
19C2×Dic652C2211yescharacteristic
20C2×Dic652C2211yescharacteristic
21#19104735C2211yescharacteristic
22#19104734—21no—
23#19104728—21no—
24#19104727C2211yescharacteristic
25#19104725—21no—
26#19104716—21no—
27#19104714C2211yescharacteristic
28#19104714C2211yescharacteristic
29#19104714C2211yescharacteristic
30#19104714C2211yescharacteristic
31#19104712—21no—
32#19104712—21no—
33C4×Dic326C2211yescharacteristic
34C4×Dic326C2211yescharacteristic
35D4×Dic163—21no—
36#19104694C2211yescharacteristic
37#19104691—21no—
38C2×C4⋊Dic163C2211yescharacteristic
39#19104669C2211yescharacteristic
40#19104667C2211yescharacteristic
41#19104667C2211yescharacteristic
42#19104660C2211yescharacteristic
43#19104660C2211yescharacteristic
44#19104660C2211yescharacteristic
45#19104660C2211yescharacteristic
46#19104660C2211yescharacteristic
47#19104660C2211yescharacteristic
48#19104660C2211yescharacteristic
49#19104660C2211yescharacteristic
50#19104660C2211yescharacteristic
51#19104660C2211yescharacteristic
52C2×M16×C163C2211yescharacteristic
53C22×C8×C163C2211yescharacteristic
54C4×D4×C163C2211yescharacteristic
55C4⋊C8×C163C2211yescharacteristic
56C22⋊C8×C163C2211yescharacteristic
57C22⋊C8×C163C2211yescharacteristic
58#97×C163C2211yescharacteristic
59C2×Dic326—22no—
60C2×Dic326—22no—
61C2×Dic326—21no—
62C2×Dic326—21no—
63C2×Dic326—41no—
64C2×Dic326—41no—
65C2×Dic326C2311yescharacteristic
66C2×Dic326C2311yescharacteristic
67C2×Dic326C2311yescharacteristic
68C2×Dic326C2311yescharacteristic
69C22×Dic163—21no—
70C22×Dic163—21no—
71C22×Dic163—21no—
72Dic652—22no—
73Dic652—41no—
74Dic652—41no—
75C4⋊Dic163—21no—
76C4⋊Dic163—21no—
77C4⋊Dic163—21no—
78C4⋊Dic163C2311yescharacteristic
79C4⋊Dic163C2311yescharacteristic
80C4⋊Dic163C2311yescharacteristic
81C4⋊Dic163C2311yescharacteristic
82Dic163⋊C4—21no—
83Dic163⋊C4—21no—
84Dic163⋊C4—21no—
85Dic163⋊C4—21no—
86Dic163⋊C4—21no—
87Dic163⋊C4—21no—
88Dic163⋊C4—21no—
89Dic163⋊C4—21no—
90C22⋊Dic163—21no—
91C22⋊Dic163—21no—
92C22⋊Dic163—21no—
93C22⋊Dic163—21no—
94C22⋊Dic163—21no—
95C4×Dic163—21no—
96C4×Dic163—21no—
97C4×Dic163—21no—
98C2×D4×C163D411yescharacteristic
99C22×C4×C163C2311yescharacteristic
100C22×C4×C163C2311yescharacteristic
101M16×C163—41no—
102C2×C8×C163—41no—
103C2×C8×C163C2311yescharacteristic
104C2×C8×C163C2311yescharacteristic
105C2×C8×C163C2311yescharacteristic
106C2×C8×C163C2311yescharacteristic
107C4⋊C4×C163D411yescharacteristic
108C22⋊C4×C163D411yescharacteristic
109C22⋊C4×C163D411yescharacteristic
110C42×C163C2311yescharacteristic
111Dic326—21no—
112Dic326—21no—
113Dic326—21no—
114Dic326—21no—
115Dic326—21no—
116Dic326—21no—
117Dic326—21no—
118Dic326—42no—
119Dic326—42no—
120Dic326#4112yes—
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—42no—
130C2×Dic163—42no—
131C2×Dic163—42no—
132C23×C163C2×D411yescharacteristic
133C23×C163C2×D411yescharacteristic
134D4×C163—22no—
135C2×C4×C163—21no—
136C2×C4×C163—21no—
137C2×C4×C163—21no—
138C2×C4×C163—21no—
139C2×C4×C163C2×D411yescharacteristic
140C2×C4×C163C2×D411yescharacteristic
141C2×C4×C163C2×D411yescharacteristic
142C2×C4×C163C2×D411yescharacteristic
143C2×C4×C163C2411yescharacteristic
144C8×C163—21no—
145C8×C163—21no—
146C8×C163—21no—
147C8×C163—21no—
148C22×C163—21no—
149C22×C163—21no—
150C22×C163—21no—
151C22×C163—21no—
152C22×C163—21no—
153C22×C163#13712yes—
154C22×C163C22×D411yescharacteristic
155C4×C163—21no—
156C4×C163—21no—
157C4×C163—21no—
158C4×C163—21no—
159C4×C163C2×#4111yescharacteristic
160C4×C163H(5,𝔽2)11yescharacteristic
161Dic163—22no—
162Dic163—41no—
163Dic163—41no—
164Dic163—41no—
165Dic163—41no—
166Dic163—41no—
167Dic163—41no—
168Dic163—41no—
169C2×C163—22no—
170C2×C163—41no—
171C2×C163#54611yescharacteristic
172C2×C163C2×#13711yescharacteristic
173C2×C163#57811yescharacteristic
174C163#331211yescharacteristic, complemented, Sylow
175#3312—1631nomaximal, Sylow
176#552—1631no—
177D4×Q8—1631no—
178#491—1631no—
179#485—1631no—
180#483—1631no—
181#479—1631no—
182#470—1631no—
183#467—1631no—
184#462—1631no—
185#451—1631no—
186#451—1631no—
187#441—1631no—
188#435D16311yescharacteristic
189#419—1631no—
190C2×#103—1631no—
191C22×Q8—1631no—
192C22×Q8—1631no—
193C2×Q16—1631no—
194C2×Q16—1631no—
195C2×Q16—1631no—
196C2×M16C2×D16311yescharacteristic
197C22×C8C2×D16311yescharacteristic
198C4⋊Q8—3261no—
199C4⋊Q8—1631no—
200#126—3261no—
201#125—3261no—
202#125—1631no—
203#123—3261no—
204C22⋊Q8—3261no—
205C22⋊Q8—3261no—
206C22⋊Q8—1631no—
207C22⋊Q8—1631no—
208C22⋊Q8—1631no—
209C22⋊Q8—1631no—
210C4×Q8—1631no—
211C4×Q8—1631no—
212C4×D4—3261no—
213C4×D4C2×D16311yescharacteristic
214#117—3261no—
215#117—1631no—
216C2×C4⋊C4—1631no—
217#107—1631no—
218#106—1631no—
219#106—1631no—
220C4⋊C8C2×D16311yescharacteristic
221#103—1631no—
222#103—1631no—
223#103—1631no—
224#103—1631no—
225#103—1631no—
226#103—1631no—
227#103—1631no—
228#103—1631no—
229#103—1631no—
230#103—1631no—
231C22⋊C8C2×D16311yescharacteristic
232C22⋊C8C2×D16311yescharacteristic
233#97C2×D16311yescharacteristic
234C2×Q8—3262no—
235C2×Q8—3262no—
236C2×Q8—3261no—
237C2×Q8—3261no—
238C2×Q8—6521no—
239C2×Q8—6521no—
240C2×Q8—1631no—
241C2×Q8—1631no—
242C2×Q8—1631no—
243C2×Q8—1631no—
244C2×D4D65211yescharacteristic
245C22×C4—3261no—
246C22×C4—3261no—
247C22×C4—3261no—
248C22×C4C22×D16311yescharacteristic
249C22×C4C22×D16311yescharacteristic
250Q16—3262no—
251Q16—6521no—
252Q16—6521no—
253M16—41no—
254C2×C8—41no—
255C2×C8C22×D16311yescharacteristic
256C2×C8C22×D16311yescharacteristic
257C2×C8C22×D16311yescharacteristic
258C2×C8C22×D16311yescharacteristic
259C4⋊C4—3261no—
260C4⋊C4—3261no—
261C4⋊C4—3261no—
262C4⋊C4—3261no—
263C4⋊C4—3261no—
264C4⋊C4—3261no—
265C4⋊C4—3261no—
266C4⋊C4—3261no—
267C4⋊C4—3261no—
268C4⋊C4—3261no—
269C4⋊C4—3261no—
270C4⋊C4—1631no—
271C4⋊C4—1631no—
272C4⋊C4—1631no—
273C4⋊C4—1631no—
274C4⋊C4D65211yescharacteristic
275C22⋊C4—3261no—
276C22⋊C4—3261no—
277C22⋊C4—3261no—
278C22⋊C4—3261no—
279C22⋊C4—3261no—
280C22⋊C4D65211yescharacteristic
281C22⋊C4D65211yescharacteristic
282C42—3261no—
283C42—3261no—
284C42—3261no—
285C42C22×D16311yescharacteristic
286C23C2×D65211yescharacteristic
287C23C2×D65211yescharacteristic
288Q8—3261no—
289Q8—3261no—
290Q8—3261no—
291Q8—3261no—
292Q8—3261no—
293Q8—3261no—
294Q8—3261no—
295Q8—6522no—
296Q8—6522no—
297Q8—1632no—
298D4—22no—
299C2×C4—21no—
300C2×C4—21no—
301C2×C4—21no—
302C2×C4—21no—
303C2×C4—3261no—
304C2×C4—3261no—
305C2×C4—3261no—
306C2×C4—3261no—
307C2×C4—3261no—
308C2×C4—3261no—
309C2×C4—3261no—
310C2×C4—3261no—
311C2×C4—6522no—
312C2×C4—6522no—
313C2×C4—6522no—
314C2×C4C2×D65211yescharacteristic
315C2×C4C2×D65211yescharacteristic
316C2×C4C2×D65211yescharacteristic
317C2×C4C2×D65211yescharacteristic
318C2×C4C23×D16311yescharacteristic
319C8—21no—
320C8—21no—
321C8—21no—
322C8—21no—
323C22—21no—
324C22—21no—
325C22—21no—
326C22—21no—
327C22—21no—
328C22#1910476812yes—
329C22C22×D65211yescharacteristic
330C4—21no—
331C4—21no—
332C4—21no—
333C4—21no—
334C4—3262no—
335C4—6521no—
336C4—6521no—
337C4—6521no—
338C4—6521no—
339C4—6521no—
340C4—6521no—
341C4—6521no—
342C4#41×D16311yescharacteristic
343C4#1910478011yescharacteristic
344C2—22no—
345C2—41no—
346C2#2378231011yescharacteristic
347C2C2×#1910476811yescharacteristic
348C2#2378248211yescharacteristic
subgroupG/H
centerC22C22×D652
commutator, cosocleC2×C4×C163C24
FrattiniC2×C4C23×D163
Fitting#435×C163C2
socleC22×C163C22×D4
prime radicalC2×D4×C163D4
hypercenter#435D163
hypocenterC163#3312
LCSG ▷ C2×C4×C163 ▷ C2×C163 ▷ C163
UCSC1 ◁ C22 ◁ C4×D4 ◁ #435

Complements

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

NH ≅ G/Ntype
3. #23782034C2semidirect
4. #23782007C2semidirect
5. #23781996C2semidirect
6. #23781973C2semidirect
7. #23781923C2semidirect
8. #23781906C2semidirect
9. #23781876C2semidirect
10. #23781823C2semidirect
11. #23781823C2semidirect
12. #23781773C2semidirect
13. #23781675C2semidirect
14. C2×#19104660C2semidirect
174. C163#3312semidirect

Maximal subgroups

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

subgroupindex
#237823552
D4×Dic3262
#237820342
#237820072
#237819962
#237819732
#237819232
#237819062
#237818762
#237818232
#237817732
#237816752
C2×#191046602
#435×C1632
#3312163