Finite Group Databasebeta

← #35224764

Subgroups — #35224764

Order 20864.

370 conjugacy classes of subgroups.

The subgroup lattice has 338 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23782284C211yesmaximal, characteristic
2#23782141C211yesmaximal, characteristic, complemented
3#23782005C211yesmaximal, characteristic, complemented
4#23781987C211yesmaximal, characteristic
5#23781961C211yesmaximal, characteristic, complemented
6#23781950C211yesmaximal, characteristic, complemented
7#23781884C211yesmaximal, characteristic, complemented
8#23781819C211yesmaximal, characteristic, complemented
9#23781802C211yesmaximal, characteristic
10#23781769C211yesmaximal, characteristic, complemented
11#23781766C211yesmaximal, characteristic, complemented
12Q16×Dic163C211yesmaximal, characteristic, complemented
13#23781756C211yesmaximal, characteristic, complemented
14#23781725C211yesmaximal, characteristic, complemented
15#464×C163C211yesmaximal, characteristic, complemented
16C2×Q8×D163C2211yescharacteristic
17C2×#14579689C2211yescharacteristic
18C2×#14579687C2211yescharacteristic
19#19104735C2211yescharacteristic
20#19104728—21no—
21#19104721—21no—
22#19104719—21no—
23#19104712—21no—
24#19104711C2211yescharacteristic
25#19104706C2211yescharacteristic
26Q8×Dic163C2211yescharacteristic
27Q8×Dic163C2211yescharacteristic
28C4×Dic326—21no—
29C4×Dic326C2211yescharacteristic
30#19104700—21no—
31#19104700C2211yescharacteristic
32D4×Dic163C2211yescharacteristic
33#19104693—21no—
34#19104693C2211yescharacteristic
35#19104692—21no—
36#19104692C2211yescharacteristic
37#19104669C2211yescharacteristic
38#19104669C2211yescharacteristic
39#19104666C2211yescharacteristic
40#19104666C2211yescharacteristic
41C4⋊(C163⋊C8)C2211yescharacteristic
42#19104660C2211yescharacteristic
43#19104660C2211yescharacteristic
44#19104659C2211yescharacteristic
45#19104659C2211yescharacteristic
46#19104659C2211yescharacteristic
47#19104656C2211yescharacteristic
48(C2×D163)⋊C8C2211yescharacteristic
49(C2×D163)⋊C8C2211yescharacteristic
50#19104645C2211yescharacteristic
51C8×Dic163C2211yescharacteristic
52C2×Q16×C163C2211yescharacteristic
53C2×SD16×C163C2211yescharacteristic
54#124×C163C2211yescharacteristic
55C4×Q8×C163C2211yescharacteristic
56C4⋊C8×C163C2211yescharacteristic
57#103×C163C2211yescharacteristic
58#102×C163C2211yescharacteristic
59C2×Dic326—21no—
60C2×Dic326—21no—
61C2×Dic326C2311yescharacteristic
62Q8×D163—22no—
63Q8×D163—22no—
64C2×C22⋊D163—21no—
65C22×Dic163—21no—
66C2×C4×D163—21no—
67C2×C4×D163C2311yescharacteristic
68#14579689—22no—
69#14579687—22no—
70C2×C163⋊C8C2311yescharacteristic
71C2×C163⋊C8C2311yescharacteristic
72C4⋊Dic163—21no—
73C4⋊Dic163—21no—
74C4⋊Dic163—21no—
75C4⋊Dic163C2311yescharacteristic
76C4⋊Dic163C2311yescharacteristic
77C4⋊Dic163C2311yescharacteristic
78Dic163⋊C4—21no—
79Dic163⋊C4—21no—
80Dic163⋊C4—21no—
81Dic163⋊C4D411yescharacteristic
82Dic163⋊C4D411yescharacteristic
83C22⋊Dic163—21no—
84C22⋊Dic163—21no—
85(C2×D163)⋊C4—21no—
86(C2×D163)⋊C4—21no—
87(C2×D163)⋊C4D411yescharacteristic
88(C2×D163)⋊C4D411yescharacteristic
89C4×Dic163—21no—
90C4×Dic163—21no—
91C4×Dic163—21no—
92C4×Dic163—21no—
93C4×Dic163—21no—
94C4×Dic163C2311yescharacteristic
95C2×Q8×C163C2311yescharacteristic
96C2×Q8×C163C2311yescharacteristic
97C2×D4×C163C2311yescharacteristic
98Q16×C163—22no—
99SD16×C163—22no—
100C2×C8×C163C2311yescharacteristic
101C2×C8×C163C2311yescharacteristic
102C4⋊C4×C163—21no—
103C4⋊C4×C163C2311yescharacteristic
104C22⋊C4×C163—21no—
105C22⋊C4×C163—21no—
106C42×C163—21no—
107C42×C163C2311yescharacteristic
108C22×D163C2×D411yescharacteristic
109Dic326—24no—
110Dic326—21no—
111Dic326—81no—
112Dic326#4112yes—
113C22⋊D163—81no—
114C2×Dic163—21no—
115C2×Dic163—21no—
116C2×Dic163—21no—
117C2×Dic163—21no—
118C2×Dic163—81no—
119C2×Dic163C2×D411yescharacteristic
120C2×Dic163C2×D411yescharacteristic
121C2×Dic163C2×D411yescharacteristic
122C4×D163—24no—
123C4×D163—21no—
124C4×D163—21no—
125C163⋊C8—21no—
126C163⋊C8—21no—
127C23×C163—21no—
128Q8×C163—21no—
129Q8×C163—21no—
130Q8×C163—21no—
131Q8×C163#4112yes—
132D4×C163—21no—
133D4×C163—21no—
134C2×C4×C163—21no—
135C2×C4×C163—21no—
136C2×C4×C163—21no—
137C2×C4×C163C2×D411yescharacteristic
138C2×C4×C163C2×D411yescharacteristic
139C2×C4×C163C2411yescharacteristic
140C8×C163—21no—
141C8×C163—21no—
142C22×C163—41no—
143C22×C163—41no—
144C22×C163—41no—
145C22×C163C22×D411yescharacteristic
146C2×D163—21no—
147C2×D163—21no—
148C2×D163—21no—
149C4×C163—22no—
150C4×C163—21no—
151C4×C163—21no—
152C4×C163—41no—
153C4×C163—41no—
154C4×C163C2×#4111yescharacteristic
155C4×C163C2×#4111yescharacteristic
156Dic163—22no—
157Dic163—21no—
158Dic163—21no—
159Dic163—21no—
160Dic163—41no—
161Dic163—41no—
162Dic163—41no—
163C2×C163—81no—
164C2×C163#52511yescharacteristic
165C2×C163#57711yescharacteristic
166C2×C163#57811yescharacteristic
167D163—41no—
168C163#312111yescharacteristic, complemented, Sylow
169#3121—1631nomaximal, Sylow
170#542—1631no—
171#519—1631no—
172#485—1631no—
173#482—1631no—
174#477—1631no—
175#475—1631no—
176#464—1631no—
177#464D16311yescharacteristic, complemented
178#451—1631no—
179#448—1631no—
180#441—1631no—
181#440—1631no—
182C4×Q16—1631no—
183C4×SD16—1631no—
184#433—1631no—
185C22×Q8—1631no—
186C2×Q16—1631no—
187C2×Q16C2×D16311yescharacteristic
188C2×SD16—1631no—
189C2×SD16C2×D16311yescharacteristic
190C4⋊Q8—1631no—
191#126—3261no—
192#124—3261no—
193#124C2×D16311yescharacteristic
194#123—3261no—
195C22⋊Q8—3261no—
196C22⋊Q8—1631no—
197#121—1631no—
198C4×Q8—3261no—
199C4×Q8—3261no—
200C4×Q8—1631no—
201C4×Q8—1631no—
202C4×Q8—1631no—
203C4×Q8—1631no—
204C4×Q8C2×D16311yescharacteristic
205C4×D4—1631no—
206#117—3261no—
207#117—3261no—
208#117—1631no—
209#117—1631no—
210#107—1631no—
211#107—1631no—
212#106—1631no—
213#106—1631no—
214C4⋊C8—1631no—
215C4⋊C8C2×D16311yescharacteristic
216#103—1631no—
217#103—1631no—
218#103—1631no—
219#103—1631no—
220#103—1631no—
221#103C2×D16311yescharacteristic
222#102—1631no—
223#102C2×D16311yescharacteristic
224C22⋊C8—1631no—
225C22⋊C8—1631no—
226#97—1631no—
227C4×C8—1631no—
228C2×Q8—3261no—
229C2×Q8—3262no—
230C2×Q8—3262no—
231C2×Q8—3261no—
232C2×Q8—1631no—
233C2×Q8C22×D16311yescharacteristic
234C2×Q8C22×D16311yescharacteristic
235C2×D4—3261no—
236C2×D4C22×D16311yescharacteristic
237C22×C4—3261no—
238C22×C4—3261no—
239C22×C4—1631no—
240Q16—22no—
241Q16—3262no—
242SD16—22no—
243SD16—3262no—
244C2×C8—1631no—
245C2×C8—1631no—
246C2×C8C22×D16311yescharacteristic
247C2×C8C22×D16311yescharacteristic
248C4⋊C4—21no—
249C4⋊C4—3261no—
250C4⋊C4—3261no—
251C4⋊C4—3261no—
252C4⋊C4—3261no—
253C4⋊C4—3261no—
254C4⋊C4—3261no—
255C4⋊C4—1631no—
256C4⋊C4—1631no—
257C4⋊C4—1631no—
258C4⋊C4—1631no—
259C4⋊C4—1631no—
260C4⋊C4C22×D16311yescharacteristic
261C22⋊C4—21no—
262C22⋊C4—21no—
263C22⋊C4—3261no—
264C22⋊C4—3261no—
265C22⋊C4—3261no—
266C22⋊C4—3261no—
267C22⋊C4—1631no—
268C22⋊C4—1631no—
269C42—21no—
270C42—3261no—
271C42—3261no—
272C42—3261no—
273C42—3261no—
274C42—3261no—
275C42—1631no—
276C42C22×D16311yescharacteristic
277C23—21no—
278C23—1631no—
279Q8—21no—
280Q8—21no—
281Q8—21no—
282Q8—13041no—
283Q8—3264no—
284Q8—3261no—
285Q8—1632no—
286Q8#1457969812yes—
287D4—21no—
288D4—21no—
289D4—13041no—
290C2×C4—21no—
291C2×C4—21no—
292C2×C4—21no—
293C2×C4—13041no—
294C2×C4—3264no—
295C2×C4—3261no—
296C2×C4—3261no—
297C2×C4—3261no—
298C2×C4—3261no—
299C2×C4—3261no—
300C2×C4—3261no—
301C2×C4—1631no—
302C2×C4—1631no—
303C2×C4—1631no—
304C2×C4D4×D16311yescharacteristic
305C2×C4D4×D16311yescharacteristic
306C2×C4C23×D16311yescharacteristic
307C8—21no—
308C8—21no—
309C8—3261no—
310C8—3261no—
311C22—41no—
312C22—41no—
313C22—41no—
314C22—3261no—
315C22—3261no—
316C22—3261no—
317C22C2×D4×D16311yescharacteristic
318C4—22no—
319C4—21no—
320C4—21no—
321C4—41no—
322C4—41no—
323C4—3262no—
324C4—3261no—
325C4—3261no—
326C4—3261no—
327C4—6521no—
328C4—6521no—
329C4—6521no—
330C4#41×D16311yescharacteristic
331C4C2×#1457969811yescharacteristic
332C2—81no—
333C2—6521no—
334C2#2378217511yescharacteristic
335C2#2378247411yescharacteristic
336C2#2378247911yescharacteristic
subgroupG/H
centerC22C2×D4×D163
commutator, cosocleC2×C4×C163C24
FrattiniC2×C4C23×D163
Fitting#464×C163C2
socleC22×C163C22×D4
prime radical#19104706C22
hypercenter#464D163
hypocenterC163#3121
LCSG ▷ C2×C4×C163 ▷ C2×C163 ▷ C163
UCSC1 ◁ C22 ◁ C42 ◁ #464

Complements

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

NH ≅ G/Ntype
2. #23782141C2semidirect
3. #23782005C2semidirect
5. #23781961C2semidirect
6. #23781950C2semidirect
7. #23781884C2semidirect
8. #23781819C2semidirect
10. #23781769C2semidirect
11. #23781766C2semidirect
12. Q16×Dic163C2semidirect
13. #23781756C2semidirect
14. #23781725C2semidirect
15. #464×C163C2semidirect
168. C163#3121semidirect
177. #464D163semidirect

Maximal subgroups

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

subgroupindex
#237822842
#237821412
#237820052
#237819872
#237819612
#237819502
#237818842
#237818192
#237818022
#237817692
#237817662
Q16×Dic1632
#237817562
#237817252
#464×C1632
#3121163