Finite Group Databasebeta

← #6465759

Subgroups — #6465759

Order 512.

1846 conjugacy classes of subgroups.

The subgroup lattice has 368 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3164C212yesmaximal
2#35167C212yesmaximal
3#28822C218yesmaximal, complemented
4#28821C218yesmaximal, complemented
5#23557C214yesmaximal
6#21170C212yesmaximal
7#21157C212yesmaximal
8#20799C211yesmaximal, characteristic
9C2×#1776C211yesmaximal, characteristic
10#20438C211yesmaximal, characteristic
11C2×#544C2214yes—
12#3164—216no—
13C2×#501C2218yes—
14C2×#500C2214yes—
15C2×#499C2214yes—
16C2×#428C2214yes—
17C2×#427C2212yes—
18C2×#427C2212yes—
19C22×#107C2211yescharacteristic
20C22×#106C2211yescharacteristic
21C2×#423C2211yescharacteristic
22C2×#405C2212yes—
23C22×#97C2211yescharacteristic
24#2857—24no—
25#2659—24no—
26#2552—22no—
27#2552C2214yes—
28#2551—22no—
29#2551C2214yes—
30#2550—24no—
31#2550C2218yes—
32#2316C2214yes—
33#2285C2212yes—
34#2285C2211yescharacteristic
35C2×C4×C4⋊C4C2212yes—
36#2059C2218yes—
37#2059C2218yes—
38#2058C2218yes—
39#2057C2218yes—
40#1949C2218yes—
41#1948C2214yes—
42#1947C2214yes—
43#1944C2214yes—
44#1944C2214yes—
45#1941C2218yes—
46#1847C2214yes—
47#1847C2214yes—
48#1776C2218yes—
49#1771C2218yes—
50#1736C2214yes—
51C2×#340C2212yes—
52#1729C2212yes—
53C22×M16C2312yes—
54C23×C8C2311yescharacteristic
55C23×C8C2311yescharacteristic
56#544—216no—
57#544—48no—
58C2×C4⋊Q8—24no—
59C2×C4⋊Q8—24no—
60C2×C4⋊Q8C2318yes—
61C2×#125—24no—
62C2×#125C2318yes—
63C2×C22⋊Q8—28no—
64C2×#117C2314yes—
65C2×#117C2312yes—
66C22×C4⋊C4C2312yes—
67C22×C4⋊C4C2312yes—
68C22×C42—22no—
69C22×C42C2311yescharacteristic
70#501—232no—
71#500—216no—
72#499—216no—
73#428—216no—
74#427—28no—
75#427—44no—
76C2×#107D418yes—
77C2×#107C2318yes—
78C2×#107C2312yes—
79C2×#107C2312yes—
80C2×#107C2312yes—
81C2×#107C2312yes—
82C2×#107C2314yes—
83C2×#106Q818yescomplemented
84C2×#106C2318yes—
85C2×#106C2314yes—
86C2×#106C2314yes—
87C2×#106C2314yes—
88#423Q818yescomplemented
89C2×C4⋊C8C2314yes—
90#405—28no—
91C2×#97Q818yescomplemented
92C2×#97C2314yes—
93C2×#97C2314yes—
94C2×C4×C8C2314yes—
95#398—28no—
96#398—28no—
97#395—28no—
98#395—28no—
99#391—28no—
100#389—24no—
101#389—24no—
102#389—24no—
103#389—24no—
104#389—28no—
105#389C2318yes—
106#389C2318yes—
107#384—24no—
108#384D411yescharacteristic
109#384D411yescharacteristic
110#384C2312yes—
111#384C2312yes—
112#382D412yes—
113#382C2314yes—
114#380—22no—
115#380—22no—
116#380—28no—
117#380D414yes—
118C4×C4⋊C4—22no—
119C4×C4⋊C4—22no—
120C4×C4⋊C4—22no—
121C4×C4⋊C4—22no—
122C4×C4⋊C4C2314yes—
123C4×C4⋊C4C2314yes—
124C4×C4⋊C4C2314yes—
125C2×#95—22no—
126#340C2318yes—
127#340C2318yes—
128#340C2318yes—
129#336C2318yes—
130C22×Q8—28no—
131C23×C4—22no—
132C23×C4—22no—
133C23×C4C2×D412yes—
134C23×C4C2411yescharacteristic
135C2×M16—24no—
136C2×M16—24no—
137C2×M16C2×Q818yes—
138C22×C8—22no—
139C22×C8—22no—
140C22×C8C2×D414yes—
141C22×C8C2×Q814yes—
142C22×C8C2×Q814yes—
143C22×C8C2×Q814yes—
144C22×C8C2412yes—
145C22×C8C2414yes—
146C22×C8C2412yes—
147C4⋊Q8—216no—
148C4⋊Q8—48no—
149C4⋊Q8—416no—
150C4⋊Q8—88no—
151#125—216no—
152#125—48no—
153#125—416no—
154C22⋊Q8—432no—
155#117—24no—
156#117—24no—
157#117—28no—
158#117—28no—
159C2×C4⋊C4—22no—
160C2×C4⋊C4—22no—
161C2×C4⋊C4—24no—
162C2×C4⋊C4—24no—
163C2×C4⋊C4—28no—
164C2×C4⋊C4—28no—
165C2×C4⋊C4—28no—
166C2×C4⋊C4—22no—
167C2×C4⋊C4—22no—
168C2×C4⋊C4—28no—
169C2×C4⋊C4—22no—
170C2×C4⋊C4—22no—
171C2×C4⋊C4—28no—
172C2×C4⋊C4C2×D412yes—
173C2×C4⋊C4C2×D412yes—
174C2×C4⋊C4C2×D418yes—
175C2×C4⋊C4C2×D414yes—
176C2×C4⋊C4C2×D414yes—
177C2×C4⋊C4C2×Q818yes—
178C2×C4⋊C4C2414yes—
179C2×C4⋊C4C2414yes—
180C2×C4⋊C4C2414yes—
181C2×C4⋊C4C2414yes—
182C2×C22⋊C4—24no—
183C2×C22⋊C4C2×D414yes—
184C2×C42—24no—
185C2×C42—24no—
186C2×C42—24no—
187C2×C42—84no—
188C2×C42C2×D411yescharacteristic
189C2×C42C2×D411yescharacteristic
190C2×C42C2×Q814yes—
191C2×C42C2×Q814yes—
192C2×C42C2414yes—
193C2×C42C2412yes—
194#107—24no—
195#107—24no—
196#107—216no—
197#107—42no—
198#107—42no—
199#107C2×D4116yes—
200#106—28no—
201#106—216no—
202#106—44no—
203#106C2×Q8116yes—
204C4⋊C8C2×Q8116yes—
205#97—28no—
206#97C2×Q8116yes—
207C4×C8—28no—
208#95—24no—
209#95—24no—
210#95—24no—
211#95—24no—
212#95—28no—
213#95—28no—
214#95C2×D418yes—
215C24C22×D411yescharacteristic
216C2×Q8—416no—
217C2×Q8—88no—
218C2×Q8—88no—
219C22×C4—24no—
220C22×C4—24no—
221C22×C4—28no—
222C22×C4—28no—
223C22×C4—24no—
224C22×C4—24no—
225C22×C4—42no—
226C22×C4—42no—
227C22×C4—82no—
228C22×C4—82no—
229C22×C4—82no—
230C22×C4—82no—
231C22×C4C4⋊Q818yes—
232C22×C4C22×D414yes—
233C22×C4C22×D414yes—
234C22×C4C22×D412yes—
235C22×C4C22×D412yes—
236C22×C4C22×D411yescharacteristic
237C22×C4C22×D412yes—
238C22×C4C22×Q812yes—
239C22×C4C22×Q812yes—
240C22×C4C22×Q812yes—
241C22×C4H(5,𝔽2)12yes—
242C22×C42−1+412yes—
243C22×C4C2511yescharacteristic
244M16—216no—
245C2×C8—28no—
246C2×C8—24no—
247C2×C8—22no—
248C2×C8—22no—
249C2×C8—24no—
250C2×C8C4⋊Q8116yescomplemented
251C2×C8C22×D414yes—
252C2×C8C22×Q814yes—
253C2×C8C22×Q818yes—
254C2×C8C22×Q814yes—
255C2×C8C22×Q814yes—
256C4⋊C4—28no—
257C4⋊C4—24no—
258C4⋊C4—24no—
259C4⋊C4—28no—
260C4⋊C4—24no—
261C4⋊C4—24no—
262C4⋊C4—24no—
263C4⋊C4—24no—
264C4⋊C4—24no—
265C4⋊C4—24no—
266C4⋊C4—44no—
267C4⋊C4—44no—
268C4⋊C4—416no—
269C4⋊C4—416no—
270C4⋊C4—416no—
271C4⋊C4—44no—
272C4⋊C4—44no—
273C4⋊C4—82no—
274C4⋊C4—82no—
275C4⋊C4—88no—
276C4⋊C4—88no—
277C4⋊C4C22×D418yes—
278C4⋊C4C22×Q818yes—
279C22⋊C4—28no—
280C22⋊C4—48no—
281C42—24no—
282C42—24no—
283C42—48no—
284C42—48no—
285C42—84no—
286C42C4⋊Q814yes—
287C42C4⋊Q814yes—
288C42C22×Q814yes—
289C42C22×Q814yes—
290C23—22no—
291C23—22no—
292C23C2×C4⋊Q812yes—
293C23D4212yes—
294C23D4×Q812yes—
295C23C23×D411yescharacteristic
296Q8—1616no—
297C2×C4—22no—
298C2×C4—22no—
299C2×C4—22no—
300C2×C4—22no—
301C2×C4—28no—
302C2×C4—28no—
303C2×C4—24no—
304C2×C4—24no—
305C2×C4—28no—
306C2×C4—28no—
307C2×C4—48no—
308C2×C4—44no—
309C2×C4—44no—
310C2×C4—44no—
311C2×C4—44no—
312C2×C4—44no—
313C2×C4—44no—
314C2×C4—44no—
315C2×C4—44no—
316C2×C4—44no—
317C2×C4—44no—
318C2×C4—88no—
319C2×C4—88no—
320C2×C4C2×C4⋊Q814yes—
321C2×C4C2×C4⋊Q814yes—
322C2×C4C2×C4⋊Q814yes—
323C2×C4C2×C4⋊Q812yes—
324C2×C4#54414yes—
325C2×C4D4212yes—
326C2×C4D4×Q814yes—
327C2×C4D4×Q812yes—
328C2×C4Q8214yes—
329C2×C4C23×D411yescharacteristic
330C2×C4C23×Q811yescharacteristic
331C2×C4C23×Q811yescharacteristic
332C2×C4C23×Q811yescharacteristic
333C2×C4C2×H(5,𝔽2)11yescharacteristic
334C2×C4C2×2−1+411yescharacteristic
335C8—24no—
336C8—28no—
337C8C2×C4⋊Q818yes—
338C22—22no—
339C22—22no—
340C22—24no—
341C22—24no—
342C22#255114yes—
343C22C22×C4⋊Q811yescharacteristic
344C22C2×D4211yescharacteristic
345C22C2×D4×Q811yescharacteristic
346C22#358512yes—
347C22#358612yes—
348C4—24no—
349C4—24no—
350C4—44no—
351C4—48no—
352C4—44no—
353C4—48no—
354C4—84no—
355C4—84no—
356C4#255114yes—
357C4C22×C4⋊Q811yescharacteristic
358C4C2×#54411yescharacteristic
359C4C2×D4×Q811yescharacteristic
360C4C2×Q8211yescharacteristic
361C2—24no—
362C2C2×#255111yescharacteristic
363C2#6066412yes—
364C2#6107911yescharacteristic
365C2#6108111yescharacteristic
366C2#6137712yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC24C22×D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ #2285 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #28822C2semidirect
4. #28821C2semidirect
83. C2×#106Q8semidirect
88. #423Q8semidirect
91. C2×#97Q8semidirect
250. C2×C8C4⋊Q8semidirect

Maximal subgroups

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

subgroupindex
C2×#31642
#351672
#288222
#288212
#235572
#211702
#211572
#207992
C2×#17762
#204382