Finite Group Databasebeta

← #391609

Subgroups — #391609

Order 512.

696 conjugacy classes of subgroups.

The subgroup lattice has 340 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23733C211yesmaximal, characteristic
2#11299C212yesmaximal, complemented
3#11213C212yesmaximal, complemented
4#11211C212yesmaximal, complemented
5#11185C212yesmaximal
6#11160C211yesmaximal, characteristic
7#11057C212yesmaximal
8#10959C211yesmaximal, characteristic
9#10904C212yesmaximal, complemented
10C2×#428C2212yes—
11C2×#421C2212yes—
12C2×#417C2212yes—
13C2×#410C2211yescharacteristic
14#2879C2211yescharacteristic
15#2873C2211yescharacteristic
16#2316C2212yes—
17#2297C2211yescharacteristic
18C2×#382C2211yescharacteristic
19#2097C414yescomplemented
20#2092C414yescomplemented
21C2×#344C2212yes—
22C2×#344C2212yes—
23C2×#343C2212yes—
24#1740C2211yescharacteristic
25#1736C2212yes—
26#1736C2212yes—
27#1733C2212yes—
28C2×#339C2212yes—
29C2×#339C2212yes—
30C2×#338C2212yes—
31C2×H(3,ℤ/4)C2212yes—
32#1728C2211yescharacteristic
33C22×#41C2311yescharacteristic
34C22×M16C2312yes—
35C22×M16C2312yes—
36C22×M16C2312yes—
37C2×#117C2312yes—
38C2×#117C2312yes—
39C2×#117C2311yescharacteristic
40C22×C4⋊C4C2312yes—
41C22×C42C2311yescharacteristic
42#428—42no—
43#428—42no—
44C2×#107C2×C414yes—
45C2×#106C2×C414yes—
46#421—24no—
47#421—42no—
48C2×#104—42no—
49#417—42no—
50#417—42no—
51C2×#103C2×C414yes—
52C2×#102C2×C414yes—
53#410—42no—
54C2×#99—41no—
55#408—42no—
56#407—41no—
57#405—42no—
58C4×M16—41no—
59#391C2×C412yes—
60#390C2×C412yes—
61#389—42no—
62#389—42no—
63#386—41no—
64#382—41no—
65#382C2×C414yes—
66#380—42no—
67C4×C22⋊C4—41no—
68C2×#95—41no—
69#344—24no—
70#344—42no—
71#343—42no—
72#342—41no—
73#340—42no—
74#340—42no—
75#339—42no—
76#339—42no—
77#339—42no—
78#338—24no—
79H(3,ℤ/4)—42no—
80#336C2×C414yes—
81C2×#41—22no—
82C2×#41—22no—
83C2×#41—22no—
84C2×#41—22no—
85C2×#41—22no—
86C2×#41—22no—
87C22×Q8C22×C411yescharacteristic
88C22×D4—21no—
89C22×D4C22×C411yescharacteristic
90C23×C4—22no—
91C23×C4—21no—
92C23×C4—21no—
93C23×C4C2411yescharacteristic
94C2×M16—22no—
95C2×M16—22no—
96C2×M16—22no—
97C2×M16—22no—
98C2×M16—22no—
99C2×M16—22no—
100C2×M16—22no—
101C2×M16—22no—
102C2×M16—22no—
103C2×M16—22no—
104C2×M16—22no—
105C2×M16—22no—
106C2×M16—22no—
107C2×M16—22no—
108C2×M16—22no—
109C2×M16—22no—
110C2×M16—22no—
111C2×M16#4114yes—
112C22×C8—22no—
113C22×C8—22no—
114C22×C8C22×C414yes—
115#117—22no—
116#117—22no—
117#117—22no—
118#117—22no—
119#117—22no—
120#117—22no—
121#117—22no—
122#117—22no—
123#117—22no—
124#117—22no—
125C2×C4⋊C4—22no—
126C2×C4⋊C4—22no—
127C2×C4⋊C4—22no—
128C2×C4⋊C4—22no—
129C2×C4⋊C4—22no—
130C2×C4⋊C4—22no—
131C2×C4⋊C4—22no—
132C2×C4⋊C4—44no—
133C2×C4⋊C4C22×C414yes—
134C2×C4⋊C4C22×C412yes—
135C2×C22⋊C4—21no—
136C2×C22⋊C4—22no—
137C2×C22⋊C4—22no—
138C2×C22⋊C4—42no—
139C2×C42—22no—
140C2×C42—22no—
141C2×C42—21no—
142C2×C42—21no—
143C2×C42—22no—
144C2×C42—22no—
145C2×C42C22×C412yes—
146#107—44no—
147#106—44no—
148#104—48no—
149#103—44no—
150#102—44no—
151#99—44no—
152C22⋊C8—44no—
153#97—42no—
154C4×C8—42no—
155#95—44no—
156#95—42no—
157#95—44no—
158C24—21no—
159C24H(5,𝔽2)11yescharacteristic
160#41—42no—
161#41—42no—
162#41—42no—
163#41—42no—
164#41—42no—
165#41—42no—
166#41—42no—
167#41—42no—
168C2×Q8—21no—
169C2×Q8—22no—
170C2×Q8—21no—
171C2×Q8—22no—
172C2×D4—21no—
173C2×D4—22no—
174C2×D4—21no—
175C2×D4—22no—
176C2×D4—42no—
177C2×D4—41no—
178C2×D4—42no—
179C2×D4—41no—
180C22×C4—22no—
181C22×C4—21no—
182C22×C4—24no—
183C22×C4—22no—
184C22×C4—42no—
185C22×C4—41no—
186C22×C4—41no—
187C22×C4—42no—
188C22×C4—42no—
189C22×C4—41no—
190C22×C4—41no—
191C22×C4—42no—
192C22×C4—42no—
193C22×C4—42no—
194C22×C4—82no—
195C22×C4—84no—
196C22×C4C23×C411yescharacteristic
197C22×C4C2×#4112yes—
198C22×C4H(5,𝔽2)12yes—
199C22×C4H(5,𝔽2)12yes—
200C22×C4H(5,𝔽2)12yes—
201C22×C4H(5,𝔽2)11yescharacteristic
202C22×C42−1+412yes—
203C22×C42−1+412yes—
204M16—24no—
205M16—24no—
206M16—42no—
207M16—42no—
208M16—42no—
209M16—42no—
210M16—42no—
211M16—42no—
212M16—42no—
213M16—42no—
214M16—42no—
215M16—42no—
216C2×C8—24no—
217C2×C8—24no—
218C2×C8—24no—
219C2×C8—24no—
220C2×C8—42no—
221C2×C8—42no—
222C2×C8—42no—
223C2×C8—42no—
224C2×C8—42no—
225C4⋊C4—24no—
226C4⋊C4—22no—
227C4⋊C4—24no—
228C4⋊C4—22no—
229C4⋊C4—42no—
230C4⋊C4—42no—
231C4⋊C4—42no—
232C4⋊C4—42no—
233C4⋊C4—42no—
234C4⋊C4—42no—
235C4⋊C4—164no—
236C22⋊C4—42no—
237C22⋊C4—42no—
238C22⋊C4—42no—
239C22⋊C4—42no—
240C22⋊C4—42no—
241C22⋊C4—84no—
242C42—24no—
243C42—41no—
244C42—41no—
245C42—42no—
246C42—42no—
247C42—42no—
248C42—42no—
249C42—42no—
250C23—42no—
251C23—41no—
252C23—41no—
253C23—41no—
254C23—82no—
255C23#51811yescharacteristic
256C23#54212yes—
257C23#56012yes—
258C23#56011yescharacteristic
259C23#56311yescharacteristic
260Q8—42no—
261Q8—42no—
262D4—42no—
263D4—42no—
264D4—82no—
265D4—82no—
266C2×C4—42no—
267C2×C4—42no—
268C2×C4—41no—
269C2×C4—42no—
270C2×C4—41no—
271C2×C4—42no—
272C2×C4—44no—
273C2×C4—41no—
274C2×C4—84no—
275C2×C4—82no—
276C2×C4—82no—
277C2×C4—82no—
278C2×C4—82no—
279C2×C4—82no—
280C2×C4—84no—
281C2×C4—82no—
282C2×C4—82no—
283C2×C4—82no—
284C2×C4—82no—
285C2×C4—82no—
286C2×C4—82no—
287C2×C4—162no—
288C2×C4C4×#4111yescharacteristic
289C2×C4#51812yes—
290C2×C4#51911yescharacteristic
291C2×C4#52012yes—
292C2×C4#53812yes—
293C2×C4#53912yes—
294C2×C4#53912yes—
295C2×C4#54012yes—
296C2×C4#54212yes—
297C2×C4#56011yescharacteristic
298C2×C4#56011yescharacteristic
299C2×C4#56212yes—
300C2×C4#56212yes—
301C2×C4#56212yes—
302C2×C4#56312yes—
303C2×C4#56312yes—
304C8—44no—
305C8—44no—
306C8—82no—
307C22—22no—
308C22—22no—
309C22—22no—
310C22—41no—
311C22—41no—
312C22—42no—
313C22—82no—
314C22—81no—
315C22—82no—
316C22—161no—
317C22#233211yescharacteristic
318C22#237611yescharacteristic
319C22#238011yescharacteristic
320C22#282912yes—
321C22#283812yes—
322C4—82no—
323C4—81no—
324C4—82no—
325C4—84no—
326C4—162no—
327C4—162no—
328C4#233512yes—
329C4#237812yes—
330C4#282712yes—
331C4#283712yes—
332C2—21no—
333C2—22no—
334C2—21no—
335C2—22no—
336C2—161no—
337C2#1464611yescharacteristic
338C2#2336212yes—
subgroupG/H
center, socleC22#2380
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC22×#41C23
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C23×C4 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #11299C2semidirect
3. #11213C2semidirect
4. #11211C2semidirect
9. #10904C2semidirect
19. #2097C4semidirect
20. #2092C4semidirect

Maximal subgroups

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

subgroupindex
#237332
#112992
#112132
#112112
#111852
#111602
#110572
#109592
#109042