Finite Group Databasebeta

← #108372

Subgroups — #108372

Order 512.

954 conjugacy classes of subgroups; only the 186 normal subgroups are shown.

normal subgroups only

This group has 954 conjugacy classes of subgroups — too many to draw. Only its 186 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34366C211yesmaximal, characteristic
2#21204C211yesmaximal, characteristic
3C2×#1533C211yesmaximal, characteristic
4#7984C212yesmaximal, complemented
5#7972C212yesmaximal, complemented
6C22×#124C2211yescharacteristic
7C2×#432C2212yes—
8C2×#419—21no—
9C2×#418—21no—
10C2×C4×M16C2211yescharacteristic
11#2379—22no—
12#2372C414yescomplemented
13#2342—24no—
14#2337—24no—
15#2310—21no—
16#2308—21no—
17#2285C2211yescharacteristic
18#2283C2212yes—
19#1959—22no—
20#1958C414yescomplemented
21#1533—22no—
22#1533—22no—
23C2×#329C414yescomplemented
24#1289—24no—
25#1288—24no—
26C23×Q8D411yescharacteristic
27C23×D4D411yescharacteristic
28C22×M16—21no—
29C2×#124—22no—
30C2×#124—22no—
31C2×#124—22no—
32C2×#124C2×C412yes—
33C2×#124C2×C412yes—
34C2×#124C2×C412yes—
35C2×#124D412yes—
36C2×#124D412yes—
37C2×#124Q812yes—
38C2×#117—22no—
39C2×#117—21no—
40C22×C22⋊C4—22no—
41C22×C22⋊C4—22no—
42C22×C42C2311yescharacteristic
43#432—22no—
44#432—22no—
45#432—22no—
46#432—22no—
47C2×C4⋊C8—22no—
48#419—44no—
49#418—44no—
50C2×#103—24no—
51C2×#102—24no—
52C2×C22⋊C8—22no—
53C4×M16—22no—
54C4×M16—22no—
55C2×#97—21no—
56C2×#97C2×C414yescomplemented
57C2×C4×C8C2×C412yescomplemented
58#391—24no—
59#391—24no—
60#388—28no—
61#388—28no—
62#386—24no—
63#386—24no—
64#385—28no—
65#384—21no—
66#383D414yes—
67#382—22no—
68#382C2×C412yescomplemented
69#381—22no—
70#381—22no—
71#380—22no—
72#380D414yes—
73#379—22no—
74C4×C4⋊C4—22no—
75C4×C4⋊C4—22no—
76C4×C22⋊C4—22no—
77C4×C22⋊C4—28no—
78#376C2×C414yes—
79#329—28no—
80C25—21no—
81C22×Q8—21no—
82C22×Q8—21no—
83C22×Q8—21no—
84C22×Q8—42no—
85C22×Q8—42no—
86C22×Q8C22⋊C412yes—
87C22×Q8C22⋊C412yes—
88C22×Q8C4⋊C412yes—
89C22×D4—21no—
90C22×D4—21no—
91C22×D4—21no—
92C22×D4—42no—
93C22×D4—42no—
94C22×D4C22⋊C412yes—
95C22×D4C22⋊C412yes—
96C22×D4C4⋊C412yes—
97C23×C4—21no—
98C23×C4—24no—
99C23×C4C2×D411yescharacteristic
100C23×C4C2×D412yes—
101C2×M16—24no—
102C2×M16—42no—
103C2×M16—41no—
104C2×M16—41no—
105C22×C8—22no—
106C22×C8—24no—
107#124—24no—
108#124—24no—
109#124—24no—
110#124—24no—
111#124—24no—
112#124—24no—
113#124C4214yes—
114#124C22⋊C414yes—
115#124C4⋊C414yes—
116#124C4⋊C414yes—
117#117—42no—
118#117—42no—
119#117—82no—
120#117—82no—
121C2×C4⋊C4—22no—
122C2×C4⋊C4—24no—
123C2×C4⋊C4—24no—
124C2×C4⋊C4—42no—
125C2×C22⋊C4—24no—
126C2×C22⋊C4—24no—
127C2×C22⋊C4—24no—
128C2×C22⋊C4—24no—
129C2×C22⋊C4—24no—
130C2×C22⋊C4—24no—
131C2×C22⋊C4—24no—
132C2×C22⋊C4—24no—
133C2×C22⋊C4—28no—
134C2×C22⋊C4—41no—
135C2×C22⋊C4—82no—
136C2×C22⋊C4—82no—
137C2×C42—24no—
138C2×C42—24no—
139C2×C42—41no—
140C2×C42C22×C411yescharacteristic
141C2×C42C22×C411yescharacteristic
142C2×C42C22×C411yescharacteristic
143C2×C42C2×D411yescharacteristic
144C2×C42C2×D411yescharacteristic
145C2×C42C2×D412yes—
146C2×C42C2×Q811yescharacteristic
147C2×C42#4112yes—
148C2×C42#4112yes—
149C4⋊C8—42no—
150C4⋊C8—42no—
151#103—48no—
152#102—48no—
153C22⋊C8—42no—
154C22⋊C8—42no—
155#97—24no—
156#97—24no—
157#97—42no—
158C4×C8—24no—
159#95—28no—
160#95—24no—
161#95—24no—
162#95—24no—
163#95—24no—
164#95C22⋊C418yes—
165C24—22no—
166C24—22no—
167C24—22no—
168C24—42no—
169C24—42no—
170C24—84no—
171C24#12011yescharacteristic
172C2×Q8—22no—
173C2×Q8—22no—
174C2×Q8—22no—
175C2×Q8—42no—
176C2×Q8—42no—
177C2×Q8—42no—
178C2×Q8—42no—
179C2×Q8—44no—
180C2×Q8—82no—
181C2×Q8—82no—
182C2×Q8—81no—
183C2×Q8—81no—
184C2×Q8#9514yes—
185C2×D4—22no—
186C2×D4—22no—
187C2×D4—22no—
188C2×D4—42no—
189C2×D4—42no—
190C2×D4—42no—
191C2×D4—42no—
192C2×D4—44no—
193C2×D4—82no—
194C2×D4—82no—
195C2×D4—81no—
196C2×D4—81no—
197C2×D4#9514yes—
198C22×C4—22no—
199C22×C4—22no—
200C22×C4—22no—
201C22×C4—22no—
202C22×C4—22no—
203C22×C4—28no—
204C22×C4—42no—
205C22×C4—42no—
206C22×C4—42no—
207C22×C4—82no—
208C22×C4—84no—
209C22×C4—88no—
210C22×C4C2×C22⋊C411yescharacteristic
211C22×C4C2×C22⋊C411yescharacteristic
212C22×C4C2×C22⋊C412yes—
213C22×C4C2×C4⋊C411yescharacteristic
214C22×C4C4×D412yes—
215C22×C4C4×D412yes—
216C22×C4#12011yescharacteristic
217C22×C4#12012yes—
218C22×C4#12112yes—
219C22×C4C22⋊Q812yes—
220C22×C4#12311yescharacteristic
221C22×C4#12311yescharacteristic
222M16—44no—
223M16—44no—
224C2×C8—24no—
225C2×C8—42no—
226C2×C8—42no—
227C2×C8—44no—
228C2×C8—84no—
229C4⋊C4—42no—
230C4⋊C4—42no—
231C4⋊C4—84no—
232C4⋊C4—84no—
233C4⋊C4—84no—
234C22⋊C4—28no—
235C22⋊C4—28no—
236C22⋊C4—28no—
237C22⋊C4—28no—
238C22⋊C4—44no—
239C22⋊C4—44no—
240C22⋊C4—88no—
241C22⋊C4—84no—
242C22⋊C4—82no—
243C42—82no—
244C42—84no—
245C42—84no—
246C42C2×C4211yescharacteristic
247C42C2×C22⋊C411yescharacteristic
248C42C2×C4⋊C411yescharacteristic
249C42C2×C4⋊C411yescharacteristic
250C42#11712yes—
251C42C4×D412yes—
252C42C4×D412yes—
253C42#12112yes—
254C42C22⋊Q812yes—
255C42#12312yes—
256C23—24no—
257C23—42no—
258C23—48no—
259C23—42no—
260C23—42no—
261C23—44no—
262C23—42no—
263C23—42no—
264C23—84no—
265C23—84no—
266C23—164no—
267C23—164no—
268C23#37911yescharacteristic
269C23#38511yescharacteristic
270C23#38611yescharacteristic
271C23#45312yes—
272C23#45512yes—
273Q8—44no—
274Q8—82no—
275Q8—82no—
276Q8—82no—
277D4—44no—
278D4—82no—
279D4—82no—
280D4—82no—
281C2×C4—22no—
282C2×C4—22no—
283C2×C4—22no—
284C2×C4—22no—
285C2×C4—21no—
286C2×C4—21no—
287C2×C4—21no—
288C2×C4—21no—
289C2×C4—24no—
290C2×C4—42no—
291C2×C4—42no—
292C2×C4—42no—
293C2×C4—42no—
294C2×C4—42no—
295C2×C4—42no—
296C2×C4—42no—
297C2×C4—42no—
298C2×C4—44no—
299C2×C4—42no—
300C2×C4—42no—
301C2×C4—48no—
302C2×C4—48no—
303C2×C4—48no—
304C2×C4—84no—
305C2×C4—82no—
306C2×C4—168no—
307C2×C4C2×#9511yescharacteristic
308C2×C4C4×C22⋊C412yes—
309C2×C4#38012yes—
310C2×C4#38111yescharacteristic
311C2×C4#38512yes—
312C2×C4#38511yescharacteristic
313C2×C4#38612yes—
314C2×C4#38611yescharacteristic
315C8—44no—
316C8—84no—
317C22—22no—
318C22—22no—
319C22—22no—
320C22—42no—
321C22—42no—
322C22—48no—
323C22—44no—
324C22—84no—
325C22—84no—
326C22—162no—
327C22—164no—
328C22—162no—
329C22C22⋊#9511yescharacteristic
330C22#179212yes—
331C22#185612yes—
332C22#188511yescharacteristic
333C22#188711yescharacteristic
334C4—22no—
335C4—22no—
336C4—22no—
337C4—22no—
338C4—21no—
339C4—21no—
340C4—21no—
341C4—21no—
342C4—44no—
343C4—88no—
344C4—162no—
345C2—22no—
346C2—21no—
347C2—22no—
348C2—21no—
349C2—84no—
350C2#1074911yescharacteristic
351C2#1079212yes—
subgroupG/H
center, socleC22C22⋊#95
commutatorC42C2×C42
Frattini, cosocleC22×C42C23
prime radicalC23×D4D4
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C22×C42 ◁ G
LECSG ▷ C22×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #7984C2semidirect
5. #7972C2semidirect
12. #2372C4semidirect
20. #1958C4semidirect
23. C2×#329C4semidirect
56. C2×#97C2×C4semidirect
57. C2×C4×C8C2×C4semidirect
68. #382C2×C4semidirect

Maximal subgroups

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

subgroupindex
#343662
#212042
C2×#15332
#79842
#79722