Finite Group Databasebeta

← #238036

Subgroups — #238036

Order 512.

1020 conjugacy classes of subgroups; only the 182 normal subgroups are shown.

normal subgroups only

This group has 1020 conjugacy classes of subgroups — too many to draw. Only its 182 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#34958C211yesmaximal, characteristic
2C2×#1826C212yesmaximal
3#20688C211yesmaximal, characteristic
4#10389C214yesmaximal, complemented
5#10387C214yesmaximal, complemented
6#8112C211yesmaximal, characteristic
7#7942C212yesmaximal
8C22×#117C2211yescharacteristic
9C2×#407C2212yes—
10#2608C2212yes—
11#2607C2212yes—
12#2569—22no—
13#2549—22no—
14#2437—22no—
15#2396—22no—
16C4×C22⋊Q8—24no—
17#2280C2211yescharacteristic
18#2280C2211yescharacteristic
19C2×C4×C22⋊C4C2212yes—
20#1826—22no—
21#1826—22no—
22#1826C2218yes—
23#1820C2212yes—
24#1820C2212yes—
25#1819—22no—
26#1740—22no—
27C2×#339—24no—
28#1306C2212yes—
29#1306C2212yes—
30#1284C2218yes—
31C24×C4C2311yescharacteristic
32C22×M16—22no—
33C2×C22⋊Q8—24no—
34C2×C4×Q8—22no—
35C2×#117—21no—
36C2×#117—21no—
37C2×#117—21no—
38C2×#117—22no—
39C2×#117—22no—
40C2×#117—22no—
41C2×#117—22no—
42C2×#117D412yes—
43C2×#117Q812yescomplemented
44C2×#117C2312yes—
45C22×C4⋊C4Q812yescomplemented
46C22×C22⋊C4—21no—
47C22×C22⋊C4—22no—
48C22×C42—21no—
49C22×C42D414yes—
50#407—22no—
51#407—22no—
52#407—22no—
53#407C2314yes—
54C2×C22⋊C8—22no—
55C2×C22⋊C8—22no—
56#400—22no—
57#399—22no—
58#398—24no—
59#396—24no—
60#395—22no—
61#393—22no—
62#391—24no—
63#389—24no—
64#387—24no—
65#387—24no—
66#385—24no—
67#385—24no—
68#381C2312yes—
69#381C2311yescharacteristic
70#381C2311yescharacteristic
71#380—24no—
72#380—22no—
73#380—22no—
74#379—21no—
75#379D412yes—
76C4×C4⋊C4—22no—
77C4×C4⋊C4—24no—
78C4×C22⋊C4—22no—
79C4×C22⋊C4—22no—
80C4×C22⋊C4—22no—
81C4×C22⋊C4—22no—
82C4×C22⋊C4D412yes—
83C4×C22⋊C4D412yes—
84C4×C22⋊C4C2314yes—
85C2×#95—22no—
86#342—24no—
87#342—24no—
88#339—28no—
89#339—28no—
90C25C2×D411yescharacteristic
91C22×Q8—22no—
92C23×C4—24no—
93C23×C4—22no—
94C23×C4—44no—
95C23×C4C2×D411yescharacteristic
96C23×C4C2×D412yes—
97C23×C4C2×D412yes—
98C23×C4C2×Q811yescharacteristic
99C23×C4C2×Q811yescharacteristic
100C23×C4#4112yes—
101C23×C4#4112yes—
102C23×C4#4111yescharacteristic
103C23×C4#4111yescharacteristic
104C23×C4C2411yescharacteristic
105C2×M16—24no—
106C2×M16—42no—
107C2×M16—48no—
108C22×C8—42no—
109C22⋊Q8—416no—
110C4×Q8—84no—
111#117—24no—
112#117—24no—
113#117—24no—
114#117—24no—
115#117—41no—
116#117—41no—
117#117—42no—
118#117—42no—
119#117—41no—
120#117—41no—
121#117—44no—
122#117—44no—
123C2×C4⋊C4—22no—
124C2×C4⋊C4—22no—
125C2×C4⋊C4—22no—
126C2×C4⋊C4—22no—
127C2×C4⋊C4—24no—
128C2×C4⋊C4—24no—
129C2×C4⋊C4—24no—
130C2×C4⋊C4—22no—
131C2×C4⋊C4—24no—
132C2×C4⋊C4—22no—
133C2×C4⋊C4—22no—
134C2×C4⋊C4—41no—
135C2×C4⋊C4C2×Q814yes—
136C2×C22⋊C4—22no—
137C2×C22⋊C4—22no—
138C2×C22⋊C4—24no—
139C2×C22⋊C4—22no—
140C2×C22⋊C4—22no—
141C2×C22⋊C4—24no—
142C2×C22⋊C4—22no—
143C2×C22⋊C4—48no—
144C2×C22⋊C4—42no—
145C2×C22⋊C4—42no—
146C2×C22⋊C4—42no—
147C2×C22⋊C4C2×D412yes—
148C2×C22⋊C4C2×D412yes—
149C2×C42—22no—
150C2×C42—24no—
151C2×C42—22no—
152C2×C42—22no—
153C2×C42—42no—
154C2×C42—42no—
155C2×C42—48no—
156C2×C42—41no—
157C2×C42C2×D414yes—
158C2×C42C2×D414yes—
159C22⋊C8—24no—
160C22⋊C8—24no—
161C22⋊C8—42no—
162C22⋊C8—42no—
163#95—22no—
164#95—24no—
165#95—24no—
166#95—22no—
167#95—22no—
168#95—22no—
169#95—24no—
170#95—22no—
171#95—22no—
172#95—24no—
173C24—44no—
174C24—42no—
175C24#12012yes—
176C24C22⋊Q811yescharacteristic
177C24C22⋊Q811yescharacteristic
178C24#12412yes—
179C24C22×D411yescharacteristic
180C2×Q8—48no—
181C2×Q8—82no—
182C22×C4—28no—
183C22×C4—24no—
184C22×C4—24no—
185C22×C4—24no—
186C22×C4—22no—
187C22×C4—22no—
188C22×C4—24no—
189C22×C4—42no—
190C22×C4—42no—
191C22×C4—44no—
192C22×C4—44no—
193C22×C4—42no—
194C22×C4—42no—
195C22×C4—44no—
196C22×C4—42no—
197C22×C4—48no—
198C22×C4—48no—
199C22×C4—42no—
200C22×C4—42no—
201C22×C4—42no—
202C22×C4—84no—
203C22×C4—84no—
204C22×C4#12011yescharacteristic
205C22×C4#12011yescharacteristic
206C22×C4C22⋊Q812yes—
207C22×C4C22⋊Q812yes—
208C22×C4C22⋊Q811yescharacteristic
209C22×C4C22⋊Q811yescharacteristic
210C22×C4C22⋊Q812yes—
211C22×C4C22⋊Q812yes—
212C22×C4#12412yes—
213C22×C4#12412yes—
214C22×C4#12411yescharacteristic
215C22×C4#12411yescharacteristic
216C22×C4#12412yes—
217C22×C4#12412yes—
218C22×C4C22×D412yes—
219C22×C4C22×Q811yescharacteristic
220C22×C4C2×#4112yes—
221C22×C4C2×#4111yescharacteristic
222M16—48no—
223M16—84no—
224C2×C8—44no—
225C2×C8—84no—
226C4⋊C4—24no—
227C4⋊C4—24no—
228C4⋊C4—48no—
229C4⋊C4—48no—
230C4⋊C4—48no—
231C4⋊C4—42no—
232C4⋊C4—42no—
233C4⋊C4—82no—
234C4⋊C4—82no—
235C4⋊C4—84no—
236C4⋊C4—82no—
237C22⋊C4—28no—
238C22⋊C4—48no—
239C22⋊C4—44no—
240C22⋊C4—48no—
241C22⋊C4—42no—
242C22⋊C4—42no—
243C22⋊C4—44no—
244C22⋊C4—42no—
245C22⋊C4—42no—
246C22⋊C4—44no—
247C22⋊C4—84no—
248C42—44no—
249C42—44no—
250C42—44no—
251C42—42no—
252C42—42no—
253C42—48no—
254C42—44no—
255C42—82no—
256C23—24no—
257C23—24no—
258C23—24no—
259C23—22no—
260C23—24no—
261C23—42no—
262C23—41no—
263C23—42no—
264C23—41no—
265C23—42no—
266C23—42no—
267C23—42no—
268C23—88no—
269C23#39312yes—
270C23#39312yes—
271C23C2×#12011yescharacteristic
272C23C2×C22⋊Q811yescharacteristic
273C23C2×#12411yescharacteristic
274Q8—88no—
275C2×C4—24no—
276C2×C4—22no—
277C2×C4—22no—
278C2×C4—22no—
279C2×C4—24no—
280C2×C4—24no—
281C2×C4—24no—
282C2×C4—24no—
283C2×C4—22no—
284C2×C4—22no—
285C2×C4—28no—
286C2×C4—42no—
287C2×C4—42no—
288C2×C4—42no—
289C2×C4—42no—
290C2×C4—41no—
291C2×C4—41no—
292C2×C4—42no—
293C2×C4—44no—
294C2×C4—44no—
295C2×C4—44no—
296C2×C4—44no—
297C2×C4—44no—
298C2×C4—44no—
299C2×C4—44no—
300C2×C4—44no—
301C2×C4—42no—
302C2×C4—42no—
303C2×C4—816no—
304C2×C4—84no—
305C2×C4—84no—
306C2×C4—84no—
307C2×C4—82no—
308C2×C4—84no—
309C2×C4#39311yescharacteristic
310C2×C4#39311yescharacteristic
311C2×C4#39311yescharacteristic
312C2×C4#39311yescharacteristic
313C2×C4#45814yes—
314C2×C4C2×C22⋊Q812yes—
315C2×C4C2×#12412yes—
316C8—84no—
317C22—22no—
318C22—22no—
319C22—22no—
320C22—21no—
321C22—21no—
322C22—22no—
323C22—22no—
324C22—22no—
325C22—22no—
326C22—22no—
327C22—48no—
328C22—44no—
329C22—82no—
330C22—84no—
331C22—82no—
332C22#203114yes—
333C22C2×#39311yescharacteristic
334C22C2×#45812yes—
335C4—22no—
336C4—22no—
337C4—21no—
338C4—21no—
339C4—48no—
340C4—82no—
341C4—84no—
342C4—82no—
343C4—84no—
344C4—162no—
345C4#203012yes—
346C4#203012yes—
347C2—22no—
348C2—24no—
349C2—41no—
350C2—41no—
351C2—42no—
352C2C2×#203011yescharacteristic
353C2C2×#203112yes—
subgroupG/H
centerC2×C4#393
commutator, Frattini, cosocleC23×C4C24
socleC22C2×#393
prime radicalC25C2×D4
LCSG ▷ C23×C4 ▷ C22 ▷ C1
UCSC1 ◁ C2×C4 ◁ C24×C4 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #10389C2semidirect
5. #10387C2semidirect
43. C2×#117Q8semidirect
45. C22×C4⋊C4Q8semidirect

Maximal subgroups

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

subgroupindex
#349582
C2×#18262
#206882
#103892
#103872
#81122
#79422