Finite Group Databasebeta

← #115774

Subgroups — #115774

Order 512.

942 conjugacy classes of subgroups; only the 142 normal subgroups are shown.

normal subgroups only

This group has 942 conjugacy classes of subgroups — too many to draw. Only its 142 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#20627C211yesmaximal, characteristic
2C2×#1511C212yesmaximal
3#14138C211yesmaximal, characteristic
4#10726C211yesmaximal, characteristic
5C2×#1284C212yesmaximal
6C22×C22⋊Q8C2211yescharacteristic
7C22×#117C2211yescharacteristic
8C2×#423—21no—
9C2×#407C2212yes—
10C2×#391C412yescomplemented
11C2×#381C2212yes—
12C2×C4×C22⋊C4C2211yescharacteristic
13#1798—24no—
14C2×#342—22no—
15#1728—21no—
16#1511—24no—
17#1511—24no—
18C2×#328—24no—
19#1440—22no—
20#1436—21no—
21#1436—21no—
22C4⋊#95C412yescomplemented
23#1284—24no—
24C23×Q8C2×C411yescharacteristic
25C24×C4C2311yescharacteristic
26C22×M16—22no—
27C2×C22⋊Q8—21no—
28C2×C22⋊Q8—21no—
29C2×C22⋊Q8—22no—
30C2×C22⋊Q8—22no—
31C2×C22⋊Q8—22no—
32C2×C22⋊Q8—22no—
33C2×C22⋊Q8D412yes—
34C2×C22⋊Q8Q812yes—
35C2×#117—22no—
36C2×#117—22no—
37C2×#117—44no—
38C22×C4⋊C4—21no—
39C22×C4⋊C4C2×C411yescharacteristic
40C22×C4⋊C4C2×C412yes—
41C22×C22⋊C4—21no—
42C22×C22⋊C4—22no—
43C22×C22⋊C4—21no—
44C22×C22⋊C4D412yes—
45C22×C42C2×C412yes—
46C22×C42D411yescharacteristic
47C22×C42D411yescharacteristic
48#423—44no—
49C2×C4⋊C8—24no—
50#407—22no—
51#407—22no—
52#407—22no—
53#407—22no—
54C2×C22⋊C8—22no—
55C2×C22⋊C8D414yes—
56#391—44no—
57#381—22no—
58#381—22no—
59#381—22no—
60#381D414yes—
61C4×C22⋊C4—22no—
62C4×C22⋊C4—22no—
63C2×#95—22no—
64C2×#95—22no—
65C2×#95—22no—
66C2×#95—22no—
67C2×#95—22no—
68C2×#95—22no—
69C2×#95—21no—
70#342—44no—
71#336—24no—
72#328—216no—
73C25C2×D411yescharacteristic
74C22×Q8—21no—
75C22×Q8—22no—
76C22×Q8—22no—
77C22×Q8—42no—
78C22×Q8—42no—
79C22×Q8C4⋊C412yes—
80C23×C4—21no—
81C23×C4—21no—
82C23×C4—22no—
83C23×C4—42no—
84C23×C4—42no—
85C23×C4—42no—
86C23×C4C22⋊C412yes—
87C23×C4C22⋊C412yes—
88C23×C4C22×C411yescharacteristic
89C23×C4C2×D411yescharacteristic
90C23×C4C2×D411yescharacteristic
91C23×C4C2×D412yes—
92C23×C4C2×D412yes—
93C23×C4C2×Q811yescharacteristic
94C23×C4#4111yescharacteristic
95C23×C4#4111yescharacteristic
96C23×C4#4112yes—
97C23×C4#4112yes—
98C2×M16—44no—
99C2×M16—44no—
100C2×M16—42no—
101C2×M16—42no—
102C22×C8—24no—
103C22⋊Q8—24no—
104C22⋊Q8—24no—
105C22⋊Q8—24no—
106C22⋊Q8—24no—
107C22⋊Q8—42no—
108C22⋊Q8—42no—
109C22⋊Q8—42no—
110C22⋊Q8—42no—
111#117—48no—
112#117—48no—
113C2×C4⋊C4—22no—
114C2×C4⋊C4—22no—
115C2×C4⋊C4—24no—
116C2×C4⋊C4—21no—
117C2×C4⋊C4—24no—
118C2×C4⋊C4—22no—
119C2×C4⋊C4—22no—
120C2×C4⋊C4—42no—
121C2×C4⋊C4—41no—
122C2×C4⋊C4—41no—
123C2×C4⋊C4—41no—
124C2×C4⋊C4—41no—
125C2×C4⋊C4C4⋊C412yes—
126C2×C22⋊C4—22no—
127C2×C22⋊C4—24no—
128C2×C22⋊C4—22no—
129C2×C22⋊C4—24no—
130C2×C22⋊C4—24no—
131C2×C22⋊C4—42no—
132C2×C22⋊C4—44no—
133C2×C22⋊C4—42no—
134C2×C22⋊C4—42no—
135C2×C22⋊C4—42no—
136C2×C22⋊C4—44no—
137C2×C22⋊C4—44no—
138C2×C22⋊C4—42no—
139C2×C42—24no—
140C2×C42—21no—
141C2×C42—21no—
142C2×C42—21no—
143C2×C42—21no—
144C2×C42—24no—
145C2×C42—42no—
146C2×C42—42no—
147C2×C42—42no—
148C4⋊C8—48no—
149C22⋊C8—24no—
150C22⋊C8—24no—
151C22⋊C8—42no—
152C22⋊C8—42no—
153#95—24no—
154#95—24no—
155#95—44no—
156#95—44no—
157#95—44no—
158#95—44no—
159#95—42no—
160#95—41no—
161#95—42no—
162#95—41no—
163C24—22no—
164C24—22no—
165C24—42no—
166C24—42no—
167C24C2×C22⋊C411yescharacteristic
168C24#12112yes—
169C24#12111yescharacteristic
170C24#12112yes—
171C24C22⋊Q811yescharacteristic
172C2×Q8—24no—
173C2×Q8—44no—
174C2×Q8—44no—
175C2×Q8—42no—
176C2×Q8—82no—
177C2×Q8—82no—
178C2×Q8—84no—
179C22×C4—22no—
180C22×C4—22no—
181C22×C4—22no—
182C22×C4—22no—
183C22×C4—22no—
184C22×C4—22no—
185C22×C4—24no—
186C22×C4—22no—
187C22×C4—22no—
188C22×C4—22no—
189C22×C4—22no—
190C22×C4—22no—
191C22×C4—42no—
192C22×C4—42no—
193C22×C4—42no—
194C22×C4—42no—
195C22×C4—44no—
196C22×C4—42no—
197C22×C4—42no—
198C22×C4—42no—
199C22×C4—42no—
200C22×C4—44no—
201C22×C4—42no—
202C22×C4—42no—
203C22×C4—42no—
204C22×C4—42no—
205C22×C4—42no—
206C22×C4—42no—
207C22×C4—42no—
208C22×C4—42no—
209C22×C4—42no—
210C22×C4—41no—
211C22×C4—41no—
212C22×C4—42no—
213C22×C4—41no—
214C22×C4—41no—
215C22×C4—42no—
216C22×C4—42no—
217C22×C4—42no—
218C22×C4—42no—
219C22×C4—44no—
220C22×C4—82no—
221C22×C4—82no—
222C22×C4—84no—
223C22×C4—84no—
224C22×C4#9914yes—
225C22×C4C2×C4⋊C411yescharacteristic
226C22×C4#11711yescharacteristic
227C22×C4C4×D412yes—
228C22×C4C4×D412yes—
229C22×C4#12011yescharacteristic
230C22×C4#12012yes—
231C22×C4#12111yescharacteristic
232C22×C4#12112yes—
233C22×C4C22⋊Q811yescharacteristic
234C22×C4C22⋊Q812yes—
235C22×C4C22⋊Q812yes—
236C22×C4#12311yescharacteristic
237C22×C4#12312yes—
238C22×C4#12312yes—
239C22×C4#12411yescharacteristic
240C22×C4#12412yes—
241C22×C4#12612yes—
242C22×C4#12711yescharacteristic
243M16—84no—
244M16—84no—
245C2×C8—44no—
246C2×C8—44no—
247C2×C8—44no—
248C4⋊C4—24no—
249C4⋊C4—48no—
250C4⋊C4—42no—
251C4⋊C4—42no—
252C4⋊C4—42no—
253C4⋊C4—42no—
254C4⋊C4—42no—
255C22⋊C4—48no—
256C22⋊C4—48no—
257C22⋊C4—44no—
258C22⋊C4—44no—
259C22⋊C4—48no—
260C22⋊C4—84no—
261C42—24no—
262C42—24no—
263C42—42no—
264C42—42no—
265C42—84no—
266C23—22no—
267C23—22no—
268C23—24no—
269C23—22no—
270C23—22no—
271C23—24no—
272C23—22no—
273C23—22no—
274C23—24no—
275C23—42no—
276C23—41no—
277C23—42no—
278C23—41no—
279C23—41no—
280C23—41no—
281C23—82no—
282C23—82no—
283C23—82no—
284C23—82no—
285C23#38011yescharacteristic
286C23#38611yescharacteristic
287C23#39011yescharacteristic
288C23#39412yes—
289C23#39612yes—
290C23C2×#9912yes—
291C23#41012yes—
292Q8—84no—
293Q8—84no—
294C2×C4—22no—
295C2×C4—22no—
296C2×C4—22no—
297C2×C4—22no—
298C2×C4—21no—
299C2×C4—21no—
300C2×C4—22no—
301C2×C4—21no—
302C2×C4—22no—
303C2×C4—21no—
304C2×C4—24no—
305C2×C4—22no—
306C2×C4—22no—
307C2×C4—24no—
308C2×C4—42no—
309C2×C4—42no—
310C2×C4—42no—
311C2×C4—42no—
312C2×C4—42no—
313C2×C4—42no—
314C2×C4—42no—
315C2×C4—42no—
316C2×C4—42no—
317C2×C4—42no—
318C2×C4—44no—
319C2×C4—44no—
320C2×C4—44no—
321C2×C4—44no—
322C2×C4—42no—
323C2×C4—42no—
324C2×C4—44no—
325C2×C4—44no—
326C2×C4—42no—
327C2×C4—42no—
328C2×C4—42no—
329C2×C4—42no—
330C2×C4—42no—
331C2×C4—82no—
332C2×C4—82no—
333C2×C4—82no—
334C2×C4—82no—
335C2×C4—82no—
336C2×C4—82no—
337C2×C4—81no—
338C2×C4—81no—
339C2×C4—82no—
340C2×C4—82no—
341C2×C4—82no—
342C2×C4—84no—
343C2×C4—84no—
344C2×C4—84no—
345C2×C4—164no—
346C2×C4—164no—
347C2×C4#38512yes—
348C2×C4#38812yes—
349C2×C4#39211yescharacteristic
350C2×C4#39311yescharacteristic
351C2×C4#39711yescharacteristic
352C2×C4#39911yescharacteristic
353C8—84no—
354C22—21no—
355C22—21no—
356C22—21no—
357C22—21no—
358C22—22no—
359C22—22no—
360C22—22no—
361C22—22no—
362C22—22no—
363C22—22no—
364C22—22no—
365C22—22no—
366C22—22no—
367C22—22no—
368C22—44no—
369C22—44no—
370C22—84no—
371C22—82no—
372C22—82no—
373C22#143911yescharacteristic
374C22#179912yes—
375C22#188412yes—
376C22#189811yescharacteristic
377C22#189911yescharacteristic
378C22#190912yes—
379C22#196412yes—
380C4—21no—
381C4—22no—
382C4—22no—
383C4—21no—
384C4—21no—
385C4—21no—
386C4—44no—
387C4—84no—
388C4—82no—
389C4—82no—
390C4—82no—
391C4—164no—
392C2—22no—
393C2—22no—
394C2—42no—
395C2—42no—
396C2#944312yes—
397C2#944412yes—
398C2#1104311yescharacteristic
399C2#1105611yescharacteristic
400C2#1107411yescharacteristic
subgroupG/H
center, socleC23#386
commutatorC23×C4C22×C4
Frattini, cosocleC24×C4C23
prime radicalC25C2×D4
LCSG ▷ C23×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C24×C4 ◁ G
LECSG ▷ C24×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
10. C2×#391C4semidirect
22. C4⋊#95C4semidirect

Maximal subgroups

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

subgroupindex
#206272
C2×#15112
#141382
#107262
C2×#12842