Finite Group Databasebeta

← #6437275

Subgroups — #6437275

Order 512.

1300 conjugacy classes of subgroups.

The subgroup lattice has 409 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35304C212yesmaximal
2#30315C214yesmaximal, complemented
3#30314C214yesmaximal, complemented
4#30294C214yesmaximal
5#30293C212yesmaximal
6#30292C212yesmaximal
7#23845C214yesmaximal, complemented
8#23843C212yesmaximal, complemented
9#23842C212yesmaximal, complemented
10#20823C212yesmaximal
11#20818C211yesmaximal, characteristic
12#20816C211yesmaximal, characteristic
13C2×#1746C211yesmaximal, characteristic
14C22×#97C2211yescharacteristic
15C22×C4×C8C2211yescharacteristic
16#2857C2214yes—
17#2851C2214yes—
18#2660C2214yes—
19#2647C2214yes—
20#2612—24no—
21#2606—22no—
22#2605—22no—
23#2569—22no—
24#2549—22no—
25#2438—24no—
26#2331C2214yes—
27#2330C2214yes—
28#2285C2212yes—
29#2283C2212yes—
30C2×#376C2211yescharacteristic
31#2085C2214yes—
32#2085C2214yes—
33#2085C2214yes—
34#2085C2214yes—
35#2084C2214yes—
36#2084C2214yes—
37#2084C2214yes—
38#2083C2214yes—
39#2059C2214yes—
40#2059C2214yes—
41#2059C2214yes—
42#2059C2214yes—
43#2058C2214yes—
44#2058C2214yes—
45#2058C2214yes—
46#2057C2214yes—
47#1847C2214yes—
48#1846C2212yes—
49#1844C2212yes—
50#1836C2214yes—
51#1835C2212yes—
52#1834C2212yes—
53#1808C2212yes—
54#1808C2214yes—
55#1807C2212yes—
56#1807C2212yes—
57#1807C2211yescharacteristic
58#1807C2211yescharacteristic
59#1807C2214yes—
60#1776C2214yes—
61#1776C2214yes—
62C4×#107C2212yes—
63C4×#106C2211yescharacteristic
64C4×#106C2211yescharacteristic
65C4×#106C2212yes—
66C4×#106C2212yes—
67#1746C2212yes—
68#1746C2212yes—
69#1746C2212yes—
70#1746C2212yes—
71C2×#336C2212yes—
72C2×#336C2212yes—
73C23×C8C2311yescharacteristic
74C23×C8C2311yescharacteristic
75C23×C8C2312yes—
76C2×C4⋊Q8C2314yes—
77C2×#125C2314yes—
78C2×C22⋊Q8—24no—
79C2×#117—22no—
80C22×C42C2312yes—
81C22×C42C2311yescharacteristic
82C2×#107C2314yes—
83C2×#107C2314yes—
84C2×#107C2314yes—
85C2×#106C2312yes—
86C2×#106C2314yes—
87C2×#106C2312yes—
88C2×#106C2314yes—
89C2×#106C2314yes—
90C2×#106C2312yes—
91C2×#106C2312yes—
92C2×#97Q818yescomplemented
93C2×#97C2312yes—
94C2×#97C2312yes—
95C2×C4×C8Q818yescomplemented
96C2×C4×C8C2311yescharacteristic
97C2×C4×C8C2311yescharacteristic
98C2×C4×C8C2311yescharacteristic
99C2×C4×C8C2311yescharacteristic
100#400—24no—
101#400—24no—
102#398—24no—
103#398—24no—
104#398—24no—
105#398—24no—
106#395—24no—
107#395—24no—
108#391—24no—
109#391—24no—
110#391—24no—
111#389—24no—
112#389—24no—
113#389—24no—
114#389C2314yes—
115#389C2314yes—
116#389C2314yes—
117#389C2314yes—
118#387—24no—
119#387—24no—
120#387—24no—
121#387—24no—
122#387—24no—
123#387—24no—
124#385—24no—
125#385—24no—
126#384C2314yes—
127#384C2314yes—
128#383—22no—
129#382—22no—
130#382C2314yes—
131#382C2314yes—
132#381—22no—
133#380—24no—
134#380—22no—
135#380C2314yes—
136#380C2312yes—
137#380C2312yes—
138C4×C4⋊C4—22no—
139C4×C4⋊C4C2312yes—
140C4×C4⋊C4C2314yes—
141C4×C4⋊C4C2312yes—
142C4×C22⋊C4—22no—
143#376—22no—
144#376D411yescharacteristic
145#376D411yescharacteristic
146#376D412yes—
147#376C2312yes—
148#376C2312yes—
149C2×#95D412yes—
150C2×#95D412yes—
151#340C2312yes—
152#340C2314yes—
153#340C2312yes—
154#340C2314yes—
155#340C2314yes—
156#340C2314yes—
157#340C2314yes—
158#340C2312yes—
159#340C2312yes—
160#340C2314yes—
161#336C2314yes—
162#336C2314yes—
163#336C2314yes—
164#336C2314yes—
165C22×Q8—24no—
166C23×C4C2×D411yescharacteristic
167C23×C4C2×D414yes—
168C23×C4C2×D411yescharacteristic
169C23×C4C2411yescharacteristic
170C22×C8—42no—
171C22×C8—42no—
172C22×C8—42no—
173C22×C8—42no—
174C22×C8C2×Q814yes—
175C22×C8C2×Q814yes—
176C22×C8C2×Q818yes—
177C22×C8C2411yescharacteristic
178C22×C8C2411yescharacteristic
179C22×C8C2411yescharacteristic
180C22×C8C2411yescharacteristic
181C22×C8C2414yes—
182C4⋊Q8—44no—
183C4⋊Q8—44no—
184#125—44no—
185#125—44no—
186C22⋊Q8—88no—
187#117—84no—
188C2×C4⋊C4—24no—
189C2×C4⋊C4—24no—
190C2×C4⋊C4—24no—
191C2×C4⋊C4—24no—
192C2×C4⋊C4—24no—
193C2×C4⋊C4—24no—
194C2×C4⋊C4—24no—
195C2×C4⋊C4—24no—
196C2×C4⋊C4—24no—
197C2×C4⋊C4—24no—
198C2×C4⋊C4—24no—
199C2×C4⋊C4C2414yes—
200C2×C4⋊C4C2414yes—
201C2×C4⋊C4C2414yes—
202C2×C4⋊C4C2414yes—
203C2×C22⋊C4—24no—
204C2×C22⋊C4—22no—
205C2×C22⋊C4—22no—
206C2×C42—22no—
207C2×C42—22no—
208C2×C42—24no—
209C2×C42—42no—
210C2×C42—42no—
211C2×C42C2×D412yes—
212C2×C42C2×D412yes—
213C2×C42C2×D411yescharacteristic
214C2×C42C2×D411yescharacteristic
215C2×C42C2×Q814yes—
216C2×C42C2×Q814yes—
217C2×C42C2414yes—
218C2×C42C2411yescharacteristic
219C2×C42C2411yescharacteristic
220#107—44no—
221#107—44no—
222#107—44no—
223#106—42no—
224#106—44no—
225#106—42no—
226#106—44no—
227#106—44no—
228#106—42no—
229#106—42no—
230#97C2×Q818yes—
231#97C2×Q818yes—
232C4×C8C2×Q814yes—
233C4×C8C2×Q814yes—
234C4×C8C2×Q814yes—
235C4×C8C2×Q814yes—
236#95—22no—
237#95—22no—
238#95—24no—
239#95—24no—
240#95—24no—
241#95—22no—
242#95—22no—
243#95—24no—
244#95—24no—
245#95—24no—
246#95—24no—
247#95—24no—
248#95—24no—
249#95—24no—
250#95C2×D412yes—
251#95C2×D414yes—
252#95C2×D412yes—
253#95C2×D412yes—
254#95C2×D414yes—
255#95C2×D412yes—
256C24C22×D411yescharacteristic
257C2×Q8—48no—
258C2×Q8—84no—
259C2×Q8—84no—
260C22×C4—24no—
261C22×C4—22no—
262C22×C4—22no—
263C22×C4—28no—
264C22×C4—24no—
265C22×C4—24no—
266C22×C4—24no—
267C22×C4—24no—
268C22×C4—42no—
269C22×C4—44no—
270C22×C4—44no—
271C22×C4C4⋊Q814yes—
272C22×C4C4⋊Q814yes—
273C22×C4C22×D414yes—
274C22×C4C22×D414yes—
275C22×C4C22×D411yescharacteristic
276C22×C4C22×D411yescharacteristic
277C22×C4C22×D411yescharacteristic
278C22×C4C22×D411yescharacteristic
279C22×C4C22×D411yescharacteristic
280C22×C4C22×Q812yes—
281C22×C4C22×Q812yes—
282C22×C4C2511yescharacteristic
283C2×C8—22no—
284C2×C8—22no—
285C2×C8—22no—
286C2×C8—24no—
287C2×C8—24no—
288C2×C8—22no—
289C2×C8—44no—
290C2×C8—44no—
291C2×C8—48no—
292C2×C8C22×Q812yes—
293C2×C8C22×Q812yes—
294C2×C8C22×Q812yes—
295C2×C8C22×Q812yes—
296C2×C8C22×Q818yes—
297C4⋊C4—24no—
298C4⋊C4—24no—
299C4⋊C4—24no—
300C4⋊C4—24no—
301C4⋊C4—24no—
302C4⋊C4—24no—
303C4⋊C4—24no—
304C4⋊C4—24no—
305C4⋊C4—48no—
306C4⋊C4—48no—
307C4⋊C4—48no—
308C4⋊C4—84no—
309C4⋊C4—84no—
310C4⋊C4—84no—
311C4⋊C4—84no—
312C4⋊C4—84no—
313C4⋊C4—84no—
314C4⋊C4—84no—
315C4⋊C4—84no—
316C22⋊C4—44no—
317C22⋊C4—48no—
318C22⋊C4—44no—
319C42—24no—
320C42—24no—
321C42—42no—
322C42—42no—
323C42—84no—
324C42—82no—
325C42—82no—
326C42C4⋊Q814yes—
327C42C4⋊Q814yes—
328C42C22×Q812yes—
329C42C22×Q812yes—
330C42C22×Q812yes—
331C42C22×Q812yes—
332C23—22no—
333C23—22no—
334C23C2×C4⋊Q812yes—
335C23#57714yes—
336C23C23×D411yescharacteristic
337Q8—168no—
338C2×C4—22no—
339C2×C4—22no—
340C2×C4—28no—
341C2×C4—22no—
342C2×C4—22no—
343C2×C4—22no—
344C2×C4—22no—
345C2×C4—22no—
346C2×C4—24no—
347C2×C4—24no—
348C2×C4—24no—
349C2×C4—24no—
350C2×C4—24no—
351C2×C4—24no—
352C2×C4—24no—
353C2×C4—44no—
354C2×C4—44no—
355C2×C4—44no—
356C2×C4—44no—
357C2×C4—44no—
358C2×C4—44no—
359C2×C4—44no—
360C2×C4—44no—
361C2×C4—44no—
362C2×C4—44no—
363C2×C4—44no—
364C2×C4—44no—
365C2×C4—44no—
366C2×C4—88no—
367C2×C4C2×C4⋊Q812yes—
368C2×C4C2×C4⋊Q812yes—
369C2×C4C2×C4⋊Q812yes—
370C2×C4C2×C4⋊Q812yes—
371C2×C4C2×C4⋊Q812yes—
372C2×C4C23×Q811yescharacteristic
373C2×C4C23×Q811yescharacteristic
374C2×C4#58512yes—
375C2×C4#58512yes—
376C8—22no—
377C8—22no—
378C8—22no—
379C8—28no—
380C8—22no—
381C22—28no—
382C22—22no—
383C22—22no—
384C22#316314yes—
385C22C22×C4⋊Q811yescharacteristic
386C22#344912yes—
387C22C2×#57712yes—
388C22#358412yes—
389C4—22no—
390C4—22no—
391C4—22no—
392C4—22no—
393C4—22no—
394C4—44no—
395C4—44no—
396C4—84no—
397C4—84no—
398C4—84no—
399C4—84no—
400C4#345811yescharacteristic
401C4#345811yescharacteristic
402C4#345812yes—
403C2—24no—
404C2#3464211yescharacteristic
405C2C2×#316311yescharacteristic
406C2#6066011yescharacteristic
407C2#6100114yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC24C22×D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×#376 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #30315C2semidirect
3. #30314C2semidirect
7. #23845C2semidirect
8. #23843C2semidirect
9. #23842C2semidirect
92. C2×#97Q8semidirect
95. C2×C4×C8Q8semidirect

Maximal subgroups

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

subgroupindex
#353042
#303152
#303142
#302942
#302932
#302922
#238452
#238432
#238422
#208232
#208182
#208162
C2×#17462