Finite Group Databasebeta

← #6498965

Subgroups — #6498965

Order 512.

1220 conjugacy classes of subgroups.

The subgroup lattice has 401 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#37567C212yesmaximal
2#33292C212yesmaximal, complemented
3#33262C212yesmaximal, complemented
4#30315C214yesmaximal, complemented
5#28795C214yesmaximal, complemented
6#28794C214yesmaximal, complemented
7#28780C212yesmaximal
8#28779C212yesmaximal
9#28773C214yesmaximal
10#21989C212yesmaximal
11#20818C211yesmaximal, characteristic
12#20760C211yesmaximal, characteristic
13#20476C211yesmaximal, characteristic
14C2×#432C2211yescharacteristic
15C22×C4×C8C2211yescharacteristic
16#2841—24no—
17#2667—22no—
18#2666—24no—
19#2665—22no—
20#2660C2214yes—
21#2647C2214yes—
22#2606—22no—
23#2605—22no—
24#2570C2214yes—
25#2550C2214yes—
26#2441C2214yes—
27#2439C2214yes—
28#2285C2212yes—
29#2283C2212yes—
30#2283C2211yescharacteristic
31#2085C2214yes—
32#2085C2214yes—
33#2084C2214yes—
34#2083C2214yes—
35#2059C2214yes—
36#2059C2214yes—
37#2058C2214yes—
38#2057C2214yes—
39#1993C2214yes—
40#1992C2214yes—
41#1991C2214yes—
42#1990C2214yes—
43#1949C2214yes—
44#1949C2214yes—
45#1944C2214yes—
46#1943C2214yes—
47#1942C2214yes—
48#1941C2214yes—
49#1854C2214yes—
50#1854C2214yes—
51#1847C2214yes—
52#1846C2212yes—
53#1836C2214yes—
54#1835C2211yescharacteristic
55#1835C2212yes—
56#1835C2212yes—
57#1835C2211yescharacteristic
58#1834C2212yes—
59#1834C2212yes—
60#1808C2212yes—
61#1807C2212yes—
62C4×#107C2212yes—
63C4×#106C2212yes—
64#1770C2212yes—
65C8×C4⋊C4C2212yes—
66#1752C2212yes—
67C8×C22⋊C4C2212yes—
68#1746C2212yes—
69#1735C2212yes—
70#1735C2211yescharacteristic
71#1735C2211yescharacteristic
72#1726C2212yes—
73C23×C8C2311yescharacteristic
74C23×C8C2311yescharacteristic
75C2×C4⋊Q8C2314yes—
76C2×#125C2314yes—
77C2×#117C2312yes—
78C2×#117C2312yes—
79C22×C42C2311yescharacteristic
80#432Q818yescomplemented
81C2×#107C2314yes—
82C2×#107C2314yes—
83C2×#107C2314yes—
84C2×#106C2314yes—
85C2×#106C2314yes—
86C2×#106C2314yes—
87C2×C4⋊C8C2312yes—
88C2×C22⋊C8C2312yes—
89C2×#97C2314yes—
90C2×#97C2312yes—
91C2×C4×C8Q818yescomplemented
92C2×C4×C8C2314yes—
93C2×C4×C8C2311yescharacteristic
94C2×C4×C8C2311yescharacteristic
95C2×C4×C8C2311yescharacteristic
96C2×C4×C8C2311yescharacteristic
97#400—24no—
98#400—24no—
99#400—24no—
100#399—24no—
101#398—24no—
102#398—24no—
103#398—24no—
104#398—24no—
105#398—24no—
106#398—24no—
107#396—24no—
108#396—24no—
109#395—24no—
110#395—24no—
111#395—24no—
112#393—24no—
113#391—24no—
114#391—24no—
115#389—24no—
116#389—24no—
117#389C2314yes—
118#389C2314yes—
119#389C2314yes—
120#389C2314yes—
121#387—24no—
122#387—24no—
123#387—24no—
124#387—24no—
125#384C2314yes—
126#384C2314yes—
127#383—22no—
128#383D412yes—
129#382D412yes—
130#382C2314yes—
131#382C2314yes—
132#382C2312yes—
133#382C2312yes—
134#381—22no—
135#381D411yescharacteristic
136#381D411yescharacteristic
137#380—22no—
138#380—22no—
139#380D412yes—
140#380C2314yes—
141C4×C4⋊C4—22no—
142C4×C4⋊C4—22no—
143C4×C4⋊C4C2314yes—
144C4×C4⋊C4C2312yes—
145C4×C22⋊C4—22no—
146C4×C22⋊C4C2312yes—
147#376—22no—
148#376C2312yes—
149#340C2314yes—
150#340C2314yes—
151#340C2312yes—
152#340C2314yes—
153#340C2312yes—
154#340C2314yes—
155#340C2314yes—
156#340C2312yes—
157#340C2312yes—
158#340C2312yes—
159#340C2312yes—
160#340C2314yes—
161#340C2312yes—
162#340C2312yes—
163#336C2314yes—
164#336C2314yes—
165C22×Q8—24no—
166C23×C4C2×D411yescharacteristic
167C23×C4C2×D411yescharacteristic
168C23×C4C2411yescharacteristic
169C22×C8—42no—
170C22×C8—42no—
171C22×C8C2×Q814yes—
172C22×C8C2×Q814yes—
173C22×C8C2411yescharacteristic
174C22×C8C2411yescharacteristic
175C22×C8C2411yescharacteristic
176C22×C8C2411yescharacteristic
177C22×C8C2414yes—
178C4⋊Q8—44no—
179C4⋊Q8—44no—
180#125—44no—
181#125—44no—
182#117—42no—
183#117—42no—
184#117—42no—
185#117—42no—
186C2×C4⋊C4—24no—
187C2×C4⋊C4—24no—
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⋊C4C2×D414yes—
198C2×C4⋊C4C2414yes—
199C2×C4⋊C4C2414yes—
200C2×C4⋊C4C2414yes—
201C2×C4⋊C4C2414yes—
202C2×C22⋊C4—22no—
203C2×C22⋊C4—22no—
204C2×C22⋊C4—24no—
205C2×C22⋊C4C2×D412yes—
206C2×C22⋊C4C2×D412yes—
207C2×C42—22no—
208C2×C42—22no—
209C2×C42—24no—
210C2×C42C2×D411yescharacteristic
211C2×C42C2×D411yescharacteristic
212C2×C42C2×Q814yes—
213C2×C42C2×Q814yes—
214C2×C42C2414yes—
215C2×C42C2411yescharacteristic
216C2×C42C2411yescharacteristic
217#107—44no—
218#107—44no—
219#107—44no—
220#106—44no—
221#106—44no—
222#106—44no—
223C4⋊C8C2×Q818yes—
224C22⋊C8C2×Q818yes—
225#97—44no—
226#97C2×Q818yes—
227C4×C8—44no—
228C4×C8C2×Q814yes—
229C4×C8C2×Q814yes—
230C4×C8C2×Q814yes—
231C4×C8C2×Q814yes—
232#95—24no—
233#95—22no—
234#95—22no—
235#95—22no—
236#95—22no—
237#95—24no—
238#95—24no—
239#95—24no—
240#95—24no—
241#95—24no—
242#95—24no—
243#95—24no—
244#95—24no—
245#95—24no—
246#95C2×D414yes—
247#95C2×D412yes—
248#95C2×D412yes—
249#95C2×D414yes—
250#95C2×D412yes—
251#95C2×D412yes—
252C24C22×D411yescharacteristic
253C2×Q8—84no—
254C2×Q8—84no—
255C2×Q8—84no—
256C22×C4—22no—
257C22×C4—22no—
258C22×C4—24no—
259C22×C4—24no—
260C22×C4—24no—
261C22×C4—24no—
262C22×C4—42no—
263C22×C4—42no—
264C22×C4C4⋊Q814yes—
265C22×C4C4⋊Q814yes—
266C22×C4C22×D414yes—
267C22×C4C22×D414yes—
268C22×C4C22×D411yescharacteristic
269C22×C4C22×D411yescharacteristic
270C22×C4C22×D411yescharacteristic
271C22×C4C22×D411yescharacteristic
272C22×C4C22×D411yescharacteristic
273C22×C4C22×Q812yes—
274C22×C4C22×Q812yes—
275C22×C4C2511yescharacteristic
276C2×C8—22no—
277C2×C8—24no—
278C2×C8—22no—
279C2×C8—24no—
280C2×C8—22no—
281C2×C8—22no—
282C2×C8—44no—
283C2×C8—44no—
284C2×C8C22×Q812yes—
285C2×C8C22×Q812yes—
286C2×C8C22×Q812yes—
287C2×C8C22×Q812yes—
288C2×C8C22×Q818yes—
289C4⋊C4—24no—
290C4⋊C4—24no—
291C4⋊C4—24no—
292C4⋊C4—24no—
293C4⋊C4—24no—
294C4⋊C4—24no—
295C4⋊C4—24no—
296C4⋊C4—24no—
297C4⋊C4—44no—
298C4⋊C4—84no—
299C4⋊C4—84no—
300C4⋊C4—84no—
301C4⋊C4—84no—
302C4⋊C4—84no—
303C4⋊C4—84no—
304C4⋊C4—84no—
305C4⋊C4—84no—
306C4⋊C4—84no—
307C4⋊C4—84no—
308C4⋊C4—84no—
309C22⋊C4—42no—
310C22⋊C4—42no—
311C22⋊C4—84no—
312C22⋊C4—82no—
313C22⋊C4—82no—
314C42—24no—
315C42—24no—
316C42—84no—
317C42—82no—
318C42—82no—
319C42C4⋊Q814yes—
320C42C4⋊Q814yes—
321C42C22×Q812yes—
322C42C22×Q812yes—
323C42C22×Q812yes—
324C42C22×Q812yes—
325C23—22no—
326C23—22no—
327C23—22no—
328C23C2×C4⋊Q812yes—
329C23C23×D411yescharacteristic
330Q8—168no—
331C2×C4—22no—
332C2×C4—22no—
333C2×C4—22no—
334C2×C4—22no—
335C2×C4—22no—
336C2×C4—22no—
337C2×C4—22no—
338C2×C4—24no—
339C2×C4—24no—
340C2×C4—24no—
341C2×C4—24no—
342C2×C4—24no—
343C2×C4—24no—
344C2×C4—24no—
345C2×C4—44no—
346C2×C4—44no—
347C2×C4—44no—
348C2×C4—44no—
349C2×C4—44no—
350C2×C4—44no—
351C2×C4—44no—
352C2×C4—44no—
353C2×C4—44no—
354C2×C4—44no—
355C2×C4—44no—
356C2×C4—44no—
357C2×C4—44no—
358C2×C4—44no—
359C2×C4C2×C4⋊Q812yes—
360C2×C4C2×C4⋊Q812yes—
361C2×C4C2×C4⋊Q812yes—
362C2×C4C2×C4⋊Q812yes—
363C2×C4C2×C4⋊Q812yes—
364C2×C4C23×Q811yescharacteristic
365C2×C4C23×Q811yescharacteristic
366C2×C4#58512yes—
367C2×C4#58512yes—
368C8—22no—
369C8—22no—
370C8—22no—
371C8—22no—
372C8—28no—
373C22—22no—
374C22—22no—
375C22—22no—
376C22—44no—
377C22C22×C4⋊Q811yescharacteristic
378C22#350012yes—
379C22#358412yes—
380C22#358412yes—
381C4—22no—
382C4—22no—
383C4—22no—
384C4—22no—
385C4—22no—
386C4—44no—
387C4—44no—
388C4—84no—
389C4—84no—
390C4—84no—
391C4—84no—
392C4#345811yescharacteristic
393C4#345811yescharacteristic
394C4#350412yes—
395C2—42no—
396C2#3533011yescharacteristic
397C2#6066011yescharacteristic
398C2#6066011yescharacteristic
399C2#6255414yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC24C22×D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ #2283 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #33292C2semidirect
3. #33262C2semidirect
4. #30315C2semidirect
5. #28795C2semidirect
6. #28794C2semidirect
80. #432Q8semidirect
91. C2×C4×C8Q8semidirect

Maximal subgroups

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

subgroupindex
#375672
#332922
#332622
#303152
#287952
#287942
#287802
#287792
#287732
#219892
#208182
#207602
#204762