Finite Group Databasebeta

← #6498966

Subgroups — #6498966

Order 512.

1276 conjugacy classes of subgroups.

The subgroup lattice has 418 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#37561C212yesmaximal
2#33292C212yesmaximal, complemented
3#33262C212yesmaximal, complemented
4#30314C214yesmaximal, complemented
5#28796C214yesmaximal, complemented
6#28794C214yesmaximal, complemented
7#28776C214yesmaximal
8#28772C212yesmaximal
9#28771C212yesmaximal
10#21986C212yesmaximal
11#20816C211yesmaximal, characteristic
12C2×#1835C211yesmaximal, characteristic
13#20476C211yesmaximal, characteristic
14C2×#432C2211yescharacteristic
15C22×C4×C8C2211yescharacteristic
16#2845—22no—
17#2844—22no—
18#2666—24no—
19#2660C2214yes—
20#2659—22no—
21#2647C2214yes—
22#2646—22no—
23#2570C2214yes—
24#2550C2214yes—
25#2442C2214yes—
26#2440C2214yes—
27#2438—24no—
28#2285C2212yes—
29#2283C2211yescharacteristic
30C2×#382C2212yes—
31#2085C2214yes—
32#2085C2214yes—
33#2084C2214yes—
34#2084C2214yes—
35#2059C2214yes—
36#2059C2214yes—
37#2058C2214yes—
38#2058C2214yes—
39#1993C2214yes—
40#1992C2214yes—
41#1991C2214yes—
42#1990C2214yes—
43#1949C2214yes—
44#1949C2214yes—
45#1944C2214yes—
46#1943C2214yes—
47#1942C2214yes—
48#1941C2214yes—
49#1853C2214yes—
50#1852C2214yes—
51#1847C2214yes—
52#1844C2212yes—
53#1836C2214yes—
54#1835C2212yes—
55#1835C2212yes—
56#1835C2212yes—
57#1835C2212yes—
58#1834C2211yescharacteristic
59#1834C2211yescharacteristic
60#1807C2212yes—
61#1807C2212yes—
62C4×#106C2212yes—
63C4×#106C2212yes—
64#1770C2212yes—
65C8×C4⋊C4C2212yes—
66#1752C2212yes—
67C8×C22⋊C4C2212yes—
68#1746C2212yes—
69C2×#340C2211yescharacteristic
70C2×#340C2211yescharacteristic
71C2×#340C2212yes—
72#1726C2212yes—
73C23×C8C2311yescharacteristic
74C23×C8C2311yescharacteristic
75C2×C4⋊Q8C2314yes—
76C2×#125C2314yes—
77C2×C22⋊Q8—24no—
78C2×#117—22no—
79C22×C4⋊C4C2312yes—
80C22×C4⋊C4C2312yes—
81C22×C42C2311yescharacteristic
82#432Q818yescomplemented
83C2×#107C2314yes—
84C2×#107C2314yes—
85C2×#106C2314yes—
86C2×#106C2314yes—
87C2×#106C2314yes—
88C2×#106C2314yes—
89C2×C4⋊C8C2312yes—
90C2×C22⋊C8C2312yes—
91C2×#97C2314yes—
92C2×#97C2312yes—
93C2×C4×C8Q818yescomplemented
94C2×C4×C8C2314yes—
95C2×C4×C8C2311yescharacteristic
96C2×C4×C8C2311yescharacteristic
97C2×C4×C8C2311yescharacteristic
98C2×C4×C8C2311yescharacteristic
99#400—24no—
100#400—24no—
101#400—24no—
102#400—24no—
103#398—24no—
104#398—24no—
105#398—24no—
106#398—24no—
107#398—24no—
108#398—24no—
109#398—24no—
110#398—24no—
111#395—24no—
112#395—24no—
113#395—24no—
114#395—24no—
115#391—24no—
116#389—24no—
117#389C2314yes—
118#389C2314yes—
119#389C2314yes—
120#389C2314yes—
121#387—24no—
122#387—24no—
123#385—24no—
124#385—24no—
125#384C2314yes—
126#384C2312yes—
127#384C2312yes—
128#383D412yes—
129#382—22no—
130#382—22no—
131#382—22no—
132#382D411yescharacteristic
133#382D411yescharacteristic
134#382C2314yes—
135#382C2314yes—
136#382C2314yes—
137#381D412yes—
138#380—22no—
139#380—24no—
140#380D412yes—
141#380C2314yes—
142C4×C4⋊C4—22no—
143C4×C4⋊C4C2314yes—
144C4×C4⋊C4C2312yes—
145C4×C22⋊C4C2312yes—
146#376C2312yes—
147C2×#95—22no—
148C2×#95—22no—
149#340C2312yes—
150#340C2314yes—
151#340C2314yes—
152#340C2314yes—
153#340C2312yes—
154#340C2314yes—
155#340C2314yes—
156#340C2312yes—
157#340C2312yes—
158#340C2312yes—
159#340C2312yes—
160#340C2312yes—
161#340C2312yes—
162#340C2314yes—
163#336C2314yes—
164#336C2314yes—
165C22×Q8—24no—
166C23×C4—22no—
167C23×C4—22no—
168C23×C4C2×D411yescharacteristic
169C23×C4C2×D411yescharacteristic
170C23×C4C2411yescharacteristic
171C22×C8—42no—
172C22×C8—42no—
173C22×C8C2×Q814yes—
174C22×C8C2×Q814yes—
175C22×C8C2411yescharacteristic
176C22×C8C2411yescharacteristic
177C22×C8C2411yescharacteristic
178C22×C8C2411yescharacteristic
179C22×C8C2414yes—
180C4⋊Q8—44no—
181C4⋊Q8—44no—
182#125—44no—
183#125—44no—
184C22⋊Q8—88no—
185#117—84no—
186C2×C4⋊C4—22no—
187C2×C4⋊C4—22no—
188C2×C4⋊C4—22no—
189C2×C4⋊C4—22no—
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—22no—
197C2×C4⋊C4—22no—
198C2×C4⋊C4—24no—
199C2×C4⋊C4—24no—
200C2×C4⋊C4—24no—
201C2×C4⋊C4—42no—
202C2×C4⋊C4—42no—
203C2×C4⋊C4—42no—
204C2×C4⋊C4—42no—
205C2×C4⋊C4C2×D412yes—
206C2×C4⋊C4C2×D412yes—
207C2×C4⋊C4C2414yes—
208C2×C4⋊C4C2414yes—
209C2×C4⋊C4C2414yes—
210C2×C4⋊C4C2414yes—
211C2×C22⋊C4—24no—
212C2×C22⋊C4C2×D414yes—
213C2×C42—24no—
214C2×C42C2×D411yescharacteristic
215C2×C42C2×D411yescharacteristic
216C2×C42C2×Q814yes—
217C2×C42C2×Q814yes—
218C2×C42C2414yes—
219C2×C42C2411yescharacteristic
220C2×C42C2411yescharacteristic
221#107—44no—
222#107—44no—
223#106—44no—
224#106—44no—
225#106—44no—
226#106—44no—
227C4⋊C8C2×Q818yes—
228C22⋊C8C2×Q818yes—
229#97—44no—
230#97C2×Q818yes—
231C4×C8—44no—
232C4×C8C2×Q814yes—
233C4×C8C2×Q814yes—
234C4×C8C2×Q814yes—
235C4×C8C2×Q814yes—
236#95—24no—
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#95—24no—
247#95—24no—
248#95C2×D414yes—
249#95C2×D412yes—
250#95C2×D412yes—
251#95C2×D412yes—
252#95C2×D412yes—
253#95C2×D414yes—
254C24C22×D411yescharacteristic
255C2×Q8—84no—
256C2×Q8—84no—
257C2×Q8—84no—
258C22×C4—24no—
259C22×C4—22no—
260C22×C4—22no—
261C22×C4—24no—
262C22×C4—24no—
263C22×C4—24no—
264C22×C4—24no—
265C22×C4—41no—
266C22×C4—41no—
267C22×C4—44no—
268C22×C4—44no—
269C22×C4—82no—
270C22×C4—82no—
271C22×C4—82no—
272C22×C4—82no—
273C22×C4C4⋊Q814yes—
274C22×C4C4⋊Q814yes—
275C22×C4C22×D414yes—
276C22×C4C22×D414yes—
277C22×C4C22×D411yescharacteristic
278C22×C4C22×D411yescharacteristic
279C22×C4C22×D411yescharacteristic
280C22×C4C22×D411yescharacteristic
281C22×C4C22×D411yescharacteristic
282C22×C4C22×Q812yes—
283C22×C4C22×Q812yes—
284C22×C4C2511yescharacteristic
285C2×C8—22no—
286C2×C8—24no—
287C2×C8—22no—
288C2×C8—24no—
289C2×C8—22no—
290C2×C8—22no—
291C2×C8—44no—
292C2×C8—44no—
293C2×C8C22×Q812yes—
294C2×C8C22×Q812yes—
295C2×C8C22×Q812yes—
296C2×C8C22×Q812yes—
297C2×C8C22×Q818yes—
298C4⋊C4—24no—
299C4⋊C4—24no—
300C4⋊C4—24no—
301C4⋊C4—24no—
302C4⋊C4—24no—
303C4⋊C4—24no—
304C4⋊C4—24no—
305C4⋊C4—24no—
306C4⋊C4—42no—
307C4⋊C4—42no—
308C4⋊C4—82no—
309C4⋊C4—82no—
310C4⋊C4—84no—
311C4⋊C4—84no—
312C4⋊C4—84no—
313C4⋊C4—82no—
314C4⋊C4—82no—
315C4⋊C4—82no—
316C4⋊C4—82no—
317C4⋊C4—84no—
318C4⋊C4—84no—
319C4⋊C4—84no—
320C4⋊C4—84no—
321C4⋊C4—84no—
322C4⋊C4—84no—
323C22⋊C4—28no—
324C22⋊C4—48no—
325C42—24no—
326C42—24no—
327C42—84no—
328C42C4⋊Q814yes—
329C42C4⋊Q814yes—
330C42C22×Q812yes—
331C42C22×Q812yes—
332C42C22×Q812yes—
333C42C22×Q812yes—
334C23—21no—
335C23—21no—
336C23—22no—
337C23C2×C4⋊Q812yes—
338C23#57714yes—
339C23C23×D411yescharacteristic
340Q8—168no—
341C2×C4—21no—
342C2×C4—21no—
343C2×C4—21no—
344C2×C4—21no—
345C2×C4—28no—
346C2×C4—22no—
347C2×C4—22no—
348C2×C4—22no—
349C2×C4—22no—
350C2×C4—22no—
351C2×C4—24no—
352C2×C4—24no—
353C2×C4—24no—
354C2×C4—24no—
355C2×C4—24no—
356C2×C4—24no—
357C2×C4—24no—
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—44no—
367C2×C4—44no—
368C2×C4—44no—
369C2×C4—44no—
370C2×C4—44no—
371C2×C4—164no—
372C2×C4—164no—
373C2×C4C2×C4⋊Q812yes—
374C2×C4C2×C4⋊Q812yes—
375C2×C4C2×C4⋊Q812yes—
376C2×C4C2×C4⋊Q812yes—
377C2×C4C2×C4⋊Q812yes—
378C2×C4C23×Q811yescharacteristic
379C2×C4C23×Q811yescharacteristic
380C2×C4#58512yes—
381C2×C4#58512yes—
382C8—22no—
383C8—22no—
384C8—22no—
385C8—22no—
386C8—28no—
387C22—22no—
388C22—24no—
389C22—24no—
390C22—22no—
391C22#316314yes—
392C22C22×C4⋊Q811yescharacteristic
393C22#350012yes—
394C22C2×#57712yes—
395C22#358411yescharacteristic
396C22#358411yescharacteristic
397C4—22no—
398C4—22no—
399C4—22no—
400C4—22no—
401C4—22no—
402C4—44no—
403C4—44no—
404C4—84no—
405C4—84no—
406C4—84no—
407C4—84no—
408C4#345811yescharacteristic
409C4#345811yescharacteristic
410C4#350412yes—
411C2—24no—
412C2#3533011yescharacteristic
413C2C2×#316311yescharacteristic
414C2#6066011yescharacteristic
415C2#6255612yes—
416C2#6255712yes—
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. #30314C2semidirect
5. #28796C2semidirect
6. #28794C2semidirect
82. #432Q8semidirect
93. C2×C4×C8Q8semidirect

Maximal subgroups

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

subgroupindex
#375612
#332922
#332622
#303142
#287962
#287942
#287762
#287722
#287712
#219862
#208162
C2×#18352
#204762