Finite Group Databasebeta

← #6499343

Subgroups — #6499343

Order 512.

1312 conjugacy classes of subgroups.

The subgroup lattice has 407 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35294C211yesmaximal, characteristic
2#35286C211yesmaximal, characteristic
3#33263C212yesmaximal, complemented
4#33256C212yesmaximal, complemented
5#30307C212yesmaximal, complemented
6#30305C212yesmaximal, complemented
7#29327C214yesmaximal, complemented
8#29323C214yesmaximal, complemented
9#29311C214yesmaximal
10#29307C214yesmaximal
11#21984C212yesmaximal
12#21981C212yesmaximal
13#20462C211yesmaximal, characteristic
14C2×#431C2212yes—
15#2868C2211yescharacteristic
16#2859C2212yes—
17#2859C2212yes—
18#2859C2212yes—
19#2859C2211yescharacteristic
20#2647C2212yes—
21#2647C2212yes—
22#2629C2212yes—
23#2628C2212yes—
24#2571—22no—
25#2570—22no—
26#2552—22no—
27#2550—22no—
28#2442—22no—
29#2442—22no—
30#2441—22no—
31#2439—22no—
32#2331C2212yes—
33#2330C2212yes—
34#2317C2212yes—
35#2316C2212yes—
36#2285C2212yes—
37#2285C2211yescharacteristic
38C2×#384C2211yescharacteristic
39C4×#117C2211yescharacteristic
40#2085C2218yes—
41#2084C2218yes—
42#2059C2218yes—
43#2058C2218yes—
44#1949C2218yes—
45#1948C2218yes—
46#1944C2218yes—
47#1943C2218yes—
48#1941C2218yes—
49#1941C2218yes—
50#1854C2218yes—
51#1853C2218yes—
52#1845C2212yes—
53#1844C2212yes—
54#1844C2212yes—
55#1844C2212yes—
56#1834C2212yes—
57#1834C2212yes—
58#1834C2212yes—
59#1834C2212yes—
60#1768C2212yes—
61#1768C2212yes—
62#1750C2212yes—
63#1750C2212yes—
64#1744C2212yes—
65C4×#97C2211yescharacteristic
66C42×C8C2211yescharacteristic
67C2×C4⋊Q8—22no—
68C2×C4⋊Q8C2312yes—
69C2×#125—22no—
70C2×#125—22no—
71C2×#125—22no—
72C2×#125C2312yes—
73C2×#117—22no—
74C2×#117C2312yes—
75C2×#117C2311yescharacteristic
76C2×#117C2311yescharacteristic
77C22×C4⋊C4—22no—
78C22×C4⋊C4C2312yes—
79C22×C42C2311yescharacteristic
80#431Q8116yescomplemented
81C2×#107C2318yes—
82C2×#106C2318yes—
83C2×#106C2318yes—
84C2×#106C2318yes—
85C2×C4⋊C8C2314yes—
86C2×C22⋊C8C2314yes—
87C2×#97C2312yes—
88C2×#97C2312yes—
89C2×C4×C8C2312yes—
90C2×C4×C8C2312yes—
91C2×C4×C8C2312yes—
92C2×C4×C8C2312yes—
93#400—22no—
94#400—24no—
95#400—24no—
96#399—22no—
97#398—24no—
98#398—24no—
99#398—24no—
100#396—24no—
101#395—22no—
102#393—22no—
103#391—24no—
104#389—24no—
105#389—22no—
106#389—22no—
107#389—22no—
108#389—22no—
109#389—22no—
110#389—22no—
111#389—22no—
112#389—22no—
113#389—24no—
114#389C2314yes—
115#389C2314yes—
116#389C2314yes—
117#389C2314yes—
118#389C2314yes—
119#389C2314yes—
120#387—24no—
121#387—24no—
122#385—22no—
123#385—22no—
124#384—22no—
125#384—22no—
126#384—22no—
127#384C2312yes—
128#384C2312yes—
129#384C2312yes—
130#384C2312yes—
131#384C2312yes—
132#384C2312yes—
133#382—22no—
134#382—22no—
135#382—22no—
136#382—22no—
137#382—22no—
138#382C2312yes—
139#382C2312yes—
140#380—22no—
141#380—22no—
142#380—24no—
143#380C2314yes—
144C4×C4⋊C4—22no—
145C4×C4⋊C4—22no—
146C4×C4⋊C4D412yes—
147C4×C4⋊C4D412yes—
148C4×C4⋊C4C2314yes—
149C4×C22⋊C4D412yes—
150C4×C22⋊C4D412yes—
151#376C2311yescharacteristic
152#376C2311yescharacteristic
153C43C2311yescharacteristic
154C43C2311yescharacteristic
155#340C2318yes—
156#340C2318yes—
157#340C2318yes—
158#340C2318yes—
159#336C2314yes—
160#336C2314yes—
161C22×Q8—22no—
162C23×C4—22no—
163C23×C4C2×D412yes—
164C23×C4C2411yescharacteristic
165C22×C8C2414yes—
166C22×C8C2414yes—
167C4⋊Q8—44no—
168C4⋊Q8—84no—
169#125—44no—
170#125—84no—
171#125—84no—
172#125—84no—
173#117—42no—
174#117—42no—
175#117—44no—
176#117—84no—
177C2×C4⋊C4—22no—
178C2×C4⋊C4—22no—
179C2×C4⋊C4—22no—
180C2×C4⋊C4—22no—
181C2×C4⋊C4—22no—
182C2×C4⋊C4—22no—
183C2×C4⋊C4—22no—
184C2×C4⋊C4—24no—
185C2×C4⋊C4—22no—
186C2×C4⋊C4—22no—
187C2×C4⋊C4—22no—
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—44no—
197C2×C4⋊C4—84no—
198C2×C4⋊C4C2×D411yescharacteristic
199C2×C4⋊C4C2×D411yescharacteristic
200C2×C4⋊C4C2×D412yes—
201C2×C4⋊C4C2414yes—
202C2×C4⋊C4C2414yes—
203C2×C4⋊C4C2414yes—
204C2×C4⋊C4C2414yes—
205C2×C22⋊C4—22no—
206C2×C22⋊C4—24no—
207C2×C22⋊C4C2×D412yes—
208C2×C22⋊C4C2×D411yescharacteristic
209C2×C22⋊C4C2×D411yescharacteristic
210C2×C42—22no—
211C2×C42—24no—
212C2×C42C2×D412yes—
213C2×C42C2×D412yes—
214C2×C42C2×D412yes—
215C2×C42C2×D412yes—
216C2×C42C2×D412yes—
217C2×C42C2×Q818yes—
218C2×C42C2411yescharacteristic
219C2×C42C2411yescharacteristic
220C2×C42C2411yescharacteristic
221C2×C42C2411yescharacteristic
222C2×C42C2411yescharacteristic
223C2×C42C2411yescharacteristic
224#107—48no—
225#106—48no—
226#106—48no—
227#106—48no—
228C4⋊C8C2×Q8116yes—
229C22⋊C8C2×Q8116yes—
230#97—42no—
231#97—42no—
232C4×C8—42no—
233C4×C8—42no—
234C4×C8C2×Q818yes—
235C4×C8C2×Q818yes—
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—22no—
246#95—22no—
247#95C2×D412yes—
248#95C2×D412yes—
249#95C2×D412yes—
250#95C2×D412yes—
251C24C22×D411yescharacteristic
252C2×Q8—44no—
253C2×Q8—84no—
254C22×C4—24no—
255C22×C4—24no—
256C22×C4—22no—
257C22×C4—22no—
258C22×C4—24no—
259C22×C4—24no—
260C22×C4—24no—
261C22×C4—24no—
262C22×C4—44no—
263C22×C4—84no—
264C22×C4C4⋊Q818yes—
265C22×C4C22×D411yescharacteristic
266C22×C4C22×D412yes—
267C22×C4C22×D411yescharacteristic
268C22×C4C22×D411yescharacteristic
269C22×C4C22×D412yes—
270C22×C4C22×D411yescharacteristic
271C22×C4C22×D412yes—
272C22×C4C22×D411yescharacteristic
273C22×C4C22×D412yes—
274C22×C4C22×Q814yes—
275C22×C4C2511yescharacteristic
276C2×C8—24no—
277C2×C8—24no—
278C2×C8—24no—
279C2×C8—24no—
280C2×C8C22×Q818yes—
281C2×C8C22×Q818yes—
282C4⋊C4—28no—
283C4⋊C4—28no—
284C4⋊C4—28no—
285C4⋊C4—28no—
286C4⋊C4—42no—
287C4⋊C4—41no—
288C4⋊C4—41no—
289C4⋊C4—44no—
290C4⋊C4—48no—
291C4⋊C4—44no—
292C4⋊C4—44no—
293C4⋊C4—44no—
294C4⋊C4—44no—
295C4⋊C4—44no—
296C4⋊C4—44no—
297C4⋊C4—44no—
298C4⋊C4—44no—
299C4⋊C4—44no—
300C4⋊C4—84no—
301C4⋊C4—84no—
302C4⋊C4—84no—
303C4⋊C4—84no—
304C4⋊C4—84no—
305C4⋊C4—84no—
306C4⋊C4—84no—
307C4⋊C4—84no—
308C22⋊C4—42no—
309C22⋊C4—41no—
310C22⋊C4—41no—
311C22⋊C4—44no—
312C22⋊C4—84no—
313C42—24no—
314C42—24no—
315C42—24no—
316C42—24no—
317C42—22no—
318C42—22no—
319C42—22no—
320C42—22no—
321C42—44no—
322C42—84no—
323C42C4⋊Q818yes—
324C42C22×Q814yes—
325C42C22×Q814yes—
326C23—21no—
327C23—21no—
328C23—22no—
329C23—22no—
330C23C2×C4⋊Q812yes—
331C23C23×D411yescharacteristic
332Q8—164no—
333C2×C4—21no—
334C2×C4—21no—
335C2×C4—22no—
336C2×C4—22no—
337C2×C4—216no—
338C2×C4—28no—
339C2×C4—21no—
340C2×C4—22no—
341C2×C4—21no—
342C2×C4—21no—
343C2×C4—22no—
344C2×C4—21no—
345C2×C4—22no—
346C2×C4—22no—
347C2×C4—22no—
348C2×C4—22no—
349C2×C4—22no—
350C2×C4—22no—
351C2×C4—22no—
352C2×C4—22no—
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—164no—
366C2×C4C2×C4⋊Q812yes—
367C2×C4C2×C4⋊Q814yes—
368C2×C4C2×C4⋊Q814yes—
369C2×C4#57714yes—
370C2×C4#57714yes—
371C2×C4C23×Q812yes—
372C2×C4#58512yes—
373C2×C4#58512yes—
374C8—28no—
375C8—28no—
376C22—22no—
377C22—24no—
378C22—41no—
379C22—41no—
380C22—42no—
381C22C22×C4⋊Q811yescharacteristic
382C22#344911yescharacteristic
383C22#344911yescharacteristic
384C22C2×#57712yes—
385C22C2×#57712yes—
386C4—22no—
387C4—24no—
388C4—41no—
389C4—41no—
390C4—41no—
391C4—41no—
392C4—42no—
393C4—42no—
394C4—84no—
395C4—84no—
396C4—84no—
397C4—84no—
398C4#316318yes—
399C4#350414yes—
400C2—42no—
401C2#3533511yescharacteristic
402C2C2×#316312yes—
403C2#6101211yescharacteristic
404C2#6101211yescharacteristic
405C2#6101512yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC24C22×D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C4×#117 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #33263C2semidirect
4. #33256C2semidirect
5. #30307C2semidirect
6. #30305C2semidirect
7. #29327C2semidirect
8. #29323C2semidirect
80. #431Q8semidirect

Maximal subgroups

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

subgroupindex
#352942
#352862
#332632
#332562
#303072
#303052
#293272
#293232
#293112
#293072
#219842
#219812
#204622