Finite Group Databasebeta

← #6467960

Subgroups — #6467960

Order 512.

1402 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#35235C212yesmaximal
2#28834C212yesmaximal, complemented
3#28815C212yesmaximal, complemented
4#28794C214yesmaximal, complemented
5#28772C212yesmaximal
6#28771C212yesmaximal
7#26068C214yesmaximal
8#23845C214yesmaximal, complemented
9#21730C214yesmaximal, complemented
10#21363C212yesmaximal
11#20818C211yesmaximal, characteristic
12#20476C211yesmaximal, characteristic
13C2×C4×#106C211yesmaximal, characteristic
14C2×#501C2214yes—
15C2×#432C2211yescharacteristic
16C22×#106C2212yes—
17C22×C4×C8C2211yescharacteristic
18#2855—24no—
19#2666—24no—
20#2606—22no—
21#2605—22no—
22#2570C2214yes—
23#2569—22no—
24#2550C2214yes—
25#2549—22no—
26#2442C2214yes—
27#2440C2214yes—
28#2331C2214yes—
29#2330C2214yes—
30#2283C2212yes—
31#2283C2211yescharacteristic
32C2×C4×C4⋊C4C2212yes—
33#2085C2214yes—
34#2085C2214yes—
35#2084C2214yes—
36#2084C2214yes—
37#2059C2214yes—
38#2059C2214yes—
39#2058C2214yes—
40#2058C2214yes—
41#1991C2214yes—
42#1990C2214yes—
43#1949C2214yes—
44#1949C2214yes—
45#1944C2214yes—
46#1943C2214yes—
47#1942C2214yes—
48#1941C2214yes—
49#1853C2214yes—
50#1852C2214yes—
51#1847C2214yes—
52#1846C2212yes—
53#1835C2212yes—
54#1808C2212yes—
55#1807C2211yescharacteristic
56#1807C2211yescharacteristic
57#1776C2214yes—
58#1776C2214yes—
59C4×#107C2212yes—
60C4×#106C2212yes—
61C4×#106C2212yes—
62C4×#106C2212yes—
63C4×#106C2211yescharacteristic
64C4×#106C2211yescharacteristic
65C4×#106C2212yes—
66#1770C2212yes—
67C8×C4⋊C4C2212yes—
68#1752C2212yes—
69C8×C22⋊C4C2212yes—
70#1746C2212yes—
71C2×#340C2212yes—
72#1726C2212yes—
73C23×C8C2311yescharacteristic
74C23×C8C2311yescharacteristic
75C2×C4⋊Q8C2314yes—
76C2×#125C2314yes—
77C2×C22⋊Q8—24no—
78C22×C4⋊C4C2314yes—
79C22×C42—22no—
80C22×C42C2311yescharacteristic
81#501—216no—
82#432Q818yescomplemented
83C2×#107C2314yes—
84C2×#107C2314yes—
85C2×#107C2314yes—
86C2×#106—28no—
87C2×#106C2314yes—
88C2×#106C2314yes—
89C2×#106C2312yes—
90C2×#106C2312yes—
91C2×#106C2314yes—
92C2×#106C2312yes—
93C2×#106C2312yes—
94C2×C4⋊C8C2312yes—
95C2×C22⋊C8C2312yes—
96C2×#97C2314yes—
97C2×#97C2312yes—
98C2×C4×C8Q818yescomplemented
99C2×C4×C8C2314yes—
100C2×C4×C8C2311yescharacteristic
101C2×C4×C8C2311yescharacteristic
102C2×C4×C8C2311yescharacteristic
103C2×C4×C8C2311yescharacteristic
104#400—24no—
105#400—24no—
106#398—24no—
107#398—24no—
108#398—24no—
109#398—24no—
110#395—24no—
111#395—24no—
112#391—24no—
113#391—24no—
114#391—24no—
115#389—24no—
116#389—24no—
117#389—24no—
118#389C2314yes—
119#389C2314yes—
120#389C2314yes—
121#389C2314yes—
122#387—24no—
123#387—24no—
124#387—24no—
125#387—24no—
126#387—24no—
127#387—24no—
128#385—24no—
129#385—24no—
130#384C2314yes—
131#383—22no—
132#383D411yescharacteristic
133#383D411yescharacteristic
134#382D412yes—
135#382C2314yes—
136#382C2314yes—
137#381—22no—
138#381D412yes—
139#380—24no—
140#380D412yes—
141#380C2312yes—
142#380C2312yes—
143C4×C4⋊C4—22no—
144C4×C4⋊C4—22no—
145C4×C4⋊C4C2312yes—
146C4×C4⋊C4C2314yes—
147C4×C4⋊C4C2314yes—
148C4×C4⋊C4C2312yes—
149C4×C4⋊C4C2312yes—
150C4×C22⋊C4—22no—
151C4×C22⋊C4C2312yes—
152#376—22no—
153#376C2312yes—
154C2×#95—22no—
155#340C2312yes—
156#340C2314yes—
157#340C2314yes—
158#340C2314yes—
159#340C2312yes—
160#340C2312yes—
161#340C2312yes—
162#340C2314yes—
163#340C2314yes—
164#340C2314yes—
165#336C2314yes—
166#336C2314yes—
167C22×Q8—24no—
168C23×C4—24no—
169C23×C4C2×D411yescharacteristic
170C23×C4C2×D411yescharacteristic
171C23×C4C2411yescharacteristic
172C22×C8—24no—
173C22×C8—42no—
174C22×C8C2×Q814yes—
175C22×C8C2×Q814yes—
176C22×C8C2411yescharacteristic
177C22×C8C2411yescharacteristic
178C22×C8C2411yescharacteristic
179C22×C8C2414yes—
180C22×C8C2411yescharacteristic
181C4⋊Q8—28no—
182C4⋊Q8—44no—
183#125—28no—
184#125—44no—
185C22⋊Q8—88no—
186C2×C4⋊C4—28no—
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⋊C4—44no—
198C2×C4⋊C4C2×D414yes—
199C2×C4⋊C4C2414yes—
200C2×C4⋊C4C2414yes—
201C2×C4⋊C4C2414yes—
202C2×C4⋊C4C2414yes—
203C2×C22⋊C4—22no—
204C2×C22⋊C4—22no—
205C2×C22⋊C4C2×D414yes—
206C2×C42—24no—
207C2×C42—22no—
208C2×C42—24no—
209C2×C42—22no—
210C2×C42—84no—
211C2×C42C2×D411yescharacteristic
212C2×C42C2×D411yescharacteristic
213C2×C42C2×Q814yes—
214C2×C42C2×Q814yes—
215C2×C42C2414yes—
216C2×C42C2411yescharacteristic
217C2×C42C2411yescharacteristic
218#107—28no—
219#107—28no—
220#107—44no—
221#106—24no—
222#106—28no—
223#106—28no—
224#106—42no—
225#106—44no—
226#106—42no—
227#106#4118yes—
228C4⋊C8C2×Q818yes—
229C22⋊C8C2×Q818yes—
230#97—28no—
231#97C2×Q818yes—
232C4×C8—44no—
233C4×C8C2×Q814yes—
234C4×C8C2×Q814yes—
235C4×C8C2×Q814yes—
236C4×C8C2×Q814yes—
237#95—24no—
238#95—22no—
239#95—22no—
240#95—22no—
241#95—22no—
242#95—24no—
243#95—24no—
244#95—24no—
245#95—24no—
246#95—24no—
247#95—24no—
248#95—24no—
249#95—24no—
250#95—24no—
251#95C2×D412yes—
252#95C2×D412yes—
253#95C2×D414yes—
254#95C2×D412yes—
255#95C2×D412yes—
256#95C2×D412yes—
257#95C2×D412yes—
258C24C22×D411yescharacteristic
259C2×Q8—48no—
260C2×Q8—84no—
261C2×Q8—84no—
262C22×C4—24no—
263C22×C4—22no—
264C22×C4—22no—
265C22×C4—24no—
266C22×C4—24no—
267C22×C4—24no—
268C22×C4—24no—
269C22×C4—42no—
270C22×C4—48no—
271C22×C4—48no—
272C22×C4—84no—
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—24no—
286C2×C8—22no—
287C2×C8—22no—
288C2×C8—24no—
289C2×C8—22no—
290C2×C8—28no—
291C2×C8—44no—
292C2×C8C22×Q812yes—
293C2×C8C22×Q812yes—
294C2×C8C22×Q812yes—
295C2×C8C22×Q818yes—
296C2×C8C22×Q812yes—
297C2×C8C2×#4114yes—
298C4⋊C4—24no—
299C4⋊C4—24no—
300C4⋊C4—24no—
301C4⋊C4—24no—
302C4⋊C4—24no—
303C4⋊C4—24no—
304C4⋊C4—24no—
305C4⋊C4—44no—
306C4⋊C4—48no—
307C4⋊C4—48no—
308C4⋊C4—48no—
309C4⋊C4—48no—
310C4⋊C4—84no—
311C4⋊C4—84no—
312C4⋊C4—84no—
313C4⋊C4—84no—
314C4⋊C4—84no—
315C4⋊C4—84no—
316C4⋊C4C2×#4118yes—
317C22⋊C4—28no—
318C22⋊C4—44no—
319C22⋊C4—44no—
320C42—24no—
321C42—24no—
322C42—84no—
323C42—82no—
324C42—84no—
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—28no—
352C2×C4—24no—
353C2×C4—24no—
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—816no—
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×C4C22×#4112yes—
375C2×C4#58512yes—
376C8—22no—
377C8—22no—
378C8—22no—
379C8—28no—
380C8#53314yes—
381C22—28no—
382C22—22no—
383C22—22no—
384C22#316314yes—
385C22C22×C4⋊Q811yescharacteristic
386C22D4×#4112yes—
387C22C2×#57712yes—
388C22#358412yes—
389C4—22no—
390C4—22no—
391C4—22no—
392C4—22no—
393C4—22no—
394C4—44no—
395C4—44no—
396C4—48no—
397C4—84no—
398C4—84no—
399C4—84no—
400C4C2×#53311yescharacteristic
401C4#345811yescharacteristic
402C4Q8×#4112yes—
403C2—24no—
404C2#3493811yescharacteristic
405C2C2×#316311yescharacteristic
406C2#6066011yescharacteristic
407C2#6149014yes—
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. #28834C2semidirect
3. #28815C2semidirect
4. #28794C2semidirect
8. #23845C2semidirect
9. #21730C2semidirect
82. #432Q8semidirect
98. C2×C4×C8Q8semidirect

Maximal subgroups

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

subgroupindex
#352352
#288342
#288152
#287942
#287722
#287712
#260682
#238452
#217302
#213632
#208182
#204762
C2×C4×#1062