Finite Group Databasebeta

← #6595445

Subgroups — #6595445

Order 512.

1450 conjugacy classes of subgroups.

The subgroup lattice has 402 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36210C212yesmaximal
2#33311C211yesmaximal, characteristic, complemented
3#33207C212yesmaximal, complemented
4#29220C212yesmaximal, complemented
5#28707C214yesmaximal
6#27878C212yesmaximal, complemented
7#27877C212yesmaximal, complemented
8#27836C218yesmaximal, complemented
9#23159C214yesmaximal, complemented
10C4×#498C211yesmaximal, characteristic, complemented
11#20816C212yesmaximal, complemented
12C4×C8×D4C211yesmaximal, characteristic
13C2×#499C2214yes—
14C2×#498C2211yescharacteristic
15C2×C8×D4C2212yes—
16C22×C4×C8C2212yes—
17#2859C2212yes—
18#2857C2212yes—
19#2857C2212yes—
20#2562C2214yes—
21#2514—22no—
22#2509—22no—
23#2495C2214yes—
24#2494—22no—
25#2494C2214yes—
26#2485—22no—
27#2421—22no—
28#2415—22no—
29#2370C2214yes—
30#2362—22no—
31#2347—22no—
32C4×C4⋊Q8C2212yes—
33#2285C2212yes—
34#2285C2212yes—
35C42×D4C2211yescharacteristic
36#2101C2218yes—
37#2076C2218yes—
38#2075C2218yes—
39#1990C2212yes—
40#1990C2212yes—
41#1990C2212yes—
42#1989C2218yes—
43#1941C2218yes—
44#1941C2214yes—
45#1921C2218yes—
46#1918C2218yes—
47#1846C2214yes—
48#1845C2211yescharacteristic
49#1845C2211yescharacteristic
50#1844C2212yes—
51#1844C2211yescharacteristic
52#1834C2212yes—
53#1834C2212yes—
54#1807C2218yes—
55#1807C2218yes—
56C4×#106C2214yes—
57C4×#106C2218yes—
58C8×C4⋊C4C2212yes—
59C4×C4⋊C8C2211yescharacteristic
60C8×C22⋊C4C2214yes—
61C4×C22⋊C8C2212yes—
62C42×C8C2211yescharacteristic
63C23×C8C2314yes—
64C2×C4⋊Q8C2312yes—
65C2×#125C2314yes—
66C2×C22⋊Q8—22no—
67C2×C22⋊Q8—22no—
68C2×C4×Q8—22no—
69C2×C4×D4D412yes—
70C2×C4×D4C2311yescharacteristic
71C2×#117—22no—
72C2×#117—22no—
73C22×C42C2312yes—
74#499—216no—
75#498—24no—
76C8×D4—28no—
77C2×#107C2314yes—
78C2×#107C2314yes—
79C2×#106C2318yes—
80C2×#106C2314yes—
81C2×#106C2314yes—
82C2×C4⋊C8C2312yes—
83C2×C22⋊C8C2314yes—
84C2×C4×C8—28no—
85C2×C4×C8C2312yes—
86C2×C4×C8C2311yescharacteristic
87C2×C4×C8C2314yes—
88C2×C4×C8C2312yes—
89C2×C4×C8C2311yescharacteristic
90C2×C4×C8C2312yes—
91#401—24no—
92#400—24no—
93#400—24no—
94#397—24no—
95#397—24no—
96#394—24no—
97#391—22no—
98#391—22no—
99#391—22no—
100#389—24no—
101#389—22no—
102#389—22no—
103#389—22no—
104#389—22no—
105#389—22no—
106#389—22no—
107#389—22no—
108#389C2314yes—
109#389C2314yes—
110#389C2314yes—
111#388—24no—
112#388—24no—
113#388—24no—
114#388—24no—
115#387—24no—
116#387—24no—
117#387—24no—
118#384C2312yes—
119#384C2312yes—
120#384C2312yes—
121#383—24no—
122#382—22no—
123#382—22no—
124#382C2314yes—
125#380—22no—
126#380—22no—
127#380—22no—
128#380—22no—
129#380C2314yes—
130#380C2314yes—
131#380C2314yes—
132#380C2314yes—
133C4×C4⋊C4—22no—
134C4×C4⋊C4—22no—
135C4×C4⋊C4—22no—
136C4×C4⋊C4—24no—
137C4×C4⋊C4D412yes—
138C4×C4⋊C4C2314yes—
139C4×C4⋊C4C2314yes—
140C4×C4⋊C4C2314yes—
141C4×C4⋊C4C2311yescharacteristic
142C4×C22⋊C4D412yes—
143C4×C22⋊C4D412yes—
144C4×C22⋊C4C2312yes—
145C43C2311yescharacteristic
146#340C2318yes—
147#340C2318yes—
148#340C2318yes—
149#340C2318yes—
150#340C2318yes—
151#336C2312yes—
152#336C2314yes—
153C22×Q8—22no—
154C22×D4C2×D411yescharacteristic
155C23×C4C2×D412yes—
156C23×C4C2×D412yes—
157C23×C4C2412yes—
158C22×C8—28no—
159C22×C8—28no—
160C22×C8—28no—
161C22×C8C2414yes—
162C22×C8C2412yes—
163C22×C8C2412yes—
164C4⋊Q8—24no—
165C4⋊Q8—24no—
166#125—216no—
167C22⋊Q8—48no—
168C22⋊Q8—84no—
169C4×Q8—84no—
170C4×D4—28no—
171C4×D4—24no—
172#117—48no—
173#117—84no—
174C2×C4⋊C4—22no—
175C2×C4⋊C4—22no—
176C2×C4⋊C4—22no—
177C2×C4⋊C4—22no—
178C2×C4⋊C4—22no—
179C2×C4⋊C4—24no—
180C2×C4⋊C4—24no—
181C2×C4⋊C4—24no—
182C2×C4⋊C4—22no—
183C2×C4⋊C4—22no—
184C2×C4⋊C4—24no—
185C2×C4⋊C4—24no—
186C2×C4⋊C4C2×D412yes—
187C2×C4⋊C4C2×D411yescharacteristic
188C2×C4⋊C4C2414yes—
189C2×C4⋊C4C2414yes—
190C2×C4⋊C4C2414yes—
191C2×C4⋊C4C2414yes—
192C2×C22⋊C4—24no—
193C2×C22⋊C4—24no—
194C2×C22⋊C4—24no—
195C2×C22⋊C4—24no—
196C2×C22⋊C4C2×D412yes—
197C2×C22⋊C4C2×D412yes—
198C2×C22⋊C4C2×D412yes—
199C2×C42—24no—
200C2×C42—24no—
201C2×C42—24no—
202C2×C42—24no—
203C2×C42—24no—
204C2×C42C2×D412yes—
205C2×C42C2×D412yes—
206C2×C42C2×D412yes—
207C2×C42C2×D412yes—
208C2×C42C2411yescharacteristic
209C2×C42C2412yes—
210C2×C42C2411yescharacteristic
211C2×C42C2411yescharacteristic
212#107—28no—
213#107—28no—
214#106—28no—
215#106—216no—
216#106—28no—
217C4⋊C8—24no—
218C22⋊C8—28no—
219C4×C8—28no—
220C4×C8—24no—
221C4×C8—24no—
222C4×C8—22no—
223C4×C8#4114yes—
224C4×C8#4118yes—
225#95—24no—
226#95—24no—
227#95—24no—
228#95—24no—
229#95—24no—
230#95—24no—
231#95—24no—
232#95—24no—
233#95—24no—
234#95—24no—
235#95—24no—
236#95—24no—
237#95C2×D411yescharacteristic
238#95C2×D411yescharacteristic
239#95C2×D412yes—
240#95C2×D412yes—
241C24C22×D412yes—
242C2×Q8—44no—
243C2×Q8—44no—
244C2×Q8—44no—
245C2×D4—24no—
246C2×D4—22no—
247C22×C4—24no—
248C22×C4—24no—
249C22×C4—24no—
250C22×C4—24no—
251C22×C4—24no—
252C22×C4—24no—
253C22×C4—22no—
254C22×C4—22no—
255C22×C4—24no—
256C22×C4—24no—
257C22×C4—24no—
258C22×C4—24no—
259C22×C4—24no—
260C22×C4#13518yes—
261C22×C4C22×D412yes—
262C22×C4C22×D412yes—
263C22×C4C22×D412yes—
264C22×C4C22×D411yescharacteristic
265C22×C4C22×D411yescharacteristic
266C22×C4C22×D412yes—
267C22×C4C22×D412yes—
268C22×C4C2511yescharacteristic
269C2×C8—22no—
270C2×C8—22no—
271C2×C8—24no—
272C2×C8—24no—
273C2×C8—24no—
274C2×C8—416no—
275C2×C8C2×#4114yes—
276C2×C8C2×#4114yes—
277C2×C8C2×#4114yes—
278C2×C82−1+414yes—
279C4⋊C4—22no—
280C4⋊C4—24no—
281C4⋊C4—24no—
282C4⋊C4—28no—
283C4⋊C4—28no—
284C4⋊C4—28no—
285C4⋊C4—24no—
286C4⋊C4—48no—
287C4⋊C4—48no—
288C4⋊C4—44no—
289C4⋊C4—44no—
290C4⋊C4—44no—
291C4⋊C4—44no—
292C4⋊C4—48no—
293C4⋊C4—44no—
294C4⋊C4—44no—
295C4⋊C4—44no—
296C4⋊C4—84no—
297C4⋊C4—84no—
298C22⋊C4—24no—
299C22⋊C4—24no—
300C22⋊C4—24no—
301C22⋊C4—48no—
302C22⋊C4—48no—
303C22⋊C4—84no—
304C22⋊C4—84no—
305C42—24no—
306C42—24no—
307C42—24no—
308C42—24no—
309C42—22no—
310C42—22no—
311C42—22no—
312C42—48no—
313C42—84no—
314C42—84no—
315C42C2×#4114yes—
316C42C2×#4112yes—
317C422−1+412yes—
318C23—22no—
319C23—22no—
320C23—22no—
321C23—22no—
322C23—22no—
323C23—22no—
324C23C2×#13514yes—
325C23C23×D411yescharacteristic
326Q8—88no—
327D4—28no—
328C2×C4—22no—
329C2×C4—22no—
330C2×C4—22no—
331C2×C4—22no—
332C2×C4—216no—
333C2×C4—22no—
334C2×C4—22no—
335C2×C4—22no—
336C2×C4—22no—
337C2×C4—22no—
338C2×C4—22no—
339C2×C4—22no—
340C2×C4—22no—
341C2×C4—22no—
342C2×C4—48no—
343C2×C4—44no—
344C2×C4—44no—
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#53212yes—
358C2×C4#53212yes—
359C2×C4#54714yes—
360C2×C4C2×#13514yes—
361C2×C4C2×#13514yes—
362C2×C4#57714yes—
363C2×C4C22×#4111yescharacteristic
364C2×C4C22×#4112yes—
365C2×C4C2×2−1+412yes—
366C2×C4C2×2−1+411yescharacteristic
367C8—24no—
368C8—44no—
369C8#55618yes—
370C22—24no—
371C22—24no—
372C22—24no—
373C22—42no—
374C22—42no—
375C22—42no—
376C22—42no—
377C22C2×#53211yescharacteristic
378C22C2×#54711yescharacteristic
379C22#349911yescharacteristic
380C22C22×#13512yes—
381C22C2×#57712yes—
382C4—22no—
383C4—22no—
384C4—24no—
385C4—42no—
386C4—42no—
387C4—42no—
388C4—84no—
389C4—84no—
390C4—84no—
391C4—84no—
392C4#333718yes—
393C4C2×#55612yes—
394C4#350912yes—
395C2—44no—
396C2#3618811yescharacteristic
397C2#6067311yescharacteristic
398C2#6071611yescharacteristic
399C2C2×#333712yes—
400C2#6247012yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC22×D4C2×D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C42×D4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #33311C2semidirect
3. #33207C2semidirect
4. #29220C2semidirect
6. #27878C2semidirect
7. #27877C2semidirect
8. #27836C2semidirect
9. #23159C2semidirect
10. C4×#498C2semidirect
11. #20816C2semidirect

Maximal subgroups

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

subgroupindex
#362102
#333112
#332072
#292202
#287072
#278782
#278772
#278362
#231592
C4×#4982
#208162
C4×C8×D42