Finite Group Databasebeta

← #6490707

Subgroups — #6490707

Order 512.

1432 conjugacy classes of subgroups.

The subgroup lattice has 421 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35284C211yesmaximal, characteristic
2#34216C211yesmaximal, characteristic
3#33157C214yesmaximal
4#30102C214yesmaximal, complemented
5#29284C214yesmaximal, complemented
6#29220C212yesmaximal, complemented
7#28687C212yesmaximal, complemented
8#27962C212yesmaximal
9#27621C214yesmaximal
10#25811C212yesmaximal
11#22246C212yesmaximal, complemented
12#22246C212yesmaximal, complemented
13#20782C211yesmaximal, characteristic
14C22×C4⋊Q8C2211yescharacteristic
15C2×#498C2212yes—
16C2×#498C2212yes—
17#2868C2211yescharacteristic
18#2859C2211yescharacteristic
19#2857C2212yes—
20#2647C2212yes—
21#2629C2212yes—
22#2507—22no—
23#2497—22no—
24#2446—22no—
25#2429—22no—
26#2369—22no—
27#2365—22no—
28#2345—22no—
29Q8×C22⋊C4—22no—
30#2331C2212yes—
31#2331C2212yes—
32#2330C2212yes—
33#2316C2212yes—
34#2314C2212yes—
35C4×C4⋊Q8C2212yes—
36C4×C4⋊Q8C2212yes—
37C4×C4⋊Q8C2212yes—
38#2285C2211yescharacteristic
39#2274C2212yes—
40C4×#117C2211yescharacteristic
41#2101C2218yes—
42#2101C2218yes—
43#2084C2218yes—
44#2058C2218yes—
45#1990C2212yes—
46#1990C2212yes—
47#1990C2212yes—
48#1990C2212yes—
49#1989C2218yes—
50#1948C2218yes—
51#1920C2218yes—
52#1920C2218yes—
53#1919C2218yes—
54#1919C2218yes—
55#1853C2218yes—
56#1840C2212yes—
57#1834C2212yes—
58#1834C2212yes—
59#1832C2212yes—
60#1825C2218yes—
61#1818C2212yes—
62#1798C2212yes—
63C4×C4⋊C8C2212yes—
64C4×C22⋊C8C2212yes—
65#1744C2211yescharacteristic
66#1744C2211yescharacteristic
67C23×Q8—21no—
68C2×C4⋊Q8—28no—
69C2×C4⋊Q8—21no—
70C2×C4⋊Q8—21no—
71C2×C4⋊Q8—22no—
72C2×C4⋊Q8C2312yes—
73C2×C4⋊Q8C2312yes—
74C2×C4⋊Q8C2312yes—
75C2×#125C2312yes—
76C2×C22⋊Q8—22no—
77C2×C4×Q8—22no—
78C2×C4×Q8—22no—
79C2×C4×Q8—22no—
80C2×#117C2312yes—
81C2×#117C2311yescharacteristic
82C2×#117C2311yescharacteristic
83C22×C4⋊C4—21no—
84C22×C4⋊C4C2312yes—
85C22×C42C2311yescharacteristic
86#498—28no—
87#498—28no—
88C2×#106C2318yes—
89C2×#106C2318yes—
90C2×C4⋊C8C2314yes—
91C2×C22⋊C8C2314yes—
92C2×C4×C8C2312yes—
93C2×C4×C8C2312yes—
94#400—24no—
95#398—24no—
96#398—24no—
97#396—24no—
98#391—22no—
99#391—22no—
100#391—22no—
101#391—22no—
102#391—24no—
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—24no—
112#389C2314yes—
113#389C2314yes—
114#389C2314yes—
115#389C2314yes—
116#388—24no—
117#387—24no—
118#387—24no—
119#387—24no—
120#387—24no—
121#384—22no—
122#384C2312yes—
123#384C2312yes—
124#384C2312yes—
125#384C2312yes—
126#382C2312yes—
127#382C2312yes—
128#380—21no—
129#380—21no—
130#380—22no—
131#380—24no—
132#380C2314yes—
133C4×C4⋊C4—24no—
134C4×C4⋊C4—22no—
135C4×C4⋊C4—22no—
136C4×C4⋊C4—22no—
137C4×C4⋊C4—21no—
138C4×C4⋊C4—21no—
139C4×C4⋊C4D412yes—
140C4×C4⋊C4D412yes—
141C4×C4⋊C4C2314yes—
142C4×C4⋊C4C2314yes—
143C4×C4⋊C4C2314yes—
144C4×C22⋊C4—24no—
145C4×C22⋊C4D412yes—
146C4×C22⋊C4D412yes—
147#376C2311yescharacteristic
148#376C2311yescharacteristic
149C43C2311yescharacteristic
150C43C2311yescharacteristic
151#340C2318yes—
152#340C2318yes—
153#340C2318yes—
154#340C2318yes—
155#340C2318yes—
156#340C2318yes—
157#336C2312yes—
158#336C2312yes—
159#336C2314yes—
160#336C2314yes—
161#336C2312yes—
162#336C2312yes—
163C22×Q8—28no—
164C22×Q8—22no—
165C22×Q8—22no—
166C22×Q8—22no—
167C22×Q8—84no—
168C23×C4—22no—
169C23×C4C2×D411yescharacteristic
170C23×C4C2×D411yescharacteristic
171C23×C4C2411yescharacteristic
172C22×C8C2414yes—
173C22×C8C2414yes—
174C4⋊Q8—28no—
175C4⋊Q8—28no—
176C4⋊Q8—28no—
177C4⋊Q8—24no—
178C4⋊Q8—24no—
179C4⋊Q8—24no—
180C4⋊Q8—84no—
181C4⋊Q8#4118yes—
182#125—28no—
183C22⋊Q8—84no—
184C4×Q8—84no—
185C4×Q8—84no—
186C4×Q8—84no—
187#117—24no—
188#117—24no—
189#117—28no—
190C2×C4⋊C4—28no—
191C2×C4⋊C4—22no—
192C2×C4⋊C4—22no—
193C2×C4⋊C4—22no—
194C2×C4⋊C4—28no—
195C2×C4⋊C4—22no—
196C2×C4⋊C4—22no—
197C2×C4⋊C4—24no—
198C2×C4⋊C4—22no—
199C2×C4⋊C4—22no—
200C2×C4⋊C4—22no—
201C2×C4⋊C4—24no—
202C2×C4⋊C4—24no—
203C2×C4⋊C4—24no—
204C2×C4⋊C4—24no—
205C2×C4⋊C4—24no—
206C2×C4⋊C4—24no—
207C2×C4⋊C4C2×D412yes—
208C2×C4⋊C4C2×D412yes—
209C2×C4⋊C4C2414yes—
210C2×C4⋊C4C2414yes—
211C2×C4⋊C4C2414yes—
212C2×C4⋊C4C2414yes—
213C2×C22⋊C4—22no—
214C2×C22⋊C4—24no—
215C2×C22⋊C4C2×D412yes—
216C2×C22⋊C4C2×D412yes—
217C2×C42—24no—
218C2×C42—22no—
219C2×C42—24no—
220C2×C42—24no—
221C2×C42—24no—
222C2×C42C2×D412yes—
223C2×C42C2×D412yes—
224C2×C42C2×D412yes—
225C2×C42C2×D412yes—
226C2×C42C2×D411yescharacteristic
227C2×C42C2×D411yescharacteristic
228C2×C42C2411yescharacteristic
229C2×C42C2411yescharacteristic
230C2×C42C2411yescharacteristic
231C2×C42C2411yescharacteristic
232C2×C42C2411yescharacteristic
233C2×C42C2411yescharacteristic
234#106—216no—
235#106—216no—
236C4⋊C8—28no—
237C22⋊C8—28no—
238C4×C8—24no—
239C4×C8—24no—
240#95—24no—
241#95—24no—
242#95—22no—
243#95—22no—
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×D412yes—
254#95C2×D412yes—
255C24C22×D411yescharacteristic
256C2×Q8—28no—
257C2×Q8—28no—
258C2×Q8—44no—
259C2×Q8—44no—
260C2×Q8—44no—
261C2×Q8—44no—
262C2×Q8—44no—
263C2×Q8—84no—
264C2×Q8—816no—
265C22×C4—22no—
266C22×C4—22no—
267C22×C4—24no—
268C22×C4—28no—
269C22×C4—22no—
270C22×C4—22no—
271C22×C4—24no—
272C22×C4—24no—
273C22×C4—24no—
274C22×C4—24no—
275C22×C4—24no—
276C22×C4—84no—
277C22×C4C22×D411yescharacteristic
278C22×C4C22×D412yes—
279C22×C4C22×D412yes—
280C22×C4C22×D412yes—
281C22×C4C22×D411yescharacteristic
282C22×C4C22×D411yescharacteristic
283C22×C4C22×D412yes—
284C22×C4C22×D411yescharacteristic
285C22×C4C22×D411yescharacteristic
286C22×C42−1+414yes—
287C22×C4C2511yescharacteristic
288C2×C8—24no—
289C2×C8—24no—
290C2×C8—24no—
291C2×C8—24no—
292C2×C8—24no—
293C2×C8—24no—
294C4⋊C4—28no—
295C4⋊C4—28no—
296C4⋊C4—28no—
297C4⋊C4—24no—
298C4⋊C4—24no—
299C4⋊C4—24no—
300C4⋊C4—24no—
301C4⋊C4—24no—
302C4⋊C4—28no—
303C4⋊C4—28no—
304C4⋊C4—44no—
305C4⋊C4—48no—
306C4⋊C4—44no—
307C4⋊C4—44no—
308C4⋊C4—44no—
309C4⋊C4—44no—
310C4⋊C4—84no—
311C4⋊C4—84no—
312C4⋊C4—84no—
313C4⋊C4—84no—
314C4⋊C4—84no—
315C4⋊C4—84no—
316C4⋊C4C2×#4118yes—
317C22⋊C4—24no—
318C22⋊C4—24no—
319C22⋊C4—44no—
320C22⋊C4—84no—
321C42—22no—
322C42—22no—
323C42—24no—
324C42—24no—
325C42—24no—
326C42—24no—
327C42—22no—
328C42—22no—
329C42—22no—
330C42—22no—
331C42—44no—
332C42—84no—
333C42—84no—
334C42—84no—
335C42C2×#4114yes—
336C422−1+414yes—
337C23—21no—
338C23—21no—
339C23—22no—
340C23—22no—
341C23#53612yes—
342C23C23×D411yescharacteristic
343Q8—416no—
344Q8—88no—
345Q8—88no—
346C2×C4—21no—
347C2×C4—21no—
348C2×C4—22no—
349C2×C4—22no—
350C2×C4—28no—
351C2×C4—21no—
352C2×C4—21no—
353C2×C4—28no—
354C2×C4—22no—
355C2×C4—22no—
356C2×C4—22no—
357C2×C4—22no—
358C2×C4—22no—
359C2×C4—22no—
360C2×C4—22no—
361C2×C4—22no—
362C2×C4—22no—
363C2×C4—22no—
364C2×C4—22no—
365C2×C4—44no—
366C2×C4—44no—
367C2×C4—44no—
368C2×C4—44no—
369C2×C4—44no—
370C2×C4—44no—
371C2×C4—44no—
372C2×C4—44no—
373C2×C4—44no—
374C2×C4—44no—
375C2×C4—44no—
376C2×C4—44no—
377C2×C4—44no—
378C2×C4—88no—
379C2×C4#52512yes—
380C2×C4#52512yes—
381C2×C4#53612yes—
382C2×C4#53614yes—
383C2×C4#57714yes—
384C2×C4#57714yes—
385C2×C4C22×#4112yes—
386C2×C4C2×2−1+412yes—
387C2×C4C2×2−1+412yes—
388C8—44no—
389C8—44no—
390C22—22no—
391C22—24no—
392C22—41no—
393C22—41no—
394C22—42no—
395C22C2×#52511yescharacteristic
396C22C2×#53611yescharacteristic
397C22C2×#53611yescharacteristic
398C22C2×#57712yes—
399C22C2×#57712yes—
400C4—22no—
401C4—24no—
402C4—41no—
403C4—48no—
404C4—41no—
405C4—42no—
406C4—42no—
407C4—42no—
408C4—84no—
409C4—84no—
410C4—84no—
411C4#312314yes—
412C4#312614yes—
413C4#352314yes—
414C2—42no—
415C2#3524811yescharacteristic
416C2C2×#312311yescharacteristic
417C2C2×#312611yescharacteristic
418C2#6088112yes—
419C2#6088112yes—
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
4. #30102C2semidirect
5. #29284C2semidirect
6. #29220C2semidirect
7. #28687C2semidirect
11. #22246C2semidirect
12. #22246C2semidirect

Maximal subgroups

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

subgroupindex
#352842
#342162
#331572
#301022
#292842
#292202
#286872
#279622
#276212
#258112
#222462
#207822