Finite Group Databasebeta

← #6667420

Subgroups — #6667420

Order 512.

1344 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#36635C212yesmaximal
2#32851C214yesmaximal, complemented
3#29942C214yesmaximal
4#29941C214yesmaximal
5#29879C214yesmaximal
6#29290C211yesmaximal, characteristic, complemented
7#29287C212yesmaximal, complemented
8#29286C211yesmaximal, characteristic, complemented
9#29222C214yesmaximal, complemented
10#28691C214yesmaximal, complemented
11#20808C211yesmaximal, characteristic
12C2×#499C2214yes—
13C2×#431C2212yes—
14#2864C2214yes—
15#2863—22no—
16#2863—22no—
17#2862C2212yes—
18#2859C2212yes—
19#2857C2212yes—
20#2857C2212yes—
21#2855—22no—
22#2855—22no—
23#2851C2214yes—
24C4×C4⋊Q8C2212yes—
25C4×C4⋊Q8C2212yes—
26C4×#125C2214yes—
27C4×C22⋊Q8—22no—
28C4×C22⋊Q8—22no—
29#2295—22no—
30#2295—22no—
31#2283C2214yes—
32C2×C43C2211yescharacteristic
33#2101C2218yes—
34#2101C2218yes—
35#2088C2218yes—
36#2087C2214yes—
37#2086C2214yes—
38#2085C2218yes—
39#2085C2218yes—
40#2077C2218yes—
41#2076C2214yes—
42#2075C2214yes—
43#1991C2218yes—
44#1988C2214yes—
45#1988C2214yes—
46#1987C2214yes—
47#1987C2214yes—
48#1948C2214yes—
49#1948C2214yes—
50#1947C2214yes—
51#1947C2214yes—
52#1842C2211yescharacteristic
53#1842C2211yescharacteristic
54#1835C2214yes—
55#1831C2212yes—
56#1797C2214yes—
57C4×C4⋊C8C2214yes—
58C2×C4⋊Q8C2312yes—
59C2×C4⋊Q8C2312yes—
60C2×#125C2314yes—
61C2×C22⋊Q8—22no—
62C2×C22⋊Q8—22no—
63C2×C4×Q8—22no—
64C2×C4×Q8—22no—
65C22×C42D411yescharacteristic
66C22×C42D411yescharacteristic
67C22×C42D412yes—
68C22×C42C2312yes—
69C22×C42C2311yescharacteristic
70#499—48no—
71#431—44no—
72C2×#107C2314yes—
73C2×#107C2314yes—
74C2×#106C2318yes—
75C2×C4⋊C8C2314yes—
76C2×C4⋊C8C2314yes—
77C2×C4⋊C8C2312yes—
78C2×C4⋊C8C2312yes—
79C2×C22⋊C8C2314yes—
80C2×C4×C8C2314yes—
81#391—22no—
82#391—22no—
83#391—22no—
84#391—22no—
85#391—22no—
86#391—22no—
87#391—22no—
88#391—22no—
89#389—22no—
90#389—22no—
91#389—22no—
92#389—22no—
93#389—22no—
94#389—22no—
95#389—22no—
96#389—22no—
97#389C2314yes—
98#389C2314yes—
99#389C2314yes—
100#389C2314yes—
101#387—24no—
102#387—24no—
103#387—24no—
104#387—24no—
105#387—24no—
106#387—24no—
107#387—24no—
108#387—24no—
109#385—24no—
110#385—24no—
111#384C2312yes—
112#384C2312yes—
113#383—24no—
114#382C2314yes—
115#382C2314yes—
116#382C2314yes—
117#381—24no—
118#380—22no—
119#380—22no—
120#380C2314yes—
121#380C2314yes—
122C4×C4⋊C4—24no—
123C4×C4⋊C4—24no—
124C4×C4⋊C4—22no—
125C4×C4⋊C4—22no—
126C4×C4⋊C4C2314yes—
127C4×C4⋊C4C2314yes—
128C4×C22⋊C4—24no—
129#376—24no—
130C43D411yescharacteristic
131C43D412yes—
132C43D411yescharacteristic
133C43C2312yes—
134C43C2311yescharacteristic
135C43C2311yescharacteristic
136#340C2318yes—
137#340C2314yes—
138#340C2318yes—
139#340C2314yes—
140#340C2318yes—
141#340C2318yes—
142#340C2318yes—
143#336C2318yes—
144C22×Q8—22no—
145C22×Q8—22no—
146C23×C4C2×D412yes—
147C23×C4C2×D411yescharacteristic
148C23×C4C2×D412yes—
149C23×C4C2×D411yescharacteristic
150C23×C4C2411yescharacteristic
151C22×C8C2414yes—
152C22×C8C2414yes—
153C4⋊Q8—24no—
154C4⋊Q8—24no—
155C4⋊Q8—24no—
156C4⋊Q8—24no—
157#125—28no—
158#125—28no—
159C22⋊Q8—48no—
160C22⋊Q8—48no—
161C4×Q8—48no—
162C4×Q8—48no—
163C2×C4⋊C4—22no—
164C2×C4⋊C4—22no—
165C2×C4⋊C4—24no—
166C2×C4⋊C4—24no—
167C2×C4⋊C4—22no—
168C2×C4⋊C4—22no—
169C2×C4⋊C4—24no—
170C2×C4⋊C4—24no—
171C2×C4⋊C4—24no—
172C2×C4⋊C4—24no—
173C2×C4⋊C4—22no—
174C2×C4⋊C4—22no—
175C2×C4⋊C4C2414yes—
176C2×C4⋊C4C2414yes—
177C2×C4⋊C4C2414yes—
178C2×C4⋊C4C2414yes—
179C2×C22⋊C4—24no—
180C2×C22⋊C4—24no—
181C2×C42—24no—
182C2×C42—22no—
183C2×C42—24no—
184C2×C42—24no—
185C2×C42—22no—
186C2×C42—24no—
187C2×C42—24no—
188C2×C42—44no—
189C2×C42—42no—
190C2×C42C2×D412yes—
191C2×C42C2×D412yes—
192C2×C42C2×D412yes—
193C2×C42C2×D411yescharacteristic
194C2×C42C2×D411yescharacteristic
195C2×C42C2×D412yes—
196C2×C42C2×D412yes—
197C2×C42C2×D411yescharacteristic
198C2×C42C2×D412yes—
199C2×C42C2×D411yescharacteristic
200C2×C42C2×D412yes—
201C2×C42C2×D412yes—
202C2×C42C2×D411yescharacteristic
203C2×C42C2×D411yescharacteristic
204C2×C42C2412yes—
205C2×C42C2411yescharacteristic
206C2×C42C2411yescharacteristic
207C2×C42C2412yes—
208#107—44no—
209#107—44no—
210#106—48no—
211C4⋊C8—28no—
212C4⋊C8—28no—
213C4⋊C8—42no—
214C4⋊C8—42no—
215C22⋊C8—44no—
216C4×C8—28no—
217#95—24no—
218#95—24no—
219#95—24no—
220#95—24no—
221#95—24no—
222#95—24no—
223#95—24no—
224#95—24no—
225#95—24no—
226#95—24no—
227#95—24no—
228#95—24no—
229#95—24no—
230#95—24no—
231C24C22×D411yescharacteristic
232C2×Q8—44no—
233C2×Q8—44no—
234C2×Q8—44no—
235C2×Q8—44no—
236C2×Q8—44no—
237C2×Q8—44no—
238C22×C4—22no—
239C22×C4—22no—
240C22×C4—22no—
241C22×C4—24no—
242C22×C4—24no—
243C22×C4—24no—
244C22×C4—22no—
245C22×C4—24no—
246C22×C4—24no—
247C22×C4—21no—
248C22×C4—22no—
249C22×C4—22no—
250C22×C4—21no—
251C22×C4—24no—
252C22×C4—24no—
253C22×C4—24no—
254C22×C4—24no—
255C22×C4—42no—
256C22×C4—42no—
257C22×C4C22×D411yescharacteristic
258C22×C4C22×D412yes—
259C22×C4C22×D412yes—
260C22×C4C22×D411yescharacteristic
261C22×C4C22×D412yes—
262C22×C4C22×D411yescharacteristic
263C22×C4C22×D411yescharacteristic
264C22×C4C22×D411yescharacteristic
265C22×C4C22×D412yes—
266C22×C4C2511yescharacteristic
267C2×C8—24no—
268C2×C8—24no—
269C2×C8—28no—
270C2×C8—28no—
271C4⋊C4—24no—
272C4⋊C4—24no—
273C4⋊C4—24no—
274C4⋊C4—24no—
275C4⋊C4—24no—
276C4⋊C4—24no—
277C4⋊C4—24no—
278C4⋊C4—24no—
279C4⋊C4—44no—
280C4⋊C4—48no—
281C4⋊C4—44no—
282C4⋊C4—48no—
283C4⋊C4—48no—
284C4⋊C4—44no—
285C4⋊C4—44no—
286C4⋊C4—48no—
287C4⋊C4—48no—
288C4⋊C4—44no—
289C4⋊C4—48no—
290C4⋊C4—44no—
291C22⋊C4—48no—
292C22⋊C4—48no—
293C42—24no—
294C42—24no—
295C42—22no—
296C42—22no—
297C42—24no—
298C42—22no—
299C42—24no—
300C42—24no—
301C42—21no—
302C42—21no—
303C42—22no—
304C42—22no—
305C42—42no—
306C42—41no—
307C42—42no—
308C42—41no—
309C42—42no—
310C42—48no—
311C42—48no—
312C42#13514yes—
313C42H(5,𝔽2)12yes—
314C422−1+414yes—
315C422−1+414yes—
316C422−1+412yes—
317C23—22no—
318C23—21no—
319C23—21no—
320C23—22no—
321C23#57512yes—
322C23C23×D411yescharacteristic
323Q8—88no—
324Q8—88no—
325C2×C4—28no—
326C2×C4—21no—
327C2×C4—21no—
328C2×C4—22no—
329C2×C4—22no—
330C2×C4—22no—
331C2×C4—21no—
332C2×C4—21no—
333C2×C4—22no—
334C2×C4—22no—
335C2×C4—22no—
336C2×C4—21no—
337C2×C4—21no—
338C2×C4—22no—
339C2×C4—22no—
340C2×C4—22no—
341C2×C4—22no—
342C2×C4—22no—
343C2×C4—48no—
344C2×C4—44no—
345C2×C4—48no—
346C2×C4—41no—
347C2×C4—41no—
348C2×C4—42no—
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—44no—
358C2×C4—44no—
359C2×C4—44no—
360C2×C4—44no—
361C2×C4#53614yes—
362C2×C4#53614yes—
363C2×C4#53712yes—
364C2×C4#53712yes—
365C2×C4C2×#13512yes—
366C2×C4C2×#13514yes—
367C2×C4#57514yes—
368C2×C4C2×H(5,𝔽2)11yescharacteristic
369C2×C4C2×2−1+412yes—
370C2×C4C2×2−1+412yes—
371C2×C4C2×2−1+411yescharacteristic
372C8—44no—
373C8—44no—
374C22—24no—
375C22—22no—
376C22—41no—
377C22—42no—
378C22—41no—
379C22C2×#53612yes—
380C22C2×#53711yescharacteristic
381C22C22×#13511yescharacteristic
382C22C2×#57512yes—
383C22C2×#57511yescharacteristic
384C4—24no—
385C4—22no—
386C4—22no—
387C4—22no—
388C4—41no—
389C4—42no—
390C4—42no—
391C4—41no—
392C4—84no—
393C4—84no—
394C4—84no—
395C4—84no—
396C4#323318yes—
397C4#355011yescharacteristic
398C4#355011yescharacteristic
399C4#355612yes—
400C2—42no—
401C2#3764111yescharacteristic
402C2C2×#323312yes—
403C2#6086612yes—
404C2#6090011yescharacteristic
405C2#6090711yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC24C22×D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C43 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #32851C2semidirect
6. #29290C2semidirect
7. #29287C2semidirect
8. #29286C2semidirect
9. #29222C2semidirect
10. #28691C2semidirect

Maximal subgroups

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

subgroupindex
#366352
#328512
#299422
#299412
#298792
#292902
#292872
#292862
#292222
#286912
#208082