Finite Group Databasebeta

← #220352

Subgroups — #220352

Order 512.

864 conjugacy classes of subgroups.

The subgroup lattice has 408 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3341C211yesmaximal, characteristic
2#22606C212yesmaximal, complemented
3#17418C212yesmaximal, complemented
4#9815C212yesmaximal, complemented
5#9242C214yesmaximal, complemented
6#8612C211yesmaximal, characteristic, complemented
7#8360C212yesmaximal, complemented
8C8×#99C211yesmaximal, characteristic, complemented
9C2×#555C2212yes—
10#3341—24no—
11#3341—22no—
12#3341—22no—
13C2×#500C411yescharacteristic, complemented
14C2×#482C412yescomplemented
15C2×#482C2212yes—
16C2×#476C412yescomplemented
17C2×#466C411yescharacteristic, complemented
18C2×#466C2211yescharacteristic
19C2×C4×D8C411yescharacteristic, complemented
20C2×C8×D4C2211yescharacteristic
21#2924C2214yes—
22C2×#427C411yescharacteristic, complemented
23C2×#427C2211yescharacteristic
24#1945—22no—
25#1922—24no—
26#1826C2214yes—
27#1810C2212yes—
28#1772—22no—
29C4×#99C2211yescharacteristic
30C8×C22⋊C4C2212yes—
31#1507C2214yes—
32#1471C2214yes—
33#1469—24no—
34#1465C2212yes—
35#1464—22no—
36#1459C2211yescharacteristic
37#1458—22no—
38#1293C2212yes—
39#1282C2212yes—
40C22×D8C2×C411yescharacteristic
41C23×C8C2×C411yescharacteristic
42C23×C8C2311yescharacteristic
43#555—24no—
44#555—24no—
45#555—44no—
46C2×C4⋊Q8C2×C412yes—
47C2×#124—22no—
48C2×#124—22no—
49C2×#121—22no—
50C2×#121C2×C412yes—
51C2×#121C2312yes—
52C2×C4×D4C2×C412yes—
53C2×C4×D4C2311yescharacteristic
54C2×#117—22no—
55C2×#117C2×C412yes—
56C2×#117C2312yes—
57#500—22no—
58#500—22no—
59#482—24no—
60#482—24no—
61#482—24no—
62#482—24no—
63#476—24no—
64#476—24no—
65#466—22no—
66#466—22no—
67#466—22no—
68#466—22no—
69C4×D8—22no—
70C4×D8—21no—
71C4×D8—21no—
72C8×D4—22no—
73C8×D4—22no—
74#433—44no—
75#432—24no—
76#431—24no—
77C2×#108—22no—
78#427—22no—
79#427—21no—
80#427—21no—
81#427D414yes—
82#427D412yes—
83#427D412yes—
84C2×#107C2×C412yes—
85C2×#107C2×C411yescharacteristic
86C2×#107C2×C411yescharacteristic
87C2×#107C2×C411yescharacteristic
88C2×#106C2×C412yes—
89C2×#106C2×C412yes—
90#424—24no—
91C2×C4⋊C8C2×C411yescharacteristic
92C2×#102C2×C412yes—
93C2×#102C2×C412yes—
94C2×#102C2×C412yes—
95C2×#99—21no—
96#407C2314yes—
97C2×C22⋊C8C2×C411yescharacteristic
98C2×C22⋊C8C2311yescharacteristic
99C2×C22⋊C8C2312yes—
100#405—24no—
101C2×C4×C8—22no—
102C2×C4×C8C2×C411yescharacteristic
103C4×C22⋊C4C2312yes—
104#342—21no—
105#341—22no—
106#339—24no—
107#339—24no—
108#336—22no—
109#327—44no—
110#324—22no—
111#324D411yescharacteristic
112#324D411yescharacteristic
113#323—22no—
114#323—24no—
115#323—24no—
116#323D412yes—
117C22×Q8—22no—
118C22×D4—22no—
119C22×D4—22no—
120C22×D4C22×C412yes—
121C22×D4#4111yescharacteristic
122C23×C4—22no—
123C23×C4C22×C411yescharacteristic
124C23×C4C2411yescharacteristic
125C2×D8—21no—
126C2×D8—21no—
127C2×D8—42no—
128C2×M16—24no—
129C22×C8—22no—
130C22×C8—21no—
131C22×C8—21no—
132C22×C8—42no—
133C22×C8—42no—
134C22×C8C22×C411yescharacteristic
135C22×C8C22×C411yescharacteristic
136C22×C8C22×C411yescharacteristic
137C22×C8C22×C411yescharacteristic
138C22×C8C2×D412yes—
139C22×C8C2×D412yes—
140C4⋊Q8—24no—
141C4⋊Q8—42no—
142#124—44no—
143#124—44no—
144#124—44no—
145#124—44no—
146#121—24no—
147#121—24no—
148#121—24no—
149#121—44no—
150#121—44no—
151#121—42no—
152C4×D4—22no—
153C4×D4—22no—
154C4×D4—22no—
155C4×D4—22no—
156C4×D4—42no—
157#117—24no—
158#117—22no—
159#117—22no—
160#117—44no—
161#117—44no—
162#117—42no—
163#117C2×D414yes—
164#117C2×D414yes—
165C2×C4⋊C4—22no—
166C2×C4⋊C4C22×C412yes—
167C2×C4⋊C4C22×C412yes—
168C2×C4⋊C4C22×C412yes—
169C2×C4⋊C4C2×D411yescharacteristic
170C2×C22⋊C4—22no—
171C2×C22⋊C4—22no—
172C2×C22⋊C4—22no—
173C2×C22⋊C4—22no—
174C2×C22⋊C4—22no—
175C2×C22⋊C4—22no—
176C2×C22⋊C4C2×D411yescharacteristic
177C2×C22⋊C4#4111yescharacteristic
178C2×C42—22no—
179C2×C42—22no—
180C2×C42—22no—
181C2×C42C22×C411yescharacteristic
182#108—44no—
183#107—21no—
184#107—21no—
185#107—24no—
186#107C22⋊C412yes—
187#107C22⋊C411yescharacteristic
188#107C22⋊C411yescharacteristic
189#107C22⋊C412yes—
190#107C22⋊C412yes—
191#106—22no—
192#106—22no—
193#106C22⋊C414yes—
194#106C22⋊C414yes—
195C4⋊C8—22no—
196C4⋊C8—44no—
197C4⋊C8—44no—
198#102—24no—
199#102—22no—
200#102—22no—
201#102—24no—
202#99—24no—
203C22⋊C8—22no—
204C22⋊C8—22no—
205C22⋊C8—22no—
206C22⋊C8—24no—
207C22⋊C8—24no—
208C22⋊C8C2×D412yes—
209C22⋊C8C2×D412yes—
210#97—44no—
211C4×C8—21no—
212C4×C8—21no—
213C4×C8—44no—
214C4×C8—44no—
215#95—22no—
216C24—22no—
217C24C4×D411yescharacteristic
218C24C2×#4111yescharacteristic
219C2×Q8—44no—
220C2×Q8—162no—
221C2×D4—22no—
222C2×D4—22no—
223C2×D4—22no—
224C2×D4—22no—
225C2×D4—41no—
226C2×D4—42no—
227C2×D4—42no—
228C2×D4—44no—
229C2×D4—44no—
230C2×D4—162no—
231C2×D4—162no—
232C22×C4—22no—
233C22×C4—22no—
234C22×C4—21no—
235C22×C4—21no—
236C22×C4—22no—
237C22×C4—41no—
238C22×C4—41no—
239C22×C4—42no—
240C22×C4—42no—
241C22×C4—41no—
242C22×C4—41no—
243C22×C4—42no—
244C22×C4—44no—
245C22×C4—82no—
246C22×C4C4×D411yescharacteristic
247C22×C4C4×D411yescharacteristic
248C22×C4C4×D411yescharacteristic
249C22×C4C23×C411yescharacteristic
250C22×C4C22×D412yes—
251C22×C4C22×D411yescharacteristic
252C22×C4H(5,𝔽2)12yes—
253D8—42no—
254D8—42no—
255M16—44no—
256M16—44no—
257C2×C8—24no—
258C2×C8—22no—
259C2×C8—21no—
260C2×C8—21no—
261C2×C8—22no—
262C2×C8—22no—
263C2×C8—41no—
264C2×C8—42no—
265C2×C8—42no—
266C2×C8—42no—
267C2×C8—42no—
268C2×C8—42no—
269C2×C8—41no—
270C2×C8—41no—
271C2×C8—44no—
272C2×C8C2×C22⋊C411yescharacteristic
273C2×C8C2×C22⋊C412yes—
274C2×C8C2×C22⋊C411yescharacteristic
275C4⋊C4—22no—
276C4⋊C4—22no—
277C4⋊C4—22no—
278C4⋊C4—22no—
279C4⋊C4—22no—
280C4⋊C4—44no—
281C4⋊C4C2×C22⋊C412yes—
282C4⋊C4C2×C22⋊C412yes—
283C4⋊C4C2×C22⋊C414yes—
284C22⋊C4—24no—
285C22⋊C4—24no—
286C22⋊C4—22no—
287C22⋊C4—22no—
288C22⋊C4—24no—
289C22⋊C4—44no—
290C22⋊C4—42no—
291C22⋊C4—42no—
292C22⋊C4—42no—
293C22⋊C4—42no—
294C22⋊C4—42no—
295C22⋊C4—44no—
296C42—21no—
297C42—21no—
298C42—24no—
299C42—24no—
300C42—42no—
301C42—42no—
302C42—42no—
303C42—42no—
304C23—21no—
305C23—21no—
306C23—21no—
307C23—21no—
308C23—21no—
309C23—41no—
310C23—41no—
311C23—41no—
312C23—41no—
313C23—42no—
314C23—42no—
315C23—42no—
316C23—82no—
317C23—82no—
318C23C2×C4×D411yescharacteristic
319C23#54612yes—
320Q8—164no—
321D4—44no—
322D4—42no—
323D4—42no—
324D4—82no—
325D4—164no—
326D4—164no—
327C2×C4—21no—
328C2×C4—21no—
329C2×C4—22no—
330C2×C4—21no—
331C2×C4—22no—
332C2×C4—21no—
333C2×C4—22no—
334C2×C4—22no—
335C2×C4—21no—
336C2×C4—24no—
337C2×C4—21no—
338C2×C4—41no—
339C2×C4—42no—
340C2×C4—42no—
341C2×C4—42no—
342C2×C4—42no—
343C2×C4—41no—
344C2×C4—41no—
345C2×C4—41no—
346C2×C4—41no—
347C2×C4—44no—
348C2×C4—42no—
349C2×C4—84no—
350C2×C4—84no—
351C2×C4—82no—
352C2×C4—82no—
353C2×C4—82no—
354C2×C4—82no—
355C2×C4#44312yes—
356C2×C4C22×C22⋊C411yescharacteristic
357C2×C4#51811yescharacteristic
358C2×C4D4212yes—
359C8—21no—
360C8—22no—
361C8—42no—
362C8—42no—
363C8—41no—
364C8—41no—
365C8—84no—
366C8#41012yes—
367C22—21no—
368C22—21no—
369C22—22no—
370C22—22no—
371C22—22no—
372C22—41no—
373C22—41no—
374C22—41no—
375C22—81no—
376C22—81no—
377C22—81no—
378C22—81no—
379C22—82no—
380C22—82no—
381C22—82no—
382C22—162no—
383C22—162no—
384C22D4×C22⋊C411yescharacteristic
385C22C2×#44311yescharacteristic
386C22#295711yescharacteristic
387C4—22no—
388C4—21no—
389C4—22no—
390C4—41no—
391C4—42no—
392C4—42no—
393C4—41no—
394C4—44no—
395C4—84no—
396C4—82no—
397C4C2×#41011yescharacteristic
398C4#288411yescharacteristic
399C2—21no—
400C2—21no—
401C2—42no—
402C2—81no—
403C2—162no—
404C2#2240611yescharacteristic
405C2#2245411yescharacteristic
406C2#2245811yescharacteristic
subgroupG/H
center, socleC22D4×C22⋊C4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC2×#3341C2
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C23×C4 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #22606C2semidirect
3. #17418C2semidirect
4. #9815C2semidirect
5. #9242C2semidirect
6. #8612C2semidirect
7. #8360C2semidirect
8. C8×#99C2semidirect
13. C2×#500C4semidirect
14. C2×#482C4semidirect
16. C2×#476C4semidirect
17. C2×#466C4semidirect
19. C2×C4×D8C4semidirect
22. C2×#427C4semidirect

Maximal subgroups

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

subgroupindex
C2×#33412
#226062
#174182
#98152
#92422
#86122
#83602
C8×#992