Finite Group Databasebeta

← #390997

Subgroups — #390997

Order 512.

858 conjugacy classes of subgroups.

The subgroup lattice has 405 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23467C211yesmaximal, characteristic
2#11395C212yesmaximal, complemented
3#11319C212yesmaximal, complemented
4#11071C211yesmaximal, characteristic
5#10937C212yesmaximal
6#10917C212yesmaximal, complemented
7#10831C212yesmaximal, complemented
8#10737C212yesmaximal
9#10729C211yesmaximal, characteristic
10C23×M16C2211yescharacteristic
11#3458C2211yescharacteristic
12#2905C2211yescharacteristic
13C2×#407C2212yes—
14C2×#407C2211yescharacteristic
15#2873C2212yes—
16#2280C2211yescharacteristic
17#2280C2212yes—
18#2030C414yes—
19#2029C414yes—
20C2×#344C2212yes—
21C2×#344C2212yes—
22C2×#344C2212yes—
23C2×#344C2212yes—
24C2×#344C2212yes—
25C2×#344C2212yes—
26C2×#344C2212yes—
27C2×#344C2212yes—
28C2×#344C2212yes—
29#1742C2211yescharacteristic
30#1742C2212yes—
31#1740C2212yes—
32#1740C2211yescharacteristic
33C24×C4C2311yescharacteristic
34C22×M16—21no—
35C22×M16—24no—
36C22×M16—42no—
37C22×M16—41no—
38C22×M16—41no—
39C22×M16C2312yes—
40C22×M16C2312yes—
41C22×M16C2312yes—
42C22×M16C2312yes—
43C23×C8—21no—
44#544—41no—
45#544—42no—
46#543C2×C414yes—
47#533—42no—
48#533—41no—
49C2×#117C2312yes—
50C2×#117C2311yescharacteristic
51C2×#117C2312yes—
52C2×#117C2311yescharacteristic
53#424—24no—
54#423—41no—
55#407—22no—
56#407—22no—
57#407—41no—
58#407—42no—
59#407Q812yescomplemented
60#407Q812yescomplemented
61C2×C22⋊C8—24no—
62C2×C22⋊C8—41no—
63#405—24no—
64#405—42no—
65C4×M16—42no—
66#381C2×C412yes—
67#381C2×C414yes—
68#380—42no—
69#380—41no—
70#379C2×C412yes—
71#379C2×C414yes—
72C4×C22⋊C4—42no—
73C4×C22⋊C4—41no—
74#344—24no—
75#344—24no—
76#344—42no—
77#344—42no—
78#344—42no—
79#344—42no—
80#344—42no—
81#344—42no—
82#344—42no—
83#343—42no—
84#343—41no—
85#342C2×C418yes—
86#342C2×C414yes—
87C25C22×C411yescharacteristic
88C23×C4—22no—
89C23×C4—22no—
90C23×C4—21no—
91C23×C4—22no—
92C23×C4—21no—
93C23×C4—22no—
94C23×C4—21no—
95C23×C4—21no—
96C23×C4C22×C411yescharacteristic
97C23×C4C2×Q811yescharacteristic
98C23×C4C2×Q811yescharacteristic
99C23×C4#4111yescharacteristic
100C23×C4#4111yescharacteristic
101C23×C4C2411yescharacteristic
102C2×M16—22no—
103C2×M16—22no—
104C2×M16—22no—
105C2×M16—22no—
106C2×M16—22no—
107C2×M16—22no—
108C2×M16—22no—
109C2×M16—22no—
110C2×M16—22no—
111C2×M16—22no—
112C2×M16—22no—
113C2×M16—22no—
114C2×M16—22no—
115C2×M16—22no—
116C2×M16—22no—
117C2×M16—22no—
118C2×M16—22no—
119C2×M16—22no—
120C2×M16—22no—
121C2×M16—22no—
122C2×M16—22no—
123C2×M16—22no—
124C2×M16—44no—
125C2×M16—44no—
126C2×M16—48no—
127C2×M16—42no—
128C2×M16—44no—
129C2×M16C2×Q814yes—
130C2×M16#4114yes—
131C22×C8—22no—
132C22×C8—22no—
133C22×C8—22no—
134C22×C8—44no—
135C22×C8—42no—
136C22×C8#4114yes—
137C4⋊Q8—44no—
138C4⋊Q8—42no—
139#125—44no—
140#125—42no—
141C22⋊Q8—44no—
142C22⋊Q8—48no—
143C4×Q8—44no—
144#117—22no—
145#117—22no—
146#117—22no—
147#117—22no—
148#117—22no—
149#117—22no—
150#117—22no—
151#117—22no—
152#117—22no—
153#117—22no—
154#117—22no—
155#117—22no—
156C2×C4⋊C4—21no—
157C2×C4⋊C4—22no—
158C2×C4⋊C4—22no—
159C2×C4⋊C4—21no—
160C2×C22⋊C4—44no—
161C2×C22⋊C4—42no—
162C2×C22⋊C4C22×C414yes—
163C2×C22⋊C4C22×C412yes—
164C2×C22⋊C4C22×C412yes—
165C2×C22⋊C4C22×C414yes—
166C2×C42—22no—
167C2×C42—21no—
168C2×C42—22no—
169C2×C42—21no—
170C4⋊C8—44no—
171C22⋊C8—42no—
172C22⋊C8—42no—
173C22⋊C8—44no—
174C22⋊C8—44no—
175#97—44no—
176C4×C8—44no—
177#95—42no—
178#95—44no—
179C24—22no—
180C24—21no—
181C24—22no—
182C24—21no—
183C24—82no—
184C24C4×Q811yescharacteristic
185C24C4×Q811yescharacteristic
186C24C23×C411yescharacteristic
187C2×Q8—44no—
188C22×C4—22no—
189C22×C4—21no—
190C22×C4—22no—
191C22×C4—21no—
192C22×C4—21no—
193C22×C4—21no—
194C22×C4—21no—
195C22×C4—21no—
196C22×C4—21no—
197C22×C4—21no—
198C22×C4—21no—
199C22×C4—21no—
200C22×C4—22no—
201C22×C4—22no—
202C22×C4—22no—
203C22×C4—22no—
204C22×C4—22no—
205C22×C4—22no—
206C22×C4—22no—
207C22×C4—22no—
208C22×C4—24no—
209C22×C4—22no—
210C22×C4—22no—
211C22×C4—24no—
212C22×C4—42no—
213C22×C4—42no—
214C22×C4—42no—
215C22×C4—42no—
216C22×C4—41no—
217C22×C4—41no—
218C22×C4—42no—
219C22×C4—42no—
220C22×C4—41no—
221C22×C4—41no—
222C22×C4C4×Q811yescharacteristic
223C22×C4C4×Q811yescharacteristic
224C22×C4C22×Q811yescharacteristic
225C22×C4C2×#4111yescharacteristic
226C22×C4C2×#4112yes—
227C22×C4C2×#4112yes—
228C22×C4H(5,𝔽2)11yescharacteristic
229C22×C4H(5,𝔽2)12yes—
230C22×C4H(5,𝔽2)12yes—
231C22×C4H(5,𝔽2)11yescharacteristic
232C22×C4H(5,𝔽2)11yescharacteristic
233C22×C42−1+411yescharacteristic
234M16—24no—
235M16—24no—
236M16—24no—
237M16—24no—
238M16—48no—
239M16—42no—
240M16—42no—
241M16—42no—
242M16—42no—
243M16—42no—
244M16—42no—
245M16—42no—
246M16—42no—
247M16—42no—
248M16—42no—
249M16—42no—
250M16—42no—
251C2×C8—24no—
252C2×C8—24no—
253C2×C8—24no—
254C2×C8—24no—
255C2×C8—24no—
256C2×C8—48no—
257C2×C8—42no—
258C2×C8—42no—
259C2×C8—42no—
260C2×C8—42no—
261C2×C8—42no—
262C2×C8—42no—
263C2×C8—42no—
264C4⋊C4—44no—
265C4⋊C4—48no—
266C4⋊C4—42no—
267C4⋊C4—42no—
268C4⋊C4—42no—
269C4⋊C4—42no—
270C4⋊C4—42no—
271C4⋊C4—42no—
272C22⋊C4—24no—
273C22⋊C4—24no—
274C22⋊C4—24no—
275C22⋊C4—24no—
276C22⋊C4—24no—
277C22⋊C4—24no—
278C22⋊C4—88no—
279C22⋊C4—84no—
280C42—42no—
281C42—42no—
282C42—41no—
283C42—41no—
284C42—42no—
285C42—41no—
286C42—42no—
287C42—41no—
288C23—21no—
289C23—22no—
290C23—21no—
291C23—22no—
292C23—42no—
293C23—42no—
294C23—44no—
295C23—44no—
296C23—42no—
297C23—42no—
298C23—41no—
299C23—41no—
300C23—81no—
301C23—81no—
302C23—82no—
303C23—81no—
304C23—82no—
305C23—81no—
306C23C2×C4×Q811yescharacteristic
307C23C4×#4112yes—
308C23#51811yescharacteristic
309C23#51812yes—
310C23#52011yescharacteristic
311Q8—164no—
312C2×C4—21no—
313C2×C4—21no—
314C2×C4—22no—
315C2×C4—22no—
316C2×C4—42no—
317C2×C4—41no—
318C2×C4—41no—
319C2×C4—41no—
320C2×C4—41no—
321C2×C4—42no—
322C2×C4—42no—
323C2×C4—41no—
324C2×C4—41no—
325C2×C4—41no—
326C2×C4—41no—
327C2×C4—41no—
328C2×C4—41no—
329C2×C4—42no—
330C2×C4—42no—
331C2×C4—44no—
332C2×C4—42no—
333C2×C4—44no—
334C2×C4—42no—
335C2×C4—84no—
336C2×C4—82no—
337C2×C4—84no—
338C2×C4—82no—
339C2×C4#53811yescharacteristic
340C2×C4#53812yes—
341C2×C4#53812yes—
342C2×C4#53911yescharacteristic
343C2×C4#54212yes—
344C2×C4#54212yes—
345C2×C4#54311yescharacteristic
346C2×C4#54411yescharacteristic
347C2×C4#55112yes—
348C2×C4#55111yescharacteristic
349C2×C4#55112yes—
350C2×C4#55111yescharacteristic
351C2×C4#55412yes—
352C2×C4#55412yes—
353C2×C4#55512yes—
354C2×C4#56012yes—
355C2×C4#56011yescharacteristic
356C2×C4H(3,𝔽4)11yescharacteristic
357C8—44no—
358C8—44no—
359C8—44no—
360C8—82no—
361C22—22no—
362C22—24no—
363C22—42no—
364C22—41no—
365C22—41no—
366C22—42no—
367C22—41no—
368C22—41no—
369C22—42no—
370C22—41no—
371C22—41no—
372C22—42no—
373C22—42no—
374C22—41no—
375C22—41no—
376C22—82no—
377C22—84no—
378C22—82no—
379C22#232911yescharacteristic
380C22#233712yes—
381C22#236812yes—
382C22#237211yescharacteristic
383C22#237511yescharacteristic
384C4—41no—
385C4—41no—
386C4—84no—
387C4—82no—
388C4—84no—
389C4—82no—
390C4#273611yescharacteristic
391C4#273712yes—
392C4#274211yescharacteristic
393C4#276011yescharacteristic
394C4#276311yescharacteristic
395C4#278412yes—
396C2—21no—
397C2—22no—
398C2—21no—
399C2—21no—
400C2—22no—
401C2—42no—
402C2—81no—
403C2#1459611yescharacteristic
subgroupG/H
centerC4#2760
commutatorC24C23×C4
Frattini, cosocleC23×C4C24
socleC2#14596
prime radicalC25C22×C4
LCSG ▷ C24 ▷ C2 ▷ C1
UCSC1 ◁ C4 ◁ C23×C4 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #11395C2semidirect
3. #11319C2semidirect
6. #10917C2semidirect
7. #10831C2semidirect
59. #407Q8semidirect
60. #407Q8semidirect

Maximal subgroups

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

subgroupindex
#234672
#113952
#113192
#110712
#109372
#109172
#108312
#107372
#107292