Finite Group Databasebeta

← #243098

Subgroups — #243098

Order 512.

547 conjugacy classes of subgroups.

The subgroup lattice has 366 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#17814C211yesmaximal, characteristic
2#17751C211yesmaximal, characteristic
3#17749C211yesmaximal, characteristic
4C2×#1497C211yesmaximal, characteristic
5#10646C218yesmaximal, complemented
6#8121C211yesmaximal, characteristic
7#7990C211yesmaximal, characteristic
8#7965C211yesmaximal, characteristic
9C2×#533C2211yescharacteristic
10#2920C2211yescharacteristic
11#2920C2211yescharacteristic
12C2×#432C2211yescharacteristic
13C2×#431C2211yescharacteristic
14C2×#423C2211yescharacteristic
15C2×#423C2211yescharacteristic
16#1571—22no—
17#1571C2214yes—
18#1567—22no—
19#1561—22no—
20#1559C2214yes—
21#1552C2214yes—
22#1550—22no—
23#1549—21no—
24#1548—21no—
25#1546—21no—
26#1546—21no—
27#1497—22no—
28#1497C2214yes—
29C2×Q8⋊C8—22no—
30#1307C2214yes—
31#1290C2214yes—
32#1287C2214yes—
33C22×M16—21no—
34C22×M16—21no—
35#533—21no—
36#533—21no—
37#533—21no—
38#533—21no—
39#533—21no—
40#533—21no—
41#533D412yes—
42#533C2312yes—
43C2×C4⋊Q8D411yescharacteristic
44C2×C4⋊Q8D411yescharacteristic
45C2×#125—21no—
46C2×C22⋊Q8—21no—
47C2×C22⋊Q8—21no—
48C2×C4×Q8—21no—
49C2×C4×Q8—21no—
50C2×#117—21no—
51C22×C42C2311yescharacteristic
52#432—21no—
53#432—21no—
54#432—21no—
55#432—21no—
56#432—21no—
57#432C2312yes—
58#432C2312yes—
59#432C2312yes—
60#431—21no—
61#431—21no—
62#431—21no—
63#431—21no—
64#431—21no—
65#431—21no—
66#431D412yes—
67#431C2312yes—
68#423—21no—
69#423—21no—
70#423—21no—
71#423—21no—
72#423—21no—
73#423—21no—
74#423—21no—
75#423—21no—
76#423—21no—
77#423C2312yes—
78C2×C4⋊C8—21no—
79C2×C4⋊C8—21no—
80C2×C4⋊C8—21no—
81C2×C4⋊C8—21no—
82C2×C4⋊C8—21no—
83C2×C4⋊C8C2312yes—
84#407—21no—
85#407—21no—
86C2×C22⋊C8—21no—
87C2×C22⋊C8—21no—
88C4×M16—21no—
89C4×M16—21no—
90C2×#97—21no—
91C2×C4×C8D412yes—
92#335—28no—
93#333—24no—
94#332—24no—
95#332—24no—
96#332—24no—
97Q8⋊C8—28no—
98C22×Q8—21no—
99C22×Q8—21no—
100C23×C4#4111yescharacteristic
101C23×C4#4111yescharacteristic
102C23×C4#4111yescharacteristic
103C2×M16—22no—
104C2×M16—41no—
105C2×M16—41no—
106C2×M16—41no—
107C2×M16—41no—
108C2×M16—41no—
109C2×M16—82no—
110C2×M16—82no—
111C22×C8—22no—
112C22×C8—22no—
113C22×C8—41no—
114C22×C8—41no—
115C4⋊Q8—21no—
116C4⋊Q8—21no—
117C4⋊Q8—21no—
118C4⋊Q8—21no—
119C4⋊Q8—21no—
120C4⋊Q8—21no—
121C4⋊Q8C2×D412yes—
122C4⋊Q8C2×D412yes—
123#125—22no—
124#125—22no—
125#125—41no—
126#125—41no—
127C22⋊Q8—22no—
128C22⋊Q8—22no—
129C22⋊Q8—22no—
130C22⋊Q8—22no—
131C22⋊Q8—41no—
132C22⋊Q8—41no—
133C22⋊Q8—41no—
134C22⋊Q8—41no—
135C4×Q8—22no—
136C4×Q8—22no—
137C4×Q8—22no—
138C4×Q8—22no—
139C4×Q8—41no—
140C4×Q8—41no—
141C4×Q8—41no—
142C4×Q8—41no—
143#117—22no—
144#117—22no—
145#117—41no—
146#117—41no—
147C2×C4⋊C4—21no—
148C2×C4⋊C4—21no—
149C2×C4⋊C4—21no—
150C2×C4⋊C4—21no—
151C2×C4⋊C4—41no—
152C2×C4⋊C4—41no—
153C2×C22⋊C4—41no—
154C2×C42—41no—
155C2×C42C2×D411yescharacteristic
156C2×C42C2×D411yescharacteristic
157C2×C42C2×D411yescharacteristic
158C2×C42C2×D411yescharacteristic
159C2×C42#4111yescharacteristic
160C2×C42#4111yescharacteristic
161C2×C42#4111yescharacteristic
162C2×C42#4111yescharacteristic
163C2×C42#4111yescharacteristic
164C2×C42#4111yescharacteristic
165C2×C42#4111yescharacteristic
166C2×C42C2411yescharacteristic
167C4⋊C8—22no—
168C4⋊C8—22no—
169C4⋊C8—22no—
170C4⋊C8—22no—
171C4⋊C8—22no—
172C4⋊C8—22no—
173C4⋊C8—22no—
174C4⋊C8—22no—
175C4⋊C8—22no—
176C4⋊C8—41no—
177C4⋊C8—41no—
178C4⋊C8—41no—
179C4⋊C8—41no—
180C4⋊C8—41no—
181C4⋊C8—41no—
182C4⋊C8—41no—
183C22⋊C8—22no—
184C22⋊C8—22no—
185C22⋊C8—22no—
186C22⋊C8—22no—
187C22⋊C8—22no—
188C22⋊C8—41no—
189C22⋊C8—41no—
190C22⋊C8—41no—
191#97—22no—
192#97—22no—
193#97—22no—
194#97—41no—
195C4×C8—22no—
196C4×C8—41no—
197C4×C8—41no—
198C4×C8C2×D414yes—
199C24#12611yescharacteristic
200C2×Q8—22no—
201C2×Q8—22no—
202C2×Q8—41no—
203C2×Q8—41no—
204C2×Q8—42no—
205C2×Q8—42no—
206C2×Q8—82no—
207C22×C4—22no—
208C22×C4—22no—
209C22×C4—42no—
210C22×C4—41no—
211C22×C4—41no—
212C22×C4—81no—
213C22×C4#12111yescharacteristic
214C22×C4#12111yescharacteristic
215C22×C4#12311yescharacteristic
216C22×C4#12311yescharacteristic
217C22×C4#12311yescharacteristic
218C22×C4#12311yescharacteristic
219C22×C4#12311yescharacteristic
220C22×C4#12311yescharacteristic
221C22×C4#12411yescharacteristic
222C22×C4#12411yescharacteristic
223C22×C4#12611yescharacteristic
224C22×C4#12611yescharacteristic
225C22×C4#12611yescharacteristic
226C22×C4#12611yescharacteristic
227C22×C4#12611yescharacteristic
228C22×C4C2×#4111yescharacteristic
229C22×C4C2×#4111yescharacteristic
230C22×C4C2×#4111yescharacteristic
231M16—82no—
232M16—82no—
233M16—82no—
234M16—82no—
235M16—82no—
236M16—82no—
237C2×C8—24no—
238C2×C8—42no—
239C2×C8—42no—
240C2×C8—42no—
241C2×C8—42no—
242C2×C8—42no—
243C2×C8—82no—
244C2×C8—82no—
245C2×C8—82no—
246C2×C8—82no—
247C4⋊C4—22no—
248C4⋊C4—22no—
249C4⋊C4—22no—
250C4⋊C4—22no—
251C4⋊C4—42no—
252C4⋊C4—42no—
253C4⋊C4—41no—
254C4⋊C4—41no—
255C4⋊C4—41no—
256C4⋊C4—41no—
257C4⋊C4—42no—
258C4⋊C4—42no—
259C22⋊C4—42no—
260C22⋊C4—42no—
261C42—42no—
262C42—42no—
263C42#12111yescharacteristic
264C42#12111yescharacteristic
265C42#12311yescharacteristic
266C42#12311yescharacteristic
267C42#12311yescharacteristic
268C42#12311yescharacteristic
269C42#12311yescharacteristic
270C42#12311yescharacteristic
271C42#12411yescharacteristic
272C42#12411yescharacteristic
273C42#12611yescharacteristic
274C42#12611yescharacteristic
275C42C22×D411yescharacteristic
276C42C22×D411yescharacteristic
277C42C2×#4111yescharacteristic
278C42C2×#4111yescharacteristic
279C23—22no—
280C23—41no—
281C23—41no—
282C23#39711yescharacteristic
283C23#39711yescharacteristic
284C23C2×#12611yescharacteristic
285Q8—42no—
286Q8—42no—
287Q8—82no—
288Q8—82no—
289Q8—82no—
290C2×C4—22no—
291C2×C4—22no—
292C2×C4—22no—
293C2×C4—22no—
294C2×C4—22no—
295C2×C4—22no—
296C2×C4—21no—
297C2×C4—21no—
298C2×C4—21no—
299C2×C4—21no—
300C2×C4—41no—
301C2×C4—41no—
302C2×C4—41no—
303C2×C4—41no—
304C2×C4—41no—
305C2×C4—41no—
306C2×C4—41no—
307C2×C4—41no—
308C2×C4—41no—
309C2×C4—42no—
310C2×C4—42no—
311C2×C4—82no—
312C2×C4—82no—
313C2×C4#39711yescharacteristic
314C2×C4#39711yescharacteristic
315C2×C4#39711yescharacteristic
316C2×C4#39711yescharacteristic
317C2×C4#39711yescharacteristic
318C2×C4#39711yescharacteristic
319C2×C4#47212yes—
320C2×C4#47312yes—
321C2×C4C2×#12111yescharacteristic
322C2×C4C2×#12311yescharacteristic
323C2×C4C2×#12311yescharacteristic
324C2×C4C2×#12311yescharacteristic
325C2×C4C2×#12411yescharacteristic
326C2×C4C2×#12611yescharacteristic
327C8—44no—
328C8—82no—
329C8—82no—
330C8—82no—
331C8—82no—
332C8—82no—
333C22—21no—
334C22—21no—
335C22—21no—
336C22—21no—
337C22—21no—
338C22—21no—
339C22—21no—
340C22—21no—
341C22—82no—
342C22#207812yes—
343C22C2×#39711yescharacteristic
344C4—21no—
345C4—21no—
346C4—21no—
347C4—21no—
348C4—21no—
349C4—21no—
350C4—21no—
351C4—21no—
352C4—21no—
353C4—82no—
354C4—82no—
355C4#207812yes—
356C4#208212yes—
357C4C2×#47211yescharacteristic
358C4C2×#47311yescharacteristic
359C2—22no—
360C2—41no—
361C2—41no—
362C2C2×#207811yescharacteristic
363C2#2364911yescharacteristic
364C2C2×#208211yescharacteristic
subgroupG/H
center, socleC22C2×#397
commutator, Frattini, cosocleC2×C42C24
prime radicalC24#126
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C22×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
5. #10646C2semidirect

Maximal subgroups

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

subgroupindex
#178142
#177512
#177492
C2×#14972
#106462
#81212
#79902
#79652