Finite Group Databasebeta

← #243144

Subgroups — #243144

Order 512.

529 conjugacy classes of subgroups.

The subgroup lattice has 377 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#17822C211yesmaximal, characteristic
2#17755C211yesmaximal, characteristic
3C2×#1559C211yesmaximal, characteristic
4#17491C211yesmaximal, characteristic
5#10648C214yesmaximal, complemented
6#10646C214yesmaximal, complemented
7#8121C211yesmaximal, characteristic
8#7989C211yesmaximal, characteristic
9#7967C211yesmaximal, characteristic
10C2×#533C2211yescharacteristic
11#2920C2211yescharacteristic
12#2920C2211yescharacteristic
13C2×#432C2211yescharacteristic
14C2×#432C2211yescharacteristic
15C2×#432C2211yescharacteristic
16C2×#423C2211yescharacteristic
17#1573—22no—
18#1571C2212yes—
19#1570C2212yes—
20#1569—22no—
21#1559—22no—
22#1559C2214yes—
23#1556—21no—
24#1556—21no—
25#1552—22no—
26#1552C2212yes—
27#1552C2212yes—
28C2×#333—21no—
29C2×#332—21no—
30#1497C2214yes—
31#1487—22no—
32#1485—22no—
33#1307C2214yes—
34#1290C2212yes—
35#1290C2212yes—
36#1287C2214yes—
37C22×M16—21no—
38#533—21no—
39#533—21no—
40#533—21no—
41#533—21no—
42#533—21no—
43#533—21no—
44#533D412yes—
45#533C2312yes—
46C2×C4⋊Q8D411yescharacteristic
47C2×C4⋊Q8D411yescharacteristic
48C2×#125—21no—
49C2×C22⋊Q8—21no—
50C2×C22⋊Q8—21no—
51C2×C4×Q8—21no—
52C2×C4×Q8—21no—
53C2×#117—21no—
54C22×C42C2311yescharacteristic
55#432—21no—
56#432—21no—
57#432—21no—
58#432—21no—
59#432—21no—
60#432—21no—
61#432—21no—
62#432—21no—
63#432—21no—
64#432—21no—
65#432—21no—
66#432—21no—
67#432—21no—
68#432C2312yes—
69#432C2312yes—
70#432C2312yes—
71#431—21no—
72#431—21no—
73#431D412yes—
74#431C2312yes—
75#423—21no—
76#423—21no—
77#423—21no—
78#423—21no—
79#423—21no—
80#423C2312yes—
81C2×C4⋊C8—21no—
82C2×C4⋊C8—21no—
83C2×C4⋊C8—21no—
84C2×C4⋊C8—21no—
85C2×C4⋊C8C2311yescharacteristic
86C2×C4⋊C8C2311yescharacteristic
87#407—21no—
88#407—21no—
89C2×C22⋊C8—21no—
90C2×C22⋊C8—21no—
91C2×C22⋊C8—21no—
92C4×M16—21no—
93C4×M16D412yes—
94C2×#97—21no—
95C2×#97—21no—
96C2×#97—21no—
97#335—24no—
98#334—24no—
99#333—24no—
100#332—24no—
101#332—22no—
102#332—22no—
103#332—22no—
104#332—22no—
105Q8⋊C8—28no—
106C22×Q8—21no—
107C22×Q8—21no—
108C23×C4#4111yescharacteristic
109C23×C4#4111yescharacteristic
110C23×C4#4111yescharacteristic
111C2×M16—22no—
112C2×M16—22no—
113C2×M16—41no—
114C2×M16—41no—
115C2×M16—82no—
116C22×C8—21no—
117C22×C8—21no—
118C22×C8—41no—
119C22×C8—41no—
120C22×C8—41no—
121C4⋊Q8—21no—
122C4⋊Q8—21no—
123C4⋊Q8—21no—
124C4⋊Q8—21no—
125C4⋊Q8—21no—
126C4⋊Q8—21no—
127C4⋊Q8C2×D412yes—
128C4⋊Q8C2×D412yes—
129#125—22no—
130#125—22no—
131#125—41no—
132#125—41no—
133C22⋊Q8—22no—
134C22⋊Q8—22no—
135C22⋊Q8—22no—
136C22⋊Q8—22no—
137C22⋊Q8—41no—
138C22⋊Q8—41no—
139C22⋊Q8—41no—
140C22⋊Q8—41no—
141C4×Q8—22no—
142C4×Q8—22no—
143C4×Q8—22no—
144C4×Q8—22no—
145C4×Q8—41no—
146C4×Q8—41no—
147C4×Q8—41no—
148C4×Q8—41no—
149#117—22no—
150#117—22no—
151#117—41no—
152#117—41no—
153C2×C4⋊C4—21no—
154C2×C4⋊C4—21no—
155C2×C4⋊C4—21no—
156C2×C4⋊C4—21no—
157C2×C4⋊C4—41no—
158C2×C4⋊C4—41no—
159C2×C22⋊C4—41no—
160C2×C42—41no—
161C2×C42C2×D411yescharacteristic
162C2×C42C2×D411yescharacteristic
163C2×C42C2×D411yescharacteristic
164C2×C42C2×D411yescharacteristic
165C2×C42#4111yescharacteristic
166C2×C42#4111yescharacteristic
167C2×C42#4111yescharacteristic
168C2×C42#4111yescharacteristic
169C2×C42#4111yescharacteristic
170C2×C42#4111yescharacteristic
171C2×C42#4111yescharacteristic
172C2×C42C2411yescharacteristic
173C4⋊C8—22no—
174C4⋊C8—22no—
175C4⋊C8—22no—
176C4⋊C8—21no—
177C4⋊C8—21no—
178C4⋊C8—22no—
179C4⋊C8—22no—
180C4⋊C8—21no—
181C4⋊C8—22no—
182C4⋊C8—21no—
183C4⋊C8—22no—
184C4⋊C8—41no—
185C4⋊C8—41no—
186C4⋊C8—41no—
187C4⋊C8—41no—
188C4⋊C8—41no—
189C22⋊C8—22no—
190C22⋊C8—22no—
191C22⋊C8—22no—
192C22⋊C8—22no—
193C22⋊C8—22no—
194C22⋊C8—41no—
195C22⋊C8—41no—
196C22⋊C8—41no—
197C22⋊C8—41no—
198C22⋊C8—41no—
199#97—22no—
200#97—22no—
201#97—22no—
202#97—22no—
203#97—41no—
204#97—41no—
205#97—41no—
206#97—41no—
207C4×C8—41no—
208C4×C8C2×D414yes—
209C24#12611yescharacteristic
210C2×Q8—22no—
211C2×Q8—22no—
212C2×Q8—41no—
213C2×Q8—41no—
214C2×Q8—81no—
215C2×Q8—81no—
216C2×Q8—82no—
217C22×C4—41no—
218C22×C4—41no—
219C22×C4—42no—
220C22×C4—41no—
221C22×C4—41no—
222C22×C4—81no—
223C22×C4#12111yescharacteristic
224C22×C4#12111yescharacteristic
225C22×C4#12311yescharacteristic
226C22×C4#12311yescharacteristic
227C22×C4#12311yescharacteristic
228C22×C4#12311yescharacteristic
229C22×C4#12311yescharacteristic
230C22×C4#12311yescharacteristic
231C22×C4#12411yescharacteristic
232C22×C4#12411yescharacteristic
233C22×C4#12611yescharacteristic
234C22×C4#12611yescharacteristic
235C22×C4#12611yescharacteristic
236C22×C4#12611yescharacteristic
237C22×C4#12611yescharacteristic
238C22×C4C2×#4111yescharacteristic
239C22×C4C2×#4111yescharacteristic
240C22×C4C2×#4111yescharacteristic
241M16—82no—
242M16—82no—
243M16—82no—
244M16—82no—
245C2×C8—24no—
246C2×C8—41no—
247C2×C8—41no—
248C2×C8—42no—
249C2×C8—42no—
250C2×C8—42no—
251C2×C8—42no—
252C2×C8—82no—
253C2×C8—82no—
254C2×C8—82no—
255C2×C8—81no—
256C2×C8—81no—
257C4⋊C4—22no—
258C4⋊C4—22no—
259C4⋊C4—22no—
260C4⋊C4—22no—
261C4⋊C4—42no—
262C4⋊C4—42no—
263C4⋊C4—41no—
264C4⋊C4—41no—
265C4⋊C4—41no—
266C4⋊C4—41no—
267C4⋊C4—42no—
268C4⋊C4—42no—
269C22⋊C4—42no—
270C22⋊C4—42no—
271C42—42no—
272C42—42no—
273C42#12111yescharacteristic
274C42#12111yescharacteristic
275C42#12311yescharacteristic
276C42#12311yescharacteristic
277C42#12311yescharacteristic
278C42#12311yescharacteristic
279C42#12311yescharacteristic
280C42#12311yescharacteristic
281C42#12411yescharacteristic
282C42#12411yescharacteristic
283C42#12611yescharacteristic
284C42#12611yescharacteristic
285C42C22×D411yescharacteristic
286C42C22×D411yescharacteristic
287C42C2×#4111yescharacteristic
288C42C2×#4111yescharacteristic
289C23—22no—
290C23—41no—
291C23—41no—
292C23#39711yescharacteristic
293C23#39711yescharacteristic
294C23C2×#12611yescharacteristic
295Q8—42no—
296Q8—42no—
297Q8—82no—
298Q8—82no—
299Q8—82no—
300C2×C4—22no—
301C2×C4—22no—
302C2×C4—21no—
303C2×C4—21no—
304C2×C4—21no—
305C2×C4—21no—
306C2×C4—21no—
307C2×C4—21no—
308C2×C4—41no—
309C2×C4—41no—
310C2×C4—41no—
311C2×C4—41no—
312C2×C4—41no—
313C2×C4—41no—
314C2×C4—41no—
315C2×C4—41no—
316C2×C4—41no—
317C2×C4—41no—
318C2×C4—41no—
319C2×C4—41no—
320C2×C4—41no—
321C2×C4—42no—
322C2×C4—42no—
323C2×C4—82no—
324C2×C4—82no—
325C2×C4#39711yescharacteristic
326C2×C4#39711yescharacteristic
327C2×C4#39711yescharacteristic
328C2×C4#39711yescharacteristic
329C2×C4#39711yescharacteristic
330C2×C4#39711yescharacteristic
331C2×C4C2×#12111yescharacteristic
332C2×C4C2×#12311yescharacteristic
333C2×C4C2×#12311yescharacteristic
334C2×C4C2×#12311yescharacteristic
335C2×C4C2×#12411yescharacteristic
336C2×C4C2×#12611yescharacteristic
337C8—44no—
338C8—82no—
339C8—82no—
340C8—82no—
341C8—81no—
342C8—82no—
343C8—81no—
344C22—21no—
345C22—21no—
346C22—21no—
347C22—21no—
348C22—21no—
349C22—21no—
350C22—21no—
351C22—21no—
352C22—82no—
353C22#208212yes—
354C22C2×#39711yescharacteristic
355C4—21no—
356C4—21no—
357C4—21no—
358C4—21no—
359C4—21no—
360C4—21no—
361C4—21no—
362C4—21no—
363C4—21no—
364C4—21no—
365C4—21no—
366C4—82no—
367C4—82no—
368C4#306811yescharacteristic
369C4#306911yescharacteristic
370C2—22no—
371C2—41no—
372C2—41no—
373C2#2364711yescharacteristic
374C2#2364911yescharacteristic
375C2C2×#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. #10648C2semidirect
6. #10646C2semidirect

Maximal subgroups

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

subgroupindex
#178222
#177552
C2×#15592
#174912
#106482
#106462
#81212
#79892
#79672