Finite Group Databasebeta

← #220482

Subgroups — #220482

Order 512.

796 conjugacy classes of subgroups.

The subgroup lattice has 380 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3336C211yesmaximal, characteristic
2#18062C212yesmaximal, complemented
3#17423C212yesmaximal
4#9792C212yesmaximal, complemented
5#9246C214yesmaximal, complemented
6#8620C211yesmaximal, characteristic
7#8371C212yesmaximal
8#8164C211yesmaximal, characteristic
9C2×#552C2212yes—
10#3336—22no—
11#3336—24no—
12#3336—22no—
13C2×#499C2211yescharacteristic
14C2×#485C412yes—
15C2×#483C412yes—
16C2×#479C2212yes—
17C2×#467C411yescharacteristic
18C2×#467C411yescharacteristic
19#2984C2214yes—
20C2×C4×Q16C2211yescharacteristic
21C2×C8×D4C2211yescharacteristic
22C2×#427C411yescharacteristic
23C22×#107C411yescharacteristic
24C4×#100C2211yescharacteristic
25#1593—22no—
26#1584—24no—
27#1579—22no—
28#1568C2212yes—
29#1528C2214yes—
30#1491C2214yes—
31#1473—24no—
32#1470C2214yes—
33#1466C2212yes—
34#1464—22no—
35#1460C2211yescharacteristic
36#1458—22no—
37#1451C2212yes—
38#1290C2212yes—
39#1283C2212yes—
40C22×Q16C2×C411yescharacteristic
41C23×C8C2×C411yescharacteristic
42C23×C8C2×C411yescharacteristic
43#552—24no—
44#552—24no—
45#552—44no—
46C2×#126—22no—
47C2×#125—22no—
48C2×#125C2312yes—
49C2×#123—22no—
50C2×C22⋊Q8C2×C412yes—
51C2×C22⋊Q8C2×C412yes—
52C2×C4×Q8C2312yes—
53C2×C4×D4C2311yescharacteristic
54C2×#117—22no—
55C2×#117C2×C412yes—
56C22×C4⋊C4C2×C412yes—
57#499—22no—
58#499—22no—
59#485—24no—
60#485—24no—
61#483—24no—
62#483—24no—
63#479—24no—
64#479—24no—
65#467—22no—
66#467—22no—
67#467—22no—
68#467—22no—
69#446—24no—
70#446—44no—
71C8×Q8—24no—
72C4×Q16D412yes—
73C4×Q16D414yes—
74C4×Q16D412yes—
75C8×D4—22no—
76C8×D4—22no—
77#432—24no—
78#432C2314yes—
79#431—24no—
80#431C2312yes—
81#427—22no—
82#427—21no—
83#427—21no—
84C2×#107—22no—
85C2×#107—21no—
86C2×#107—21no—
87C2×#107C2×C412yes—
88C2×#107C2×C411yescharacteristic
89C2×#107C2×C411yescharacteristic
90C2×#107C2×C411yescharacteristic
91C2×#106C2×C412yes—
92C2×#106C2×C412yes—
93C2×C4⋊C8C2311yescharacteristic
94C2×#103C2×C412yes—
95C2×#103C2×C412yes—
96C2×#103C2×C412yes—
97C2×#100—21no—
98C2×C22⋊C8C2×C411yescharacteristic
99C2×C22⋊C8C2×C411yescharacteristic
100C4×M16C2312yes—
101C2×C4×C8C2311yescharacteristic
102#343—21no—
103#335—22no—
104#334—22no—
105#330—44no—
106Q8⋊C8—24no—
107Q8⋊C8—24no—
108#324—24no—
109#324—24no—
110#323—22no—
111#323—22no—
112#323D412yes—
113#323D412yes—
114#322—22no—
115#322—22no—
116C22×Q8C22×C412yes—
117C22×D4#4111yescharacteristic
118C23×C4—22no—
119C23×C4C22×C411yescharacteristic
120C23×C4C22×C411yescharacteristic
121C2×Q16—42no—
122C2×Q16C22⋊C412yes—
123C2×Q16C22⋊C412yes—
124C2×M16—22no—
125C22×C8—21no—
126C22×C8—21no—
127C22×C8—21no—
128C22×C8—21no—
129C22×C8—42no—
130C22×C8—42no—
131C22×C8C22×C411yescharacteristic
132C22×C8C22×C411yescharacteristic
133C22×C8C22×C411yescharacteristic
134C22×C8C22×C411yescharacteristic
135#126—44no—
136#126—44no—
137#125—24no—
138#125—24no—
139#125—44no—
140#125—44no—
141#123—44no—
142#123—44no—
143C22⋊Q8—24no—
144C22⋊Q8—24no—
145C22⋊Q8—42no—
146C22⋊Q8—42no—
147C4×Q8—24no—
148C4×Q8C2×D414yes—
149C4×Q8C2×D414yes—
150C4×D4—22no—
151C4×D4—22no—
152#117—22no—
153#117—22no—
154#117—44no—
155#117—44no—
156#117—42no—
157C2×C4⋊C4—22no—
158C2×C4⋊C4—22no—
159C2×C4⋊C4—22no—
160C2×C4⋊C4—22no—
161C2×C4⋊C4—22no—
162C2×C4⋊C4—22no—
163C2×C4⋊C4—22no—
164C2×C4⋊C4—22no—
165C2×C4⋊C4—42no—
166C2×C4⋊C4C22×C412yes—
167C2×C4⋊C4C22×C412yes—
168C2×C4⋊C4C22×C412yes—
169C2×C4⋊C4#4111yescharacteristic
170C2×C22⋊C4—22no—
171C2×C22⋊C4—22no—
172C2×C22⋊C4—22no—
173C2×C22⋊C4C2×D411yescharacteristic
174C2×C22⋊C4C2×D411yescharacteristic
175C2×C42—22no—
176C2×C42—22no—
177C2×C42C2411yescharacteristic
178#107—21no—
179#107—21no—
180#107—21no—
181#107—21no—
182#107—22no—
183#107—22no—
184#107C22⋊C412yes—
185#107C22⋊C411yescharacteristic
186#107C22⋊C411yescharacteristic
187#106—24no—
188#106—22no—
189#106—22no—
190C4⋊C8—22no—
191C4⋊C8—24no—
192C4⋊C8—22no—
193C4⋊C8—44no—
194C4⋊C8—44no—
195#103—24no—
196#103—24no—
197#103C22⋊C414yes—
198#103C22⋊C414yes—
199#100—42no—
200C22⋊C8—22no—
201C22⋊C8—22no—
202C22⋊C8—24no—
203C22⋊C8C2×D414yes—
204#97—22no—
205#97—24no—
206#97—44no—
207C4×C8—22no—
208C4×C8—22no—
209C4×C8—44no—
210C4×C8C2×D412yes—
211C4×C8C2×D412yes—
212C24C4×D411yescharacteristic
213C24C4×D411yescharacteristic
214C2×Q8—22no—
215C2×Q8—42no—
216C2×Q8—42no—
217C2×Q8C2×C22⋊C414yes—
218C2×D4—22no—
219C2×D4—22no—
220C2×D4—41no—
221C22×C4—22no—
222C22×C4—22no—
223C22×C4—21no—
224C22×C4—21no—
225C22×C4—21no—
226C22×C4—21no—
227C22×C4—22no—
228C22×C4—22no—
229C22×C4—41no—
230C22×C4—41no—
231C22×C4—41no—
232C22×C4—41no—
233C22×C4—42no—
234C22×C4—42no—
235C22×C4—44no—
236C22×C4—82no—
237C22×C4C4×D411yescharacteristic
238C22×C4C4×D411yescharacteristic
239C22×C4C23×C411yescharacteristic
240C22×C4C22×D411yescharacteristic
241C22×C4C2×#4111yescharacteristic
242Q16—24no—
243Q16—42no—
244M16—42no—
245M16—42no—
246C2×C8—22no—
247C2×C8—21no—
248C2×C8—21no—
249C2×C8—24no—
250C2×C8—41no—
251C2×C8—42no—
252C2×C8—42no—
253C2×C8—42no—
254C2×C8—42no—
255C2×C8—41no—
256C2×C8—41no—
257C2×C8—44no—
258C2×C8—42no—
259C2×C8C2×C22⋊C412yes—
260C2×C8C2×C22⋊C411yescharacteristic
261C2×C8C2×C22⋊C411yescharacteristic
262C4⋊C4—22no—
263C4⋊C4—24no—
264C4⋊C4—24no—
265C4⋊C4—22no—
266C4⋊C4—22no—
267C4⋊C4—22no—
268C4⋊C4—22no—
269C4⋊C4—22no—
270C4⋊C4—42no—
271C4⋊C4—42no—
272C4⋊C4—44no—
273C4⋊C4—44no—
274C4⋊C4—44no—
275C4⋊C4—44no—
276C4⋊C4C2×C22⋊C412yes—
277C4⋊C4C2×C22⋊C412yes—
278C22⋊C4—22no—
279C22⋊C4—22no—
280C22⋊C4—42no—
281C22⋊C4—42no—
282C22⋊C4—44no—
283C22⋊C4—44no—
284C42—24no—
285C42—24no—
286C42—42no—
287C42—42no—
288C42C22×D412yes—
289C42H(5,𝔽2)12yes—
290C23—21no—
291C23—21no—
292C23—21no—
293C23—21no—
294C23—21no—
295C23—41no—
296C23—41no—
297C23—41no—
298C23—41no—
299C23#44412yes—
300C23C2×C4×D411yescharacteristic
301Q8—24no—
302Q8—24no—
303Q8—82no—
304D4—44no—
305C2×C4—21no—
306C2×C4—21no—
307C2×C4—21no—
308C2×C4—21no—
309C2×C4—22no—
310C2×C4—22no—
311C2×C4—24no—
312C2×C4—21no—
313C2×C4—21no—
314C2×C4—41no—
315C2×C4—42no—
316C2×C4—42no—
317C2×C4—42no—
318C2×C4—42no—
319C2×C4—41no—
320C2×C4—41no—
321C2×C4—41no—
322C2×C4—41no—
323C2×C4—41no—
324C2×C4—41no—
325C2×C4—42no—
326C2×C4—42no—
327C2×C4—42no—
328C2×C4—42no—
329C2×C4—82no—
330C2×C4—84no—
331C2×C4—84no—
332C2×C4—82no—
333C2×C4C22×C22⋊C411yescharacteristic
334C2×C4#51811yescharacteristic
335C2×C4D4212yes—
336C2×C4#54612yes—
337C8—21no—
338C8—22no—
339C8—44no—
340C8—41no—
341C8—41no—
342C8—84no—
343C8—82no—
344C8#41312yes—
345C22—21no—
346C22—21no—
347C22—22no—
348C22—42no—
349C22—42no—
350C22—41no—
351C22—41no—
352C22—41no—
353C22—41no—
354C22—41no—
355C22—81no—
356C22—81no—
357C22D4×C22⋊C411yescharacteristic
358C22C2×#44411yescharacteristic
359C22#295711yescharacteristic
360C4—21no—
361C4—22no—
362C4—22no—
363C4—41no—
364C4—41no—
365C4—44no—
366C4—42no—
367C4—42no—
368C4—82no—
369C4—82no—
370C4C2×#41311yescharacteristic
371C4#288911yescharacteristic
372C2—21no—
373C2—21no—
374C2—42no—
375C2—81no—
376C2#2241111yescharacteristic
377C2#2245211yescharacteristic
378C2#2245711yescharacteristic
subgroupG/H
center, socleC22D4×C22⋊C4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
prime radicalC22×D4#41
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C2×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #18062C2semidirect
4. #9792C2semidirect
5. #9246C2semidirect

Maximal subgroups

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

subgroupindex
C2×#33362
#180622
#174232
#97922
#92462
#86202
#83712
#81642