Finite Group Databasebeta

← #220215

Subgroups — #220215

Order 512.

800 conjugacy classes of subgroups.

The subgroup lattice has 386 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3177C211yesmaximal, characteristic
2#17880C212yesmaximal, complemented
3#17391C212yesmaximal
4#9671C212yesmaximal, complemented
5#9092C212yesmaximal, complemented
6#9088C212yesmaximal, complemented
7#8602C211yesmaximal, characteristic
8#8327C212yesmaximal
9#8157C211yesmaximal, characteristic
10C2×#541C2212yes—
11#3177—24no—
12#3177—24no—
13C2×#499C411yescharacteristic
14C2×#494C411yescharacteristic
15C2×#491C412yes—
16C2×#470C412yes—
17C2×#467C411yescharacteristic
18C2×#467C411yescharacteristic
19#2983C2212yes—
20#2982C2212yes—
21C2×#441C2212yes—
22C2×C4×Q16C2211yescharacteristic
23C2×C4×Q16C2211yescharacteristic
24C2×#405C2211yescharacteristic
25#1587—24no—
26#1584—24no—
27#1564C2212yes—
28#1479C2212yes—
29#1479C2212yes—
30#1479C2212yes—
31#1479C2212yes—
32#1473—24no—
33#1470—24no—
34#1461C2212yes—
35#1461C2212yes—
36#1458C2211yescharacteristic
37#1458C2212yes—
38#1458C2211yescharacteristic
39#1452C2212yes—
40#1281C2212yes—
41#1275C2211yescharacteristic
42#1275C2211yescharacteristic
43C22×Q16C2×C412yes—
44C22×Q16C2×C411yescharacteristic
45C22×Q16C2×C411yescharacteristic
46C22×M16C2×C411yescharacteristic
47C23×C8C2×C411yescharacteristic
48#541—24no—
49#541—24no—
50#541—42no—
51#541—42no—
52C2×C4⋊Q8C2×C412yes—
53C2×#126—22no—
54C2×#126—22no—
55C2×#125—22no—
56C2×#125—22no—
57C2×#125C2×C412yes—
58C2×C22⋊Q8C2×C412yes—
59C2×C22⋊Q8C2×C412yes—
60C2×C4×Q8C2312yes—
61C2×C4×Q8C2312yes—
62C2×#117C2311yescharacteristic
63#499—22no—
64#499—22no—
65#494—22no—
66#494—22no—
67#491—28no—
68#470—28no—
69#467—22no—
70#467—22no—
71#467—22no—
72#467—22no—
73#446—24no—
74#446—42no—
75C8×Q8—24no—
76C8×Q8—42no—
77#441—24no—
78#441—24no—
79C4×Q16—22no—
80C4×Q16—22no—
81C4×Q16D414yes—
82C4×Q16D414yes—
83#432—24no—
84#431—24no—
85#431C2312yes—
86#431C2312yes—
87#431C2312yes—
88#431C2312yes—
89C2×#107C2×C411yescharacteristic
90C2×#107C2×C411yescharacteristic
91C2×#106C2×C412yes—
92C2×#103C2×C412yes—
93C2×#103C2×C412yes—
94C2×#103C2×C412yes—
95C2×#103C2×C412yes—
96#405—22no—
97#405—22no—
98C2×#97C2×C411yescharacteristic
99C2×#97C2311yescharacteristic
100C2×C4×C8C2×C411yescharacteristic
101C2×C4×C8C2311yescharacteristic
102#335—22no—
103#334—22no—
104Q8⋊C8—24no—
105Q8⋊C8—24no—
106Q8⋊C8—42no—
107Q8⋊C8—42no—
108#324—22no—
109#324—21no—
110#324—24no—
111#324—24no—
112#324D412yes—
113#323—22no—
114#323—21no—
115#323D412yes—
116#322—22no—
117#322—22no—
118C22×Q8C22×C412yes—
119C22×Q8C22×C412yes—
120C23×C4C22×C411yescharacteristic
121C2×Q16—24no—
122C2×Q16—21no—
123C2×Q16—21no—
124C2×Q16—24no—
125C2×Q16—41no—
126C2×Q16—41no—
127C2×Q16—41no—
128C2×Q16—41no—
129C2×Q16—42no—
130C2×Q16C22⋊C412yes—
131C2×Q16C22⋊C412yes—
132C2×M16—22no—
133C2×M16—22no—
134C2×M16—41no—
135C22×C8—21no—
136C22×C8—21no—
137C22×C8—41no—
138C22×C8—41no—
139C22×C8C22×C411yescharacteristic
140C22×C8C22×C411yescharacteristic
141C22×C8C22×C411yescharacteristic
142C22×C8C22×C411yescharacteristic
143C4⋊Q8—24no—
144C4⋊Q8—42no—
145#126—44no—
146#126—44no—
147#126—44no—
148#126—44no—
149#125—24no—
150#125—42no—
151#125—44no—
152#125—44no—
153#125—44no—
154#125—44no—
155C22⋊Q8—24no—
156C22⋊Q8—24no—
157C22⋊Q8—42no—
158C22⋊Q8—42no—
159C4×Q8—22no—
160C4×Q8—22no—
161C4×Q8—22no—
162C4×Q8—22no—
163C4×Q8—22no—
164C4×Q8—22no—
165C4×Q8C2×D414yes—
166C4×Q8C2×D414yes—
167#117—22no—
168#117—22no—
169C2×C4⋊C4—22no—
170C2×C4⋊C4—22no—
171C2×C4⋊C4—22no—
172C2×C4⋊C4—22no—
173C2×C4⋊C4—22no—
174C2×C4⋊C4—22no—
175C2×C4⋊C4—22no—
176C2×C4⋊C4—22no—
177C2×C4⋊C4C22×C412yes—
178C2×C4⋊C4C22×C412yes—
179C2×C4⋊C4C2×D411yescharacteristic
180C2×C4⋊C4#4111yescharacteristic
181C2×C22⋊C4—22no—
182C2×C22⋊C4—22no—
183C2×C22⋊C4C2×D411yescharacteristic
184C2×C22⋊C4#4111yescharacteristic
185C2×C42—22no—
186C2×C42—22no—
187C2×C42C22×C411yescharacteristic
188C2×C42C2411yescharacteristic
189#107—21no—
190#107—21no—
191#107C22⋊C412yes—
192#107C22⋊C412yes—
193#106—24no—
194C4⋊C8—22no—
195C4⋊C8—22no—
196C4⋊C8—22no—
197C4⋊C8—22no—
198C4⋊C8—44no—
199C4⋊C8—44no—
200#103—24no—
201#103—24no—
202#103—24no—
203#103C22⋊C418yes—
204C22⋊C8—22no—
205C22⋊C8—22no—
206C22⋊C8—22no—
207C22⋊C8C2×D414yes—
208#97—22no—
209#97—22no—
210#97—44no—
211C4×C8—21no—
212C4×C8—21no—
213C4×C8—22no—
214C4×C8—22no—
215C4×C8—22no—
216C4×C8—22no—
217C4×C8—44no—
218C4×C8C2×D412yes—
219C4×C8C2×D412yes—
220C24C4×D411yescharacteristic
221C2×Q8—22no—
222C2×Q8—22no—
223C2×Q8—22no—
224C2×Q8—22no—
225C2×Q8—22no—
226C2×Q8—22no—
227C2×Q8—22no—
228C2×Q8—22no—
229C2×Q8—22no—
230C2×Q8—22no—
231C2×Q8—22no—
232C2×Q8C2×C22⋊C414yes—
233C22×C4—22no—
234C22×C4—22no—
235C22×C4—21no—
236C22×C4—21no—
237C22×C4—21no—
238C22×C4—21no—
239C22×C4—21no—
240C22×C4—21no—
241C22×C4—22no—
242C22×C4—22no—
243C22×C4C4×D411yescharacteristic
244C22×C4C4×D411yescharacteristic
245C22×C4C4×D411yescharacteristic
246C22×C4C23×C411yescharacteristic
247C22×C4C22×D411yescharacteristic
248C22×C4C2×#4111yescharacteristic
249Q16—42no—
250Q16—42no—
251Q16—48no—
252Q16—42no—
253Q16—42no—
254M16—44no—
255C2×C8—21no—
256C2×C8—21no—
257C2×C8—21no—
258C2×C8—21no—
259C2×C8—21no—
260C2×C8—21no—
261C2×C8—21no—
262C2×C8—24no—
263C2×C8—21no—
264C2×C8—21no—
265C2×C8—42no—
266C2×C8—42no—
267C2×C8—42no—
268C2×C8—42no—
269C2×C8—42no—
270C2×C8C2×C22⋊C411yescharacteristic
271C2×C8C2×C22⋊C411yescharacteristic
272C2×C8C2×C22⋊C412yes—
273C2×C8#13112yes—
274C4⋊C4—22no—
275C4⋊C4—22no—
276C4⋊C4—24no—
277C4⋊C4—24no—
278C4⋊C4—22no—
279C4⋊C4—22no—
280C4⋊C4—22no—
281C4⋊C4—44no—
282C4⋊C4—44no—
283C4⋊C4—44no—
284C4⋊C4—44no—
285C4⋊C4—44no—
286C4⋊C4—44no—
287C4⋊C4—42no—
288C4⋊C4—42no—
289C4⋊C4C2×C22⋊C414yes—
290C22⋊C4—22no—
291C22⋊C4—22no—
292C22⋊C4—44no—
293C22⋊C4—44no—
294C42—21no—
295C42—21no—
296C42—24no—
297C42—24no—
298C42—42no—
299C42—42no—
300C42C22×D412yes—
301C42H(5,𝔽2)12yes—
302C23—21no—
303C23—21no—
304C23—21no—
305C23—41no—
306C23—41no—
307C23C2×C4×D411yescharacteristic
308Q8—24no—
309Q8—24no—
310Q8—42no—
311Q8—42no—
312Q8—42no—
313Q8—42no—
314Q8—42no—
315Q8—42no—
316C2×C4—21no—
317C2×C4—21no—
318C2×C4—24no—
319C2×C4—24no—
320C2×C4—21no—
321C2×C4—21no—
322C2×C4—21no—
323C2×C4—41no—
324C2×C4—41no—
325C2×C4—42no—
326C2×C4—42no—
327C2×C4—41no—
328C2×C4—41no—
329C2×C4—41no—
330C2×C4—41no—
331C2×C4—41no—
332C2×C4—41no—
333C2×C4—42no—
334C2×C4—42no—
335C2×C4—42no—
336C2×C4—42no—
337C2×C4—82no—
338C2×C4—82no—
339C2×C4C22×C22⋊C411yescharacteristic
340C2×C4#51811yescharacteristic
341C2×C4D4212yes—
342C2×C4#54612yes—
343C2×C4C2×#13111yescharacteristic
344C2×C4#56811yescharacteristic
345C2×C4#56811yescharacteristic
346C2×C4#56811yescharacteristic
347C8—21no—
348C8—21no—
349C8—22no—
350C8—22no—
351C8—44no—
352C8—41no—
353C8—82no—
354C8—82no—
355C8—82no—
356C22—21no—
357C22—21no—
358C22—41no—
359C22—41no—
360C22—41no—
361C22—81no—
362C22—81no—
363C22D4×C22⋊C411yescharacteristic
364C22#293211yescharacteristic
365C22#293511yescharacteristic
366C4—22no—
367C4—22no—
368C4—41no—
369C4—44no—
370C4—41no—
371C4—41no—
372C4—44no—
373C4—82no—
374C4—82no—
375C4#287911yescharacteristic
376C4#288011yescharacteristic
377C4#298011yescharacteristic
378C4#298111yescharacteristic
379C2—21no—
380C2—21no—
381C2—81no—
382C2#2240311yescharacteristic
383C2#2245711yescharacteristic
384C2#2246011yescharacteristic
subgroupG/H
center, socleC22D4×C22⋊C4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
prime radicalC24C4×D4
LCSG ▷ C22×C4 ▷ C2 ▷ 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. #17880C2semidirect
4. #9671C2semidirect
5. #9092C2semidirect
6. #9088C2semidirect

Maximal subgroups

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

subgroupindex
C2×#31772
#178802
#173912
#96712
#90922
#90882
#86022
#83272
#81572