Finite Group Databasebeta

← #220522

Subgroups — #220522

Order 512.

802 conjugacy classes of subgroups.

The subgroup lattice has 384 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3336C211yesmaximal, characteristic
2#18058C212yesmaximal, complemented
3#17423C212yesmaximal
4#9788C212yesmaximal, complemented
5#9246C214yesmaximal, complemented
6#8622C211yesmaximal, characteristic
7#8371C212yesmaximal
8C8×#100C211yesmaximal, 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#1591—22no—
26#1587—24no—
27C8×Q16—22no—
28#1568C2212yes—
29#1528C2214yes—
30#1491C2214yes—
31#1473—24no—
32#1470C2214yes—
33#1466C2212yes—
34#1464—22no—
35#1460C2211yescharacteristic
36#1458—22no—
37C8×M16C2212yes—
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#323D411yescharacteristic
113#323D411yescharacteristic
114#323D412yes—
115#322—22no—
116C82—22no—
117C22×Q8C22×C412yes—
118C22×D4#4111yescharacteristic
119C23×C4—22no—
120C23×C4C22×C411yescharacteristic
121C23×C4C22×C411yescharacteristic
122C2×Q16—42no—
123C2×Q16C22⋊C412yes—
124C2×Q16C22⋊C412yes—
125C2×M16—22no—
126C22×C8—21no—
127C22×C8—21no—
128C22×C8—21no—
129C22×C8—21no—
130C22×C8—42no—
131C22×C8—42no—
132C22×C8C22×C411yescharacteristic
133C22×C8C22×C411yescharacteristic
134C22×C8C22×C411yescharacteristic
135C22×C8C22×C411yescharacteristic
136#126—44no—
137#126—44no—
138#125—24no—
139#125—24no—
140#125—44no—
141#125—44no—
142#123—44no—
143#123—44no—
144C22⋊Q8—24no—
145C22⋊Q8—24no—
146C22⋊Q8—42no—
147C22⋊Q8—42no—
148C4×Q8—24no—
149C4×Q8C2×D414yes—
150C4×Q8C2×D414yes—
151C4×D4—22no—
152C4×D4—22no—
153#117—22no—
154#117—22no—
155#117—44no—
156#117—44no—
157#117—42no—
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—22no—
166C2×C4⋊C4—42no—
167C2×C4⋊C4C22×C412yes—
168C2×C4⋊C4C22×C412yes—
169C2×C4⋊C4C22×C412yes—
170C2×C4⋊C4#4111yescharacteristic
171C2×C22⋊C4—22no—
172C2×C22⋊C4—22no—
173C2×C22⋊C4—22no—
174C2×C22⋊C4C2×D411yescharacteristic
175C2×C22⋊C4C2×D411yescharacteristic
176C2×C42—22no—
177C2×C42—22no—
178C2×C42C2411yescharacteristic
179#107—21no—
180#107—21no—
181#107—21no—
182#107—21no—
183#107—22no—
184#107—22no—
185#107C22⋊C412yes—
186#107C22⋊C411yescharacteristic
187#107C22⋊C411yescharacteristic
188#106—24no—
189#106—22no—
190#106—22no—
191C4⋊C8—22no—
192C4⋊C8—24no—
193C4⋊C8—22no—
194C4⋊C8—44no—
195C4⋊C8—44no—
196#103—24no—
197#103—24no—
198#103C22⋊C414yes—
199#103C22⋊C414yes—
200#100—24no—
201C22⋊C8—22no—
202C22⋊C8—22no—
203C22⋊C8—24no—
204C22⋊C8C2×D412yes—
205C22⋊C8C2×D412yes—
206#97—22no—
207#97—24no—
208#97—44no—
209C4×C8—22no—
210C4×C8—22no—
211C4×C8—44no—
212C4×C8C2×D412yes—
213C4×C8C2×D412yes—
214C24C4×D411yescharacteristic
215C24C4×D411yescharacteristic
216C2×Q8—22no—
217C2×Q8—42no—
218C2×Q8—42no—
219C2×Q8C2×C22⋊C414yes—
220C2×D4—22no—
221C2×D4—22no—
222C2×D4—41no—
223C22×C4—22no—
224C22×C4—22no—
225C22×C4—21no—
226C22×C4—21no—
227C22×C4—21no—
228C22×C4—21no—
229C22×C4—22no—
230C22×C4—22no—
231C22×C4—41no—
232C22×C4—41no—
233C22×C4—41no—
234C22×C4—41no—
235C22×C4—42no—
236C22×C4—42no—
237C22×C4—44no—
238C22×C4—82no—
239C22×C4C4×D411yescharacteristic
240C22×C4C4×D411yescharacteristic
241C22×C4C23×C411yescharacteristic
242C22×C4C22×D411yescharacteristic
243C22×C4C2×#4111yescharacteristic
244Q16—24no—
245Q16—42no—
246M16—24no—
247M16—42no—
248C2×C8—22no—
249C2×C8—21no—
250C2×C8—21no—
251C2×C8—22no—
252C2×C8—22no—
253C2×C8—41no—
254C2×C8—42no—
255C2×C8—42no—
256C2×C8—42no—
257C2×C8—42no—
258C2×C8—42no—
259C2×C8—41no—
260C2×C8—41no—
261C2×C8—44no—
262C2×C8C2×C22⋊C412yes—
263C2×C8C2×C22⋊C411yescharacteristic
264C2×C8C2×C22⋊C411yescharacteristic
265C4⋊C4—22no—
266C4⋊C4—24no—
267C4⋊C4—24no—
268C4⋊C4—22no—
269C4⋊C4—22no—
270C4⋊C4—22no—
271C4⋊C4—22no—
272C4⋊C4—22no—
273C4⋊C4—42no—
274C4⋊C4—42no—
275C4⋊C4—44no—
276C4⋊C4—44no—
277C4⋊C4—44no—
278C4⋊C4—44no—
279C4⋊C4C2×C22⋊C412yes—
280C4⋊C4C2×C22⋊C412yes—
281C22⋊C4—22no—
282C22⋊C4—22no—
283C22⋊C4—42no—
284C22⋊C4—42no—
285C22⋊C4—44no—
286C22⋊C4—44no—
287C42—24no—
288C42—24no—
289C42—42no—
290C42—42no—
291C42C22×D412yes—
292C42H(5,𝔽2)12yes—
293C23—21no—
294C23—21no—
295C23—21no—
296C23—21no—
297C23—21no—
298C23—41no—
299C23—41no—
300C23—41no—
301C23—41no—
302C23#44312yes—
303C23C2×C4×D411yescharacteristic
304Q8—24no—
305Q8—24no—
306Q8—82no—
307D4—44no—
308C2×C4—21no—
309C2×C4—21no—
310C2×C4—21no—
311C2×C4—21no—
312C2×C4—22no—
313C2×C4—22no—
314C2×C4—24no—
315C2×C4—21no—
316C2×C4—21no—
317C2×C4—41no—
318C2×C4—42no—
319C2×C4—42no—
320C2×C4—42no—
321C2×C4—42no—
322C2×C4—41no—
323C2×C4—41no—
324C2×C4—41no—
325C2×C4—41no—
326C2×C4—41no—
327C2×C4—41no—
328C2×C4—42no—
329C2×C4—42no—
330C2×C4—42no—
331C2×C4—42no—
332C2×C4—82no—
333C2×C4—84no—
334C2×C4—84no—
335C2×C4—82no—
336C2×C4C22×C22⋊C411yescharacteristic
337C2×C4#51811yescharacteristic
338C2×C4D4212yes—
339C2×C4#54612yes—
340C8—21no—
341C8—22no—
342C8—42no—
343C8—42no—
344C8—44no—
345C8—41no—
346C8—41no—
347C8—84no—
348C8#41312yes—
349C22—21no—
350C22—21no—
351C22—22no—
352C22—42no—
353C22—42no—
354C22—41no—
355C22—41no—
356C22—41no—
357C22—41no—
358C22—41no—
359C22—81no—
360C22—81no—
361C22D4×C22⋊C411yescharacteristic
362C22C2×#44311yescharacteristic
363C22#295711yescharacteristic
364C4—21no—
365C4—22no—
366C4—22no—
367C4—41no—
368C4—41no—
369C4—44no—
370C4—42no—
371C4—42no—
372C4—82no—
373C4—82no—
374C4C2×#41311yescharacteristic
375C4#288911yescharacteristic
376C2—21no—
377C2—21no—
378C2—42no—
379C2—81no—
380C2#2241111yescharacteristic
381C2#2245411yescharacteristic
382C2#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. #18058C2semidirect
4. #9788C2semidirect
5. #9246C2semidirect

Maximal subgroups

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

subgroupindex
C2×#33362
#180582
#174232
#97882
#92462
#86222
#83712
C8×#1002