Finite Group Databasebeta

← #290230

Subgroups — #290230

Order 512.

535 conjugacy classes of subgroups.

The subgroup lattice has 365 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#29287C211yesmaximal, characteristic
2#29208C211yesmaximal, characteristic
3#29141C211yesmaximal, characteristic
4#23250C211yesmaximal, characteristic
5#10655C212yesmaximal
6#10588C212yesmaximal
7#9896C211yesmaximal, characteristic
8#9622C212yesmaximal
9#9602C211yesmaximal, characteristic
10#9569C212yesmaximal
11C4×#332C211yesmaximal, characteristic
12#2868C2211yescharacteristic
13C4×C4⋊Q8C2211yescharacteristic
14C4×C4⋊Q8C2211yescharacteristic
15#2088—22no—
16#2057—22no—
17#2024—22no—
18#1990—21no—
19#1988C2212yes—
20#1987—21no—
21#1985—22no—
22#1956C2212yes—
23#1948—21no—
24#1948C2212yes—
25#1947C2212yes—
26#1947C2212yes—
27#1939—21no—
28#1939C2212yes—
29#1845—21no—
30#1842C2211yescharacteristic
31#1834C2212yes—
32#1803—21no—
33#1803—21no—
34#1770—21no—
35#1770C2212yes—
36C4×C4⋊C8C2211yescharacteristic
37C4×C4⋊C8C2211yescharacteristic
38C4×C4⋊C8C2211yescharacteristic
39C2×#334C2212yes—
40C2×#332C2211yescharacteristic
41C2×#332C2211yescharacteristic
42#1296C2211yescharacteristic
43#1296C2211yescharacteristic
44#1296C2212yes—
45#1296C2212yes—
46#1277C2212yes—
47C2×C4⋊Q8—21no—
48C2×C4⋊Q8—21no—
49C2×C4⋊Q8—21no—
50C2×C4⋊Q8D411yescharacteristic
51C2×C4⋊Q8C2311yescharacteristic
52C2×C4×Q8—21no—
53C2×C4×Q8—21no—
54C2×#107—22no—
55C2×#107—22no—
56C2×#106—22no—
57C2×C4⋊C8—21no—
58C2×C4⋊C8—22no—
59C2×C4⋊C8—21no—
60C2×C4⋊C8D411yescharacteristic
61C2×C4⋊C8D411yescharacteristic
62C2×C4⋊C8C2311yescharacteristic
63C2×C4⋊C8C2312yes—
64C2×C4⋊C8C2311yescharacteristic
65C2×C4⋊C8C2311yescharacteristic
66C2×C4⋊C8C2311yescharacteristic
67C2×#103—22no—
68C2×#103—22no—
69C2×#97—22no—
70C2×C4×C8—21no—
71C2×C4×C8—21no—
72C2×C4×C8C2312yes—
73#391—21no—
74#391—21no—
75#389—21no—
76#389—21no—
77#389—21no—
78#389D412yes—
79#384—21no—
80#384—21no—
81#384—21no—
82#384D411yescharacteristic
83#384C2311yescharacteristic
84#384C2312yes—
85C4×C4⋊C4—21no—
86C4×C4⋊C4—21no—
87C4×C4⋊C4—21no—
88C4×C4⋊C4C2312yes—
89C43C2311yescharacteristic
90#340—22no—
91#340—22no—
92#340—22no—
93#340—22no—
94#340—22no—
95#340—22no—
96#340—22no—
97#336—22no—
98#336—22no—
99#336—21no—
100#336—21no—
101#336D412yes—
102#334—24no—
103#332—22no—
104#332Q814yescomplemented
105C22×Q8—21no—
106C22×Q8—21no—
107C22×Q8—41no—
108C22×C8—21no—
109C22×C8—21no—
110C22×C8—21no—
111C22×C8—21no—
112C22×C8—22no—
113C22×C8C2×D412yes—
114C22×C8C2×D412yes—
115C4⋊Q8—22no—
116C4⋊Q8—22no—
117C4⋊Q8—21no—
118C4⋊Q8—21no—
119C4⋊Q8—21no—
120C4⋊Q8—81no—
121C4⋊Q8—81no—
122C4⋊Q8—81no—
123C4⋊Q8—81no—
124C4⋊Q8—82no—
125C4⋊Q8C2×Q812yes—
126C4×Q8—42no—
127C4×Q8—42no—
128C4×Q8—81no—
129C4×Q8—81no—
130C2×C4⋊C4—21no—
131C2×C4⋊C4—21no—
132C2×C4⋊C4—21no—
133C2×C4⋊C4—21no—
134C2×C4⋊C4—21no—
135C2×C4⋊C4—22no—
136C2×C4⋊C4—22no—
137C2×C4⋊C4—21no—
138C2×C4⋊C4—21no—
139C2×C4⋊C4—22no—
140C2×C4⋊C4—22no—
141C2×C4⋊C4—21no—
142C2×C4⋊C4—41no—
143C2×C4⋊C4—41no—
144C2×C4⋊C4—41no—
145C2×C4⋊C4—41no—
146C2×C4⋊C4C2×D412yes—
147C2×C4⋊C4C2×D412yes—
148C2×C42—22no—
149C2×C42—41no—
150C2×C42C2×D411yescharacteristic
151C2×C42C2×D411yescharacteristic
152C2×C42C2×D411yescharacteristic
153C2×C42C2×D411yescharacteristic
154C2×C42#4111yescharacteristic
155C2×C42#4111yescharacteristic
156C2×C42C2411yescharacteristic
157#107—82no—
158#107—82no—
159#106—82no—
160C4⋊C8—22no—
161C4⋊C8—22no—
162C4⋊C8—21no—
163C4⋊C8—22no—
164C4⋊C8—21no—
165C4⋊C8—21no—
166C4⋊C8—21no—
167C4⋊C8—21no—
168C4⋊C8—22no—
169C4⋊C8—42no—
170C4⋊C8—81no—
171C4⋊C8—82no—
172C4⋊C8C2×Q812yes—
173C4⋊C8C2×Q812yes—
174C4⋊C8#4112yes—
175#103—82no—
176#103—82no—
177#97—82no—
178C4×C8—22no—
179C4×C8—22no—
180C4×C8—42no—
181C4×C8—42no—
182#95—22no—
183#95—22no—
184#95—41no—
185#95—41no—
186C2×Q8—22no—
187C2×Q8—41no—
188C2×Q8—42no—
189C2×Q8—41no—
190C2×Q8—41no—
191C2×Q8—41no—
192C2×Q8—41no—
193C2×Q8—81no—
194C2×Q8—81no—
195C2×Q8—82no—
196C2×Q8—82no—
197C2×Q8—82no—
198C22×C4—22no—
199C22×C4—22no—
200C22×C4—41no—
201C22×C4—41no—
202C22×C4—42no—
203C22×C4#12111yescharacteristic
204C22×C4#12111yescharacteristic
205C22×C4#12111yescharacteristic
206C22×C4#12111yescharacteristic
207C22×C4C22×D411yescharacteristic
208C22×C4C22×D411yescharacteristic
209C22×C4C2×#4111yescharacteristic
210C2×C8—22no—
211C2×C8—22no—
212C2×C8—22no—
213C2×C8—22no—
214C2×C8—22no—
215C2×C8—22no—
216C2×C8—22no—
217C2×C8—22no—
218C2×C8—22no—
219C2×C8—41no—
220C2×C8—41no—
221C2×C8—41no—
222C2×C8—41no—
223C2×C8—41no—
224C2×C8—41no—
225C2×C8—41no—
226C2×C8—42no—
227C2×C8—41no—
228C2×C8—42no—
229C2×C8—41no—
230C2×C8—42no—
231C4⋊C4—22no—
232C4⋊C4—22no—
233C4⋊C4—22no—
234C4⋊C4—22no—
235C4⋊C4—22no—
236C4⋊C4—22no—
237C4⋊C4—41no—
238C4⋊C4—41no—
239C4⋊C4—42no—
240C4⋊C4—42no—
241C4⋊C4—41no—
242C4⋊C4—41no—
243C4⋊C4—42no—
244C4⋊C4—42no—
245C4⋊C4—41no—
246C4⋊C4—42no—
247C4⋊C4—42no—
248C4⋊C4—41no—
249C4⋊C4—81no—
250C4⋊C4—82no—
251C4⋊C4—81no—
252C4⋊C4—82no—
253C4⋊C4—82no—
254C4⋊C4—82no—
255C4⋊C4—82no—
256C4⋊C4—82no—
257C42—21no—
258C42—21no—
259C42—21no—
260C42—21no—
261C42—22no—
262C42—41no—
263C42—42no—
264C42—42no—
265C42—82no—
266C42#13512yes—
267C42#13712yes—
268C42#13712yes—
269C42#13712yes—
270C42C22×Q811yescharacteristic
271C42C2×#4111yescharacteristic
272C422−1+411yescharacteristic
273C422−1+411yescharacteristic
274C23C2×#12111yescharacteristic
275Q8—42no—
276Q8—42no—
277Q8—84no—
278Q8—81no—
279Q8—81no—
280Q8—164no—
281C2×C4—21no—
282C2×C4—21no—
283C2×C4—21no—
284C2×C4—21no—
285C2×C4—21no—
286C2×C4—21no—
287C2×C4—21no—
288C2×C4—21no—
289C2×C4—21no—
290C2×C4—42no—
291C2×C4—42no—
292C2×C4—42no—
293C2×C4—42no—
294C2×C4—42no—
295C2×C4—42no—
296C2×C4—42no—
297C2×C4—81no—
298C2×C4—81no—
299C2×C4—81no—
300C2×C4—82no—
301C2×C4—82no—
302C2×C4—81no—
303C2×C4—82no—
304C2×C4—81no—
305C2×C4#46412yes—
306C2×C4#46412yes—
307C2×C4#47012yes—
308C2×C4#47012yes—
309C2×C4#47112yes—
310C2×C4#53311yescharacteristic
311C2×C4#54111yescharacteristic
312C2×C4#54711yescharacteristic
313C2×C4#54811yescharacteristic
314C2×C4D4×Q811yescharacteristic
315C2×C4D4×Q811yescharacteristic
316C2×C4C2×#13511yescharacteristic
317C2×C4C2×#13711yescharacteristic
318C2×C4C2×#13711yescharacteristic
319C2×C4C2×#13711yescharacteristic
320C2×C4#57611yescharacteristic
321C2×C4#57611yescharacteristic
322C2×C4#57611yescharacteristic
323C2×C4#57711yescharacteristic
324C8—42no—
325C8—42no—
326C8—42no—
327C8—42no—
328C8—42no—
329C8—81no—
330C8—82no—
331C22#242911yescharacteristic
332C22C2×#46411yescharacteristic
333C22#304111yescharacteristic
334C22C2×#47011yescharacteristic
335C22#306211yescharacteristic
336C22C2×#47111yescharacteristic
337C22#306911yescharacteristic
338C4—22no—
339C4—21no—
340C4—21no—
341C4—22no—
342C4—41no—
343C4—82no—
344C4—82no—
345C4—82no—
346C4—161no—
347C4—162no—
348C4#170414yes—
349C4#334111yescharacteristic
350C4#335711yescharacteristic
351C4#337211yescharacteristic
352C4#337511yescharacteristic
353C4#338011yescharacteristic
354C4#338511yescharacteristic
355C4#338511yescharacteristic
356C4#338711yescharacteristic
357C2C2×#170411yescharacteristic
358C2#1924511yescharacteristic
359C2#1942311yescharacteristic
360C2#1955211yescharacteristic
361C2#2545011yescharacteristic
362C2#2548011yescharacteristic
363C2#2549211yescharacteristic
subgroupG/H
center, socle, prime radicalC23C2×#121
commutator, Frattini, cosocleC2×C42C24
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C43 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
104. #332Q8semidirect

Maximal subgroups

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

subgroupindex
#292872
#292082
#291412
#232502
#106552
#105882
#98962
#96222
#96022
#95692
C4×#3322