Finite Group Databasebeta

← #323344

Subgroups — #323344

Order 512.

680 conjugacy classes of subgroups.

The subgroup lattice has 349 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#33289C211yesmaximal, characteristic
2#29206C211yesmaximal, characteristic
3#29206C211yesmaximal, characteristic
4#29141C211yesmaximal, characteristic
5#28511C211yesmaximal, characteristic
6C4×#498C211yesmaximal, characteristic
7#10686C212yesmaximal
8#10675C212yesmaximal
9#9828C212yesmaximal
10#8819C212yesmaximal
11#8261C211yesmaximal, characteristic
12C2×#498C2211yescharacteristic
13C2×#477—22no—
14#2868C2211yescharacteristic
15#2868C2211yescharacteristic
16C4×C4⋊Q8C2211yescharacteristic
17C4×C4⋊Q8C2211yescharacteristic
18#2095—22no—
19#2058—22no—
20#2058—22no—
21#2024—22no—
22#1990—22no—
23#1990—22no—
24#1990C2212yes—
25#1990C2212yes—
26#1990C2211yescharacteristic
27#1990C2211yescharacteristic
28#1956C2212yes—
29#1956C2212yes—
30#1948—22no—
31#1948—22no—
32#1941—22no—
33#1845C2212yes—
34#1844C2212yes—
35#1844C2212yes—
36#1844C2211yescharacteristic
37#1834C2212yes—
38#1834C2212yes—
39#1834C2212yes—
40#1803—22no—
41C4×#106—22no—
42C4×C4⋊C8C2211yescharacteristic
43C4×C4⋊C8C2211yescharacteristic
44C42×C8C2211yescharacteristic
45C2×#322C2212yes—
46#1273C2212yes—
47C2×C4⋊Q8—21no—
48C2×C4⋊Q8—21no—
49C2×C4⋊Q8—21no—
50C2×C4⋊Q8—21no—
51C2×C4⋊Q8—21no—
52C2×C4⋊Q8—21no—
53C2×C4⋊Q8C2311yescharacteristic
54C2×C4⋊Q8C2311yescharacteristic
55C2×C4×Q8—21no—
56C2×C4×Q8—21no—
57#498—22no—
58#498Q814yescomplemented
59#477—48no—
60C2×#107—24no—
61C2×#106—22no—
62C2×#106—22no—
63C2×#106—24no—
64C2×#106—24no—
65C2×C4⋊C8D412yes—
66C2×C4⋊C8D412yes—
67C2×#103—24no—
68C2×#103—24no—
69C2×C4×C8D412yes—
70C2×C4×C8D412yes—
71C2×C4×C8C2311yescharacteristic
72C2×C4×C8C2312yes—
73C2×C4×C8C2312yes—
74C2×C4×C8C2311yescharacteristic
75#391—21no—
76#391—21no—
77#389—21no—
78#389—21no—
79#389—21no—
80#389—21no—
81#384—21no—
82#384—21no—
83#384—21no—
84#384—21no—
85#384—21no—
86#384—21no—
87#384C2311yescharacteristic
88#384C2311yescharacteristic
89#384C2312yes—
90#384C2312yes—
91C4×C4⋊C4—21no—
92C4×C4⋊C4—21no—
93C4×C4⋊C4—21no—
94C4×C4⋊C4—21no—
95C43C2311yescharacteristic
96#340—24no—
97#340—24no—
98#340—24no—
99#340—24no—
100#340—24no—
101#340—22no—
102#340—22no—
103#336D412yes—
104#336D412yes—
105#322—24no—
106C22×Q8—21no—
107C22×Q8—21no—
108C22×Q8—41no—
109C22×Q8—41no—
110C22×C8C2×D414yes—
111C22×C8C2×D412yes—
112C22×C8C2×D412yes—
113C22×C8C2×D414yes—
114C4⋊Q8—21no—
115C4⋊Q8—21no—
116C4⋊Q8—21no—
117C4⋊Q8—21no—
118C4⋊Q8—21no—
119C4⋊Q8—21no—
120C4⋊Q8—44no—
121C4⋊Q8—44no—
122C4⋊Q8—44no—
123C4⋊Q8—82no—
124C4⋊Q8—82no—
125C4⋊Q8—82no—
126C4⋊Q8C2×Q812yes—
127C4⋊Q8C2×Q812yes—
128C4×Q8—44no—
129C4×Q8—44no—
130C2×C4⋊C4—21no—
131C2×C4⋊C4—21no—
132C2×C4⋊C4—21no—
133C2×C4⋊C4—21no—
134C2×C4⋊C4—21no—
135C2×C4⋊C4—21no—
136C2×C4⋊C4—22no—
137C2×C4⋊C4—22no—
138C2×C4⋊C4—22no—
139C2×C4⋊C4—22no—
140C2×C4⋊C4—22no—
141C2×C4⋊C4—22no—
142C2×C4⋊C4—21no—
143C2×C4⋊C4—21no—
144C2×C4⋊C4—21no—
145C2×C4⋊C4—21no—
146C2×C4⋊C4—41no—
147C2×C4⋊C4—41no—
148C2×C4⋊C4—41no—
149C2×C4⋊C4—41no—
150C2×C4⋊C4—41no—
151C2×C4⋊C4—41no—
152C2×C4⋊C4—41no—
153C2×C4⋊C4—41no—
154C2×C42—41no—
155C2×C42—41no—
156C2×C42C2×D411yescharacteristic
157C2×C42C2×D411yescharacteristic
158C2×C42C2×D411yescharacteristic
159C2×C42C2×D411yescharacteristic
160C2×C42C2×D411yescharacteristic
161C2×C42C2×D411yescharacteristic
162C2×C42C2411yescharacteristic
163#107—48no—
164#106—28no—
165#106—44no—
166#106—84no—
167#106—84no—
168C4⋊C8—24no—
169C4⋊C8SD1618yes—
170#103—48no—
171#103—84no—
172C4×C8—24no—
173C4×C8—22no—
174C4×C8—22no—
175C4×C8—22no—
176C4×C8—21no—
177C4×C8—21no—
178C4×C8—22no—
179C4×C8—42no—
180C4×C8C2×Q812yes—
181C4×C8#4112yes—
182#95—41no—
183#95—41no—
184#95—41no—
185#95—41no—
186C2×Q8—22no—
187C2×Q8—22no—
188C2×Q8—41no—
189C2×Q8—41no—
190C2×Q8—42no—
191C2×Q8—42no—
192C2×Q8—42no—
193C2×Q8—42no—
194C2×Q8—82no—
195C2×Q8—82no—
196C2×Q8—82no—
197C2×Q8—82no—
198C2×Q8—82no—
199C2×Q8—82no—
200C22×C4—41no—
201C22×C4—41no—
202C22×C4—41no—
203C22×C4—41no—
204C22×C4—42no—
205C22×C4—42no—
206C22×C4#12711yescharacteristic
207C22×C4#12711yescharacteristic
208C22×C4#12711yescharacteristic
209C22×C4#12711yescharacteristic
210C22×C4C22×D411yescharacteristic
211C22×C4C22×D411yescharacteristic
212C22×C4C22×D411yescharacteristic
213C2×C8—24no—
214C2×C8—22no—
215C2×C8—22no—
216C2×C8—24no—
217C2×C8—22no—
218C2×C8—24no—
219C2×C8—22no—
220C2×C8—24no—
221C2×C8—22no—
222C2×C8—24no—
223C2×C8C22⋊Q814yes—
224C2×C8C2×SD1618yes—
225C4⋊C4—22no—
226C4⋊C4—22no—
227C4⋊C4—22no—
228C4⋊C4—22no—
229C4⋊C4—41no—
230C4⋊C4—41no—
231C4⋊C4—41no—
232C4⋊C4—41no—
233C4⋊C4—44no—
234C4⋊C4—42no—
235C4⋊C4—42no—
236C4⋊C4—44no—
237C4⋊C4—42no—
238C4⋊C4—42no—
239C4⋊C4—44no—
240C4⋊C4—44no—
241C4⋊C4—42no—
242C4⋊C4—44no—
243C4⋊C4—42no—
244C4⋊C4—44no—
245C4⋊C4—82no—
246C4⋊C4—82no—
247C4⋊C4—82no—
248C4⋊C4—82no—
249C4⋊C4—82no—
250C4⋊C4—82no—
251C4⋊C4—82no—
252C4⋊C4—82no—
253C42—22no—
254C42—22no—
255C42—22no—
256C42—82no—
257C42—82no—
258C42C2×SD1614yes—
259C42#13714yes—
260C42#13714yes—
261C42C22×Q811yescharacteristic
262C42C2×#4111yescharacteristic
263C422−1+411yescharacteristic
264C422−1+411yescharacteristic
265C23C2×#12711yescharacteristic
266Q8—44no—
267Q8—44no—
268Q8—82no—
269Q8—82no—
270Q8—164no—
271Q8—164no—
272C2×C4—21no—
273C2×C4—21no—
274C2×C4—21no—
275C2×C4—21no—
276C2×C4—21no—
277C2×C4—21no—
278C2×C4—42no—
279C2×C4—42no—
280C2×C4—81no—
281C2×C4—81no—
282C2×C4—81no—
283C2×C4—81no—
284C2×C4—81no—
285C2×C4—82no—
286C2×C4—81no—
287C2×C4—82no—
288C2×C4—82no—
289C2×C4—81no—
290C2×C4—81no—
291C2×C4—82no—
292C2×C4—81no—
293C2×C4—82no—
294C2×C4—82no—
295C2×C4—81no—
296C2×C4#49212yes—
297C2×C4#49212yes—
298C2×C4#49712yes—
299C2×C4#49712yes—
300C2×C4#49712yes—
301C2×C4#49712yes—
302C2×C4C2×C22⋊Q811yescharacteristic
303C2×C4#53611yescharacteristic
304C2×C4#54711yescharacteristic
305C2×C4#54711yescharacteristic
306C2×C4D4×Q811yescharacteristic
307C2×C4D4×Q811yescharacteristic
308C2×C4C22×SD1612yes—
309C2×C4C2×#13712yes—
310C2×C4C2×#13712yes—
311C2×C4#57512yes—
312C2×C4#57612yes—
313C2×C4#57712yes—
314C8—24no—
315C8—28no—
316C8—44no—
317C8—42no—
318C22#254211yescharacteristic
319C22C2×#49211yescharacteristic
320C22C2×#49711yescharacteristic
321C22C2×#49711yescharacteristic
322C22#315311yescharacteristic
323C22#315411yescharacteristic
324C22#315511yescharacteristic
325C4—22no—
326C4—22no—
327C4—22no—
328C4—82no—
329C4—82no—
330C4—161no—
331C4—162no—
332C4—161no—
333C4—162no—
334C4#171614yes—
335C4#307812yes—
336C4#323712yes—
337C4#333512yes—
338C4#335712yes—
339C4Q8×SD1612yes—
340C4#338512yes—
341C2C2×#171611yescharacteristic
342C2#1962311yescharacteristic
343C2#1968511yescharacteristic
344C2#1970511yescharacteristic
345C2#2852211yescharacteristic
346C2#2853011yescharacteristic
347C2#2853911yescharacteristic
subgroupG/H
center, socle, prime radicalC23C2×#127
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
58. #498Q8semidirect

Maximal subgroups

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

subgroupindex
#332892
#292062
#291412
#285112
C4×#4982
#106862
#106752
#98282
#88192
#82612