Finite Group Databasebeta

← #315133

Subgroups — #315133

Order 512.

559 conjugacy classes of subgroups.

The subgroup lattice has 323 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#28811C211yesmaximal, characteristic
2#28683C211yesmaximal, characteristic
3#27749C212yesmaximal
4#10419C212yesmaximal
5#10357C212yesmaximal
6#9836C211yesmaximal, characteristic
7#9731C211yesmaximal, characteristic
8#9561C212yesmaximal
9#8651C211yesmaximal, characteristic
10#8285C211yesmaximal, characteristic
11#8211C211yesmaximal, characteristic
12C2×#500C2211yescharacteristic
13C2×#499—21no—
14C2×#446C2211yescharacteristic
15#2552C2212yes—
16#2546C2212yes—
17#2087—22no—
18#2072—22no—
19#2059—22no—
20#2043—22no—
21#1990C2212yes—
22#1988C2212yes—
23#1981C2212yes—
24#1956—22no—
25#1956C2212yes—
26#1947C2212yes—
27#1944—21no—
28#1944—21no—
29#1941—21no—
30#1933—22no—
31#1844C2211yescharacteristic
32#1803C2212yes—
33C4×#106C2211yescharacteristic
34C8×C4⋊C4C2211yescharacteristic
35C8×C4⋊C4C2212yes—
36C4×#103C2211yescharacteristic
37C2×#335C2211yescharacteristic
38C2×#332C2212yes—
39C2×Q8⋊C8C2211yescharacteristic
40C2×#322C2211yescharacteristic
41#1296C2212yes—
42#1279C2211yescharacteristic
43#1279C2212yes—
44#1277C2211yescharacteristic
45C2×C4⋊Q8—22no—
46C2×C4⋊Q8C2312yes—
47C2×#125—22no—
48C2×C4×Q8C2311yescharacteristic
49#500—22no—
50#500—22no—
51#499—44no—
52#446—22no—
53#446—22no—
54C2×#107—21no—
55C2×#107—21no—
56C2×#107—22no—
57C2×#106—21no—
58C2×#106—21no—
59C2×#106D411yescharacteristic
60C2×#106D411yescharacteristic
61C2×C4⋊C8—21no—
62C2×C4⋊C8C2312yes—
63C2×C4⋊C8C2311yescharacteristic
64C2×C4⋊C8C2311yescharacteristic
65C2×#103—22no—
66C2×#103—22no—
67C2×#103—22no—
68C2×#103—22no—
69C2×#103—22no—
70C2×#103D412yes—
71C2×#97—21no—
72C2×C4×C8—21no—
73C2×C4×C8—22no—
74C2×C4×C8C2311yescharacteristic
75C2×C4×C8C2311yescharacteristic
76C2×C4×C8C2311yescharacteristic
77#398—42no—
78#398—42no—
79#391—22no—
80#389—22no—
81#389—22no—
82#389—22no—
83#389—22no—
84#384—22no—
85#384C2312yes—
86C4×C4⋊C4C2312yes—
87C4×C4⋊C4C2311yescharacteristic
88#340—22no—
89#340—22no—
90#340—22no—
91#340—22no—
92#340—42no—
93#340D412yes—
94#340D412yes—
95#336—21no—
96#336—22no—
97#335—22no—
98#332—24no—
99Q8⋊C8—22no—
100#322—22no—
101C22×Q8—22no—
102C22×Q8C2×D411yescharacteristic
103C22×C8—21no—
104C22×C8—22no—
105C22×C8—21no—
106C22×C8—21no—
107C22×C8C2×D412yes—
108C22×C8C2×D411yescharacteristic
109C22×C8C2×D411yescharacteristic
110C4⋊Q8—22no—
111C4⋊Q8—22no—
112C4⋊Q8—22no—
113C4⋊Q8—22no—
114C4⋊Q8—44no—
115C4⋊Q8—82no—
116#125—44no—
117#125—82no—
118C4×Q8—21no—
119C4×Q8—21no—
120C4×Q8—21no—
121C4×Q8—21no—
122C2×C4⋊C4—21no—
123C2×C4⋊C4—22no—
124C2×C4⋊C4—22no—
125C2×C4⋊C4—22no—
126C2×C4⋊C4—22no—
127C2×C4⋊C4—22no—
128C2×C4⋊C4—42no—
129C2×C4⋊C4—42no—
130C2×C4⋊C4—42no—
131C2×C4⋊C4—42no—
132C2×C4⋊C4—42no—
133C2×C4⋊C4C2×D411yescharacteristic
134C2×C4⋊C4C2×D412yes—
135C2×C4⋊C4C2×D412yes—
136C2×C4⋊C4C2×D412yes—
137C2×C42—21no—
138C2×C42—21no—
139C2×C42—22no—
140C2×C42C2411yescharacteristic
141#107—42no—
142#107—42no—
143#107—44no—
144#106—42no—
145#106—81no—
146#106SD1614yes—
147#106Q1614yes—
148C4⋊C8—22no—
149C4⋊C8—22no—
150C4⋊C8—21no—
151C4⋊C8—21no—
152C4⋊C8—21no—
153C4⋊C8—42no—
154C4⋊C8#4112yes—
155#103—42no—
156#103—44no—
157#103—82no—
158#103—82no—
159#103—82no—
160#103—82no—
161#97—42no—
162C4×C8—21no—
163C4×C8—21no—
164C4×C8—21no—
165C4×C8—21no—
166C4×C8—42no—
167C4×C8—82no—
168C4×C8#4112yes—
169C4×C8#4112yes—
170#95—22no—
171#95—21no—
172#95—42no—
173#95—42no—
174#95—42no—
175C2×Q8—21no—
176C2×Q8—21no—
177C2×Q8—22no—
178C2×Q8—22no—
179C2×Q8—42no—
180C2×Q8—42no—
181C2×Q8—42no—
182C2×Q8—42no—
183C2×Q8—42no—
184C2×Q8—42no—
185C22×C4—22no—
186C22×C4—22no—
187C22×C4—22no—
188C22×C4—21no—
189C22×C4—21no—
190C22×C4—42no—
191C22×C4—42no—
192C22×C4C22×D411yescharacteristic
193C22×C4C22×D411yescharacteristic
194C22×C4H(5,𝔽2)11yescharacteristic
195C2×C8—21no—
196C2×C8—21no—
197C2×C8—22no—
198C2×C8—22no—
199C2×C8—22no—
200C2×C8—22no—
201C2×C8—41no—
202C2×C8—41no—
203C2×C8—42no—
204C2×C8—41no—
205C2×C8—42no—
206C2×C8—41no—
207C2×C8—42no—
208C2×C8—42no—
209C2×C8—41no—
210C2×C8—41no—
211C2×C8#12412yes—
212C2×C8#12412yes—
213C2×C8C2×SD1612yes—
214C2×C8C2×Q1612yes—
215C4⋊C4—22no—
216C4⋊C4—22no—
217C4⋊C4—21no—
218C4⋊C4—21no—
219C4⋊C4—22no—
220C4⋊C4—22no—
221C4⋊C4—41no—
222C4⋊C4—41no—
223C4⋊C4—42no—
224C4⋊C4—42no—
225C4⋊C4—42no—
226C4⋊C4—42no—
227C4⋊C4—42no—
228C4⋊C4—42no—
229C4⋊C4—42no—
230C4⋊C4—42no—
231C4⋊C4—84no—
232C4⋊C4—84no—
233C4⋊C4—82no—
234C4⋊C4—84no—
235C4⋊C4—84no—
236C4⋊C4—162no—
237C4⋊C4C2×SD1614yes—
238C4⋊C4C2×Q1614yes—
239C42—41no—
240C42—41no—
241C42—41no—
242C42—42no—
243C42—42no—
244C42—41no—
245C42C2×#4111yescharacteristic
246C42C2×#4111yescharacteristic
247C42C2×#4111yescharacteristic
248C422−1+411yescharacteristic
249C23D4211yescharacteristic
250Q8—42no—
251Q8—41no—
252Q8—41no—
253Q8—82no—
254Q8—82no—
255Q8—82no—
256Q8—82no—
257C2×C4—22no—
258C2×C4—24no—
259C2×C4—24no—
260C2×C4—42no—
261C2×C4—42no—
262C2×C4—42no—
263C2×C4—42no—
264C2×C4—41no—
265C2×C4—41no—
266C2×C4—42no—
267C2×C4—42no—
268C2×C4—42no—
269C2×C4—42no—
270C2×C4—42no—
271C2×C4—82no—
272C2×C4—82no—
273C2×C4—82no—
274C2×C4—82no—
275C2×C4—82no—
276C2×C4—82no—
277C2×C4C2×#12411yescharacteristic
278C2×C4#53211yescharacteristic
279C2×C4#54211yescharacteristic
280C2×C4#54711yescharacteristic
281C2×C4#54811yescharacteristic
282C2×C4#55511yescharacteristic
283C2×C4C22×SD1611yescharacteristic
284C2×C4C22×Q1611yescharacteristic
285C2×C4#57512yes—
286C2×C4#57612yes—
287C2×C4#57711yescharacteristic
288C2×C4#57811yescharacteristic
289C8—22no—
290C8—42no—
291C8—42no—
292C8—81no—
293C8—81no—
294C8—82no—
295C8#48714yes—
296C22#249111yescharacteristic
297C22#326812yes—
298C22D4×SD1611yescharacteristic
299C22D4×Q1611yescharacteristic
300C22#328912yes—
301C4—21no—
302C4—21no—
303C4—44no—
304C4—44no—
305C4—82no—
306C4—81no—
307C4—81no—
308C4—162no—
309C4—162no—
310C4C2×#48711yescharacteristic
311C4#313711yescharacteristic
312C4#317611yescharacteristic
313C4#318311yescharacteristic
314C4#335512yes—
315C4#335911yescharacteristic
316C4#336311yescharacteristic
317C2#1976712yes—
318C2#1981512yes—
319C2#2776911yescharacteristic
320C2#2780511yescharacteristic
321C2#2781011yescharacteristic
subgroupG/H
center, socle, prime radicalC23D42
commutator, Frattini, cosocleC2×C42C24
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C2×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Maximal subgroups

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

subgroupindex
#288112
#286832
#277492
#104192
#103572
#98362
#97312
#95612
#86512
#82852
#82112