Finite Group Databasebeta

← #410799

Subgroups — #410799

Order 512.

738 conjugacy classes of subgroups.

The subgroup lattice has 332 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23747C212yesmaximal
2C2×#1942C211yesmaximal, characteristic
3#20437C211yesmaximal, characteristic
4#11365C212yesmaximal
5#11270C212yesmaximal
6#11204C212yesmaximal
7#10827C214yesmaximal
8#10822C211yesmaximal, characteristic
9C22×#107C2211yescharacteristic
10C22×#107C2211yescharacteristic
11C2×#423C2211yescharacteristic
12C2×#405C2212yes—
13C22×C4×C8C2211yescharacteristic
14C2×#389C2212yes—
15#2285C2211yescharacteristic
16#2101—24no—
17#2100—22no—
18#2099—22no—
19#1942—21no—
20#1942—21no—
21#1942—21no—
22#1942—21no—
23#1770—21no—
24#1770—21no—
25C8×C4⋊C4—21no—
26C8×C4⋊C4—21no—
27C2×#343C2214yes—
28C2×#343C2214yes—
29C2×#341C2212yes—
30C2×#341C2212yes—
31#1736C2214yes—
32#1736C2214yes—
33C2×#340C2212yes—
34#1729C2212yes—
35#1728C2212yes—
36C22×M16C2314yes—
37C22×M16C2312yes—
38C23×C8C2311yescharacteristic
39C23×C8C2311yescharacteristic
40C2×#117C2312yes—
41C22×C4⋊C4—22no—
42C22×C4⋊C4C2312yes—
43C22×C4⋊C4C2312yes—
44C22×C42C2311yescharacteristic
45C2×#107—21no—
46C2×#107—21no—
47C2×#107—24no—
48C2×#107—21no—
49C2×#107—21no—
50C2×#107—24no—
51#423—24no—
52C2×C4⋊C8—21no—
53C2×C4⋊C8—21no—
54#405—28no—
55C2×#97—22no—
56C2×C4×C8—24no—
57C2×C4×C8—22no—
58C2×C4×C8D411yescharacteristic
59C2×C4×C8D411yescharacteristic
60C2×C4×C8D411yescharacteristic
61C2×C4×C8D411yescharacteristic
62#389—22no—
63#389—22no—
64#389—22no—
65#389—22no—
66#384D411yescharacteristic
67#384D411yescharacteristic
68#382D412yes—
69#380—21no—
70#380—21no—
71C4×C4⋊C4—21no—
72C4×C4⋊C4—21no—
73C2×#95—22no—
74#343—28no—
75#343—28no—
76#341—24no—
77#341—24no—
78#340—22no—
79#340—22no—
80#340—24no—
81#340—24no—
82#340—24no—
83#340—24no—
84#336—22no—
85#336—22no—
86#336—22no—
87#336—22no—
88C23×C4—22no—
89C23×C4—22no—
90C23×C4#4111yescharacteristic
91C23×C4#4111yescharacteristic
92C23×C4C2411yescharacteristic
93C2×M16—22no—
94C2×M16—24no—
95C2×M16—24no—
96C2×M16—24no—
97C2×M16—24no—
98C2×M16—24no—
99C2×M16—24no—
100C2×M16—24no—
101C2×M16—24no—
102C2×M16#4114yes—
103C22×C8—22no—
104C22×C8—22no—
105C22×C8—22no—
106C22×C8—22no—
107C22×C8—22no—
108C22×C8—24no—
109C22×C8C2×D411yescharacteristic
110C22×C8C2×D411yescharacteristic
111C22×C8C2×D411yescharacteristic
112C22×C8C2×D411yescharacteristic
113C22×C8#4114yes—
114C22×C8#4114yes—
115#117—24no—
116#117—24no—
117C2×C4⋊C4—22no—
118C2×C4⋊C4—22no—
119C2×C4⋊C4—22no—
120C2×C4⋊C4—22no—
121C2×C4⋊C4—21no—
122C2×C4⋊C4—21no—
123C2×C4⋊C4—21no—
124C2×C4⋊C4—21no—
125C2×C4⋊C4—22no—
126C2×C4⋊C4—22no—
127C2×C4⋊C4—22no—
128C2×C4⋊C4—24no—
129C2×C4⋊C4—24no—
130C2×C4⋊C4—42no—
131C2×C4⋊C4—42no—
132C2×C4⋊C4—84no—
133C2×C4⋊C4C2×D412yes—
134C2×C4⋊C4C2×D412yes—
135C2×C4⋊C4#4114yes—
136C2×C22⋊C4—22no—
137C2×C42—22no—
138C2×C42—22no—
139C2×C42—22no—
140C2×C42C2×D411yescharacteristic
141C2×C42C2×D411yescharacteristic
142C2×C42C2×D411yescharacteristic
143C2×C42C2×D411yescharacteristic
144#107—24no—
145#107—24no—
146#107—42no—
147#107—42no—
148C4⋊C8—24no—
149C4⋊C8—24no—
150#97—44no—
151C4×C8—22no—
152C4×C8—22no—
153C4×C8—22no—
154C4×C8—22no—
155C4×C8—44no—
156#95—22no—
157#95—22no—
158#95—42no—
159#95—42no—
160C24C2×#4111yescharacteristic
161C22×C4—22no—
162C22×C4—22no—
163C22×C4—22no—
164C22×C4—22no—
165C22×C4—22no—
166C22×C4—22no—
167C22×C4—24no—
168C22×C4—41no—
169C22×C4—41no—
170C22×C4—42no—
171C22×C4—42no—
172C22×C4—42no—
173C22×C4—44no—
174C22×C4—82no—
175C22×C4—82no—
176C22×C4—82no—
177C22×C4—82no—
178C22×C4#12111yescharacteristic
179C22×C4#12111yescharacteristic
180C22×C4#12111yescharacteristic
181C22×C4#12111yescharacteristic
182C22×C4C22×D411yescharacteristic
183C22×C4C22×D411yescharacteristic
184C22×C4C2×#4112yes—
185C22×C4C2×#4112yes—
186C22×C4C2×#4112yes—
187C22×C4C2×#4112yes—
188C22×C42−1+412yes—
189C22×C42−1+412yes—
190M16—28no—
191M16—44no—
192M16—44no—
193M16—44no—
194M16—44no—
195M16—44no—
196C2×C8—21no—
197C2×C8—21no—
198C2×C8—21no—
199C2×C8—21no—
200C2×C8—21no—
201C2×C8—21no—
202C2×C8—21no—
203C2×C8—21no—
204C2×C8—28no—
205C2×C8—28no—
206C2×C8—24no—
207C2×C8—42no—
208C2×C8—42no—
209C2×C8—44no—
210C2×C8—44no—
211C2×C8—44no—
212C2×C8#12312yes—
213C2×C8#12312yes—
214C2×C8#12412yes—
215C2×C8#12412yes—
216C4⋊C4—24no—
217C4⋊C4—24no—
218C4⋊C4—24no—
219C4⋊C4—24no—
220C4⋊C4—42no—
221C4⋊C4—42no—
222C4⋊C4—42no—
223C4⋊C4—42no—
224C4⋊C4—42no—
225C4⋊C4—42no—
226C4⋊C4—42no—
227C4⋊C4—42no—
228C4⋊C4—44no—
229C4⋊C4—84no—
230C4⋊C4—84no—
231C22⋊C4—44no—
232C42—21no—
233C42—21no—
234C42—21no—
235C42—21no—
236C42—22no—
237C42—22no—
238C42—42no—
239C42—42no—
240C23—21no—
241C23—21no—
242C23—21no—
243C23—21no—
244C23C2×#12111yescharacteristic
245C23#52912yes—
246C23#55212yes—
247C23#55612yes—
248C2×C4—21no—
249C2×C4—21no—
250C2×C4—21no—
251C2×C4—21no—
252C2×C4—21no—
253C2×C4—21no—
254C2×C4—21no—
255C2×C4—21no—
256C2×C4—21no—
257C2×C4—21no—
258C2×C4—44no—
259C2×C4—44no—
260C2×C4—44no—
261C2×C4—42no—
262C2×C4—42no—
263C2×C4—42no—
264C2×C4—42no—
265C2×C4—42no—
266C2×C4—42no—
267C2×C4—84no—
268C2×C4—82no—
269C2×C4—82no—
270C2×C4—84no—
271C2×C4—82no—
272C2×C4—82no—
273C2×C4—82no—
274C2×C4—164no—
275C2×C4#47212yes—
276C2×C4#47312yes—
277C2×C4C2×#12311yescharacteristic
278C2×C4#52511yescharacteristic
279C2×C4#52511yescharacteristic
280C2×C4C2×#12411yescharacteristic
281C2×C4#52914yes—
282C2×C4#53212yes—
283C2×C4#53611yescharacteristic
284C2×C4#53611yescharacteristic
285C2×C4#54114yes—
286C2×C4#54712yes—
287C2×C4#54812yes—
288C2×C4#55214yes—
289C2×C4#55214yes—
290C8—22no—
291C8—22no—
292C8—22no—
293C8—22no—
294C8—44no—
295C8—84no—
296C22—22no—
297C22—22no—
298C22—22no—
299C22—22no—
300C22—22no—
301C22—22no—
302C22#253314yes—
303C22#253611yescharacteristic
304C22#254711yescharacteristic
305C22#285411yescharacteristic
306C22C2×#47211yescharacteristic
307C22C2×#47311yescharacteristic
308C22#306711yescharacteristic
309C22#306711yescharacteristic
310C4—22no—
311C4—22no—
312C4—41no—
313C4—41no—
314C4—84no—
315C4—82no—
316C4—82no—
317C4—162no—
318C4#239811yescharacteristic
319C4#248411yescharacteristic
320C4#250311yescharacteristic
321C4#251914yes—
322C4#263411yescharacteristic
323C2—24no—
324C2#1509011yescharacteristic
325C2#2821711yescharacteristic
326C2#2821911yescharacteristic
327C2#2873111yescharacteristic
328C2#2873311yescharacteristic
329C2#3312311yescharacteristic
330C2#3312411yescharacteristic
subgroupG/H
center, socleC23C2×#121
commutator, Frattini, cosocleC23×C4C24
prime radicalC24C2×#41
LCSG ▷ C23×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Maximal subgroups

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

subgroupindex
#237472
C2×#19422
#204372
#113652
#112702
#112042
#108272
#108222