Finite Group Databasebeta

← #475849

Subgroups — #475849

Order 512.

784 conjugacy classes of subgroups.

The subgroup lattice has 338 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2089C212yesmaximal
2#23657C212yesmaximal
3C2×#1942C211yesmaximal, characteristic
4#21143C211yesmaximal, characteristic
5#11249C214yesmaximal
6#11239C212yesmaximal
7#11204C212yesmaximal
8#10864C211yesmaximal, characteristic
9C22×#108C2212yes—
10C2×#428C2214yes—
11C22×#107C2211yescharacteristic
12C22×#107C2211yescharacteristic
13C2×#424C2212yes—
14C2×#423C2211yescharacteristic
15C22×C4×C8C2211yescharacteristic
16C2×#389C2212yes—
17#2285C2211yescharacteristic
18#2089—28no—
19#2085—24no—
20#2084—22no—
21#2083—22no—
22#1942—22no—
23#1942—22no—
24#1939—22no—
25#1939—22no—
26C2×#343C2214yes—
27C2×#343C2214yes—
28C2×#341C2212yes—
29#1736C2214yes—
30C2×#340C2212yes—
31#1729C2212yes—
32#1728C2212yes—
33C22×M16C2312yes—
34C22×M16C2314yes—
35C23×C8C2311yescharacteristic
36C23×C8C2311yescharacteristic
37C2×#117C2312yes—
38C22×C4⋊C4—22no—
39C22×C4⋊C4C2312yes—
40C22×C4⋊C4C2312yes—
41C22×C42C2311yescharacteristic
42C2×#108—42no—
43C2×#108—42no—
44C2×#108Q818yes—
45#428—28no—
46#428—44no—
47C2×#107—22no—
48C2×#107—22no—
49C2×#107—22no—
50C2×#107—24no—
51C2×#107—42no—
52C2×#107D414yes—
53C2×#106—24no—
54#424—24no—
55#424—24no—
56#423—42no—
57C2×C4⋊C8—24no—
58C2×C4⋊C8—21no—
59C2×C4⋊C8—21no—
60C2×C4×C8—42no—
61C2×C4×C8D412yes—
62C2×C4×C8D412yes—
63#389—22no—
64#389—22no—
65#389—22no—
66#389—22no—
67#384D411yescharacteristic
68#384D411yescharacteristic
69#382D412yes—
70#380—21no—
71#380—21no—
72C4×C4⋊C4—21no—
73C4×C4⋊C4—21no—
74C2×#95—22no—
75#343—28no—
76#343—44no—
77#341—24no—
78#340—24no—
79#340—24no—
80#340—24no—
81#336—24no—
82#336—22no—
83#336—22no—
84C23×C4—22no—
85C23×C4—22no—
86C23×C4#4111yescharacteristic
87C23×C4#4111yescharacteristic
88C23×C4C2411yescharacteristic
89C2×M16—24no—
90C2×M16—24no—
91C2×M16—24no—
92C2×M16—24no—
93C2×M16—24no—
94C2×M16—22no—
95C2×M16—22no—
96C2×M16—24no—
97C2×M16—24no—
98C2×M16C2×Q818yes—
99C22×C8—22no—
100C22×C8—22no—
101C22×C8—21no—
102C22×C8—22no—
103C22×C8—21no—
104C22×C8—24no—
105C22×C8—22no—
106C22×C8C2×D412yes—
107C22×C8C2×D412yes—
108C22×C8C2×D412yes—
109C22×C8C2×Q814yes—
110C22×C8#4112yes—
111#117—22no—
112#117—22no—
113#117—22no—
114#117—22no—
115C2×C4⋊C4—22no—
116C2×C4⋊C4—22no—
117C2×C4⋊C4—22no—
118C2×C4⋊C4—22no—
119C2×C4⋊C4—21no—
120C2×C4⋊C4—21no—
121C2×C4⋊C4—21no—
122C2×C4⋊C4—21no—
123C2×C4⋊C4—22no—
124C2×C4⋊C4—22no—
125C2×C4⋊C4—22no—
126C2×C4⋊C4—24no—
127C2×C4⋊C4—24no—
128C2×C4⋊C4—42no—
129C2×C4⋊C4—42no—
130C2×C4⋊C4—84no—
131C2×C4⋊C4C2×D414yes—
132C2×C4⋊C4C2×D412yes—
133C2×C4⋊C4C2×D412yes—
134C2×C22⋊C4—22no—
135C2×C42—22no—
136C2×C42—22no—
137C2×C42—22no—
138C2×C42C2×D411yescharacteristic
139C2×C42C2×D411yescharacteristic
140C2×C42C2×D411yescharacteristic
141C2×C42C2×D411yescharacteristic
142#108—48no—
143#107—28no—
144#107—48no—
145#107—44no—
146#106—48no—
147C4⋊C8—42no—
148C4⋊C8—42no—
149C4⋊C8—48no—
150C4×C8—42no—
151C4×C8—42no—
152#95—22no—
153#95—22no—
154#95—42no—
155#95—42no—
156C24C2×#4111yescharacteristic
157C22×C4—22no—
158C22×C4—22no—
159C22×C4—22no—
160C22×C4—22no—
161C22×C4—22no—
162C22×C4—22no—
163C22×C4—24no—
164C22×C4—41no—
165C22×C4—41no—
166C22×C4—42no—
167C22×C4—42no—
168C22×C4—42no—
169C22×C4—44no—
170C22×C4—82no—
171C22×C4—82no—
172C22×C4—82no—
173C22×C4—82no—
174C22×C4#12111yescharacteristic
175C22×C4#12111yescharacteristic
176C22×C4#12111yescharacteristic
177C22×C4#12111yescharacteristic
178C22×C4C22×D411yescharacteristic
179C22×C4C22×D411yescharacteristic
180C22×C4C22×D411yescharacteristic
181C22×C4C22×Q812yes—
182C22×C4C2×#4111yescharacteristic
183C22×C4H(5,𝔽2)12yes—
184C22×C4H(5,𝔽2)11yescharacteristic
185C22×C42−1+412yes—
186C22×C42−1+412yes—
187C22×C42−1+411yescharacteristic
188M16—28no—
189M16—28no—
190M16—44no—
191M16—44no—
192M16—44no—
193M16—44no—
194C2×C8—22no—
195C2×C8—22no—
196C2×C8—22no—
197C2×C8—24no—
198C2×C8—22no—
199C2×C8—24no—
200C2×C8—24no—
201C2×C8—24no—
202C2×C8—28no—
203C2×C8—44no—
204C2×C8—44no—
205C2×C8—42no—
206C2×C8—42no—
207C2×C8—42no—
208C2×C8#12114yes—
209C2×C8C4⋊Q814yes—
210C4⋊C4—24no—
211C4⋊C4—22no—
212C4⋊C4—24no—
213C4⋊C4—22no—
214C4⋊C4—22no—
215C4⋊C4—22no—
216C4⋊C4—42no—
217C4⋊C4—42no—
218C4⋊C4—42no—
219C4⋊C4—42no—
220C4⋊C4—42no—
221C4⋊C4—42no—
222C4⋊C4—42no—
223C4⋊C4—42no—
224C4⋊C4—44no—
225C4⋊C4—84no—
226C4⋊C4—84no—
227C22⋊C4—42no—
228C22⋊C4—42no—
229C42—21no—
230C42—21no—
231C42—21no—
232C42—21no—
233C42—22no—
234C42—22no—
235C42—42no—
236C42—42no—
237C23—21no—
238C23—21no—
239C23—21no—
240C23—21no—
241C23C2×#12111yescharacteristic
242C23#53912yes—
243C23#54711yescharacteristic
244C23#55511yescharacteristic
245C23#55712yes—
246C2×C4—21no—
247C2×C4—21no—
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—44no—
257C2×C4—44no—
258C2×C4—42no—
259C2×C4—42no—
260C2×C4—44no—
261C2×C4—42no—
262C2×C4—42no—
263C2×C4—42no—
264C2×C4—42no—
265C2×C4—84no—
266C2×C4—82no—
267C2×C4—82no—
268C2×C4—84no—
269C2×C4—82no—
270C2×C4—82no—
271C2×C4—82no—
272C2×C4—164no—
273C2×C4#47212yes—
274C2×C4#47312yes—
275C2×C4C2×#12111yescharacteristic
276C2×C4C2×C4⋊Q811yescharacteristic
277C2×C4#53411yescharacteristic
278C2×C4#53611yescharacteristic
279C2×C4#53611yescharacteristic
280C2×C4#53711yescharacteristic
281C2×C4#53712yes—
282C2×C4#53712yes—
283C2×C4#53912yes—
284C2×C4#54312yes—
285C2×C4#54412yes—
286C2×C4D4211yescharacteristic
287C2×C4#54711yescharacteristic
288C2×C4D4×Q812yes—
289C2×C4D4×Q812yes—
290C2×C4#55712yes—
291C2×C4#56312yes—
292C2×C4#56412yes—
293C8—24no—
294C8—24no—
295C8—48no—
296C8—82no—
297C22—22no—
298C22—22no—
299C22—21no—
300C22—21no—
301C22—21no—
302C22—21no—
303C22—22no—
304C22—22no—
305C22#253711yescharacteristic
306C22#254611yescharacteristic
307C22#261811yescharacteristic
308C22#272612yes—
309C22#280912yes—
310C22C2×#47211yescharacteristic
311C22C2×#47311yescharacteristic
312C22#306711yescharacteristic
313C22#306711yescharacteristic
314C4—22no—
315C4—22no—
316C4—41no—
317C4—41no—
318C4—82no—
319C4—84no—
320C4—82no—
321C4—162no—
322C4#254611yescharacteristic
323C4#261811yescharacteristic
324C4#265911yescharacteristic
325C4#268312yes—
326C4#272612yes—
327C4#282211yescharacteristic
328C2—22no—
329C2—22no—
330C2#1707011yescharacteristic
331C2#2824011yescharacteristic
332C2#2824111yescharacteristic
333C2#2869911yescharacteristic
334C2#2870111yescharacteristic
335C2#2978811yescharacteristic
336C2#2978911yescharacteristic
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
C2×#20892
#236572
C2×#19422
#211432
#112492
#112392
#112042
#108642