Finite Group Databasebeta

← #120232

Subgroups — #120232

Order 512.

531 conjugacy classes of subgroups.

The subgroup lattice has 359 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21949C211yesmaximal, characteristic
2#21949C211yesmaximal, characteristic
3#8115C211yesmaximal, characteristic
4#8108C211yesmaximal, characteristic
5#8104C211yesmaximal, characteristic
6#8009C211yesmaximal, characteristic
7#8009C211yesmaximal, characteristic
8C2×#407C2211yescharacteristic
9C2×#407C2211yescharacteristic
10C2×#407C2211yescharacteristic
11C2×#407C2211yescharacteristic
12C2×#407C2211yescharacteristic
13C2×#407C2211yescharacteristic
14#2280C2211yescharacteristic
15#1855—22no—
16#1855—22no—
17#1855—22no—
18#1855C414yescomplemented
19#1811—22no—
20#1811—22no—
21#1306—22no—
22#1305—22no—
23#1305—22no—
24#1293—22no—
25#1293—22no—
26C24×C4C2311yescharacteristic
27C22×M16—21no—
28C22×M16—21no—
29C22×M16—21no—
30C22×M16—21no—
31C22×M16C2×C412yes—
32C22×M16D412yes—
33C2×#117—21no—
34C2×#108—24no—
35C2×#108—24no—
36#407—21no—
37#407—21no—
38#407—21no—
39#407—21no—
40#407—21no—
41#407—21no—
42#407—21no—
43#407—21no—
44#407—21no—
45#407—21no—
46#407—21no—
47#407—21no—
48#407—21no—
49#407—21no—
50#407—21no—
51#407—21no—
52#407—21no—
53#407—21no—
54#407C2×C412yes—
55#407D412yes—
56#407D412yes—
57#407D412yes—
58#407D412yes—
59#407Q812yes—
60C2×C22⋊C8—21no—
61C2×C22⋊C8—21no—
62C2×C22⋊C8—21no—
63C2×C22⋊C8—21no—
64C2×C22⋊C8—21no—
65C2×C22⋊C8—21no—
66C2×C22⋊C8—21no—
67C2×C22⋊C8—21no—
68C2×C22⋊C8—21no—
69C2×C22⋊C8—21no—
70C2×C22⋊C8C2×C412yes—
71C2×C22⋊C8Q812yes—
72#381—21no—
73#380—21no—
74#380—21no—
75#379Q812yes—
76C4×C22⋊C4—21no—
77C4×C22⋊C4—21no—
78#341—24no—
79#341—24no—
80C25C2×Q811yescharacteristic
81C23×C4—82no—
82C23×C4C22×C411yescharacteristic
83C23×C4C2×D411yescharacteristic
84C23×C4C2×D411yescharacteristic
85C23×C4C2×D411yescharacteristic
86C23×C4C2×D411yescharacteristic
87C23×C4C2×D411yescharacteristic
88C23×C4C2×Q811yescharacteristic
89C23×C4C2×Q811yescharacteristic
90C23×C4#4111yescharacteristic
91C23×C4#4111yescharacteristic
92C23×C4#4111yescharacteristic
93C23×C4#4111yescharacteristic
94C23×C4#4111yescharacteristic
95C23×C4#4111yescharacteristic
96C2×M16—22no—
97C2×M16—22no—
98C2×M16—22no—
99C2×M16—22no—
100C2×M16—22no—
101C2×M16—22no—
102C2×M16—22no—
103C2×M16—41no—
104C2×M16—41no—
105C2×M16—41no—
106C2×M16—41no—
107C2×M16—82no—
108C2×M16—82no—
109C2×M16—82no—
110C2×M16—82no—
111C2×M16—82no—
112C2×M16—82no—
113C2×M16C22⋊C414yes—
114C22×C8—22no—
115C22×C8—22no—
116C22×C8—22no—
117C22×C8—22no—
118C22×C8—41no—
119C22×C8—41no—
120#117—82no—
121C2×C4⋊C4—41no—
122C2×C22⋊C4—22no—
123C2×C22⋊C4—22no—
124C2×C22⋊C4—22no—
125C2×C42—41no—
126#108—84no—
127#108—84no—
128C22⋊C8—22no—
129C22⋊C8—22no—
130C22⋊C8—22no—
131C22⋊C8—22no—
132C22⋊C8—22no—
133C22⋊C8—22no—
134C22⋊C8—22no—
135C22⋊C8—22no—
136C22⋊C8—22no—
137C22⋊C8—22no—
138C22⋊C8—22no—
139C22⋊C8—22no—
140C22⋊C8—22no—
141C22⋊C8—41no—
142C22⋊C8—41no—
143C22⋊C8—41no—
144C22⋊C8—41no—
145C22⋊C8—41no—
146C22⋊C8—41no—
147C22⋊C8—41no—
148C22⋊C8—41no—
149C22⋊C8—41no—
150C22⋊C8—41no—
151C22⋊C8C4⋊C414yes—
152#95—41no—
153#95—41no—
154C24—81no—
155C24—81no—
156C24—81no—
157C24C4×Q811yescharacteristic
158C24C22⋊Q811yescharacteristic
159C24C22⋊Q811yescharacteristic
160C24C22⋊Q811yescharacteristic
161C24#12511yescharacteristic
162C24C4⋊Q811yescharacteristic
163C24C4⋊Q811yescharacteristic
164C22×C4—42no—
165C22×C4—42no—
166C22×C4—42no—
167C22×C4—42no—
168C22×C4—42no—
169C22×C4—42no—
170C22×C4—42no—
171C22×C4—42no—
172C22×C4—81no—
173C22×C4—81no—
174C22×C4—81no—
175C22×C4—81no—
176C22×C4—81no—
177C22×C4—81no—
178C22×C4—81no—
179C22×C4C2×C22⋊C411yescharacteristic
180C22×C4C2×C4⋊C411yescharacteristic
181C22×C4#11711yescharacteristic
182C22×C4C4×D411yescharacteristic
183C22×C4C4×D411yescharacteristic
184C22×C4C4×Q811yescharacteristic
185C22×C4#12011yescharacteristic
186C22×C4#12111yescharacteristic
187C22×C4#12111yescharacteristic
188C22×C4#12111yescharacteristic
189C22×C4#12111yescharacteristic
190C22×C4#12111yescharacteristic
191C22×C4#12111yescharacteristic
192C22×C4C22⋊Q811yescharacteristic
193C22×C4C22⋊Q811yescharacteristic
194C22×C4C22⋊Q811yescharacteristic
195C22×C4C22⋊Q811yescharacteristic
196C22×C4C22⋊Q811yescharacteristic
197C22×C4#12311yescharacteristic
198C22×C4#12311yescharacteristic
199C22×C4#12311yescharacteristic
200C22×C4#12411yescharacteristic
201C22×C4#12511yescharacteristic
202C22×C4#12511yescharacteristic
203C22×C4#12511yescharacteristic
204C22×C4#12611yescharacteristic
205C22×C4#12611yescharacteristic
206C22×C4C4⋊Q811yescharacteristic
207M16—44no—
208M16—44no—
209M16—44no—
210M16—44no—
211M16—82no—
212M16—82no—
213M16—82no—
214M16—82no—
215M16—82no—
216M16—82no—
217M16—82no—
218M16—82no—
219C2×C8—24no—
220C2×C8—24no—
221C2×C8—44no—
222C2×C8—44no—
223C2×C8—42no—
224C2×C8—42no—
225C2×C8—42no—
226C2×C8—42no—
227C2×C8—82no—
228C2×C8—82no—
229C2×C8—82no—
230C2×C8—82no—
231C4⋊C4—82no—
232C22⋊C4—82no—
233C22⋊C4—82no—
234C22⋊C4—82no—
235C42—82no—
236C23—41no—
237C23—41no—
238C23—41no—
239C23—41no—
240C23—41no—
241C23—41no—
242C23—41no—
243C23—41no—
244C23—41no—
245C23—41no—
246C23—41no—
247C23—41no—
248C23—41no—
249C23—41no—
250C23—41no—
251C23—41no—
252C23—41no—
253C23—41no—
254C23—41no—
255C23—41no—
256C23—41no—
257C23—88no—
258C23#38711yescharacteristic
259C23#38911yescharacteristic
260C23#39111yescharacteristic
261C23#39511yescharacteristic
262C23#39611yescharacteristic
263C23#39811yescharacteristic
264C23#39811yescharacteristic
265C2×C4—22no—
266C2×C4—22no—
267C2×C4—22no—
268C2×C4—22no—
269C2×C4—22no—
270C2×C4—22no—
271C2×C4—22no—
272C2×C4—41no—
273C2×C4—41no—
274C2×C4—41no—
275C2×C4—41no—
276C2×C4—41no—
277C2×C4—41no—
278C2×C4—41no—
279C2×C4—41no—
280C2×C4—41no—
281C2×C4—41no—
282C2×C4—41no—
283C2×C4—41no—
284C2×C4—41no—
285C2×C4—41no—
286C2×C4—41no—
287C2×C4—41no—
288C2×C4—41no—
289C2×C4—41no—
290C2×C4—41no—
291C2×C4—41no—
292C2×C4—41no—
293C2×C4—82no—
294C2×C4—82no—
295C2×C4—82no—
296C2×C4#38011yescharacteristic
297C2×C4#38611yescharacteristic
298C2×C4#38711yescharacteristic
299C2×C4#38911yescharacteristic
300C2×C4#39411yescharacteristic
301C2×C4#39611yescharacteristic
302C2×C4#39811yescharacteristic
303C2×C4#40011yescharacteristic
304C8—44no—
305C8—44no—
306C8—82no—
307C8—82no—
308C8—82no—
309C8—82no—
310C22—21no—
311C22—21no—
312C22—21no—
313C22—21no—
314C22—21no—
315C22—21no—
316C22—21no—
317C22—21no—
318C22—21no—
319C22—21no—
320C22—21no—
321C22—21no—
322C22—21no—
323C22—21no—
324C22—21no—
325C22—21no—
326C22—21no—
327C22—21no—
328C22—21no—
329C22—21no—
330C22—21no—
331C22—82no—
332C22—82no—
333C22—82no—
334C22—82no—
335C22—82no—
336C22—82no—
337C22—82no—
338C22#144111yescharacteristic
339C4—21no—
340C4—21no—
341C4—21no—
342C4—21no—
343C4—21no—
344C4—21no—
345C4—21no—
346C4—162no—
347C4#191312yes—
348C2—41no—
349C2—41no—
350C2—41no—
351C2—41no—
352C2—41no—
353C2—41no—
354C2—41no—
355C2#1120511yescharacteristic
356C2#1124811yescharacteristic
357C2#1125411yescharacteristic
subgroupG/H
center, socleC22#1441
commutatorC23×C4C22×C4
Frattini, cosocleC24×C4C23
prime radicalC25C2×Q8
LCSG ▷ C23×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C24×C4 ◁ G
LECSG ▷ C24×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
18. #1855C4semidirect

Maximal subgroups

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

subgroupindex
#219492
#81152
#81082
#81042
#80092