Finite Group Databasebeta

← #108752

Subgroups — #108752

Order 512.

745 conjugacy classes of subgroups.

The subgroup lattice has 357 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20523C211yesmaximal, characteristic
2#10854C212yesmaximal
3#8023C212yesmaximal
4C2×#1289C211yesmaximal, characteristic
5C2×#1289C211yesmaximal, characteristic
6C22×C22⋊C8C2211yescharacteristic
7C22×C22⋊C8C2211yescharacteristic
8C2×#380C2212yes—
9C2×#380C2212yes—
10C2×C4×C22⋊C4C2211yescharacteristic
11C8×C22⋊C4—22no—
12C8×C22⋊C4C414yescomplemented
13C4×C22⋊C8—22no—
14C2×#342—22no—
15C2×#342—22no—
16#1294C418yescomplemented
17#1289—22no—
18#1289—22no—
19C24×C4C2311yescharacteristic
20C23×C8—21no—
21C23×C8C2×C412yes—
22C22×C4⋊C4—22no—
23C22×C4⋊C4—22no—
24C22×C22⋊C4—22no—
25C22×C22⋊C4—22no—
26C22×C22⋊C4D411yescharacteristic
27C22×C22⋊C4D411yescharacteristic
28C22×C42—21no—
29C2×C22⋊C8—21no—
30C2×C22⋊C8—22no—
31C2×C22⋊C8—22no—
32C2×C22⋊C8—22no—
33C2×C22⋊C8C2×C414yes—
34C2×C22⋊C8C2×C412yes—
35C2×C22⋊C8D412yes—
36C2×C22⋊C8D412yes—
37C2×C4×C8—24no—
38#380—22no—
39#380—22no—
40#380—22no—
41#380—22no—
42#380—22no—
43#380—22no—
44#380—22no—
45#380C2×C414yes—
46C4×C22⋊C4—21no—
47C4×C22⋊C4—21no—
48C4×C22⋊C4C2×C412yes—
49C4×C22⋊C4Q812yes—
50C2×#95—21no—
51C2×#95—22no—
52C2×#95C2×C414yes—
53#342—44no—
54#342—44no—
55#336—24no—
56C25C2×D411yescharacteristic
57C23×C4—21no—
58C23×C4—21no—
59C23×C4—24no—
60C23×C4—42no—
61C23×C4—82no—
62C23×C4C22×C411yescharacteristic
63C23×C4C22×C412yes—
64C23×C4C2×D411yescharacteristic
65C23×C4C2×D411yescharacteristic
66C23×C4C2×Q811yescharacteristic
67C23×C4#4112yes—
68C23×C4#4112yes—
69C23×C4#4111yescharacteristic
70C23×C4#4111yescharacteristic
71C23×C4#4112yes—
72C22×C8—22no—
73C22×C8—22no—
74C22×C8—22no—
75C22×C8—24no—
76C22×C8—82no—
77C22×C8—82no—
78C22×C8C22⋊C414yes—
79C2×C4⋊C4—24no—
80C2×C4⋊C4—42no—
81C2×C4⋊C4—42no—
82C2×C4⋊C4—42no—
83C2×C4⋊C4—84no—
84C2×C4⋊C4—84no—
85C2×C22⋊C4—21no—
86C2×C22⋊C4—21no—
87C2×C22⋊C4—22no—
88C2×C22⋊C4—21no—
89C2×C22⋊C4—21no—
90C2×C22⋊C4—24no—
91C2×C22⋊C4—42no—
92C2×C22⋊C4—42no—
93C2×C22⋊C4—42no—
94C2×C22⋊C4—42no—
95C2×C22⋊C4—42no—
96C2×C22⋊C4—42no—
97C2×C22⋊C4—42no—
98C2×C22⋊C4—42no—
99C2×C22⋊C4—42no—
100C2×C22⋊C4—42no—
101C2×C22⋊C4—42no—
102C2×C22⋊C4C4⋊C412yes—
103C2×C22⋊C4C4⋊C412yes—
104C2×C22⋊C4D812yes—
105C2×C22⋊C4SD1614yes—
106C2×C22⋊C4Q1612yes—
107C2×C42—22no—
108C2×C42—22no—
109C2×C42—82no—
110C22⋊C8—22no—
111C22⋊C8—24no—
112C22⋊C8—24no—
113C22⋊C8—24no—
114C22⋊C8C4214yes—
115C4×C8—84no—
116#95—22no—
117#95—22no—
118#95—24no—
119#95—24no—
120#95—24no—
121#95—42no—
122C24—81no—
123C24—81no—
124C24C2×C4⋊C411yescharacteristic
125C24C4×D412yes—
126C24#12111yescharacteristic
127C24#12111yescharacteristic
128C24#12412yes—
129C24C2×D811yescharacteristic
130C24C2×SD1612yes—
131C24C2×Q1611yescharacteristic
132C24#13512yes—
133C24#13511yescharacteristic
134C24#13511yescharacteristic
135C22×C4—22no—
136C22×C4—22no—
137C22×C4—24no—
138C22×C4—22no—
139C22×C4—22no—
140C22×C4—42no—
141C22×C4—44no—
142C22×C4—44no—
143C22×C4—42no—
144C22×C4—42no—
145C22×C4—44no—
146C22×C4—44no—
147C22×C4—44no—
148C22×C4—44no—
149C22×C4—82no—
150C22×C4—81no—
151C22×C4—81no—
152C22×C4—81no—
153C22×C4—82no—
154C22×C4—81no—
155C22×C4—81no—
156C22×C4—81no—
157C22×C4—81no—
158C22×C4—84no—
159C22×C4—84no—
160C22×C4—82no—
161C22×C4—82no—
162C22×C4—84no—
163C22×C4—84no—
164C22×C4C2×C4211yescharacteristic
165C22×C4C2×C22⋊C411yescharacteristic
166C22×C4#11711yescharacteristic
167C22×C4#11711yescharacteristic
168C22×C4#11711yescharacteristic
169C22×C4#11712yes—
170C22×C4#11712yes—
171C22×C4#11711yescharacteristic
172C22×C4C4×D412yes—
173C22×C4C4×D412yes—
174C22×C4C4×Q812yes—
175C22×C4C22⋊Q811yescharacteristic
176C22×C4C22⋊Q811yescharacteristic
177C22×C4#12312yes—
178C22×C4#12312yes—
179C22×C4#12512yes—
180C22×C4#12612yes—
181C22×C4#12612yes—
182C2×C8—24no—
183C2×C8—24no—
184C2×C8—44no—
185C2×C8—44no—
186C2×C8—82no—
187C2×C8—82no—
188C2×C8—84no—
189C4⋊C4—84no—
190C4⋊C4—84no—
191C4⋊C4—84no—
192C4⋊C4—84no—
193C22⋊C4—22no—
194C22⋊C4—22no—
195C22⋊C4—24no—
196C22⋊C4—22no—
197C22⋊C4—22no—
198C22⋊C4—44no—
199C22⋊C4—44no—
200C22⋊C4—82no—
201C22⋊C4—82no—
202C22⋊C4—82no—
203C22⋊C4—82no—
204C22⋊C4—82no—
205C22⋊C4—82no—
206C22⋊C4#10614yes—
207C22⋊C4#10714yes—
208C42—44no—
209C42—82no—
210C23—42no—
211C23—41no—
212C23—42no—
213C23—41no—
214C23—41no—
215C23—42no—
216C23—41no—
217C23—42no—
218C23—41no—
219C23—41no—
220C23—81no—
221C23—82no—
222C23—81no—
223C23—82no—
224C23—81no—
225C23—81no—
226C23C4×C4⋊C411yescharacteristic
227C23#38011yescharacteristic
228C23#38211yescharacteristic
229C23#38812yes—
230C23#38812yes—
231C23C2×#10611yescharacteristic
232C23C2×#10711yescharacteristic
233C23#42711yescharacteristic
234C23#42711yescharacteristic
235C23C4×D812yes—
236C23C4×SD1612yes—
237C23C4×SD1612yes—
238C23C4×Q1612yes—
239C23#46512yes—
240C23#46512yes—
241C23#46611yescharacteristic
242C23#46611yescharacteristic
243C23#46711yescharacteristic
244C23#46711yescharacteristic
245C23#48612yes—
246C23#48712yes—
247C23#48812yes—
248C23#48812yes—
249C2×C4—22no—
250C2×C4—24no—
251C2×C4—24no—
252C2×C4—22no—
253C2×C4—22no—
254C2×C4—24no—
255C2×C4—24no—
256C2×C4—41no—
257C2×C4—42no—
258C2×C4—42no—
259C2×C4—42no—
260C2×C4—41no—
261C2×C4—41no—
262C2×C4—41no—
263C2×C4—42no—
264C2×C4—42no—
265C2×C4—42no—
266C2×C4—41no—
267C2×C4—42no—
268C2×C4—42no—
269C2×C4—42no—
270C2×C4—42no—
271C2×C4—42no—
272C2×C4—42no—
273C2×C4—84no—
274C2×C4—82no—
275C2×C4—82no—
276C2×C4—84no—
277C2×C4—84no—
278C2×C4—84no—
279C2×C4—84no—
280C2×C4—84no—
281C2×C4—84no—
282C2×C4—164no—
283C2×C4—164no—
284C2×C4—164no—
285C2×C4—164no—
286C2×C4#37611yescharacteristic
287C2×C4C4×C22⋊C411yescharacteristic
288C2×C4#38111yescharacteristic
289C2×C4#38311yescharacteristic
290C2×C4#38712yes—
291C2×C4#38712yes—
292C8—44no—
293C8—44no—
294C22—21no—
295C22—22no—
296C22—22no—
297C22—21no—
298C22—21no—
299C22—21no—
300C22—22no—
301C22—22no—
302C22—21no—
303C22—21no—
304C22—42no—
305C22—41no—
306C22—41no—
307C22—42no—
308C22—42no—
309C22—42no—
310C22—42no—
311C22—41no—
312C22—41no—
313C22—41no—
314C22—41no—
315C22—42no—
316C22—42no—
317C22—42no—
318C22—41no—
319C22—41no—
320C22—42no—
321C22—42no—
322C22#143411yescharacteristic
323C22C4×#10611yescharacteristic
324C22C4×#10711yescharacteristic
325C22#180711yescharacteristic
326C22#180811yescharacteristic
327C22#183411yescharacteristic
328C22#183511yescharacteristic
329C22#193312yes—
330C22#193312yes—
331C22#193412yes—
332C22#193412yes—
333C4—21no—
334C4—21no—
335C4—21no—
336C4—22no—
337C4—22no—
338C4—44no—
339C4—44no—
340C4—164no—
341C4—164no—
342C4#182812yes—
343C2—21no—
344C2—21no—
345C2—22no—
346C2—22no—
347C2—21no—
348C2—21no—
349C2—22no—
350C2—22no—
351C2—21no—
352C2—21no—
353C2#1081111yescharacteristic
354C2#1082011yescharacteristic
355C2#1082911yescharacteristic
subgroupG/H
center, socleC22#1434
commutatorC22×C4C2×C42
Frattini, cosocleC24×C4C23
prime radicalC25C2×D4
LCSG ▷ C22×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
12. C8×C22⋊C4C4semidirect
16. #1294C4semidirect

Maximal subgroups

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

subgroupindex
#205232
#108542
#80232
C2×#12892