Finite Group Databasebeta

← #120408

Subgroups — #120408

Order 512.

477 conjugacy classes of subgroups.

The subgroup lattice has 350 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21985C211yesmaximal, characteristic
2#17809C211yesmaximal, characteristic
3#10836C211yesmaximal, characteristic
4#8109C211yesmaximal, characteristic
5#8092C211yesmaximal, characteristic
6#8052C211yesmaximal, characteristic
7#7960C211yesmaximal, characteristic
8#2920C2211yescharacteristic
9#2920C2211yescharacteristic
10C2×#431C2211yescharacteristic
11C2×#423C2211yescharacteristic
12#2285C2211yescharacteristic
13#2283C2211yescharacteristic
14C4×#117C2211yescharacteristic
15#1948—21no—
16#1947—21no—
17#1854—22no—
18#1846—21no—
19#1835—21no—
20C2×H(3,ℤ/4)—22no—
21#1574—22no—
22#1571—21no—
23#1570—21no—
24#1565C414yescomplemented
25#1305—22no—
26#1303—22no—
27#1297—22no—
28#1286—21no—
29#1286—21no—
30C22×M16—21no—
31C2×#117—21no—
32C2×#117D412yes—
33C22×C42C2311yescharacteristic
34#432—21no—
35#432—21no—
36#432—21no—
37#432—21no—
38#431—21no—
39#431—21no—
40#431—21no—
41#431—21no—
42#431—21no—
43#431C2×C412yes—
44#431D412yes—
45#431Q812yes—
46C2×#107—22no—
47C2×#106—22no—
48#423—21no—
49#423—21no—
50#423—21no—
51#423—21no—
52#423—21no—
53#423Q812yes—
54C2×C4⋊C8—21no—
55C2×C4⋊C8C2×C412yes—
56C2×C4⋊C8D411yescharacteristic
57C2×C4⋊C8D411yescharacteristic
58#407—21no—
59#407D412yes—
60C2×C22⋊C8—21no—
61C4×M16—21no—
62C4×M16C2×C412yes—
63C2×C4×C8—21no—
64#384—21no—
65#383—21no—
66#382—21no—
67#382D411yescharacteristic
68#382D411yescharacteristic
69#381—21no—
70#380—21no—
71#380—21no—
72#380—21no—
73C4×C4⋊C4—21no—
74C4×C4⋊C4—21no—
75C4×C4⋊C4—21no—
76C4×C4⋊C4—21no—
77C4×C4⋊C4—21no—
78C4×C22⋊C4—21no—
79C4×C22⋊C4—21no—
80C4×C22⋊C4—21no—
81#376—21no—
82#376Q812yes—
83C43—21no—
84#340—22no—
85#340—22no—
86H(3,ℤ/4)—44no—
87#335—24no—
88#334—24no—
89C23×C4C2×D411yescharacteristic
90C23×C4C2×D411yescharacteristic
91C23×C4#4111yescharacteristic
92C2×M16—42no—
93C2×M16—42no—
94C2×M16—41no—
95C2×M16—41no—
96C2×M16—41no—
97C2×M16C4⋊C414yes—
98C22×C8—21no—
99C22×C8—21no—
100C22×C8—22no—
101#117—22no—
102#117—22no—
103#117—41no—
104#117—41no—
105#117—42no—
106#117—42no—
107C2×C4⋊C4—22no—
108C2×C4⋊C4—21no—
109C2×C4⋊C4—21no—
110C2×C4⋊C4—41no—
111C2×C4⋊C4—41no—
112C2×C4⋊C4—41no—
113C2×C22⋊C4—22no—
114C2×C22⋊C4—41no—
115C2×C22⋊C4—41no—
116C2×C42—22no—
117C2×C42—22no—
118C2×C42—41no—
119C2×C42—41no—
120C2×C42—41no—
121C2×C42C22×C411yescharacteristic
122C2×C42C2×D411yescharacteristic
123C2×C42C2×D411yescharacteristic
124C2×C42C2×D411yescharacteristic
125C2×C42C2×Q811yescharacteristic
126C2×C42C2×Q811yescharacteristic
127C2×C42C2×Q811yescharacteristic
128C2×C42#4111yescharacteristic
129C2×C42#4111yescharacteristic
130C2×C42#4111yescharacteristic
131C2×C42#4111yescharacteristic
132C2×C42#4111yescharacteristic
133#107—82no—
134#106—82no—
135C4⋊C8—22no—
136C4⋊C8—22no—
137C4⋊C8—22no—
138C4⋊C8—22no—
139C4⋊C8—22no—
140C4⋊C8—21no—
141C4⋊C8—21no—
142C4⋊C8—41no—
143C4⋊C8—41no—
144C4⋊C8—41no—
145C4⋊C8C22⋊C412yes—
146C4⋊C8C22⋊C412yes—
147C22⋊C8—22no—
148C22⋊C8—22no—
149C22⋊C8—22no—
150C22⋊C8—22no—
151C22⋊C8—22no—
152C22⋊C8—41no—
153#97—22no—
154#97—41no—
155C4×C8—22no—
156C4×C8—22no—
157C4×C8—22no—
158C4×C8—22no—
159#95—21no—
160#95—21no—
161#95—22no—
162#95—22no—
163#95—21no—
164#95—21no—
165#95—41no—
166#95—41no—
167#95—41no—
168#95—41no—
169C24#12111yescharacteristic
170C22×C4—21no—
171C22×C4—21no—
172C22×C4—21no—
173C22×C4—21no—
174C22×C4—42no—
175C22×C4—42no—
176C22×C4—41no—
177C22×C4—41no—
178C22×C4—42no—
179C22×C4—81no—
180C22×C4C2×C4⋊C411yescharacteristic
181C22×C4C4×D411yescharacteristic
182C22×C4C4×Q811yescharacteristic
183C22×C4#12011yescharacteristic
184C22×C4#12111yescharacteristic
185C22×C4#12111yescharacteristic
186C22×C4#12111yescharacteristic
187C22×C4#12111yescharacteristic
188C22×C4C22⋊Q811yescharacteristic
189C22×C4C22⋊Q811yescharacteristic
190C22×C4C22⋊Q811yescharacteristic
191C22×C4C22⋊Q811yescharacteristic
192C22×C4#12311yescharacteristic
193C22×C4#12311yescharacteristic
194C22×C4#12411yescharacteristic
195C22×C4#12511yescharacteristic
196C22×C4#12611yescharacteristic
197C22×C4C4⋊Q811yescharacteristic
198M16—44no—
199M16—82no—
200M16—82no—
201M16—81no—
202M16—81no—
203C2×C8—24no—
204C2×C8—22no—
205C2×C8—22no—
206C2×C8—41no—
207C2×C8—41no—
208C2×C8—41no—
209C2×C8—41no—
210C2×C8—44no—
211C2×C8—42no—
212C2×C8—42no—
213C4⋊C4—42no—
214C4⋊C4—42no—
215C4⋊C4—41no—
216C4⋊C4—42no—
217C4⋊C4—41no—
218C4⋊C4—41no—
219C4⋊C4—42no—
220C4⋊C4—41no—
221C4⋊C4—82no—
222C4⋊C4—82no—
223C22⋊C4—42no—
224C22⋊C4—42no—
225C22⋊C4—82no—
226C22⋊C4—82no—
227C42—24no—
228C42—42no—
229C42—44no—
230C42—82no—
231C42—82no—
232C42—82no—
233C42C2×C22⋊C411yescharacteristic
234C42#11711yescharacteristic
235C42C4×D411yescharacteristic
236C42C4×Q811yescharacteristic
237C42#12111yescharacteristic
238C42C22⋊Q811yescharacteristic
239C42C22⋊Q811yescharacteristic
240C42C22⋊Q811yescharacteristic
241C42C22⋊Q811yescharacteristic
242C42#12311yescharacteristic
243C42#12511yescharacteristic
244C42#12511yescharacteristic
245C42#12511yescharacteristic
246C42#12611yescharacteristic
247C42C4⋊Q811yescharacteristic
248C42C4⋊Q811yescharacteristic
249C23—21no—
250C23—21no—
251C23—41no—
252C23—41no—
253C23#38911yescharacteristic
254C23#39411yescharacteristic
255C23#39611yescharacteristic
256C2×C4—21no—
257C2×C4—21no—
258C2×C4—21no—
259C2×C4—21no—
260C2×C4—21no—
261C2×C4—21no—
262C2×C4—22no—
263C2×C4—22no—
264C2×C4—22no—
265C2×C4—21no—
266C2×C4—21no—
267C2×C4—21no—
268C2×C4—21no—
269C2×C4—41no—
270C2×C4—41no—
271C2×C4—41no—
272C2×C4—41no—
273C2×C4—41no—
274C2×C4—41no—
275C2×C4—41no—
276C2×C4—41no—
277C2×C4—41no—
278C2×C4—44no—
279C2×C4—82no—
280C2×C4—82no—
281C2×C4—81no—
282C2×C4—81no—
283C2×C4—81no—
284C2×C4—81no—
285C2×C4—82no—
286C2×C4—82no—
287C2×C4—82no—
288C2×C4—81no—
289C2×C4—81no—
290C2×C4#38011yescharacteristic
291C2×C4#38611yescharacteristic
292C2×C4#38711yescharacteristic
293C2×C4#38711yescharacteristic
294C2×C4#38911yescharacteristic
295C2×C4#39111yescharacteristic
296C2×C4#39511yescharacteristic
297C2×C4#39611yescharacteristic
298C2×C4#39811yescharacteristic
299C2×C4#39811yescharacteristic
300C2×C4#39811yescharacteristic
301C2×C4#40011yescharacteristic
302C2×C4#45311yescharacteristic
303C2×C4#45411yescharacteristic
304C2×C4#45511yescharacteristic
305C2×C4#45611yescharacteristic
306C8—44no—
307C8—42no—
308C8—42no—
309C8—82no—
310C8—82no—
311C22—21no—
312C22—21no—
313C22—21no—
314C22—21no—
315C22—21no—
316C22—21no—
317C22—21no—
318C22—21no—
319C22—81no—
320C22—81no—
321C22#144111yescharacteristic
322C22#203711yescharacteristic
323C22#203811yescharacteristic
324C4—21no—
325C4—21no—
326C4—21no—
327C4—21no—
328C4—21no—
329C4—21no—
330C4—21no—
331C4—21no—
332C4—21no—
333C4—84no—
334C4—161no—
335C4—162no—
336C4—161no—
337C4#181212yes—
338C4#188211yescharacteristic
339C4#188311yescharacteristic
340C4#206011yescharacteristic
341C4#206111yescharacteristic
342C2—21no—
343C2—21no—
344C2—41no—
345C2—41no—
346C2#1121711yescharacteristic
347C2#1122911yescharacteristic
348C2#1125111yescharacteristic
subgroupG/H
center, socleC22#1441
commutatorC2×C42C22×C4
Frattini, cosocleC22×C42C23
prime radicalC24#121
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C22×C42 ◁ G
LECSG ▷ C22×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
24. #1565C4semidirect

Maximal subgroups

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

subgroupindex
#219852
#178092
#108362
#81092
#80922
#80522
#79602