Finite Group Databasebeta

← #120916

Subgroups — #120916

Order 512.

474 conjugacy classes of subgroups.

The subgroup lattice has 355 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21969C211yesmaximal, characteristic
2#10850C211yesmaximal, characteristic
3#8098C211yesmaximal, characteristic
4#7959C211yesmaximal, characteristic
5#7959C211yesmaximal, characteristic
6#7950C211yesmaximal, characteristic
7#7935C211yesmaximal, characteristic
8#2920C2211yescharacteristic
9#2920C2211yescharacteristic
10#2920C2211yescharacteristic
11#2920C2211yescharacteristic
12C22×C4⋊C8C2211yescharacteristic
13C2×#382C2211yescharacteristic
14C2×C4×C4⋊C4C2211yescharacteristic
15#1950—21no—
16#1950—21no—
17#1950—21no—
18#1950—21no—
19#1810—22no—
20C2×#340—22no—
21#1303—22no—
22#1286—21no—
23#1286—21no—
24#1286—21no—
25#1286—21no—
26#1285—22no—
27#1283C414yescomplemented
28C23×C8D412yes—
29C22×C4⋊C4—21no—
30C22×C4⋊C4—21no—
31C22×C42—21no—
32C22×C42C2311yescharacteristic
33#432—21no—
34#432—21no—
35#432—21no—
36#432—21no—
37#432—21no—
38#432—21no—
39#432—21no—
40#432—21no—
41#431—21no—
42#431—21no—
43#431—21no—
44#431—21no—
45#431—21no—
46#431—21no—
47#431—21no—
48#431C2×C412yes—
49C2×#108—22no—
50C2×#108—22no—
51#423—21no—
52#423—21no—
53#423D411yescharacteristic
54#423D411yescharacteristic
55#423D411yescharacteristic
56#423D411yescharacteristic
57C2×C4⋊C8—21no—
58C2×C4⋊C8—21no—
59C2×C4⋊C8—21no—
60C2×C4⋊C8—21no—
61C2×C4⋊C8C2×C412yes—
62C2×C4⋊C8Q812yes—
63#407—21no—
64#407—21no—
65#407—21no—
66#407—21no—
67C4×M16—21no—
68C4×M16—21no—
69C4×M16—21no—
70C4×M16C2×C412yes—
71#382—21no—
72#382—21no—
73#382—21no—
74#382Q812yes—
75C4×C4⋊C4—21no—
76C4×C4⋊C4—21no—
77C4×C4⋊C4—21no—
78C4×C4⋊C4D412yes—
79C2×#95—21no—
80C2×#95—21no—
81C2×#95—21no—
82#341—22no—
83#341—22no—
84#340—44no—
85C23×C4—41no—
86C23×C4—41no—
87C23×C4C2×D411yescharacteristic
88C23×C4#4111yescharacteristic
89C23×C4#4111yescharacteristic
90C2×M16—22no—
91C2×M16—21no—
92C2×M16—21no—
93C2×M16—21no—
94C2×M16—21no—
95C2×M16—41no—
96C22×C8—22no—
97C22×C8—22no—
98C22×C8—42no—
99C22×C8—42no—
100C22×C8C4⋊C414yes—
101C2×C4⋊C4—22no—
102C2×C4⋊C4—22no—
103C2×C4⋊C4—22no—
104C2×C4⋊C4—22no—
105C2×C4⋊C4—41no—
106C2×C4⋊C4—41no—
107C2×C4⋊C4—41no—
108C2×C4⋊C4—42no—
109C2×C4⋊C4—42no—
110C2×C4⋊C4—41no—
111C2×C42—22no—
112C2×C42—41no—
113C2×C42—82no—
114C2×C42C22×C411yescharacteristic
115C2×C42C2×D411yescharacteristic
116C2×C42C2×D411yescharacteristic
117C2×C42C2×D411yescharacteristic
118C2×C42C2×Q811yescharacteristic
119C2×C42C2×Q811yescharacteristic
120C2×C42#4111yescharacteristic
121C2×C42#4111yescharacteristic
122C2×C42#4111yescharacteristic
123C2×C42#4111yescharacteristic
124C2×C42#4111yescharacteristic
125C2×C42#4111yescharacteristic
126#108—82no—
127#108—82no—
128C4⋊C8—22no—
129C4⋊C8—22no—
130C4⋊C8—22no—
131C4⋊C8—21no—
132C4⋊C8—21no—
133C4⋊C8—21no—
134C4⋊C8—21no—
135C4⋊C8—22no—
136C4⋊C8—22no—
137C4⋊C8—21no—
138C4⋊C8—21no—
139C4⋊C8—21no—
140C4⋊C8—21no—
141C4⋊C8—41no—
142C4⋊C8—41no—
143C4⋊C8—41no—
144C22⋊C8—22no—
145C22⋊C8—41no—
146C22⋊C8—41no—
147C22⋊C8—41no—
148C22⋊C8—41no—
149C22⋊C8—41no—
150C22⋊C8—41no—
151C22⋊C8—41no—
152#97—22no—
153#97—41no—
154#97—41no—
155#97—41no—
156C4×C8—22no—
157C4×C8—22no—
158C4×C8—41no—
159C4×C8—41no—
160#95—22no—
161#95—22no—
162#95—22no—
163#95—41no—
164#95—41no—
165#95—41no—
166C24#12411yescharacteristic
167C22×C4—21no—
168C22×C4—21no—
169C22×C4—21no—
170C22×C4—21no—
171C22×C4—41no—
172C22×C4—41no—
173C22×C4—42no—
174C22×C4—42no—
175C22×C4—42no—
176C22×C4—42no—
177C22×C4—42no—
178C22×C4—82no—
179C22×C4—82no—
180C22×C4—82no—
181C22×C4—81no—
182C22×C4—81no—
183C22×C4—82no—
184C22×C4C2×C4⋊C411yescharacteristic
185C22×C4#11711yescharacteristic
186C22×C4#11711yescharacteristic
187C22×C4#12011yescharacteristic
188C22×C4#12111yescharacteristic
189C22×C4C22⋊Q811yescharacteristic
190C22×C4C22⋊Q811yescharacteristic
191C22×C4C22⋊Q811yescharacteristic
192C22×C4#12311yescharacteristic
193C22×C4#12311yescharacteristic
194C22×C4#12311yescharacteristic
195C22×C4#12411yescharacteristic
196C22×C4#12411yescharacteristic
197C22×C4#12411yescharacteristic
198C22×C4#12511yescharacteristic
199C22×C4#12511yescharacteristic
200C22×C4#12611yescharacteristic
201C22×C4#12611yescharacteristic
202M16—44no—
203M16—41no—
204M16—41no—
205M16—41no—
206M16—41no—
207M16—41no—
208M16—41no—
209M16—41no—
210M16—41no—
211M16—81no—
212M16—81no—
213C2×C8—24no—
214C2×C8—42no—
215C2×C8—42no—
216C2×C8—44no—
217C2×C8—42no—
218C2×C8—42no—
219C2×C8—41no—
220C2×C8—41no—
221C2×C8—41no—
222C2×C8—41no—
223C2×C8—82no—
224C4⋊C4—24no—
225C4⋊C4—42no—
226C4⋊C4—42no—
227C4⋊C4—42no—
228C4⋊C4—44no—
229C4⋊C4—81no—
230C4⋊C4—81no—
231C42—82no—
232C42—82no—
233C42C4×D411yescharacteristic
234C42C4×D411yescharacteristic
235C42C4×D411yescharacteristic
236C42C4×Q811yescharacteristic
237C42#12111yescharacteristic
238C42#12111yescharacteristic
239C42C22⋊Q811yescharacteristic
240C42C22⋊Q811yescharacteristic
241C42#12311yescharacteristic
242C42#12311yescharacteristic
243C42#12511yescharacteristic
244C42#12511yescharacteristic
245C42#12511yescharacteristic
246C42#12611yescharacteristic
247C42#12611yescharacteristic
248C42C4⋊Q811yescharacteristic
249C23—21no—
250C23—21no—
251C23—22no—
252C23—21no—
253C23—21no—
254C23#38211yescharacteristic
255C23#39311yescharacteristic
256C23#39711yescharacteristic
257C2×C4—21no—
258C2×C4—21no—
259C2×C4—21no—
260C2×C4—21no—
261C2×C4—22no—
262C2×C4—22no—
263C2×C4—21no—
264C2×C4—21no—
265C2×C4—21no—
266C2×C4—21no—
267C2×C4—21no—
268C2×C4—21no—
269C2×C4—21no—
270C2×C4—21no—
271C2×C4—21no—
272C2×C4—21no—
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—44no—
282C2×C4—84no—
283C2×C4—84no—
284C2×C4—82no—
285C2×C4—82no—
286C2×C4—82no—
287C2×C4—82no—
288C2×C4—82no—
289C2×C4—162no—
290C2×C4—162no—
291C2×C4#38511yescharacteristic
292C2×C4#38711yescharacteristic
293C2×C4#38711yescharacteristic
294C2×C4#38811yescharacteristic
295C2×C4#38811yescharacteristic
296C2×C4#38911yescharacteristic
297C2×C4#39411yescharacteristic
298C2×C4#39811yescharacteristic
299C2×C4#39811yescharacteristic
300C2×C4#40011yescharacteristic
301C2×C4#40011yescharacteristic
302C2×C4#40011yescharacteristic
303C2×C4#45311yescharacteristic
304C2×C4#45411yescharacteristic
305C2×C4#45511yescharacteristic
306C2×C4#45611yescharacteristic
307C8—44no—
308C8—82no—
309C8—82no—
310C8—81no—
311C8—81no—
312C8—81no—
313C8—81no—
314C22—22no—
315C22—22no—
316C22—22no—
317C22—22no—
318C22—21no—
319C22—21no—
320C22—21no—
321C22—21no—
322C22—21no—
323C22—21no—
324C22#144211yescharacteristic
325C22#183312yes—
326C22#202711yescharacteristic
327C22#202811yescharacteristic
328C22#208111yescharacteristic
329C22#208211yescharacteristic
330C4—21no—
331C4—21no—
332C4—21no—
333C4—21no—
334C4—21no—
335C4—21no—
336C4—21no—
337C4—21no—
338C4—21no—
339C4—21no—
340C4—84no—
341C4—162no—
342C4#185611yescharacteristic
343C4#185711yescharacteristic
344C4#203911yescharacteristic
345C4#204011yescharacteristic
346C2—21no—
347C2—21no—
348C2—21no—
349C2—22no—
350C2—21no—
351C2#1125911yescharacteristic
352C2#1128311yescharacteristic
353C2#1129611yescharacteristic
subgroupG/H
center, socleC22#1442
commutatorC2×C42C22×C4
Frattini, cosocleC22×C42C23
prime radicalC24#124
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
27. #1283C4semidirect

Maximal subgroups

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

subgroupindex
#219692
#108502
#80982
#79592
#79502
#79352