Finite Group Databasebeta

← #118256

Subgroups — #118256

Order 512.

520 conjugacy classes of subgroups.

The subgroup lattice has 358 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#17803C211yesmaximal, characteristic
2#17496C211yesmaximal, characteristic
3C2×#1497C211yesmaximal, characteristic
4#17385C211yesmaximal, characteristic
5#8124C211yesmaximal, characteristic
6#8089C211yesmaximal, characteristic
7#7933C211yesmaximal, characteristic
8C2×#533C2211yescharacteristic
9#2920C2211yescharacteristic
10#2920C2211yescharacteristic
11C2×#432C2211yescharacteristic
12C2×#432C2211yescharacteristic
13C2×#431C2211yescharacteristic
14C2×#431C2211yescharacteristic
15#1572—22no—
16#1571—21no—
17#1570—21no—
18#1564—22no—
19#1497—22no—
20#1497—22no—
21#1497—22no—
22#1491—22no—
23#1481—22no—
24C2×Q8⋊C8—22no—
25#1472C414yes—
26#1470—21no—
27#1470—21no—
28#1461—22no—
29#1308—22no—
30#1302—22no—
31#1283—22no—
32#533—21no—
33#533—21no—
34#533—21no—
35#533—21no—
36#533—21no—
37#533—21no—
38#533—21no—
39#533—21no—
40C2×C4⋊Q8Q811yescharacteristic
41C2×C4⋊Q8Q811yescharacteristic
42C2×#125—21no—
43C2×C22⋊Q8—21no—
44C2×C22⋊Q8C2×C412yes—
45C2×C4×Q8—21no—
46C2×C4×Q8D412yes—
47C2×#117—21no—
48C22×C42C2311yescharacteristic
49#432—21no—
50#432—21no—
51#432—21no—
52#432—21no—
53#432—21no—
54#432—21no—
55#432—21no—
56#432—21no—
57#432—21no—
58#432—21no—
59#432—21no—
60#432D412yes—
61#431—21no—
62#431—21no—
63#431—21no—
64#431—21no—
65#431—21no—
66#431—21no—
67#431—21no—
68#431—21no—
69#431D412yes—
70#431D412yes—
71#431D412yes—
72#431D411yescharacteristic
73#431D411yescharacteristic
74#423—21no—
75#423—21no—
76C2×C4⋊C8—21no—
77C2×C4⋊C8—21no—
78C2×C4⋊C8—21no—
79C2×C4⋊C8D412yes—
80#407—21no—
81#407C2×C412yes—
82C2×C22⋊C8—21no—
83C2×C22⋊C8—21no—
84C2×C22⋊C8—21no—
85C2×C22⋊C8C2×C412yes—
86C4×M16—21no—
87C4×M16—21no—
88C2×#97—21no—
89C2×#97—21no—
90C2×C4×C8—21no—
91C2×C4×C8—21no—
92#335—24no—
93#334—24no—
94Q8⋊C8—28no—
95Q8⋊C8—28no—
96#324—24no—
97#324—24no—
98C22×Q8—21no—
99C22×Q8C4⋊C412yes—
100C23×C4C22×C411yescharacteristic
101C23×C4#4111yescharacteristic
102C23×C4#4111yescharacteristic
103C2×M16—22no—
104C2×M16—41no—
105C22×C8—22no—
106C22×C8—22no—
107C22×C8—41no—
108C22×C8—41no—
109C4⋊Q8—21no—
110C4⋊Q8—21no—
111C4⋊Q8—21no—
112C4⋊Q8—21no—
113C4⋊Q8—21no—
114C4⋊Q8—21no—
115C4⋊Q8—21no—
116C4⋊Q8—21no—
117#125—41no—
118#125—41no—
119#125—41no—
120#125—41no—
121C22⋊Q8—22no—
122C22⋊Q8—22no—
123C22⋊Q8—22no—
124C22⋊Q8—22no—
125C22⋊Q8—41no—
126C22⋊Q8—41no—
127C22⋊Q8—41no—
128C22⋊Q8—41no—
129C4×Q8—22no—
130C4×Q8—22no—
131C4×Q8—22no—
132C4×Q8—22no—
133C4×Q8—41no—
134C4×Q8—41no—
135C4×Q8—41no—
136C4×Q8—41no—
137#117—41no—
138#117—41no—
139#117—41no—
140#117—41no—
141C2×C4⋊C4—22no—
142C2×C4⋊C4—22no—
143C2×C4⋊C4—21no—
144C2×C4⋊C4—21no—
145C2×C4⋊C4—21no—
146C2×C4⋊C4C4⋊C412yes—
147C2×C22⋊C4—22no—
148C2×C42—22no—
149C2×C42C2×D411yescharacteristic
150C2×C42C2×D411yescharacteristic
151C2×C42C2×D411yescharacteristic
152C2×C42C2×D411yescharacteristic
153C2×C42C2×D411yescharacteristic
154C2×C42C2×D411yescharacteristic
155C2×C42C2×D411yescharacteristic
156C2×C42C2×Q811yescharacteristic
157C2×C42#4111yescharacteristic
158C2×C42#4111yescharacteristic
159C2×C42#4111yescharacteristic
160C2×C42#4111yescharacteristic
161C4⋊C8—22no—
162C4⋊C8—22no—
163C4⋊C8—22no—
164C4⋊C8—22no—
165C4⋊C8—22no—
166C4⋊C8—22no—
167C4⋊C8—21no—
168C4⋊C8—21no—
169C4⋊C8—41no—
170C4⋊C8—41no—
171C4⋊C8—41no—
172C4⋊C8—41no—
173C4⋊C8—41no—
174C22⋊C8—22no—
175C22⋊C8—22no—
176C22⋊C8—22no—
177C22⋊C8—22no—
178C22⋊C8—22no—
179C22⋊C8—22no—
180C22⋊C8—41no—
181C22⋊C8—41no—
182C22⋊C8—41no—
183C22⋊C8—41no—
184C22⋊C8—41no—
185C22⋊C8C22⋊C412yes—
186C22⋊C8C22⋊C412yes—
187#97—22no—
188#97—41no—
189#97—41no—
190#97—41no—
191#97—41no—
192#97—41no—
193C4×C8—21no—
194C4×C8—21no—
195C4×C8—22no—
196C4×C8—22no—
197C4×C8—22no—
198C4×C8—41no—
199C4×C8—41no—
200C24#11711yescharacteristic
201C2×Q8—22no—
202C2×Q8—22no—
203C2×Q8—44no—
204C2×Q8—44no—
205C2×Q8—41no—
206C2×Q8—41no—
207C22×C4—22no—
208C22×C4—22no—
209C22×C4—42no—
210C22×C4—42no—
211C22×C4—42no—
212C22×C4C2×C22⋊C411yescharacteristic
213C22×C4C2×C4⋊C411yescharacteristic
214C22×C4C4×D411yescharacteristic
215C22×C4C4×D411yescharacteristic
216C22×C4C4×D411yescharacteristic
217C22×C4C4×D411yescharacteristic
218C22×C4#12111yescharacteristic
219C22×C4#12111yescharacteristic
220C22×C4#12111yescharacteristic
221C22×C4#12111yescharacteristic
222C22×C4C22⋊Q811yescharacteristic
223C22×C4C22⋊Q811yescharacteristic
224C22×C4#12311yescharacteristic
225C22×C4#12311yescharacteristic
226C22×C4#12411yescharacteristic
227C22×C4#12411yescharacteristic
228C22×C4#12611yescharacteristic
229C22×C4#12611yescharacteristic
230M16—44no—
231M16—82no—
232C2×C8—22no—
233C2×C8—22no—
234C2×C8—44no—
235C2×C8—42no—
236C2×C8—42no—
237C2×C8—42no—
238C2×C8—42no—
239C2×C8—42no—
240C2×C8—82no—
241C2×C8—82no—
242C2×C8—82no—
243C4⋊C4—22no—
244C4⋊C4—22no—
245C4⋊C4—42no—
246C4⋊C4—42no—
247C4⋊C4—42no—
248C4⋊C4—41no—
249C4⋊C4—41no—
250C4⋊C4—41no—
251C4⋊C4—41no—
252C4⋊C4—42no—
253C4⋊C4—41no—
254C4⋊C4—41no—
255C22⋊C4—42no—
256C22⋊C4—42no—
257C42—42no—
258C42—42no—
259C42#12011yescharacteristic
260C42#12011yescharacteristic
261C42#12011yescharacteristic
262C42#12111yescharacteristic
263C42#12111yescharacteristic
264C42#12111yescharacteristic
265C42#12111yescharacteristic
266C42C22⋊Q811yescharacteristic
267C42C22⋊Q811yescharacteristic
268C42C22⋊Q811yescharacteristic
269C42C22⋊Q811yescharacteristic
270C42#12311yescharacteristic
271C42#12311yescharacteristic
272C42#12311yescharacteristic
273C42#12411yescharacteristic
274C42#12711yescharacteristic
275C23—22no—
276C23—41no—
277C23—41no—
278C23#38011yescharacteristic
279C23#38811yescharacteristic
280C23#38811yescharacteristic
281Q8—44no—
282Q8—44no—
283Q8—82no—
284Q8—82no—
285C2×C4—22no—
286C2×C4—22no—
287C2×C4—22no—
288C2×C4—22no—
289C2×C4—22no—
290C2×C4—22no—
291C2×C4—22no—
292C2×C4—22no—
293C2×C4—41no—
294C2×C4—41no—
295C2×C4—41no—
296C2×C4—41no—
297C2×C4—41no—
298C2×C4—41no—
299C2×C4—41no—
300C2×C4—41no—
301C2×C4—41no—
302C2×C4—41no—
303C2×C4—44no—
304C2×C4—42no—
305C2×C4—42no—
306C2×C4—82no—
307C2×C4#38511yescharacteristic
308C2×C4#38511yescharacteristic
309C2×C4#38611yescharacteristic
310C2×C4#39011yescharacteristic
311C2×C4#39211yescharacteristic
312C2×C4#39311yescharacteristic
313C2×C4#39411yescharacteristic
314C2×C4#39411yescharacteristic
315C2×C4#39611yescharacteristic
316C2×C4#39611yescharacteristic
317C2×C4#39711yescharacteristic
318C2×C4#39911yescharacteristic
319C8—42no—
320C8—42no—
321C8—82no—
322C8—82no—
323C8—82no—
324C8—82no—
325C8—82no—
326C22—21no—
327C22—21no—
328C22—21no—
329C22—21no—
330C22—21no—
331C22—21no—
332C22—21no—
333C22—21no—
334C22—82no—
335C22#143911yescharacteristic
336C22#181412yes—
337C4—21no—
338C4—21no—
339C4—21no—
340C4—21no—
341C4—21no—
342C4—21no—
343C4—21no—
344C4—21no—
345C4—21no—
346C4—21no—
347C4—82no—
348C4—82no—
349C4#186812yes—
350C4#204912yes—
351C2—22no—
352C2—41no—
353C2—41no—
354C2#1108811yescharacteristic
355C2#1110411yescharacteristic
356C2#1114311yescharacteristic
subgroupG/H
centerC2×C4#394
commutatorC23×C4C22×C4
Frattini, cosocleC22×C42C23
socleC22#1439
prime radicalC24#117
LCSG ▷ C23×C4 ▷ C22 ▷ C1
UCSC1 ◁ C2×C4 ◁ C22×C42 ◁ G
LECSG ▷ C22×C42 ▷ C22 ▷ C1

Maximal subgroups

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

subgroupindex
#178032
#174962
C2×#14972
#173852
#81242
#80892
#79332