Finite Group Databasebeta

← #110530

Subgroups — #110530

Order 512.

564 conjugacy classes of subgroups.

The subgroup lattice has 381 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#17708C211yesmaximal, characteristic
2#17464C211yesmaximal, characteristic
3#17385C211yesmaximal, characteristic
4#17326C211yesmaximal, characteristic
5#8119C211yesmaximal, characteristic
6#8076C211yesmaximal, characteristic
7#7880C211yesmaximal, characteristic
8C2×#533C2211yescharacteristic
9#2920C2211yescharacteristic
10#2920C2211yescharacteristic
11C2×#431C2211yescharacteristic
12C2×#431C2211yescharacteristic
13C2×#431C2211yescharacteristic
14C2×#423C2211yescharacteristic
15#1550—22no—
16#1549C412yes—
17#1548C412yes—
18#1542—21no—
19#1542—21no—
20#1485—21no—
21#1485—21no—
22#1481—22no—
23#1479C414yes—
24#1472C414yes—
25#1470—22no—
26#1461—22no—
27#1454—22no—
28#1452—21no—
29#1451—21no—
30#1450—21no—
31C8×M16—21no—
32#1307—22no—
33#1301—22no—
34#1277—22no—
35C22×M16—21no—
36#533—21no—
37#533—21no—
38#533—21no—
39#533—21no—
40#533—21no—
41#533—21no—
42#533—21no—
43#533—21no—
44C2×C4⋊Q8C2×C411yescharacteristic
45C2×C4⋊Q8C2×C411yescharacteristic
46C2×#125—21no—
47C2×C22⋊Q8—21no—
48C2×C22⋊Q8C2×C412yes—
49C2×C4×Q8—21no—
50C2×C4×Q8C2×C412yes—
51C2×#117—21no—
52C22×C42C2311yescharacteristic
53#432—21no—
54#432—21no—
55#432—21no—
56#432D412yes—
57#431—21no—
58#431—21no—
59#431—21no—
60#431—21no—
61#431—21no—
62#431—21no—
63#431—21no—
64#431—21no—
65#431—21no—
66#431—21no—
67#431C2×C412yes—
68#431C2×C412yes—
69#431C2×C412yes—
70#431D411yescharacteristic
71#431D411yescharacteristic
72#431D411yescharacteristic
73#431D411yescharacteristic
74#431D412yes—
75#423—21no—
76#423—21no—
77#423—21no—
78#423—21no—
79#423—21no—
80#423C2×C412yes—
81C2×C4⋊C8—21no—
82C2×C4⋊C8—21no—
83C2×C4⋊C8—21no—
84C2×C4⋊C8D412yes—
85C2×C4⋊C8D412yes—
86#407—21no—
87#407C2×C412yes—
88C2×C22⋊C8—21no—
89C2×C22⋊C8—21no—
90C2×C22⋊C8C2×C412yes—
91C4×M16—21no—
92C4×M16—21no—
93C2×C4×C8—21no—
94C2×C4×C8—21no—
95C2×C4×C8—21no—
96#333—22no—
97#333—22no—
98#332—22no—
99#332—22no—
100Q8⋊C8—24no—
101Q8⋊C8—24no—
102#324—28no—
103#322—22no—
104#322—22no—
105#322—22no—
106C82—22no—
107C22×Q8—21no—
108C22×Q8C4212yes—
109C23×C4C22×C411yescharacteristic
110C23×C4#4111yescharacteristic
111C23×C4#4111yescharacteristic
112C2×M16—22no—
113C2×M16—22no—
114C2×M16—41no—
115C2×M16—41no—
116C2×M16—82no—
117C22×C8—22no—
118C22×C8—22no—
119C22×C8—22no—
120C22×C8—41no—
121C4⋊Q8—21no—
122C4⋊Q8—21no—
123C4⋊Q8—21no—
124C4⋊Q8—21no—
125C4⋊Q8—21no—
126C4⋊Q8—21no—
127C4⋊Q8—21no—
128C4⋊Q8—21no—
129#125—41no—
130#125—41no—
131#125—41no—
132#125—41no—
133C22⋊Q8—22no—
134C22⋊Q8—22no—
135C22⋊Q8—22no—
136C22⋊Q8—22no—
137C22⋊Q8—41no—
138C22⋊Q8—41no—
139C22⋊Q8—41no—
140C22⋊Q8—41no—
141C4×Q8—22no—
142C4×Q8—22no—
143C4×Q8—22no—
144C4×Q8—22no—
145C4×Q8—41no—
146C4×Q8—41no—
147C4×Q8—41no—
148C4×Q8—41no—
149#117—41no—
150#117—41no—
151#117—41no—
152#117—41no—
153C2×C4⋊C4—22no—
154C2×C4⋊C4—21no—
155C2×C4⋊C4—21no—
156C2×C4⋊C4—21no—
157C2×C4⋊C4—22no—
158C2×C4⋊C4C4212yes—
159C2×C22⋊C4—22no—
160C2×C42—22no—
161C2×C42C22×C411yescharacteristic
162C2×C42C22×C411yescharacteristic
163C2×C42C2×D411yescharacteristic
164C2×C42C2×D411yescharacteristic
165C2×C42C2×D411yescharacteristic
166C2×C42C2×D411yescharacteristic
167C2×C42C2×D411yescharacteristic
168C2×C42C2×D411yescharacteristic
169C2×C42#4111yescharacteristic
170C2×C42#4111yescharacteristic
171C2×C42#4111yescharacteristic
172C2×C42#4111yescharacteristic
173C4⋊C8—22no—
174C4⋊C8—22no—
175C4⋊C8—22no—
176C4⋊C8—22no—
177C4⋊C8—22no—
178C4⋊C8—22no—
179C4⋊C8—22no—
180C4⋊C8—41no—
181C4⋊C8—41no—
182C4⋊C8—41no—
183C4⋊C8—41no—
184C4⋊C8—41no—
185C4⋊C8C22⋊C412yes—
186C4⋊C8C22⋊C412yes—
187C4⋊C8C22⋊C412yes—
188C4⋊C8C22⋊C412yes—
189C22⋊C8—22no—
190C22⋊C8—22no—
191C22⋊C8—22no—
192C22⋊C8—21no—
193C22⋊C8—21no—
194C22⋊C8—21no—
195C22⋊C8—21no—
196C22⋊C8—22no—
197C22⋊C8—22no—
198C22⋊C8—22no—
199C22⋊C8—41no—
200C22⋊C8C22⋊C414yes—
201#97—22no—
202#97—41no—
203C4×C8—22no—
204C4×C8—22no—
205C4×C8—22no—
206C4×C8—21no—
207C4×C8—21no—
208C4×C8—22no—
209C4×C8—21no—
210C4×C8—21no—
211C4×C8—41no—
212C4×C8—41no—
213C24#11711yescharacteristic
214C2×Q8—22no—
215C2×Q8—22no—
216C2×Q8—44no—
217C2×Q8—44no—
218C2×Q8—41no—
219C2×Q8—41no—
220C22×C4—22no—
221C22×C4—22no—
222C22×C4—22no—
223C22×C4—22no—
224C22×C4—42no—
225C22×C4—42no—
226C22×C4C2×C4211yescharacteristic
227C22×C4C2×C22⋊C411yescharacteristic
228C22×C4C4×D411yescharacteristic
229C22×C4C4×D411yescharacteristic
230C22×C4C4×D411yescharacteristic
231C22×C4C4×D411yescharacteristic
232C22×C4C4×D411yescharacteristic
233C22×C4C4×D411yescharacteristic
234C22×C4C4×D411yescharacteristic
235C22×C4C4×D411yescharacteristic
236C22×C4#12111yescharacteristic
237C22×C4#12111yescharacteristic
238C22×C4#12311yescharacteristic
239C22×C4#12311yescharacteristic
240C22×C4#12411yescharacteristic
241C22×C4#12411yescharacteristic
242C22×C4#12611yescharacteristic
243C22×C4#12611yescharacteristic
244M16—44no—
245M16—82no—
246M16—82no—
247M16—82no—
248C2×C8—22no—
249C2×C8—22no—
250C2×C8—24no—
251C2×C8—22no—
252C2×C8—22no—
253C2×C8—42no—
254C2×C8—42no—
255C2×C8—44no—
256C2×C8—42no—
257C2×C8—42no—
258C2×C8—42no—
259C2×C8—82no—
260C2×C8—82no—
261C4⋊C4—22no—
262C4⋊C4—22no—
263C4⋊C4—42no—
264C4⋊C4—41no—
265C4⋊C4—41no—
266C4⋊C4—41no—
267C4⋊C4—42no—
268C4⋊C4—41no—
269C4⋊C4—41no—
270C4⋊C4—41no—
271C4⋊C4—42no—
272C4⋊C4—42no—
273C22⋊C4—42no—
274C22⋊C4—42no—
275C42—42no—
276C42—42no—
277C42C2×C22⋊C411yescharacteristic
278C42C2×C22⋊C411yescharacteristic
279C42#11711yescharacteristic
280C42#11711yescharacteristic
281C42C4×D411yescharacteristic
282C42C4×D411yescharacteristic
283C42C4×D411yescharacteristic
284C42C4×D411yescharacteristic
285C42#12011yescharacteristic
286C42#12111yescharacteristic
287C42#12111yescharacteristic
288C42#12111yescharacteristic
289C42#12111yescharacteristic
290C42#12311yescharacteristic
291C42#12411yescharacteristic
292C42#12711yescharacteristic
293C23—22no—
294C23—41no—
295C23—41no—
296C23C4×C22⋊C411yescharacteristic
297C23#38811yescharacteristic
298C23#38811yescharacteristic
299Q8—44no—
300Q8—82no—
301Q8—82no—
302Q8—82no—
303C2×C4—22no—
304C2×C4—22no—
305C2×C4—22no—
306C2×C4—22no—
307C2×C4—22no—
308C2×C4—22no—
309C2×C4—21no—
310C2×C4—21no—
311C2×C4—21no—
312C2×C4—21no—
313C2×C4—22no—
314C2×C4—22no—
315C2×C4—22no—
316C2×C4—41no—
317C2×C4—41no—
318C2×C4—41no—
319C2×C4—41no—
320C2×C4—41no—
321C2×C4—41no—
322C2×C4—44no—
323C2×C4—42no—
324C2×C4—42no—
325C2×C4—82no—
326C2×C4C4×C22⋊C411yescharacteristic
327C2×C4C4×C22⋊C411yescharacteristic
328C2×C4#38611yescharacteristic
329C2×C4#38611yescharacteristic
330C2×C4#38611yescharacteristic
331C2×C4#38811yescharacteristic
332C2×C4#38811yescharacteristic
333C2×C4#38811yescharacteristic
334C2×C4#38811yescharacteristic
335C2×C4#39011yescharacteristic
336C2×C4#39011yescharacteristic
337C2×C4#39011yescharacteristic
338C2×C4#47112yes—
339C2×C4#47312yes—
340C8—44no—
341C8—42no—
342C8—42no—
343C8—42no—
344C8—42no—
345C8—82no—
346C8—82no—
347C8—82no—
348C22—21no—
349C22—21no—
350C22—21no—
351C22—21no—
352C22—21no—
353C22—21no—
354C22—21no—
355C22—21no—
356C22—82no—
357C22#143711yescharacteristic
358C22#193012yes—
359C4—21no—
360C4—21no—
361C4—21no—
362C4—21no—
363C4—21no—
364C4—21no—
365C4—21no—
366C4—21no—
367C4—82no—
368C4—82no—
369C4#175712yes—
370C4#188112yes—
371C4#190812yes—
372C4#197711yescharacteristic
373C4#197911yescharacteristic
374C2—22no—
375C2—41no—
376C2—41no—
377C2#1094211yescharacteristic
378C2#1095611yescharacteristic
379C2#1096111yescharacteristic
subgroupG/H
centerC2×C4#386
commutatorC22×C4C2×C42
Frattini, cosocleC22×C42C23
socleC22#1437
prime radicalC24#117
LCSG ▷ C22×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
#177082
#174642
#173852
#173262
#81192
#80762
#78802