Finite Group Databasebeta

← #110419

Subgroups — #110419

Order 512.

626 conjugacy classes of subgroups.

The subgroup lattice has 380 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21820C211yesmaximal, characteristic
2#20727C211yesmaximal, characteristic
3#17365C211yesmaximal, characteristic
4#17324C211yesmaximal, characteristic
5#8124C211yesmaximal, characteristic
6#8118C211yesmaximal, characteristic
7#7936C211yesmaximal, characteristic
8C22×C4×Q8C2211yescharacteristic
9#2920C2211yescharacteristic
10#2920C2211yescharacteristic
11#2920C2211yescharacteristic
12#2920C2211yescharacteristic
13C2×#432C2211yescharacteristic
14C2×#431C2211yescharacteristic
15#1985—22no—
16#1984—21no—
17#1984—21no—
18#1821C412yes—
19#1817C414yes—
20#1801—22no—
21#1784C412yes—
22C4×#101—22no—
23#1467—22no—
24#1463C414yes—
25#1461—22no—
26#1454—22no—
27#1453—22no—
28#1451—21no—
29C8×M16—21no—
30#1308—22no—
31#1307—22no—
32#1283—22no—
33C23×Q8C2×C411yescharacteristic
34C2×C4×Q8—21no—
35C2×C4×Q8—21no—
36C2×C4×Q8—21no—
37C2×C4×Q8—21no—
38C2×C4×Q8—21no—
39C2×C4×Q8—21no—
40C2×C4×Q8—21no—
41C2×C4×Q8—21no—
42C2×C4×Q8—21no—
43C2×C4×Q8—21no—
44C2×C4×Q8—21no—
45C2×C4×Q8—21no—
46C22×C4⋊C4C2×C411yescharacteristic
47C22×C4⋊C4C2×C412yes—
48C22×C42C2×C412yes—
49C22×C42C2311yescharacteristic
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#432—21no—
61#432D412yes—
62#431—21no—
63#431—21no—
64#431—21no—
65#431—21no—
66#431—21no—
67#431—21no—
68#431—21no—
69#431—21no—
70#431—21no—
71#431—21no—
72#431—21no—
73#431D412yes—
74#423—21no—
75#423—21no—
76#423—21no—
77#423D412yes—
78C2×C4⋊C8—21no—
79C2×C4⋊C8D412yes—
80C2×#101—24no—
81#407C2×C412yes—
82#407C2×C412yes—
83#407C2×C412yes—
84#407C2×C412yes—
85C2×C22⋊C8C2×C412yes—
86C2×C22⋊C8C2×C412yes—
87C4×M16—21no—
88C4×M16—21no—
89C4×M16—21no—
90C4×M16D412yes—
91C2×#97—21no—
92C2×C4×C8D411yescharacteristic
93C2×C4×C8D411yescharacteristic
94#343—24no—
95#336—24no—
96#336—24no—
97#324—28no—
98#322—22no—
99#322—24no—
100C82—22no—
101C22×Q8—21no—
102C22×Q8—21no—
103C22×Q8—21no—
104C22×Q8—21no—
105C22×Q8—21no—
106C22×Q8—21no—
107C22×Q8—44no—
108C23×C4C4212yes—
109C23×C4C4212yes—
110C23×C4C22×C411yescharacteristic
111C23×C4C22×C411yescharacteristic
112C23×C4C22×C411yescharacteristic
113C2×M16—22no—
114C2×M16—22no—
115C2×M16—22no—
116C2×M16C22⋊C414yes—
117C22×C8—22no—
118C22×C8C22⋊C412yes—
119C22×C8C22⋊C412yes—
120C4×Q8—41no—
121C4×Q8—41no—
122C4×Q8—41no—
123C4×Q8—41no—
124C4×Q8—41no—
125C4×Q8—41no—
126C4×Q8—41no—
127C4×Q8—41no—
128C4×Q8—41no—
129C4×Q8—41no—
130C4×Q8—41no—
131C4×Q8—41no—
132C4×Q8—41no—
133C4×Q8—41no—
134C4×Q8—41no—
135C4×Q8—41no—
136C2×C4⋊C4—22no—
137C2×C4⋊C4—21no—
138C2×C4⋊C4—22no—
139C2×C4⋊C4—21no—
140C2×C4⋊C4—22no—
141C2×C4⋊C4—22no—
142C2×C4⋊C4—21no—
143C2×C4⋊C4—21no—
144C2×C4⋊C4—22no—
145C2×C4⋊C4—22no—
146C2×C4⋊C4—21no—
147C2×C4⋊C4—21no—
148C2×C42—22no—
149C2×C42—22no—
150C2×C42—22no—
151C2×C42—22no—
152C2×C42—22no—
153C2×C42—22no—
154C2×C42C2×D411yescharacteristic
155C2×C42C2×D411yescharacteristic
156C2×C42C2×D411yescharacteristic
157C2×C42C2×D411yescharacteristic
158C2×C42C2×D411yescharacteristic
159C2×C42C2×D411yescharacteristic
160C2×C42#4111yescharacteristic
161C2×C42#4111yescharacteristic
162C2×C42#4111yescharacteristic
163C2×C42#4111yescharacteristic
164C2×C42#4111yescharacteristic
165C2×C42#4111yescharacteristic
166C4⋊C8—22no—
167C4⋊C8—22no—
168C4⋊C8—22no—
169C4⋊C8—22no—
170C4⋊C8—22no—
171C4⋊C8—22no—
172C4⋊C8—41no—
173C4⋊C8—41no—
174C4⋊C8—41no—
175C4⋊C8—41no—
176C4⋊C8—41no—
177C4⋊C8—41no—
178#101—48no—
179C22⋊C8—22no—
180C22⋊C8—22no—
181C22⋊C8—22no—
182C22⋊C8—22no—
183C22⋊C8—22no—
184C22⋊C8—22no—
185C22⋊C8—22no—
186C22⋊C8—22no—
187C22⋊C8—22no—
188C22⋊C8—22no—
189C22⋊C8—22no—
190C22⋊C8C22⋊C414yes—
191#97—22no—
192#97—22no—
193#97—41no—
194#97—41no—
195#97—41no—
196#97—41no—
197C4×C8—21no—
198C4×C8—22no—
199C4×C8—22no—
200C4×C8—21no—
201C4×C8—21no—
202C4×C8—21no—
203C4×C8—41no—
204C4×C8—41no—
205C24C2×C4211yescharacteristic
206C2×Q8—24no—
207C2×Q8—24no—
208C2×Q8—24no—
209C2×Q8—42no—
210C2×Q8—42no—
211C2×Q8—42no—
212C2×Q8—44no—
213C2×Q8—44no—
214C2×Q8—44no—
215C2×Q8—41no—
216C2×Q8—41no—
217C2×Q8—41no—
218C2×Q8—41no—
219C22×C4—22no—
220C22×C4—22no—
221C22×C4—22no—
222C22×C4—22no—
223C22×C4—22no—
224C22×C4—22no—
225C22×C4—44no—
226C22×C4—42no—
227C22×C4—42no—
228C22×C4—42no—
229C22×C4—44no—
230C22×C4C2×C22⋊C411yescharacteristic
231C22×C4C2×C22⋊C411yescharacteristic
232C22×C4C2×C22⋊C411yescharacteristic
233C22×C4#11711yescharacteristic
234C22×C4#11711yescharacteristic
235C22×C4#11711yescharacteristic
236C22×C4C4×D411yescharacteristic
237C22×C4C4×D411yescharacteristic
238C22×C4C4×D411yescharacteristic
239C22×C4C4×D411yescharacteristic
240C22×C4C4×D411yescharacteristic
241C22×C4C4×D411yescharacteristic
242C22×C4C4×D411yescharacteristic
243C22×C4C4×D411yescharacteristic
244C22×C4C4×D411yescharacteristic
245C22×C4C4×D411yescharacteristic
246C22×C4C4×D411yescharacteristic
247C22×C4C4×D411yescharacteristic
248M16—44no—
249M16—44no—
250M16—44no—
251M16—44no—
252C2×C8—24no—
253C2×C8—24no—
254C2×C8—22no—
255C2×C8—22no—
256C2×C8—42no—
257C2×C8—42no—
258C2×C8—44no—
259C2×C8—42no—
260C2×C8—42no—
261C2×C8—42no—
262C4⋊C4—42no—
263C4⋊C4—41no—
264C4⋊C4—42no—
265C4⋊C4—42no—
266C4⋊C4—41no—
267C4⋊C4—41no—
268C4⋊C4—42no—
269C4⋊C4—41no—
270C42—42no—
271C42—42no—
272C42—42no—
273C42—42no—
274C42#12011yescharacteristic
275C42#12111yescharacteristic
276C42#12111yescharacteristic
277C42#12111yescharacteristic
278C42#12111yescharacteristic
279C42#12111yescharacteristic
280C42#12111yescharacteristic
281C42#12311yescharacteristic
282C42#12311yescharacteristic
283C42#12311yescharacteristic
284C42#12411yescharacteristic
285C42#12411yescharacteristic
286C42#12411yescharacteristic
287C42#12611yescharacteristic
288C42#12611yescharacteristic
289C42#12711yescharacteristic
290C23—22no—
291C23—41no—
292C23—41no—
293C23C4×C22⋊C411yescharacteristic
294C23C4×C22⋊C411yescharacteristic
295C23C4×C22⋊C411yescharacteristic
296Q8—44no—
297Q8—44no—
298Q8—44no—
299Q8—44no—
300C2×C4—22no—
301C2×C4—22no—
302C2×C4—22no—
303C2×C4—22no—
304C2×C4—22no—
305C2×C4—22no—
306C2×C4—22no—
307C2×C4—22no—
308C2×C4—24no—
309C2×C4—24no—
310C2×C4—24no—
311C2×C4—41no—
312C2×C4—41no—
313C2×C4—41no—
314C2×C4—41no—
315C2×C4—41no—
316C2×C4—41no—
317C2×C4—41no—
318C2×C4—41no—
319C2×C4—41no—
320C2×C4—41no—
321C2×C4—42no—
322C2×C4—42no—
323C2×C4—42no—
324C2×C4—42no—
325C2×C4—42no—
326C2×C4—44no—
327C2×C4—82no—
328C2×C4—82no—
329C2×C4#38611yescharacteristic
330C2×C4#38611yescharacteristic
331C2×C4#38611yescharacteristic
332C2×C4#38811yescharacteristic
333C2×C4#38811yescharacteristic
334C2×C4#38811yescharacteristic
335C2×C4#38811yescharacteristic
336C2×C4#38811yescharacteristic
337C2×C4#38811yescharacteristic
338C2×C4#39011yescharacteristic
339C2×C4#39011yescharacteristic
340C2×C4#39011yescharacteristic
341C8—44no—
342C8—44no—
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—44no—
357C22#143711yescharacteristic
358C22#175712yes—
359C4—21no—
360C4—21no—
361C4—21no—
362C4—21no—
363C4—21no—
364C4—21no—
365C4—21no—
366C4—21no—
367C4—21no—
368C4—21no—
369C4—44no—
370C4—82no—
371C4#189112yes—
372C4#191512yes—
373C2—22no—
374C2—41no—
375C2—41no—
376C2#1093711yescharacteristic
377C2#1094011yescharacteristic
378C2#1096111yescharacteristic
subgroupG/H
centerC2×C4#386
commutator, prime radicalC24C2×C42
Frattini, cosocleC22×C42C23
socleC22#1437
LCSG ▷ C24 ▷ 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
#218202
#207272
#173652
#173242
#81242
#81182
#79362