Finite Group Databasebeta

← #219172

Subgroups — #219172

Order 512.

530 conjugacy classes of subgroups.

The subgroup lattice has 366 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23820C211yesmaximal, characteristic
2#21712C211yesmaximal, characteristic
3#21673C211yesmaximal, characteristic
4#9876C211yesmaximal, characteristic
5#9647C211yesmaximal, characteristic
6#9548C212yesmaximal
7#9547C212yesmaximal
8#8850C212yesmaximal
9#8848C212yesmaximal
10#8812C211yesmaximal, characteristic
11#8132C211yesmaximal, characteristic
12C2×#446C2212yes—
13C4×C4⋊Q8C2211yescharacteristic
14C4×#125C2211yescharacteristic
15#2099C411yescharacteristic
16#2099C411yescharacteristic
17#2074C412yes—
18#2043C411yescharacteristic
19#2043C411yescharacteristic
20#2026C412yes—
21#1984C2212yes—
22#1939—21no—
23#1939—21no—
24#1939C2211yescharacteristic
25#1939C2211yescharacteristic
26#1916—21no—
27#1916—21no—
28#1842C2211yescharacteristic
29#1831C2211yescharacteristic
30#1771—21no—
31#1771—21no—
32#1770—21no—
33#1770—21no—
34#1770C2211yescharacteristic
35#1770C2211yescharacteristic
36C4×#103C2212yes—
37C4×#103C2212yes—
38#1744C2211yescharacteristic
39C4×#97C2211yescharacteristic
40C42×C8C2211yescharacteristic
41C2×#335C2212yes—
42C2×#334C2212yes—
43C2×Q8⋊C8C2212yes—
44C2×Q8⋊C8C2212yes—
45#1279C2212yes—
46#1279C2212yes—
47#1273C2212yes—
48#1273C2212yes—
49C2×C4⋊Q8C2×C411yescharacteristic
50C2×#125C2×C411yescharacteristic
51C2×C4×Q8C2312yes—
52#446—42no—
53#446—42no—
54C2×C4⋊C8—21no—
55C2×C4⋊C8—21no—
56C2×C4⋊C8—22no—
57C2×C4⋊C8—21no—
58C2×C4⋊C8—21no—
59C2×C4⋊C8C2311yescharacteristic
60C2×C4⋊C8C2312yes—
61C2×C4⋊C8C2311yescharacteristic
62C2×#103C2×C412yes—
63C2×#103C2×C412yes—
64C2×#103C2×C412yes—
65C2×#103C2×C412yes—
66C2×#97—21no—
67C2×#97—22no—
68C2×#97—21no—
69C2×#97—21no—
70C2×#97—21no—
71C2×C4×C8D411yescharacteristic
72C2×C4×C8D411yescharacteristic
73C2×C4×C8D411yescharacteristic
74C2×C4×C8D411yescharacteristic
75C2×C4×C8C2312yes—
76C2×C4×C8C2312yes—
77C2×C4×C8C2311yescharacteristic
78C2×C4×C8C2311yescharacteristic
79#391C2×C412yes—
80#389—21no—
81#389—21no—
82#389C2×C411yescharacteristic
83#389C2×C411yescharacteristic
84#387—21no—
85#387—21no—
86#384C2×C411yescharacteristic
87#382C2×C411yescharacteristic
88C4×C4⋊C4—21no—
89C4×C4⋊C4—21no—
90C4×C4⋊C4—21no—
91C4×C4⋊C4—21no—
92C4×C4⋊C4C2311yescharacteristic
93C4×C4⋊C4C2311yescharacteristic
94C43C2311yescharacteristic
95#340C2×C412yes—
96#340C2×C412yes—
97#340C2×C412yes—
98#340C2×C412yes—
99#336—21no—
100#336—21no—
101#336—21no—
102#336—21no—
103#336—22no—
104#336—22no—
105#336—21no—
106#336—21no—
107#336C2×C412yes—
108#336C2×C411yescharacteristic
109#336C2×C411yescharacteristic
110#335—42no—
111#334—42no—
112Q8⋊C8—42no—
113Q8⋊C8—42no—
114C22×Q8C22×C412yes—
115C22×C8—22no—
116C22×C8—21no—
117C22×C8—21no—
118C22×C8—22no—
119C22×C8C22×C412yes—
120C22×C8C22×C412yes—
121C22×C8C2×D411yescharacteristic
122C22×C8C2×D411yescharacteristic
123C22×C8C2×D411yescharacteristic
124C22×C8C2×D411yescharacteristic
125C4⋊Q8—42no—
126#125—41no—
127#125—41no—
128C4×Q8—22no—
129C4×Q8—22no—
130C4×Q8—22no—
131C4×Q8—22no—
132C2×C4⋊C4—21no—
133C2×C4⋊C4—21no—
134C2×C4⋊C4—21no—
135C2×C4⋊C4—21no—
136C2×C4⋊C4—21no—
137C2×C4⋊C4—21no—
138C2×C4⋊C4—21no—
139C2×C4⋊C4—21no—
140C2×C4⋊C4—22no—
141C2×C4⋊C4C22×C412yes—
142C2×C4⋊C4C22×C411yescharacteristic
143C2×C4⋊C4C22×C411yescharacteristic
144C2×C4⋊C4C22×C411yescharacteristic
145C2×C4⋊C4C22×C411yescharacteristic
146C2×C42—22no—
147C2×C42—21no—
148C2×C42—21no—
149C2×C42C22×C411yescharacteristic
150C2×C42C22×C411yescharacteristic
151C2×C42C2×D411yescharacteristic
152C2×C42C2×D411yescharacteristic
153C2×C42#4111yescharacteristic
154C2×C42#4111yescharacteristic
155C2×C42C2411yescharacteristic
156C4⋊C8—22no—
157C4⋊C8—22no—
158C4⋊C8—22no—
159C4⋊C8—22no—
160C4⋊C8—42no—
161C4⋊C8—42no—
162C4⋊C8—42no—
163C4⋊C8—44no—
164C4⋊C8—42no—
165#103—42no—
166#103—42no—
167#103—42no—
168#103—42no—
169#97—24no—
170#97—42no—
171#97—42no—
172#97—44no—
173#97—42no—
174C4×C8—22no—
175C4×C8—22no—
176C4×C8—22no—
177C4×C8—22no—
178C4×C8—22no—
179C4×C8—22no—
180C4×C8—22no—
181C4×C8—22no—
182C4×C8—41no—
183C4×C8—41no—
184#95—21no—
185#95—21no—
186#95—21no—
187#95—21no—
188#95—22no—
189#95—22no—
190#95—21no—
191#95—21no—
192C2×Q8—22no—
193C2×Q8—22no—
194C2×Q8—22no—
195C2×Q8—22no—
196C2×Q8—22no—
197C2×Q8—22no—
198C22×C4—21no—
199C22×C4—21no—
200C22×C4—22no—
201C22×C4—21no—
202C22×C4—21no—
203C22×C4—22no—
204C22×C4C4×D411yescharacteristic
205C22×C4C4×D411yescharacteristic
206C22×C4C4×D411yescharacteristic
207C22×C4C4×D411yescharacteristic
208C22×C4C23×C411yescharacteristic
209C22×C4C22×D411yescharacteristic
210C22×C4C2×#4111yescharacteristic
211C2×C8—21no—
212C2×C8—21no—
213C2×C8—21no—
214C2×C8—21no—
215C2×C8—24no—
216C2×C8—22no—
217C2×C8—22no—
218C2×C8—21no—
219C2×C8—21no—
220C2×C8—22no—
221C2×C8—22no—
222C2×C8—22no—
223C2×C8—22no—
224C2×C8—21no—
225C2×C8—21no—
226C2×C8—21no—
227C2×C8—21no—
228C2×C8—42no—
229C2×C8—42no—
230C2×C8—42no—
231C2×C8—41no—
232C2×C8—41no—
233C2×C8—41no—
234C2×C8—41no—
235C2×C8—41no—
236C2×C8—42no—
237C2×C8—42no—
238C2×C8—41no—
239C2×C8#13712yes—
240C2×C8#13712yes—
241C4⋊C4—22no—
242C4⋊C4—22no—
243C4⋊C4—21no—
244C4⋊C4—21no—
245C4⋊C4—22no—
246C4⋊C4—22no—
247C4⋊C4—21no—
248C4⋊C4—21no—
249C4⋊C4—42no—
250C4⋊C4—42no—
251C4⋊C4—42no—
252C4⋊C4—42no—
253C4⋊C4—42no—
254C4⋊C4—42no—
255C4⋊C4—42no—
256C4⋊C4—42no—
257C4⋊C4—42no—
258C4⋊C4—42no—
259C42—21no—
260C42—21no—
261C42—22no—
262C42—21no—
263C42—21no—
264C42—21no—
265C42—21no—
266C42—22no—
267C42—22no—
268C42—42no—
269C42—42no—
270C42—41no—
271C42—41no—
272C42—41no—
273C42—41no—
274C42H(5,𝔽2)12yes—
275C422−1+412yes—
276C23C2×C4×D411yescharacteristic
277Q8—42no—
278Q8—42no—
279Q8—42no—
280Q8—42no—
281C2×C4—21no—
282C2×C4—21no—
283C2×C4—21no—
284C2×C4—21no—
285C2×C4—21no—
286C2×C4—21no—
287C2×C4—21no—
288C2×C4—21no—
289C2×C4—21no—
290C2×C4—21no—
291C2×C4—21no—
292C2×C4—21no—
293C2×C4—42no—
294C2×C4—42no—
295C2×C4—41no—
296C2×C4—41no—
297C2×C4—41no—
298C2×C4—41no—
299C2×C4—42no—
300C2×C4—42no—
301C2×C4—41no—
302C2×C4—41no—
303C2×C4—41no—
304C2×C4—41no—
305C2×C4—41no—
306C2×C4—41no—
307C2×C4—42no—
308C2×C4—41no—
309C2×C4—41no—
310C2×C4—42no—
311C2×C4#44312yes—
312C2×C4#44412yes—
313C2×C4#52012yes—
314C2×C4#53511yescharacteristic
315C2×C4#53611yescharacteristic
316C2×C4#54212yes—
317C2×C4#56812yes—
318C2×C4#56811yescharacteristic
319C2×C4#56811yescharacteristic
320C2×C4C2×#13711yescharacteristic
321C2×C4C2×#13711yescharacteristic
322C2×C4#57511yescharacteristic
323C2×C4#57511yescharacteristic
324C8—22no—
325C8—22no—
326C8—44no—
327C8—41no—
328C8—41no—
329C8—42no—
330C8—42no—
331C8—81no—
332C8—82no—
333C8—81no—
334C22#232611yescharacteristic
335C22#293011yescharacteristic
336C22#293011yescharacteristic
337C22C4×#13711yescharacteristic
338C22C4×#13711yescharacteristic
339C22C2×#44311yescharacteristic
340C22C2×#44411yescharacteristic
341C4—22no—
342C4—22no—
343C4—22no—
344C4—41no—
345C4—41no—
346C4—81no—
347C4—81no—
348C4—81no—
349C4—81no—
350C4—82no—
351C4—82no—
352C4#299512yes—
353C4#299512yes—
354C4#321811yescharacteristic
355C4#321911yescharacteristic
356C4#323611yescharacteristic
357C4#323711yescharacteristic
358C2#1793911yescharacteristic
359C2#1794111yescharacteristic
360C2#1794311yescharacteristic
361C2#1794411yescharacteristic
362C2#2221511yescharacteristic
363C2#2222511yescharacteristic
364C2#2223411yescharacteristic
subgroupG/H
centerC22×C4C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
socle, prime radicalC23C2×C4×D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C4 ◁ C42×C8 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Maximal subgroups

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

subgroupindex
#238202
#217122
#216732
#98762
#96472
#95482
#95472
#88502
#88482
#88122
#81322