Finite Group Databasebeta

← #219163

Subgroups — #219163

Order 512.

544 conjugacy classes of subgroups.

The subgroup lattice has 369 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21712C211yesmaximal, characteristic
2#21673C211yesmaximal, characteristic
3C4×#491C211yesmaximal, characteristic
4#9872C211yesmaximal, characteristic
5#9641C211yesmaximal, characteristic
6#9548C212yesmaximal
7#9547C212yesmaximal
8#8850C212yesmaximal
9#8848C212yesmaximal
10#8808C211yesmaximal, characteristic
11#8132C211yesmaximal, characteristic
12C2×#491C411yescharacteristic
13C2×#446C2212yes—
14C4×C4⋊Q8C2211yescharacteristic
15C4×#125C2211yescharacteristic
16#1992C411yescharacteristic
17#1984C2212yes—
18#1960C411yescharacteristic
19#1939—21no—
20#1939—21no—
21#1939C2211yescharacteristic
22#1939C2211yescharacteristic
23#1935C412yes—
24#1935C412yes—
25#1916—21no—
26#1916—21no—
27#1842C2211yescharacteristic
28#1836C411yescharacteristic
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#491—24no—
53#446—42no—
54#446—42no—
55C2×C4⋊C8—21no—
56C2×C4⋊C8—21no—
57C2×C4⋊C8—22no—
58C2×C4⋊C8—21no—
59C2×C4⋊C8—21no—
60C2×C4⋊C8C2311yescharacteristic
61C2×C4⋊C8C2312yes—
62C2×C4⋊C8C2311yescharacteristic
63C2×#103C2×C412yes—
64C2×#103C2×C412yes—
65C2×#103C2×C412yes—
66C2×#103C2×C412yes—
67C2×#97—22no—
68C2×#97—21no—
69C2×#97—21no—
70C2×#97C2×C411yescharacteristic
71C2×#97C2×C412yes—
72C2×#97C2×C411yescharacteristic
73C2×C4×C8D411yescharacteristic
74C2×C4×C8D411yescharacteristic
75C2×C4×C8D411yescharacteristic
76C2×C4×C8D411yescharacteristic
77C2×C4×C8C2312yes—
78C2×C4×C8C2312yes—
79C2×C4×C8C2311yescharacteristic
80C2×C4×C8C2311yescharacteristic
81#391C2×C412yes—
82#389—21no—
83#389—21no—
84#389C2×C411yescharacteristic
85#389C2×C411yescharacteristic
86#387—21no—
87#387—21no—
88#384C2×C411yescharacteristic
89#382C2×C411yescharacteristic
90C4×C4⋊C4—21no—
91C4×C4⋊C4—21no—
92C4×C4⋊C4—21no—
93C4×C4⋊C4—21no—
94C4×C4⋊C4C2311yescharacteristic
95C4×C4⋊C4C2311yescharacteristic
96C43C2311yescharacteristic
97#340C2×C412yes—
98#340C2×C412yes—
99#340C2×C412yes—
100#340C2×C412yes—
101#336—21no—
102#336—21no—
103#336—21no—
104#336—21no—
105#336—22no—
106#336—22no—
107#336—21no—
108#336—21no—
109#336—21no—
110#336—21no—
111#335—42no—
112#334—42no—
113Q8⋊C8—42no—
114Q8⋊C8—42no—
115C22×Q8C22×C412yes—
116C22×C8—21no—
117C22×C8—21no—
118C22×C8—22no—
119C22×C8—22no—
120C22×C8C22×C412yes—
121C22×C8C22×C412yes—
122C22×C8C2×D411yescharacteristic
123C22×C8C2×D411yescharacteristic
124C22×C8C2×D411yescharacteristic
125C22×C8C2×D411yescharacteristic
126C4⋊Q8—22no—
127C4⋊Q8—22no—
128#125—22no—
129#125—22no—
130C4×Q8—22no—
131C4×Q8—22no—
132C4×Q8—22no—
133C4×Q8—22no—
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—21no—
141C2×C4⋊C4—21no—
142C2×C4⋊C4—22no—
143C2×C4⋊C4C22×C412yes—
144C2×C4⋊C4C22×C411yescharacteristic
145C2×C4⋊C4C22×C411yescharacteristic
146C2×C4⋊C4C22×C411yescharacteristic
147C2×C4⋊C4C22×C411yescharacteristic
148C2×C42—22no—
149C2×C42—21no—
150C2×C42—21no—
151C2×C42C22×C411yescharacteristic
152C2×C42C22×C411yescharacteristic
153C2×C42C2×D411yescharacteristic
154C2×C42C2×D411yescharacteristic
155C2×C42#4111yescharacteristic
156C2×C42#4111yescharacteristic
157C2×C42C2411yescharacteristic
158C4⋊C8—22no—
159C4⋊C8—22no—
160C4⋊C8—22no—
161C4⋊C8—22no—
162C4⋊C8—42no—
163C4⋊C8—42no—
164C4⋊C8—42no—
165C4⋊C8—44no—
166C4⋊C8—42no—
167#103—24no—
168#103—24no—
169#103—42no—
170#103—42no—
171#97—22no—
172#97—42no—
173#97—42no—
174#97—44no—
175#97—42no—
176#97M1614yes—
177C4×C8—22no—
178C4×C8—22no—
179C4×C8—22no—
180C4×C8—22no—
181C4×C8—22no—
182C4×C8—22no—
183C4×C8—22no—
184C4×C8—22no—
185C4×C8—41no—
186C4×C8—41no—
187#95—21no—
188#95—21no—
189#95—21no—
190#95—21no—
191#95—22no—
192#95—22no—
193#95—21no—
194#95—21no—
195C2×Q8—22no—
196C2×Q8—22no—
197C2×Q8—22no—
198C2×Q8—22no—
199C2×Q8—22no—
200C2×Q8—22no—
201C22×C4—21no—
202C22×C4—21no—
203C22×C4—22no—
204C22×C4—21no—
205C22×C4—21no—
206C22×C4—22no—
207C22×C4C4×D411yescharacteristic
208C22×C4C4×D411yescharacteristic
209C22×C4C4×D411yescharacteristic
210C22×C4C4×D411yescharacteristic
211C22×C4C23×C411yescharacteristic
212C22×C4C22×D411yescharacteristic
213C22×C4C2×#4111yescharacteristic
214C2×C8—22no—
215C2×C8—21no—
216C2×C8—21no—
217C2×C8—22no—
218C2×C8—21no—
219C2×C8—21no—
220C2×C8—22no—
221C2×C8—21no—
222C2×C8—21no—
223C2×C8—22no—
224C2×C8—22no—
225C2×C8—21no—
226C2×C8—21no—
227C2×C8—21no—
228C2×C8—21no—
229C2×C8—42no—
230C2×C8—42no—
231C2×C8—42no—
232C2×C8—41no—
233C2×C8—41no—
234C2×C8—42no—
235C2×C8—41no—
236C2×C8—41no—
237C2×C8—41no—
238C2×C8—42no—
239C2×C8—42no—
240C2×C8—41no—
241C2×C8C2×M1614yes—
242C2×C8#13712yes—
243C2×C8#13712yes—
244C4⋊C4—22no—
245C4⋊C4—22no—
246C4⋊C4—21no—
247C4⋊C4—21no—
248C4⋊C4—21no—
249C4⋊C4—21no—
250C4⋊C4—22no—
251C4⋊C4—22no—
252C4⋊C4—42no—
253C4⋊C4—42no—
254C4⋊C4—42no—
255C4⋊C4—42no—
256C4⋊C4—42no—
257C4⋊C4—42no—
258C4⋊C4—42no—
259C4⋊C4—42no—
260C4⋊C4—42no—
261C4⋊C4—42no—
262C42—22no—
263C42—21no—
264C42—21no—
265C42—21no—
266C42—21no—
267C42—41no—
268C42—41no—
269C42—42no—
270C42—42no—
271C42—41no—
272C42—41no—
273C42—41no—
274C42—41no—
275C42C2×M1612yes—
276C42#13112yes—
277C42H(5,𝔽2)12yes—
278C422−1+412yes—
279C23C2×C4×D411yescharacteristic
280Q8—42no—
281Q8—42no—
282Q8—42no—
283Q8—42no—
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—21no—
294C2×C4—21no—
295C2×C4—21no—
296C2×C4—42no—
297C2×C4—42no—
298C2×C4—41no—
299C2×C4—41no—
300C2×C4—42no—
301C2×C4—41no—
302C2×C4—41no—
303C2×C4—42no—
304C2×C4—41no—
305C2×C4—41no—
306C2×C4—41no—
307C2×C4—41no—
308C2×C4—41no—
309C2×C4—41no—
310C2×C4—42no—
311C2×C4—42no—
312C2×C4—41no—
313C2×C4—41no—
314C2×C4#43612yes—
315C2×C4#43612yes—
316C2×C4#52012yes—
317C2×C4#53511yescharacteristic
318C2×C4#53611yescharacteristic
319C2×C4#54212yes—
320C2×C4C22×M1611yescharacteristic
321C2×C4C2×#13111yescharacteristic
322C2×C4#56812yes—
323C2×C4C2×#13711yescharacteristic
324C2×C4C2×#13711yescharacteristic
325C2×C4#57511yescharacteristic
326C2×C4#57511yescharacteristic
327C8—22no—
328C8—22no—
329C8—24no—
330C8—41no—
331C8—41no—
332C8—42no—
333C8—82no—
334C8—81no—
335C8—82no—
336C8—81no—
337C22#232611yescharacteristic
338C22C2×#43611yescharacteristic
339C22#293211yescharacteristic
340C22C4×#13711yescharacteristic
341C22C4×#13711yescharacteristic
342C22#295511yescharacteristic
343C22#295511yescharacteristic
344C4—22no—
345C4—22no—
346C4—22no—
347C4—41no—
348C4—41no—
349C4—81no—
350C4—81no—
351C4—82no—
352C4—82no—
353C4—81no—
354C4—81no—
355C4#298912yes—
356C4#299312yes—
357C4#321811yescharacteristic
358C4#321911yescharacteristic
359C4#323611yescharacteristic
360C4#323711yescharacteristic
361C2#1789211yescharacteristic
362C2#1789211yescharacteristic
363C2#1798411yescharacteristic
364C2#1798611yescharacteristic
365C2#2221411yescharacteristic
366C2#2222511yescharacteristic
367C2#2223511yescharacteristic
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
#217122
#216732
C4×#4912
#98722
#96412
#95482
#95472
#88502
#88482
#88082
#81322