Finite Group Databasebeta

← #99481

Subgroups — #99481

Order 512.

390 conjugacy classes of subgroups; only the 94 normal subgroups are shown.

normal subgroups only

This group has 390 conjugacy classes of subgroups — too many to draw. Only its 94 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22289C211yesmaximal, characteristic
2#11269C211yesmaximal, characteristic
3#8942C211yesmaximal, characteristic
4#8832C211yesmaximal, characteristic
5#7566C211yesmaximal, characteristic
6#7029C211yesmaximal, characteristic
7#7026C211yesmaximal, characteristic
8C2×#423C2211yescharacteristic
9C4×C22⋊Q8C2211yescharacteristic
10#1984—21no—
11#1940—21no—
12#1917—21no—
13#1916—21no—
14#1851—21no—
15#1797C2211yescharacteristic
16#1753—21no—
17C4×C22⋊C8C2211yescharacteristic
18C4×C22⋊C8C2211yescharacteristic
19C2×#341—21no—
20C2×#341—21no—
21#1728C2211yescharacteristic
22C2×#336C2211yescharacteristic
23C2×#335—21no—
24C2×Q8⋊C8—21no—
25#1279—21no—
26#1277—21no—
27#1272C411yescharacteristic
28#1272C411yescharacteristic
29#1271C411yescharacteristic
30#1271C411yescharacteristic
31C22×M16—21no—
32C22×M16D411yescharacteristic
33C22×M16D411yescharacteristic
34C23×C8—21no—
35C2×C22⋊Q8Q811yescharacteristic
36C2×C4×Q8D411yescharacteristic
37C22×C42C2311yescharacteristic
38#423—42no—
39C2×C4⋊C8—21no—
40C2×C4⋊C8—21no—
41C2×C4⋊C8—21no—
42C2×C4⋊C8—21no—
43C2×C4⋊C8D411yescharacteristic
44C2×C4⋊C8D411yescharacteristic
45C2×C22⋊C8—21no—
46C2×C22⋊C8—21no—
47C2×C22⋊C8—21no—
48C2×#97—21no—
49C2×C4×C8—21no—
50C2×C4×C8—21no—
51C2×C4×C8—21no—
52#391C2×C411yescharacteristic
53#389C2×C411yescharacteristic
54#387—21no—
55#385—21no—
56#380Q811yescharacteristic
57C4×C4⋊C4—21no—
58C4×C4⋊C4D411yescharacteristic
59C4×C22⋊C4—21no—
60#341—42no—
61#341—42no—
62#336—21no—
63#336—21no—
64#336—21no—
65#336—21no—
66#336—21no—
67#336—21no—
68#336—21no—
69#336—21no—
70#336—21no—
71#336C2×C411yescharacteristic
72#336C2×C411yescharacteristic
73#336C2×C411yescharacteristic
74#336C2×C411yescharacteristic
75#335—42no—
76Q8⋊C8—42no—
77C22×Q8C4⋊C411yescharacteristic
78C23×C4C2×D411yescharacteristic
79C23×C4C2×Q811yescharacteristic
80C23×C4#4111yescharacteristic
81C2×M16—21no—
82C2×M16—22no—
83C2×M16—21no—
84C2×M16—21no—
85C2×M16—41no—
86C2×M16—41no—
87C2×M16—41no—
88C2×M16—41no—
89C2×M16—41no—
90C2×M16—41no—
91C2×M16—41no—
92C2×M16—41no—
93C2×M16—41no—
94C2×M16Q1612yes—
95C22×C8—21no—
96C22×C8—21no—
97C22×C8—21no—
98C22×C8—21no—
99C22×C8—21no—
100C22×C8—21no—
101C22×C8—21no—
102C22×C8—21no—
103C22×C8—41no—
104C22×C8—41no—
105C22×C8—41no—
106C22×C8—41no—
107C22×C8—41no—
108C22×C8—41no—
109C22×C8—41no—
110C22×C8—41no—
111C22⋊Q8—22no—
112C22⋊Q8—41no—
113C4×Q8—41no—
114C4×Q8—41no—
115C2×C4⋊C4—21no—
116C2×C4⋊C4—21no—
117C2×C4⋊C4—21no—
118C2×C4⋊C4C4⋊C411yescharacteristic
119C2×C4⋊C4C4⋊C411yescharacteristic
120C2×C4⋊C4C4⋊C411yescharacteristic
121C2×C22⋊C4—21no—
122C2×C22⋊C4—21no—
123C2×C42—21no—
124C2×C42—21no—
125C2×C42—41no—
126C2×C42—41no—
127C2×C42C22×C411yescharacteristic
128C2×C42C2×D411yescharacteristic
129C2×C42C2×D411yescharacteristic
130C2×C42#4111yescharacteristic
131C4⋊C8—22no—
132C4⋊C8—22no—
133C4⋊C8—41no—
134C4⋊C8—41no—
135C4⋊C8—41no—
136C4⋊C8—41no—
137C4⋊C8—81no—
138C4⋊C8—81no—
139C22⋊C8—22no—
140C22⋊C8—42no—
141C22⋊C8—41no—
142C22⋊C8—41no—
143C22⋊C8—41no—
144#97—81no—
145C4×C8—41no—
146C4×C8—41no—
147C4×C8—41no—
148C4×C8—41no—
149C4×C8—81no—
150#95—21no—
151#95—21no—
152#95—21no—
153#95—21no—
154#95—21no—
155C24C22⋊Q811yescharacteristic
156C2×Q8—21no—
157C2×Q8—41no—
158C2×Q8—41no—
159C2×Q8#10812yes—
160C22×C4—21no—
161C22×C4—21no—
162C22×C4—21no—
163C22×C4—21no—
164C22×C4—21no—
165C22×C4—21no—
166C22×C4—21no—
167C22×C4—21no—
168C22×C4—21no—
169C22×C4—21no—
170C22×C4—21no—
171C22×C4—21no—
172C22×C4—41no—
173C22×C4—41no—
174C22×C4—41no—
175C22×C4—41no—
176C22×C4C2×C4⋊C411yescharacteristic
177C22×C4C4×D411yescharacteristic
178C22×C4C4×D411yescharacteristic
179C22×C4#12011yescharacteristic
180C22×C4C22⋊Q811yescharacteristic
181C22×C4#12311yescharacteristic
182C22×C4C2×Q1611yescharacteristic
183C22×C4#13511yescharacteristic
184C22×C4AGL(1,ℤ/8)11yescharacteristic
185C22×C4#13711yescharacteristic
186M16—42no—
187M16—42no—
188M16—42no—
189M16—42no—
190M16—81no—
191M16—81no—
192M16—81no—
193M16—81no—
194C2×C8—22no—
195C2×C8—22no—
196C2×C8—41no—
197C2×C8—41no—
198C2×C8—41no—
199C2×C8—41no—
200C2×C8—41no—
201C2×C8—42no—
202C2×C8—41no—
203C2×C8—41no—
204C2×C8—41no—
205C2×C8—41no—
206C2×C8—41no—
207C2×C8—41no—
208C2×C8—41no—
209C2×C8—41no—
210C2×C8—41no—
211C2×C8—41no—
212C2×C8—41no—
213C2×C8—41no—
214C2×C8—41no—
215C2×C8—41no—
216C2×C8—41no—
217C2×C8—42no—
218C2×C8—41no—
219C2×C8—41no—
220C2×C8—81no—
221C2×C8—81no—
222C2×C8—81no—
223C2×C8—81no—
224C2×C8—81no—
225C2×C8—81no—
226C2×C8—81no—
227C2×C8—81no—
228C4⋊C4—21no—
229C4⋊C4—21no—
230C4⋊C4—42no—
231C4⋊C4—41no—
232C4⋊C4—41no—
233C4⋊C4—41no—
234C4⋊C4—41no—
235C4⋊C4—81no—
236C22⋊C4—42no—
237C22⋊C4—81no—
238C42—21no—
239C42—21no—
240C42—21no—
241C42—21no—
242C42—41no—
243C42—41no—
244C42—41no—
245C42—41no—
246C42—41no—
247C42—41no—
248C23—21no—
249C23—21no—
250C23—21no—
251C23—21no—
252C23—21no—
253C23—21no—
254C23#38511yescharacteristic
255C23#47711yescharacteristic
256C23#47911yescharacteristic
257Q8—42no—
258Q8—81no—
259C2×C4—21no—
260C2×C4—21no—
261C2×C4—21no—
262C2×C4—21no—
263C2×C4—21no—
264C2×C4—21no—
265C2×C4—21no—
266C2×C4—21no—
267C2×C4—21no—
268C2×C4—21no—
269C2×C4—21no—
270C2×C4—21no—
271C2×C4—21no—
272C2×C4—21no—
273C2×C4—22no—
274C2×C4—21no—
275C2×C4—21no—
276C2×C4—21no—
277C2×C4—21no—
278C2×C4—21no—
279C2×C4—21no—
280C2×C4—41no—
281C2×C4—41no—
282C2×C4—41no—
283C2×C4—41no—
284C2×C4—41no—
285C2×C4—41no—
286C2×C4—41no—
287C2×C4—41no—
288C2×C4—41no—
289C2×C4—41no—
290C2×C4—41no—
291C2×C4—81no—
292C2×C4—81no—
293C2×C4—81no—
294C2×C4—81no—
295C2×C4C2×#10811yescharacteristic
296C2×C4#43011yescharacteristic
297C2×C4C4×Q1611yescharacteristic
298C2×C4#44011yescharacteristic
299C2×C4#44311yescharacteristic
300C2×C4#44411yescharacteristic
301C2×C4#44911yescharacteristic
302C2×C4#45111yescharacteristic
303C2×C4#45411yescharacteristic
304C2×C4#45511yescharacteristic
305C2×C4B(4,𝔽2)11yescharacteristic
306C2×C4#45811yescharacteristic
307C8—42no—
308C8—42no—
309C8—81no—
310C8—81no—
311C8—81no—
312C8—82no—
313C8—81no—
314C8—82no—
315C8—81no—
316C8—81no—
317C22—21no—
318C22—21no—
319C22—21no—
320C22—21no—
321C22—21no—
322C22—21no—
323C22—41no—
324C22—41no—
325C22—41no—
326C22—41no—
327C22#185511yescharacteristic
328C22#185811yescharacteristic
329C22#186011yescharacteristic
330C22#186111yescharacteristic
331C22#186311yescharacteristic
332C22#186611yescharacteristic
333C22#186711yescharacteristic
334C4—21no—
335C4—21no—
336C4—21no—
337C4—21no—
338C4—21no—
339C4—21no—
340C4—41no—
341C4—42no—
342C4—41no—
343C4—81no—
344C4—81no—
345C4—81no—
346C4#159711yescharacteristic
347C4#160211yescharacteristic
348C4#161211yescharacteristic
349C4#161711yescharacteristic
350C2—41no—
351C2—41no—
352C2#885911yescharacteristic
353C2#886711yescharacteristic
354C2#887411yescharacteristic
355C2#890111yescharacteristic
356C2#890711yescharacteristic
357C2#891711yescharacteristic
358C2#893311yescharacteristic
subgroupG/H
centerC22×C4#120
commutatorC2×C42C22×C4
Frattini, cosocleC22×C42C23
socleC23#385
prime radicalC24C22⋊Q8
LCSG ▷ C2×C42 ▷ C23 ▷ C1
UCSC1 ◁ C22×C4 ◁ C22×C42 ◁ G
LECSG ▷ C22×C42 ▷ C23 ▷ C1

Maximal subgroups

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

subgroupindex
#222892
#112692
#89422
#88322
#75662
#70292
#70262