Finite Group Databasebeta

← #99409

Subgroups — #99409

Order 512.

427 conjugacy classes of subgroups; only the 98 normal subgroups are shown.

normal subgroups only

This group has 427 conjugacy classes of subgroups — too many to draw. Only its 98 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#21957C211yesmaximal, characteristic
2#11209C211yesmaximal, characteristic
3#8832C211yesmaximal, characteristic
4#8832C211yesmaximal, characteristic
5#7564C211yesmaximal, characteristic
6#7028C211yesmaximal, characteristic
7#7028C211yesmaximal, characteristic
8C2×#433—21no—
9C2×#423C2211yescharacteristic
10#2296C2211yescharacteristic
11#1940—21no—
12#1940—21no—
13#1939—21no—
14#1939—21no—
15#1798—21no—
16#1797C2211yescharacteristic
17#1797C2211yescharacteristic
18C4×C22⋊C8C2211yescharacteristic
19C4×C22⋊C8C2211yescharacteristic
20C2×#339—21no—
21C2×#339—21no—
22C2×#335—21no—
23C2×#335—21no—
24C4×#95C2211yescharacteristic
25#1277—21no—
26#1277—21no—
27#1271C411yescharacteristic, complemented
28#1271C411yescharacteristic, complemented
29#1271C411yescharacteristic, complemented
30#1271C411yescharacteristic, complemented
31C22×M16—21no—
32C2×#117D411yescharacteristic
33C2×#117D411yescharacteristic
34C22×C42—21no—
35C22×C42C2311yescharacteristic
36#433—42no—
37#433—42no—
38#423—42no—
39C2×C4⋊C8—21no—
40C2×C4⋊C8—21no—
41C2×C4⋊C8—21no—
42C2×C4⋊C8—21no—
43C2×C4⋊C8—21no—
44C2×C4⋊C8—21no—
45C2×C4⋊C8D411yescharacteristic
46C2×C4⋊C8D411yescharacteristic
47C2×C4⋊C8D411yescharacteristic
48C2×C4⋊C8D411yescharacteristic
49C2×C22⋊C8—21no—
50C2×C22⋊C8—21no—
51C2×C22⋊C8—21no—
52C2×C22⋊C8—21no—
53C2×#97—21no—
54C2×C4×C8—21no—
55C2×C4×C8—21no—
56C2×C4×C8—21no—
57#389C2×C411yescharacteristic
58#389C2×C411yescharacteristic
59#389Q811yescharacteristic
60#389Q811yescharacteristic
61#385—21no—
62#385—21no—
63C4×C4⋊C4—21no—
64C4×C4⋊C4—21no—
65C2×#95—21no—
66C2×#95—21no—
67#339—42no—
68#339—42no—
69#336—21no—
70#336—21no—
71#336—21no—
72#336—21no—
73#336C2×C411yescharacteristic
74#336C2×C411yescharacteristic
75#336C2×C411yescharacteristic
76#336C2×C411yescharacteristic
77#335—42no—
78#335—42no—
79C23×C4—41no—
80C23×C4C2×D411yescharacteristic
81C23×C4#4111yescharacteristic
82C23×C4#4111yescharacteristic
83C2×M16—22no—
84C2×M16—41no—
85C2×M16—41no—
86C2×M16—41no—
87C2×M16—41no—
88C2×M16—41no—
89C22×C8—21no—
90C22×C8—21no—
91C22×C8—21no—
92C22×C8—21no—
93C22×C8—21no—
94C22×C8—21no—
95C22×C8—21no—
96C22×C8—21no—
97C22×C8—41no—
98#117—21no—
99#117—21no—
100#117—41no—
101#117—41no—
102#117D812yes—
103#117Q1612yes—
104C2×C4⋊C4—21no—
105C2×C4⋊C4—21no—
106C2×C4⋊C4—21no—
107C2×C4⋊C4—21no—
108C2×C4⋊C4C4⋊C411yescharacteristic
109C2×C4⋊C4C4⋊C411yescharacteristic
110C2×C4⋊C4C4⋊C411yescharacteristic
111C2×C4⋊C4C4⋊C411yescharacteristic
112C2×C22⋊C4—21no—
113C2×C22⋊C4—21no—
114C2×C42—21no—
115C2×C42—21no—
116C2×C42—22no—
117C2×C42—41no—
118C2×C42—41no—
119C2×C42—41no—
120C2×C42—41no—
121C2×C42—41no—
122C2×C42—41no—
123C2×C42—41no—
124C2×C42C22×C411yescharacteristic
125C2×C42C2×D411yescharacteristic
126C2×C42C2×D411yescharacteristic
127C2×C42C2×Q811yescharacteristic
128C4⋊C8—22no—
129C4⋊C8—22no—
130C4⋊C8—41no—
131C4⋊C8—41no—
132C4⋊C8—41no—
133C4⋊C8—41no—
134C4⋊C8—41no—
135C4⋊C8—41no—
136C4⋊C8—42no—
137C4⋊C8—42no—
138C4⋊C8—81no—
139C4⋊C8—81no—
140C22⋊C8—42no—
141C22⋊C8—42no—
142C22⋊C8—41no—
143C22⋊C8—41no—
144C22⋊C8—41no—
145C22⋊C8—41no—
146#97—42no—
147C4×C8—41no—
148C4×C8—41no—
149C4×C8—41no—
150C4×C8—41no—
151C4×C8—42no—
152#95—21no—
153#95—21no—
154#95—21no—
155#95—21no—
156#95—22no—
157#95—22no—
158#95—41no—
159#95—41no—
160C24#12311yescharacteristic
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—21no—
173C22×C4—41no—
174C22×C4—41no—
175C22×C4—41no—
176C22×C4—41no—
177C22×C4—42no—
178C22×C4—42no—
179C22×C4—42no—
180C22×C4—81no—
181C22×C4—81no—
182C22×C4—81no—
183C22×C4—81no—
184C22×C4C2×C4⋊C411yescharacteristic
185C22×C4C4×D411yescharacteristic
186C22×C4C4×D411yescharacteristic
187C22×C4#12011yescharacteristic
188C22×C4C22⋊Q811yescharacteristic
189C22×C4C22⋊Q811yescharacteristic
190C22×C4C2×D811yescharacteristic
191C22×C4C2×Q1611yescharacteristic
192C22×C4AGL(1,ℤ/8)11yescharacteristic
193C22×C4#13711yescharacteristic
194M16—42no—
195M16—42no—
196M16—81no—
197M16—81no—
198C2×C8—22no—
199C2×C8—22no—
200C2×C8—41no—
201C2×C8—41no—
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—41no—
218C2×C8—42no—
219C2×C8—41no—
220C2×C8—41no—
221C2×C8—41no—
222C2×C8—41no—
223C2×C8—81no—
224C2×C8—81no—
225C4⋊C4—21no—
226C4⋊C4—21no—
227C4⋊C4—41no—
228C4⋊C4—41no—
229C4⋊C4—41no—
230C4⋊C4—41no—
231C4⋊C4—41no—
232C4⋊C4—41no—
233C4⋊C4—81no—
234C4⋊C4—81no—
235C4⋊C4#10712yes—
236C4⋊C4#10712yes—
237C22⋊C4—22no—
238C22⋊C4—22no—
239C22⋊C4—81no—
240C42—22no—
241C42—22no—
242C42—21no—
243C42—21no—
244C42—21no—
245C42—21no—
246C42—41no—
247C42—41no—
248C42—42no—
249C42—42no—
250C42—41no—
251C42—41no—
252C42—81no—
253C42—81no—
254C23—21no—
255C23—21no—
256C23—21no—
257C23—21no—
258C23—21no—
259C23—21no—
260C23#38511yescharacteristic
261C23#48011yescharacteristic
262C23#48411yescharacteristic
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—21no—
274C2×C4—21no—
275C2×C4—21no—
276C2×C4—21no—
277C2×C4—22no—
278C2×C4—22no—
279C2×C4—21no—
280C2×C4—21no—
281C2×C4—21no—
282C2×C4—21no—
283C2×C4—21no—
284C2×C4—21no—
285C2×C4—41no—
286C2×C4—41no—
287C2×C4—41no—
288C2×C4—41no—
289C2×C4—44no—
290C2×C4—41no—
291C2×C4—41no—
292C2×C4—41no—
293C2×C4—41no—
294C2×C4—41no—
295C2×C4—41no—
296C2×C4—81no—
297C2×C4—81no—
298C2×C4—81no—
299C2×C4—81no—
300C2×C4—82no—
301C2×C4—82no—
302C2×C4—82no—
303C2×C4—82no—
304C2×C4—82no—
305C2×C4—82no—
306C2×C4C2×#10711yescharacteristic
307C2×C4#42811yescharacteristic
308C2×C4#44311yescharacteristic
309C2×C4#44311yescharacteristic
310C2×C4#44411yescharacteristic
311C2×C4#44411yescharacteristic
312C2×C4#44711yescharacteristic
313C2×C4#45111yescharacteristic
314C2×C4#45411yescharacteristic
315C2×C4#45611yescharacteristic
316C2×C4B(4,𝔽2)11yescharacteristic
317C2×C4#45811yescharacteristic
318C8—42no—
319C8—42no—
320C8—81no—
321C8—81no—
322C8—81no—
323C8—81no—
324C8—82no—
325C8—81no—
326C8—81no—
327C22—21no—
328C22—21no—
329C22—21no—
330C22—21no—
331C22—21no—
332C22—21no—
333C22—41no—
334C22—41no—
335C22—41no—
336C22—41no—
337C22#185211yescharacteristic
338C22#185711yescharacteristic
339C22#186111yescharacteristic
340C22#186111yescharacteristic
341C22#186611yescharacteristic
342C22#186611yescharacteristic
343C22#186711yescharacteristic
344C4—21no—
345C4—21no—
346C4—21no—
347C4—21no—
348C4—21no—
349C4—21no—
350C4—41no—
351C4—41no—
352C4—42no—
353C4—42no—
354C4—84no—
355C4—81no—
356C4—81no—
357C4#159811yescharacteristic
358C4#160511yescharacteristic
359C4#161011yescharacteristic
360C4#161711yescharacteristic
361C2—41no—
362C2—41no—
363C2#885611yescharacteristic
364C2#887411yescharacteristic
365C2#887511yescharacteristic
366C2#890211yescharacteristic
367C2#890211yescharacteristic
368C2#891911yescharacteristic
369C2#893611yescharacteristic
subgroupG/H
centerC22×C4#120
commutatorC2×C42C22×C4
Frattini, cosocleC22×C42C23
socleC23#385
prime radicalC24#123
LCSG ▷ C2×C42 ▷ C23 ▷ C1
UCSC1 ◁ C22×C4 ◁ C22×C42 ◁ G
LECSG ▷ C22×C42 ▷ C23 ▷ C1

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
27. #1271C4semidirect
28. #1271C4semidirect
29. #1271C4semidirect
30. #1271C4semidirect

Maximal subgroups

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

subgroupindex
#219572
#112092
#88322
#75642
#70282