Finite Group Databasebeta

← #100570

Subgroups — #100570

Order 512.

407 conjugacy classes of subgroups; only the 78 normal subgroups are shown.

normal subgroups only

This group has 407 conjugacy classes of subgroups — too many to draw. Only its 78 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#30176C211yesmaximal, characteristic
2#23343C211yesmaximal, characteristic
3#9909C211yesmaximal, characteristic
4#8219C211yesmaximal, characteristic
5#7707C211yesmaximal, characteristic
6#7038C211yesmaximal, characteristic
7#7017C211yesmaximal, characteristic
8#2857C2211yescharacteristic
9C4×C22⋊Q8C2211yescharacteristic
10#2084C411yescharacteristic, complemented
11#2075—21no—
12#2072C411yescharacteristic, complemented
13#2057C411yescharacteristic, complemented
14#2035—21no—
15#2024C411yescharacteristic, complemented
16#1984—21no—
17#1940—21no—
18#1924—21no—
19#1923—21no—
20#1920—21no—
21#1917—21no—
22#1916—21no—
23#1863—21no—
24#1817C2211yescharacteristic
25#1817C2211yescharacteristic
26#1797C2211yescharacteristic
27#1749C2211yescharacteristic
28C2×#328—21no—
29C4×#95C2211yescharacteristic
30#1294—21no—
31#1294—21no—
32#1294—21no—
33#1272—21no—
34#1272—21no—
35C2×C4⋊Q8C2×C411yescharacteristic
36C2×C22⋊Q8—21no—
37C2×C22⋊Q8D411yescharacteristic
38C2×C4×Q8D411yescharacteristic
39C22×C42—21no—
40C22×C42C2311yescharacteristic
41C2×#107—21no—
42C2×#106—21no—
43C2×C4⋊C8—21no—
44C2×C4⋊C8—21no—
45C2×C4⋊C8—21no—
46C2×#103—21no—
47C2×#103—21no—
48C2×C22⋊C8—21no—
49C2×C22⋊C8—21no—
50C2×C22⋊C8—21no—
51C2×C22⋊C8D411yescharacteristic
52C2×C22⋊C8D411yescharacteristic
53C2×#97—21no—
54C2×#97—21no—
55C2×C4×C8—21no—
56#391—21no—
57#391C2×C411yescharacteristic
58#389—21no—
59#389—21no—
60#389C2×C411yescharacteristic
61#387—21no—
62#385—21no—
63#384C2×C411yescharacteristic
64#380—21no—
65#380—21no—
66#380D411yescharacteristic
67C4×C4⋊C4—21no—
68C4×C4⋊C4D411yescharacteristic
69C4×C22⋊C4—21no—
70C2×#95—21no—
71C2×#95—21no—
72#340—21no—
73#340—21no—
74#340—21no—
75#340—21no—
76#336—21no—
77#336—21no—
78#336—21no—
79#336—21no—
80#336—21no—
81#336—21no—
82#336—21no—
83#336—21no—
84#336C2×C411yescharacteristic
85#336C2×C411yescharacteristic
86#336D411yescharacteristic
87#336D411yescharacteristic
88#328—24no—
89C22×Q8—21no—
90C22×Q8C22⋊C411yescharacteristic
91C23×C4—41no—
92C23×C4C2×D411yescharacteristic
93C23×C4C2×D411yescharacteristic
94C23×C4#4111yescharacteristic
95C22×C8—21no—
96C22×C8—21no—
97C22×C8—21no—
98C22×C8—21no—
99C22×C8—21no—
100C22×C8—21no—
101C22×C8—41no—
102C4⋊Q8—81no—
103C22⋊Q8—22no—
104C22⋊Q8—41no—
105C22⋊Q8—81no—
106C22⋊Q8—81no—
107C4×Q8—81no—
108C2×C4⋊C4—21no—
109C2×C4⋊C4—21no—
110C2×C4⋊C4—21no—
111C2×C4⋊C4—21no—
112C2×C4⋊C4—21no—
113C2×C4⋊C4—21no—
114C2×C4⋊C4—21no—
115C2×C4⋊C4—21no—
116C2×C4⋊C4—41no—
117C2×C4⋊C4C22⋊C411yescharacteristic
118C2×C4⋊C4C22⋊C411yescharacteristic
119C2×C4⋊C4C22⋊C411yescharacteristic
120C2×C22⋊C4—21no—
121C2×C22⋊C4—21no—
122C2×C22⋊C4—41no—
123C2×C42—21no—
124C2×C42—21no—
125C2×C42—22no—
126C2×C42—41no—
127C2×C42—81no—
128C2×C42—81no—
129C2×C42—81no—
130C2×C42C22×C411yescharacteristic
131C2×C42C2×D411yescharacteristic
132C2×C42C2×D411yescharacteristic
133C2×C42#4111yescharacteristic
134#107—81no—
135#106—42no—
136C4⋊C8—81no—
137C4⋊C8—81no—
138C4⋊C8—81no—
139#103—42no—
140#103—81no—
141C22⋊C8—22no—
142C22⋊C8—41no—
143C22⋊C8—41no—
144C22⋊C8—41no—
145C22⋊C8—81no—
146C22⋊C8—81no—
147#97—81no—
148#97—81no—
149C4×C8—81no—
150#95—21no—
151#95—21no—
152#95—21no—
153#95—21no—
154#95—21no—
155#95—22no—
156#95—22no—
157#95—41no—
158#95—41no—
159#95—41no—
160#95—41no—
161C24#12111yescharacteristic
162C2×Q8—21no—
163C2×Q8—41no—
164C2×Q8—41no—
165C2×Q8—41no—
166C2×Q8—41no—
167C2×Q8—81no—
168C2×Q8—81no—
169C2×Q8#10412yes—
170C22×C4—21no—
171C22×C4—21no—
172C22×C4—21no—
173C22×C4—21no—
174C22×C4—21no—
175C22×C4—21no—
176C22×C4—21no—
177C22×C4—21no—
178C22×C4—21no—
179C22×C4—21no—
180C22×C4—21no—
181C22×C4—21no—
182C22×C4—41no—
183C22×C4—41no—
184C22×C4—41no—
185C22×C4—41no—
186C22×C4—41no—
187C22×C4—42no—
188C22×C4—42no—
189C22×C4—42no—
190C22×C4—41no—
191C22×C4—41no—
192C22×C4—81no—
193C22×C4—81no—
194C22×C4—81no—
195C22×C4—81no—
196C22×C4C2×C22⋊C411yescharacteristic
197C22×C4C4×D411yescharacteristic
198C22×C4C4×D411yescharacteristic
199C22×C4#12011yescharacteristic
200C22×C4#12111yescharacteristic
201C22×C4#12311yescharacteristic
202C2×C8—41no—
203C2×C8—41no—
204C2×C8—42no—
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—41no—
219C2×C8—41no—
220C2×C8—41no—
221C2×C8—81no—
222C2×C8—81no—
223C4⋊C4—21no—
224C4⋊C4—21no—
225C4⋊C4—42no—
226C4⋊C4—41no—
227C4⋊C4—41no—
228C4⋊C4—41no—
229C4⋊C4—41no—
230C4⋊C4—41no—
231C4⋊C4—41no—
232C4⋊C4—41no—
233C4⋊C4—41no—
234C4⋊C4—81no—
235C4⋊C4—81no—
236C4⋊C4—81no—
237C4⋊C4—81no—
238C4⋊C4—82no—
239C22⋊C4—42no—
240C22⋊C4—41no—
241C22⋊C4—41no—
242C22⋊C4—81no—
243C22⋊C4—81no—
244C42—41no—
245C42—41no—
246C42—41no—
247C42—41no—
248C42—41no—
249C42—41no—
250C42—41no—
251C42—41no—
252C42—82no—
253C42—82no—
254C23—21no—
255C23—21no—
256C23—21no—
257C23—21no—
258C23—21no—
259C23—21no—
260C23#38611yescharacteristic
261C23#47111yescharacteristic
262C23#47211yescharacteristic
263Q8—42no—
264Q8—81no—
265Q8—161no—
266Q8—161no—
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—21no—
281C2×C4—21no—
282C2×C4—21no—
283C2×C4—41no—
284C2×C4—41no—
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—41no—
297C2×C4—41no—
298C2×C4—41no—
299C2×C4—81no—
300C2×C4—81no—
301C2×C4—81no—
302C2×C4—81no—
303C2×C4—82no—
304C2×C4—82no—
305C2×C4—82no—
306C2×C4—82no—
307C2×C4—82no—
308C2×C4—82no—
309C2×C4—81no—
310C2×C4—81no—
311C2×C4—81no—
312C2×C4—81no—
313C2×C4—81no—
314C2×C4—81no—
315C2×C4C2×#10411yescharacteristic
316C2×C4#42111yescharacteristic
317C2×C4#44311yescharacteristic
318C2×C4#44411yescharacteristic
319C2×C4#45511yescharacteristic
320C2×C4#45611yescharacteristic
321C8—81no—
322C8—81no—
323C8—81no—
324C8—81no—
325C8—81no—
326C8—81no—
327C8—82no—
328C22—21no—
329C22—21no—
330C22—21no—
331C22—21no—
332C22—21no—
333C22—21no—
334C22—41no—
335C22—41no—
336C22—41no—
337C22—41no—
338C22#187111yescharacteristic
339C22#187611yescharacteristic
340C22#188211yescharacteristic
341C22#190511yescharacteristic
342C22#190711yescharacteristic
343C22#190911yescharacteristic
344C22#191311yescharacteristic
345C4—21no—
346C4—21no—
347C4—21no—
348C4—21no—
349C4—21no—
350C4—21no—
351C4—41no—
352C4—41no—
353C4—41no—
354C4—42no—
355C4—84no—
356C4—81no—
357C4—81no—
358C4—81no—
359C4—161no—
360C4—161no—
361C2—41no—
362C2—41no—
363C2#897211yescharacteristic
364C2#916411yescharacteristic
365C2#918311yescharacteristic
366C2#926411yescharacteristic
367C2#928011yescharacteristic
368C2#934211yescharacteristic
369C2#937511yescharacteristic
subgroupG/H
center, socleC23#386
commutatorC2×C42C22×C4
Frattini, cosocleC22×C42C23
prime radicalC24#121
LCSG ▷ C2×C42 ▷ C23 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C22×C42 ▷ C23 ▷ C1

Complements

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

NH ≅ G/Ntype
10. #2084C4semidirect
12. #2072C4semidirect
13. #2057C4semidirect
15. #2024C4semidirect

Maximal subgroups

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

subgroupindex
#301762
#233432
#99092
#82192
#77072
#70382
#70172