Finite Group Databasebeta

← #100556

Subgroups — #100556

Order 512.

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

normal subgroups only

This group has 415 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#30175C211yesmaximal, characteristic
2#23337C211yesmaximal, characteristic
3#9711C211yesmaximal, characteristic
4#8219C211yesmaximal, characteristic
5#7708C211yesmaximal, characteristic
6#7038C211yesmaximal, characteristic
7#7017C211yesmaximal, characteristic
8#2857C2211yescharacteristic
9C4×C22⋊Q8C2211yescharacteristic
10#2083C411yescharacteristic, complemented
11#2077—21no—
12#2072C411yescharacteristic, complemented
13#2058C411yescharacteristic, complemented
14#2036—21no—
15#2024C411yescharacteristic, complemented
16#1985—21no—
17#1939—21no—
18#1924—21no—
19#1923—21no—
20#1918—21no—
21#1917—21no—
22#1916—21no—
23#1865—21no—
24#1817C2211yescharacteristic
25#1817C2211yescharacteristic
26#1797C2211yescharacteristic
27#1749C2211yescharacteristic
28C2×#328—21no—
29C2×#328—21no—
30C2×#328—21no—
31C4×#95C2211yescharacteristic
32#1294—21no—
33#1271—21no—
34#1271—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—
89#328—42no—
90#328—42no—
91C22×Q8—21no—
92C22×Q8C22⋊C411yescharacteristic
93C23×C4—41no—
94C23×C4C2×D411yescharacteristic
95C23×C4C2×D411yescharacteristic
96C23×C4#4111yescharacteristic
97C22×C8—21no—
98C22×C8—21no—
99C22×C8—21no—
100C22×C8—21no—
101C22×C8—21no—
102C22×C8—21no—
103C22×C8—41no—
104C4⋊Q8—81no—
105C22⋊Q8—22no—
106C22⋊Q8—41no—
107C22⋊Q8—41no—
108C22⋊Q8—41no—
109C22⋊Q8—41no—
110C22⋊Q8—41no—
111C4×Q8—81no—
112C2×C4⋊C4—21no—
113C2×C4⋊C4—21no—
114C2×C4⋊C4—21no—
115C2×C4⋊C4—21no—
116C2×C4⋊C4—21no—
117C2×C4⋊C4—21no—
118C2×C4⋊C4—21no—
119C2×C4⋊C4—21no—
120C2×C4⋊C4—41no—
121C2×C4⋊C4C22⋊C411yescharacteristic
122C2×C4⋊C4C22⋊C411yescharacteristic
123C2×C4⋊C4C22⋊C411yescharacteristic
124C2×C22⋊C4—21no—
125C2×C22⋊C4—21no—
126C2×C22⋊C4—41no—
127C2×C42—21no—
128C2×C42—21no—
129C2×C42—22no—
130C2×C42—41no—
131C2×C42—81no—
132C2×C42—81no—
133C2×C42—81no—
134C2×C42C22×C411yescharacteristic
135C2×C42C2×D411yescharacteristic
136C2×C42C2×D411yescharacteristic
137C2×C42#4111yescharacteristic
138#107—42no—
139#106—81no—
140C4⋊C8—81no—
141C4⋊C8—81no—
142C4⋊C8—81no—
143#103—42no—
144#103—81no—
145C22⋊C8—22no—
146C22⋊C8—41no—
147C22⋊C8—41no—
148C22⋊C8—41no—
149C22⋊C8—81no—
150C22⋊C8—81no—
151#97—81no—
152#97—81no—
153C4×C8—81no—
154#95—21no—
155#95—21no—
156#95—21no—
157#95—21no—
158#95—21no—
159#95—22no—
160#95—22no—
161#95—41no—
162#95—41no—
163#95—41no—
164#95—41no—
165C24#12111yescharacteristic
166C2×Q8—21no—
167C2×Q8—21no—
168C2×Q8—41no—
169C2×Q8—41no—
170C2×Q8—41no—
171C2×Q8—41no—
172C2×Q8—81no—
173C2×Q8—81no—
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—21no—
183C22×C4—21no—
184C22×C4—21no—
185C22×C4—21no—
186C22×C4—41no—
187C22×C4—41no—
188C22×C4—41no—
189C22×C4—41no—
190C22×C4—41no—
191C22×C4—42no—
192C22×C4—42no—
193C22×C4—42no—
194C22×C4—41no—
195C22×C4—41no—
196C22×C4—81no—
197C22×C4—81no—
198C22×C4—81no—
199C22×C4—81no—
200C22×C4C2×C22⋊C411yescharacteristic
201C22×C4C4×D411yescharacteristic
202C22×C4C4×D411yescharacteristic
203C22×C4#12011yescharacteristic
204C22×C4#12111yescharacteristic
205C22×C4#12311yescharacteristic
206C2×C8—41no—
207C2×C8—41no—
208C2×C8—42no—
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—41no—
222C2×C8—41no—
223C2×C8—41no—
224C2×C8—41no—
225C2×C8—81no—
226C2×C8—81no—
227C4⋊C4—21no—
228C4⋊C4—42no—
229C4⋊C4—41no—
230C4⋊C4—41no—
231C4⋊C4—41no—
232C4⋊C4—41no—
233C4⋊C4—41no—
234C4⋊C4—41no—
235C4⋊C4—41no—
236C4⋊C4—41no—
237C4⋊C4—41no—
238C4⋊C4—41no—
239C4⋊C4—41no—
240C4⋊C4—41no—
241C4⋊C4—81no—
242C4⋊C4—81no—
243C4⋊C4—81no—
244C4⋊C4—81no—
245C4⋊C4#10412yes—
246C22⋊C4—42no—
247C22⋊C4—41no—
248C22⋊C4—41no—
249C22⋊C4—81no—
250C22⋊C4—81no—
251C42—41no—
252C42—41no—
253C42—41no—
254C42—41no—
255C42—41no—
256C42—41no—
257C42—41no—
258C42—41no—
259C42—82no—
260C42—82no—
261C23—21no—
262C23—21no—
263C23—21no—
264C23—21no—
265C23—21no—
266C23—21no—
267C23#38611yescharacteristic
268C23#47111yescharacteristic
269C23#47311yescharacteristic
270Q8—42no—
271Q8—81no—
272Q8—161no—
273Q8—161no—
274C2×C4—21no—
275C2×C4—21no—
276C2×C4—21no—
277C2×C4—21no—
278C2×C4—21no—
279C2×C4—21no—
280C2×C4—22no—
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—41no—
291C2×C4—41no—
292C2×C4—41no—
293C2×C4—41no—
294C2×C4—41no—
295C2×C4—41no—
296C2×C4—44no—
297C2×C4—41no—
298C2×C4—41no—
299C2×C4—41no—
300C2×C4—41no—
301C2×C4—41no—
302C2×C4—41no—
303C2×C4—41no—
304C2×C4—41no—
305C2×C4—41no—
306C2×C4—81no—
307C2×C4—81no—
308C2×C4—81no—
309C2×C4—81no—
310C2×C4—82no—
311C2×C4—82no—
312C2×C4—82no—
313C2×C4—82no—
314C2×C4—82no—
315C2×C4—82no—
316C2×C4—81no—
317C2×C4—81no—
318C2×C4—81no—
319C2×C4—81no—
320C2×C4—81no—
321C2×C4—81no—
322C2×C4C2×#10411yescharacteristic
323C2×C4#42111yescharacteristic
324C2×C4#44311yescharacteristic
325C2×C4#44411yescharacteristic
326C2×C4#45311yescharacteristic
327C2×C4#45411yescharacteristic
328C8—81no—
329C8—81no—
330C8—81no—
331C8—81no—
332C8—81no—
333C8—81no—
334C8—82no—
335C22—21no—
336C22—21no—
337C22—21no—
338C22—21no—
339C22—21no—
340C22—21no—
341C22—41no—
342C22—41no—
343C22—41no—
344C22—41no—
345C22#187111yescharacteristic
346C22#187611yescharacteristic
347C22#188311yescharacteristic
348C22#190511yescharacteristic
349C22#190811yescharacteristic
350C22#190911yescharacteristic
351C22#191311yescharacteristic
352C4—21no—
353C4—21no—
354C4—21no—
355C4—21no—
356C4—21no—
357C4—21no—
358C4—42no—
359C4—41no—
360C4—41no—
361C4—41no—
362C4—84no—
363C4—81no—
364C4—81no—
365C4—81no—
366C4—161no—
367C4—161no—
368C2—41no—
369C2—41no—
370C2#896911yescharacteristic
371C2#916611yescharacteristic
372C2#918411yescharacteristic
373C2#926311yescharacteristic
374C2#928011yescharacteristic
375C2#934411yescharacteristic
376C2#937711yescharacteristic
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. #2083C4semidirect
12. #2072C4semidirect
13. #2058C4semidirect
15. #2024C4semidirect

Maximal subgroups

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

subgroupindex
#301752
#233372
#97112
#82192
#77082
#70382
#70172