Finite Group Databasebeta

← #100669

Subgroups — #100669

Order 512.

403 conjugacy classes of subgroups; only the 84 normal subgroups are shown.

normal subgroups only

This group has 403 conjugacy classes of subgroups — too many to draw. Only its 84 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#30169C211yesmaximal, characteristic
2#23240C211yesmaximal, characteristic
3#9889C211yesmaximal, characteristic
4#8221C211yesmaximal, characteristic
5#7687C211yesmaximal, characteristic
6#7037C211yesmaximal, characteristic
7#7021C211yesmaximal, characteristic
8#2857C2211yescharacteristic
9C4×C22⋊Q8C2211yescharacteristic
10#2083C411yescharacteristic, complemented
11#2076—21no—
12#2072C411yescharacteristic, complemented
13#2057C411yescharacteristic, complemented
14#2036—21no—
15#2024C411yescharacteristic, complemented
16#1985—21no—
17#1921—21no—
18#1917—21no—
19#1916—21no—
20#1863—21no—
21#1851—21no—
22#1817C2211yescharacteristic
23#1770—21no—
24#1753—21no—
25#1749C2211yescharacteristic
26C4×C22⋊C8C2211yescharacteristic
27#1728C2211yescharacteristic
28C2×#328—21no—
29#1434C2211yescharacteristic
30#1294—21no—
31#1294—21no—
32#1294—21no—
33#1272—21no—
34#1272—21no—
35C22×M16—21no—
36C2×C4⋊Q8C2×C411yescharacteristic
37C2×C22⋊Q8—21no—
38C2×C22⋊Q8D411yescharacteristic
39C2×C4×Q8D411yescharacteristic
40C22×C42C2311yescharacteristic
41C2×#107—21no—
42C2×#107—21no—
43C2×C4⋊C8—21no—
44C2×C4⋊C8—21no—
45C2×#103—21no—
46C2×#103—21no—
47C2×C22⋊C8—21no—
48C2×C22⋊C8—21no—
49C2×C22⋊C8D411yescharacteristic
50C2×C22⋊C8D411yescharacteristic
51C2×#97—21no—
52C2×#97—21no—
53C2×C4×C8—21no—
54C2×C4×C8—21no—
55#391—21no—
56#391C2×C411yescharacteristic
57#389—21no—
58#389—21no—
59#389C2×C411yescharacteristic
60#387—21no—
61#385—21no—
62#384C2×C411yescharacteristic
63#380—21no—
64#380—21no—
65#380D411yescharacteristic
66C4×C4⋊C4—21no—
67C4×C4⋊C4D411yescharacteristic
68C4×C22⋊C4—21no—
69C2×#95—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
95C2×M16—42no—
96C2×M16—41no—
97C2×M16—41no—
98C2×M16—81no—
99C22×C8—21no—
100C22×C8—21no—
101C22×C8—21no—
102C22×C8—21no—
103C22×C8—21no—
104C22×C8—21no—
105C22×C8—41no—
106C4⋊Q8—81no—
107C22⋊Q8—22no—
108C22⋊Q8—41no—
109C22⋊Q8—82no—
110C4×Q8—81no—
111C2×C4⋊C4—21no—
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—41no—
120C2×C4⋊C4C22⋊C411yescharacteristic
121C2×C4⋊C4C22⋊C411yescharacteristic
122C2×C4⋊C4C22⋊C411yescharacteristic
123C2×C22⋊C4—21no—
124C2×C22⋊C4—21no—
125C2×C22⋊C4—41no—
126C2×C42—21no—
127C2×C42—21no—
128C2×C42—81no—
129C2×C42C22×C411yescharacteristic
130C2×C42C2×D411yescharacteristic
131C2×C42C2×D411yescharacteristic
132C2×C42#4111yescharacteristic
133#107—42no—
134#107—81no—
135C4⋊C8—42no—
136C4⋊C8—81no—
137#103—42no—
138#103—81no—
139C22⋊C8—21no—
140C22⋊C8—22no—
141C22⋊C8—21no—
142C22⋊C8—21no—
143C22⋊C8—41no—
144C22⋊C8—81no—
145C22⋊C8Q1612yes—
146#97—81no—
147#97—81no—
148C4×C8—42no—
149C4×C8—81no—
150#95—21no—
151#95—21no—
152#95—21no—
153#95—21no—
154#95—21no—
155#95—22no—
156#95—41no—
157#95—41no—
158#95—41no—
159#95—41no—
160#95—41no—
161#95—81no—
162C24#12111yescharacteristic
163C2×Q8—21no—
164C2×Q8—21no—
165C2×Q8—41no—
166C2×Q8—41no—
167C2×Q8—42no—
168C2×Q8—81no—
169C2×Q8—81no—
170C22×C4—21no—
171C22×C4—21no—
172C22×C4—21no—
173C22×C4—21no—
174C22×C4—21no—
175C22×C4—21no—
176C22×C4—41no—
177C22×C4—41no—
178C22×C4—41no—
179C22×C4—41no—
180C22×C4—41no—
181C22×C4—41no—
182C22×C4—41no—
183C22×C4—42no—
184C22×C4—41no—
185C22×C4—41no—
186C22×C4—81no—
187C22×C4—81no—
188C22×C4—81no—
189C22×C4—81no—
190C22×C4—81no—
191C22×C4—81no—
192C22×C4C2×C22⋊C411yescharacteristic
193C22×C4C4×D411yescharacteristic
194C22×C4C4×D411yescharacteristic
195C22×C4#12011yescharacteristic
196C22×C4#12111yescharacteristic
197C22×C4#12311yescharacteristic
198C22×C4C2×Q1611yescharacteristic
199C22×C4#13511yescharacteristic
200C22×C4#13711yescharacteristic
201C22×C4#13711yescharacteristic
202M16—82no—
203M16—82no—
204C2×C8—22no—
205C2×C8—22no—
206C2×C8—41no—
207C2×C8—41no—
208C2×C8—41no—
209C2×C8—41no—
210C2×C8—41no—
211C2×C8—42no—
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—81no—
224C2×C8—81no—
225C4⋊C4—21no—
226C4⋊C4—42no—
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—81no—
236C4⋊C4—81no—
237C4⋊C4—81no—
238C4⋊C4—82no—
239C4⋊C4#10412yes—
240C22⋊C4—42no—
241C22⋊C4—81no—
242C22⋊C4—82no—
243C42—41no—
244C42—41no—
245C42—41no—
246C42—41no—
247C42—41no—
248C42—41no—
249C42—41no—
250C42—41no—
251C23—21no—
252C23—21no—
253C23—21no—
254C23—21no—
255C23—21no—
256C23—21no—
257C23#38611yescharacteristic
258C23#46211yescharacteristic
259C23#46411yescharacteristic
260Q8—42no—
261Q8—81no—
262Q8—162no—
263C2×C4—21no—
264C2×C4—21no—
265C2×C4—21no—
266C2×C4—21no—
267C2×C4—21no—
268C2×C4—21no—
269C2×C4—22no—
270C2×C4—21no—
271C2×C4—21no—
272C2×C4—21no—
273C2×C4—21no—
274C2×C4—21no—
275C2×C4—21no—
276C2×C4—21no—
277C2×C4—21no—
278C2×C4—21no—
279C2×C4—41no—
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—41no—
292C2×C4—41no—
293C2×C4—41no—
294C2×C4—81no—
295C2×C4—81no—
296C2×C4—82no—
297C2×C4—82no—
298C2×C4—82no—
299C2×C4—82no—
300C2×C4—81no—
301C2×C4—81no—
302C2×C4—81no—
303C2×C4—81no—
304C2×C4—81no—
305C2×C4—81no—
306C2×C4—161no—
307C2×C4—161no—
308C2×C4—161no—
309C2×C4—161no—
310C2×C4C2×#10411yescharacteristic
311C2×C4#42111yescharacteristic
312C2×C4C4×Q1611yescharacteristic
313C2×C4#44111yescharacteristic
314C2×C4#45111yescharacteristic
315C2×C4#45211yescharacteristic
316C8—42no—
317C8—42no—
318C8—81no—
319C8—81no—
320C8—81no—
321C8—81no—
322C8—82no—
323C22—21no—
324C22—21no—
325C22—21no—
326C22—21no—
327C22—21no—
328C22—21no—
329C22—41no—
330C22—41no—
331C22—41no—
332C22—41no—
333C22#187111yescharacteristic
334C22#187811yescharacteristic
335C22#188111yescharacteristic
336C22#190311yescharacteristic
337C22#190611yescharacteristic
338C22#190911yescharacteristic
339C22#191311yescharacteristic
340C4—21no—
341C4—21no—
342C4—21no—
343C4—21no—
344C4—21no—
345C4—21no—
346C4—41no—
347C4—41no—
348C4—42no—
349C4—41no—
350C4—81no—
351C4—81no—
352C4—81no—
353C4—162no—
354C4—161no—
355C4—161no—
356C2—41no—
357C2—41no—
358C2#899011yescharacteristic
359C2#916111yescharacteristic
360C2#918411yescharacteristic
361C2#927611yescharacteristic
362C2#928811yescharacteristic
363C2#933211yescharacteristic
364C2#936911yescharacteristic
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. #2057C4semidirect
15. #2024C4semidirect

Maximal subgroups

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

subgroupindex
#301692
#232402
#98892
#82212
#76872
#70372
#70212