Finite Group Databasebeta

← #101580

Subgroups — #101580

Order 512.

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

normal subgroups only

This group has 440 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#27648C211yesmaximal, characteristic
2#22289C211yesmaximal, characteristic
3#11354C211yesmaximal, characteristic
4#8724C211yesmaximal, characteristic
5#7617C211yesmaximal, characteristic
6#7045C211yesmaximal, characteristic
7#7034C211yesmaximal, characteristic
8C2×#484C412yes—
9C4×C22⋊Q8C2211yescharacteristic
10C2×#384C2211yescharacteristic
11#2083—21no—
12#2072—21no—
13#1984—21no—
14#1947—21no—
15#1940—21no—
16#1917—21no—
17#1916—21no—
18#1851—21no—
19#1824C412yes—
20#1803—21no—
21#1797C2211yescharacteristic
22#1797C2211yescharacteristic
23#1797C2211yescharacteristic
24#1753—21no—
25C2×#343—21no—
26C2×#343—21no—
27#1728C2211yescharacteristic
28#1728C2211yescharacteristic
29#1296—21no—
30#1296—21no—
31#1272—21no—
32#1271—21no—
33C22×M16—21no—
34C22×M16D411yescharacteristic
35C22×M16D411yescharacteristic
36C2×C22⋊Q8C2×C411yescharacteristic
37C2×C4×Q8D411yescharacteristic
38C22×C4⋊C4—21no—
39C22×C4⋊C4—21no—
40C22×C4⋊C4C2×C412yes—
41C22×C42C2311yescharacteristic
42#484—44no—
43C2×#107—22no—
44C2×C4⋊C8—21no—
45C2×C4⋊C8—21no—
46C2×C4⋊C8—21no—
47C2×C4⋊C8—21no—
48C2×C4⋊C8D411yescharacteristic
49C2×C4⋊C8D411yescharacteristic
50C2×#103—22no—
51C2×C22⋊C8—21no—
52C2×C22⋊C8—21no—
53C2×C22⋊C8C2×C412yes—
54C2×#97—21no—
55C2×C4×C8—21no—
56#391D411yescharacteristic
57#389D411yescharacteristic
58#387—21no—
59#385—21no—
60#384—21no—
61#384—21no—
62#384—21no—
63#384—21no—
64#380C2×C411yescharacteristic
65C4×C4⋊C4—21no—
66C4×C4⋊C4D411yescharacteristic
67C4×C22⋊C4—21no—
68#343—42no—
69#343—42no—
70#340—22no—
71#340—22no—
72#336—21no—
73#336—21no—
74#336—21no—
75#336—21no—
76#336—21no—
77#336—21no—
78#336—21no—
79#336—21no—
80#336—21no—
81#336—21no—
82#336—21no—
83C22×Q8C22⋊C411yescharacteristic
84C23×C4—22no—
85C23×C4C22×C411yescharacteristic
86C23×C4C2×D411yescharacteristic
87C23×C4#4111yescharacteristic
88C2×M16—21no—
89C2×M16—22no—
90C2×M16—21no—
91C2×M16—21no—
92C2×M16—41no—
93C2×M16—41no—
94C2×M16—41no—
95C2×M16—41no—
96C2×M16—41no—
97C2×M16—41no—
98C2×M16—41no—
99C2×M16—41no—
100C2×M16—41no—
101C2×M16Q1612yes—
102C22×C8—21no—
103C22×C8—22no—
104C22×C8—21no—
105C22×C8—21no—
106C22×C8—21no—
107C22×C8—41no—
108C22×C8—41no—
109C22⋊Q8—22no—
110C22⋊Q8—41no—
111C4×Q8—41no—
112C4×Q8—41no—
113C2×C4⋊C4—21no—
114C2×C4⋊C4—21no—
115C2×C4⋊C4—22no—
116C2×C4⋊C4—21no—
117C2×C4⋊C4—22no—
118C2×C4⋊C4—42no—
119C2×C4⋊C4—42no—
120C2×C4⋊C4—41no—
121C2×C4⋊C4—41no—
122C2×C4⋊C4—41no—
123C2×C4⋊C4—41no—
124C2×C4⋊C4—41no—
125C2×C4⋊C4—41no—
126C2×C4⋊C4—41no—
127C2×C4⋊C4—41no—
128C2×C4⋊C4—81no—
129C2×C4⋊C4—81no—
130C2×C4⋊C4C22⋊C411yescharacteristic
131C2×C4⋊C4C22⋊C411yescharacteristic
132C2×C4⋊C4C22⋊C411yescharacteristic
133C2×C22⋊C4—21no—
134C2×C22⋊C4—21no—
135C2×C42—21no—
136C2×C42—21no—
137C2×C42—41no—
138C2×C42—41no—
139C2×C42C2×D411yescharacteristic
140C2×C42C2×D411yescharacteristic
141C2×C42C2×D411yescharacteristic
142C2×C42#4111yescharacteristic
143#107—44no—
144C4⋊C8—41no—
145C4⋊C8—41no—
146C4⋊C8—41no—
147C4⋊C8—41no—
148C4⋊C8—41no—
149C4⋊C8—41no—
150C4⋊C8—81no—
151C4⋊C8—81no—
152#103—44no—
153C22⋊C8—41no—
154C22⋊C8—41no—
155C22⋊C8—42no—
156C22⋊C8—81no—
157#97—81no—
158C4×C8—81no—
159#95—21no—
160#95—21no—
161#95—21no—
162#95—21no—
163#95—21no—
164C24C4×D411yescharacteristic
165C2×Q8—21no—
166C2×Q8—41no—
167C2×Q8—41no—
168C2×Q8#10012yes—
169C22×C4—21no—
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—41no—
182C22×C4—41no—
183C22×C4—41no—
184C22×C4—41no—
185C22×C4—42no—
186C22×C4—42no—
187C22×C4—42no—
188C22×C4—82no—
189C22×C4—82no—
190C22×C4C2×C22⋊C411yescharacteristic
191C22×C4C4×D411yescharacteristic
192C22×C4#12011yescharacteristic
193C22×C4#12111yescharacteristic
194C22×C4#12111yescharacteristic
195C22×C4#12311yescharacteristic
196C22×C4C2×Q1611yescharacteristic
197C22×C4#13511yescharacteristic
198C22×C4#13711yescharacteristic
199C22×C4#13711yescharacteristic
200M16—42no—
201M16—42no—
202M16—42no—
203M16—42no—
204M16—81no—
205M16—81no—
206M16—81no—
207M16—81no—
208C2×C8—22no—
209C2×C8—41no—
210C2×C8—41no—
211C2×C8—42no—
212C2×C8—42no—
213C2×C8—42no—
214C2×C8—42no—
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—81no—
225C2×C8—81no—
226C2×C8—81no—
227C2×C8—81no—
228C2×C8—81no—
229C4⋊C4—21no—
230C4⋊C4—21no—
231C4⋊C4—42no—
232C4⋊C4—41no—
233C4⋊C4—41no—
234C4⋊C4—41no—
235C4⋊C4—41no—
236C4⋊C4—42no—
237C4⋊C4—42no—
238C4⋊C4—81no—
239C4⋊C4—82no—
240C4⋊C4—81no—
241C4⋊C4—81no—
242C4⋊C4—82no—
243C4⋊C4—81no—
244C4⋊C4—81no—
245C4⋊C4—81no—
246C4⋊C4—81no—
247C22⋊C4—42no—
248C22⋊C4—81no—
249C42—21no—
250C42—21no—
251C42—21no—
252C42—21no—
253C42—41no—
254C42—41no—
255C42—41no—
256C42—41no—
257C42—41no—
258C42—41no—
259C23—21no—
260C23—21no—
261C23—21no—
262C23—21no—
263C23—21no—
264C23—21no—
265C23#38611yescharacteristic
266C23C4×Q1611yescharacteristic
267C23#44111yescharacteristic
268Q8—42no—
269Q8—81no—
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—21no—
280C2×C4—21no—
281C2×C4—21no—
282C2×C4—21no—
283C2×C4—21no—
284C2×C4—21no—
285C2×C4—21no—
286C2×C4—22no—
287C2×C4—21no—
288C2×C4—21no—
289C2×C4—21no—
290C2×C4—21no—
291C2×C4—42no—
292C2×C4—42no—
293C2×C4—42no—
294C2×C4—42no—
295C2×C4—41no—
296C2×C4—41no—
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—82no—
307C2×C4—82no—
308C2×C4—82no—
309C2×C4—162no—
310C2×C4—162no—
311C2×C4C2×#10011yescharacteristic
312C2×C4#41311yescharacteristic
313C2×C4#45111yescharacteristic
314C2×C4#45211yescharacteristic
315C2×C4#45311yescharacteristic
316C2×C4#45411yescharacteristic
317C2×C4#45811yescharacteristic
318C2×C4#45811yescharacteristic
319C2×C4#46211yescharacteristic
320C2×C4#46411yescharacteristic
321C2×C4#47111yescharacteristic
322C2×C4#47211yescharacteristic
323C8—42no—
324C8—82no—
325C8—82no—
326C8—81no—
327C8—81no—
328C8—81no—
329C8—161no—
330C22—21no—
331C22—21no—
332C22—21no—
333C22—21no—
334C22—21no—
335C22—21no—
336C22—41no—
337C22—41no—
338C22—41no—
339C22—41no—
340C22#187811yescharacteristic
341C22#188311yescharacteristic
342C22#189011yescharacteristic
343C22#190011yescharacteristic
344C22#190211yescharacteristic
345C22#190711yescharacteristic
346C22#191511yescharacteristic
347C4—21no—
348C4—21no—
349C4—21no—
350C4—21no—
351C4—21no—
352C4—21no—
353C4—41no—
354C4—42no—
355C4—41no—
356C4—81no—
357C4—81no—
358C4—81no—
359C4—162no—
360C4#161411yescharacteristic
361C4#161511yescharacteristic
362C4#161711yescharacteristic
363C4#161811yescharacteristic
364C2—41no—
365C2—41no—
366C2#904611yescharacteristic
367C2#924811yescharacteristic
368C2#927011yescharacteristic
369C2#933911yescharacteristic
370C2#936011yescharacteristic
371C2#950711yescharacteristic
372C2#952211yescharacteristic
subgroupG/H
center, socleC23#386
commutatorC23×C4C22×C4
Frattini, cosocleC22×C42C23
prime radicalC24C4×D4
LCSG ▷ C23×C4 ▷ C23 ▷ C1
UCSC1 ◁ C23 ◁ 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
#276482
#222892
#113542
#87242
#76172
#70452
#70342