Finite Group Databasebeta

← #103565

Subgroups — #103565

Order 512.

547 conjugacy classes of subgroups; only the 96 normal subgroups are shown.

normal subgroups only

This group has 547 conjugacy classes of subgroups — too many to draw. Only its 96 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#29755C211yesmaximal, characteristic
2#27973C211yesmaximal, characteristic
3C2×#2083C211yesmaximal, characteristic, complemented
4#9879C211yesmaximal, characteristic, complemented
5#8724C211yesmaximal, characteristic, complemented
6#7664C211yesmaximal, characteristic
7#7029C211yesmaximal, characteristic, complemented
8C2×#481—21no—
9C22×#107—21no—
10#2856C2211yescharacteristic
11C2×#389C2211yescharacteristic
12C2×#384C2211yescharacteristic
13#2084—21no—
14#2083—22no—
15#2083—22no—
16#2071—22no—
17#2065—22no—
18#2041—21no—
19#1957—21no—
20#1874—22no—
21#1874—21no—
22#1834—21no—
23#1797C2211yescharacteristic
24#1797C2211yescharacteristic
25C4×C22⋊C8C2211yescharacteristic
26C2×#340—21no—
27C2×#336C2211yescharacteristic
28#1296—22no—
29#1280—22no—
30#1270—22no—
31C23×C8—21no—
32C2×#127D411yescharacteristic
33C2×#121—21no—
34C2×#121—21no—
35C2×#121—21no—
36C22×C4⋊C4—21no—
37C22×C4⋊C4—21no—
38C22×C4⋊C4—21no—
39C22×C4⋊C4—21no—
40C22×C4⋊C4D411yescharacteristic
41C22×C4⋊C4D411yescharacteristic
42C22×C42C2311yescharacteristic
43#481—42no—
44#481—42no—
45C2×#107—24no—
46C2×#107—42no—
47C2×#107—42no—
48C2×#106—22no—
49C2×C4⋊C8—21no—
50C2×C4⋊C8D412yes—
51C2×#102—22no—
52C2×#102—22no—
53C2×#102—24no—
54C2×C22⋊C8—21no—
55C2×C22⋊C8—21no—
56C2×C22⋊C8D412yes—
57C2×C4×C8—21no—
58#390—21no—
59#390—21no—
60#390—21no—
61#389—21no—
62#389—21no—
63#389—21no—
64#389—21no—
65#384—21no—
66#384—21no—
67#384—21no—
68#384D412yes—
69#384D411yescharacteristic
70C2×#95—21no—
71#340—22no—
72#340—24no—
73#340—24no—
74#336—21no—
75#336—21no—
76#336—21no—
77#336—21no—
78#336D412yes—
79C22×D4—21no—
80C22×D4—21no—
81C22×D4—21no—
82C22×D4—41no—
83C23×C4—21no—
84C23×C4—21no—
85C23×C4—41no—
86C23×C4C2×D411yescharacteristic
87C23×C4C2×D411yescharacteristic
88C23×C4#4111yescharacteristic
89C22×C8—24no—
90C22×C8—22no—
91C22×C8—22no—
92C22×C8—22no—
93C22×C8—42no—
94C22×C8—41no—
95C22×C8—41no—
96C22×C8—41no—
97#127—42no—
98#121—41no—
99#121—41no—
100#121—42no—
101#121—41no—
102#121—41no—
103#121—42no—
104#121—82no—
105C2×C4⋊C4—21no—
106C2×C4⋊C4—21no—
107C2×C4⋊C4—21no—
108C2×C4⋊C4—21no—
109C2×C4⋊C4—21no—
110C2×C4⋊C4—22no—
111C2×C4⋊C4—22no—
112C2×C4⋊C4—21no—
113C2×C4⋊C4—21no—
114C2×C4⋊C4—21no—
115C2×C4⋊C4—21no—
116C2×C4⋊C4—22no—
117C2×C4⋊C4—21no—
118C2×C4⋊C4—42no—
119C2×C4⋊C4—41no—
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—82no—
129C2×C4⋊C4—82no—
130C2×C4⋊C4—82no—
131C2×C4⋊C4D812yes—
132C2×C22⋊C4—41no—
133C2×C22⋊C4—41no—
134C2×C22⋊C4—41no—
135C2×C42—42no—
136C2×C42C2×D411yescharacteristic
137C2×C42C2×D411yescharacteristic
138C2×C42C2×D411yescharacteristic
139C2×C42C2×D411yescharacteristic
140#107—48no—
141#106—44no—
142C4⋊C8—42no—
143C4⋊C8—42no—
144#102—44no—
145#102—84no—
146#102—82no—
147C22⋊C8—22no—
148C22⋊C8—22no—
149C22⋊C8—41no—
150C22⋊C8—41no—
151C22⋊C8—42no—
152C4×C8—42no—
153#95—41no—
154#95—41no—
155C24—41no—
156C24#12111yescharacteristic
157C2×D4—42no—
158C2×D4—42no—
159C2×D4—41no—
160C2×D4—41no—
161C2×D4—42no—
162C2×D4—42no—
163C2×D4—42no—
164C2×D4—42no—
165C2×D4—81no—
166C2×D4—81no—
167C2×D4—82no—
168C2×D4—81no—
169C2×D4—81no—
170C2×D4—82no—
171C22×C4—22no—
172C22×C4—22no—
173C22×C4—22no—
174C22×C4—22no—
175C22×C4—22no—
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—42no—
185C22×C4—81no—
186C22×C4—81no—
187C22×C4—82no—
188C22×C4—82no—
189C22×C4—81no—
190C22×C4—81no—
191C22×C4—81no—
192C22×C4—81no—
193C22×C4#12011yescharacteristic
194C22×C4#12011yescharacteristic
195C22×C4#12011yescharacteristic
196C22×C4#12111yescharacteristic
197C22×C4#12111yescharacteristic
198C22×C4#12711yescharacteristic
199C22×C4C2×D811yescharacteristic
200C22×C4#13511yescharacteristic
201C22×C4#13512yes—
202C22×C4AGL(1,ℤ/8)11yescharacteristic
203C22×C4AGL(1,ℤ/8)11yescharacteristic
204C22×C4#13712yes—
205C2×C8—24no—
206C2×C8—42no—
207C2×C8—44no—
208C2×C8—44no—
209C2×C8—42no—
210C2×C8—42no—
211C2×C8—42no—
212C2×C8—42no—
213C2×C8—42no—
214C2×C8—42no—
215C2×C8—42no—
216C2×C8—81no—
217C2×C8—81no—
218C2×C8—81no—
219C4⋊C4—22no—
220C4⋊C4—22no—
221C4⋊C4—44no—
222C4⋊C4—42no—
223C4⋊C4—41no—
224C4⋊C4—41no—
225C4⋊C4—41no—
226C4⋊C4—42no—
227C4⋊C4—42no—
228C4⋊C4—41no—
229C4⋊C4—42no—
230C4⋊C4—42no—
231C4⋊C4—82no—
232C4⋊C4—81no—
233C4⋊C4—82no—
234C4⋊C4—82no—
235C4⋊C4—81no—
236C4⋊C4—81no—
237C4⋊C4—82no—
238C4⋊C4—82no—
239C22⋊C4—82no—
240C22⋊C4—81no—
241C22⋊C4—81no—
242C22⋊C4—81no—
243C22⋊C4—81no—
244C42—22no—
245C42—22no—
246C42—41no—
247C42—41no—
248C23—21no—
249C23—21no—
250C23—21no—
251C23—21no—
252C23—81no—
253C23—81no—
254C23—81no—
255C23—81no—
256C23—81no—
257C23—81no—
258C23—81no—
259C23#39211yescharacteristic
260C23#46412yes—
261C23#46611yescharacteristic
262C23#46911yescharacteristic
263C23#47112yes—
264D4—84no—
265D4—82no—
266D4—82no—
267D4—162no—
268D4—162no—
269D4—162no—
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—22no—
279C2×C4—22no—
280C2×C4—22no—
281C2×C4—22no—
282C2×C4—21no—
283C2×C4—21no—
284C2×C4—21no—
285C2×C4—42no—
286C2×C4—42no—
287C2×C4—44no—
288C2×C4—42no—
289C2×C4—82no—
290C2×C4—81no—
291C2×C4—81no—
292C2×C4—82no—
293C2×C4—81no—
294C2×C4—82no—
295C2×C4—82no—
296C2×C4—81no—
297C2×C4—81no—
298C2×C4—82no—
299C2×C4—81no—
300C2×C4—81no—
301C2×C4—81no—
302C2×C4—162no—
303C2×C4—162no—
304C2×C4—162no—
305C2×C4#44711yescharacteristic
306C2×C4#44711yescharacteristic
307C2×C4#44911yescharacteristic
308C2×C4#44911yescharacteristic
309C2×C4#45212yes—
310C2×C4#45612yes—
311C2×C4#45812yes—
312C2×C4#47212yes—
313C2×C4#49511yescharacteristic
314C2×C4#49711yescharacteristic
315C8—44no—
316C8—82no—
317C8—82no—
318C8—161no—
319C22—22no—
320C22—22no—
321C22—22no—
322C22—21no—
323C22—21no—
324C22—21no—
325C22—21no—
326C22—22no—
327C22—161no—
328C22—161no—
329C22—161no—
330C22—161no—
331C22—161no—
332C22—161no—
333C22—161no—
334C22#168212yes—
335C22#170212yes—
336C22#199911yescharacteristic
337C22#200511yescharacteristic
338C22#200811yescharacteristic
339C22#201511yescharacteristic
340C22#201811yescharacteristic
341C22#201911yescharacteristic
342C22#202211yescharacteristic
343C4—22no—
344C4—21no—
345C4—21no—
346C4—22no—
347C4—21no—
348C4—21no—
349C4—41no—
350C4—82no—
351C4—161no—
352C4—161no—
353C4—162no—
354C4#160912yes—
355C4#161012yes—
356C2—22no—
357C2—22no—
358C2—321no—
359C2#991911yescharacteristic
360C2#995811yescharacteristic
361C2#999211yescharacteristic
362C2#1013811yescharacteristic
363C2#1016411yescharacteristic
364C2#1017611yescharacteristic
365C2#1026311yescharacteristic
subgroupG/H
center, socleC23#392
commutator, Frattini, cosocleC22×C42C23
prime radical#2856C22
LCSG ▷ C22×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
3. C2×#2083C2semidirect
4. #9879C2semidirect
5. #8724C2semidirect
7. #7029C2semidirect

Maximal subgroups

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

subgroupindex
#297552
#279732
C2×#20832
#98792
#87242
#76642
#70292