Finite Group Databasebeta

← #101707

Subgroups — #101707

Order 512.

389 conjugacy classes of subgroups; only the 86 normal subgroups are shown.

normal subgroups only

This group has 389 conjugacy classes of subgroups — too many to draw. Only its 86 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#29574C211yesmaximal, characteristic
2#21269C211yesmaximal, characteristic
3#9891C211yesmaximal, characteristic, complemented
4#7660C211yesmaximal, characteristic, complemented
5#7614C211yesmaximal, characteristic
6#7030C211yesmaximal, characteristic, complemented
7#7027C211yesmaximal, characteristic, complemented
8C2×#433—21no—
9#2859C2211yescharacteristic
10#2293C2211yescharacteristic
11#2100C411yescharacteristic
12#2094C411yescharacteristic
13#2084C411yescharacteristic
14#2071C411yescharacteristic
15#1955—21no—
16#1941—21no—
17#1940—21no—
18#1934—21no—
19#1923—22no—
20#1920—21no—
21#1864—21no—
22#1831C2211yescharacteristic
23#1767C2211yescharacteristic
24C4×C4⋊C8C2211yescharacteristic
25#1750—21no—
26#1744C2211yescharacteristic
27C4×#97C2211yescharacteristic
28#1295—21no—
29#1295—21no—
30#1271—21no—
31#1271—22no—
32#1270—21no—
33C2×#125—21no—
34C2×C4×D4D411yescharacteristic
35C2×#117—21no—
36#433—42no—
37#433—42no—
38C2×#106—21no—
39C2×#106—21no—
40C2×C4⋊C8—21no—
41C2×C4⋊C8—22no—
42C2×C4⋊C8—21no—
43C2×C4⋊C8—21no—
44C2×#102—21no—
45C2×#102—21no—
46C2×C22⋊C8—22no—
47C2×C22⋊C8—22no—
48C2×#97—21no—
49C2×#97—21no—
50C2×#97D411yescharacteristic
51C2×#97D411yescharacteristic
52C2×C4×C8—21no—
53C2×C4×C8—21no—
54C2×C4×C8—21no—
55#390C2×C411yescharacteristic
56#390D411yescharacteristic
57#389—21no—
58#389—21no—
59#389—21no—
60#389—21no—
61#389C2×C411yescharacteristic
62#389D411yescharacteristic
63#388—21no—
64#388—21no—
65#384C2×C411yescharacteristic
66#384C2×C411yescharacteristic
67#382—21no—
68C4×C4⋊C4D411yescharacteristic
69C4×C22⋊C4—21no—
70C43C2311yescharacteristic
71#340—21no—
72#340—21no—
73#340—21no—
74#340—21no—
75#336—22no—
76#336—21no—
77#336—21no—
78#336—21no—
79#336—21no—
80#336C2×C411yescharacteristic
81#336C2×C411yescharacteristic
82#336D412yes—
83C22×D4C22⋊C411yescharacteristic
84C23×C4—21no—
85C22×C8—21no—
86C22×C8—21no—
87C22×C8—21no—
88C22×C8—21no—
89C22×C8—22no—
90C22×C8—41no—
91C22×C8—41no—
92#125—81no—
93#125—81no—
94C4×D4—41no—
95C4×D4—41no—
96#117—41no—
97#117—41no—
98#117—41no—
99#117—41no—
100C2×C4⋊C4—21no—
101C2×C4⋊C4—21no—
102C2×C4⋊C4—21no—
103C2×C4⋊C4—21no—
104C2×C4⋊C4—21no—
105C2×C4⋊C4—21no—
106C2×C4⋊C4—21no—
107C2×C4⋊C4—21no—
108C2×C4⋊C4—41no—
109C2×C4⋊C4—41no—
110C2×C4⋊C4C22⋊C411yescharacteristic
111C2×C4⋊C4C22⋊C411yescharacteristic
112C2×C4⋊C4C22⋊C411yescharacteristic
113C2×C22⋊C4—21no—
114C2×C22⋊C4—21no—
115C2×C22⋊C4—21no—
116C2×C22⋊C4—21no—
117C2×C22⋊C4—21no—
118C2×C42—21no—
119C2×C42C22×C411yescharacteristic
120C2×C42C2×D411yescharacteristic
121C2×C42C2×D411yescharacteristic
122C2×C42C2×D411yescharacteristic
123C2×C42C2×D411yescharacteristic
124C2×C42#4111yescharacteristic
125C2×C42#4111yescharacteristic
126#106—42no—
127#106—42no—
128C4⋊C8—22no—
129C4⋊C8—41no—
130C4⋊C8—41no—
131C4⋊C8—41no—
132C4⋊C8—44no—
133C4⋊C8—81no—
134#102—42no—
135#102—42no—
136C22⋊C8—44no—
137C22⋊C8—44no—
138#97—21no—
139#97—21no—
140#97—21no—
141#97—41no—
142#97—41no—
143#97—41no—
144#97—41no—
145#97SD1612yes—
146C4×C8—42no—
147C4×C8—42no—
148C4×C8—41no—
149C4×C8—41no—
150#95—21no—
151#95—21no—
152#95—21no—
153#95—41no—
154#95—41no—
155C24—21no—
156C2×D4—21no—
157C2×D4—21no—
158C2×D4—41no—
159C2×D4—41no—
160C22×C4—21no—
161C22×C4—21no—
162C22×C4—21no—
163C22×C4—42no—
164C22×C4—42no—
165C22×C4—41no—
166C22×C4—41no—
167C22×C4—41no—
168C22×C4—41no—
169C22×C4C2×C22⋊C411yescharacteristic
170C22×C4C4×D411yescharacteristic
171C22×C4C4×D411yescharacteristic
172C22×C4#12011yescharacteristic
173C22×C4#12111yescharacteristic
174C22×C4#12111yescharacteristic
175C22×C4#12311yescharacteristic
176C2×C8—22no—
177C2×C8—22no—
178C2×C8—22no—
179C2×C8—41no—
180C2×C8—41no—
181C2×C8—41no—
182C2×C8—41no—
183C2×C8—41no—
184C2×C8—41no—
185C2×C8—41no—
186C2×C8—42no—
187C2×C8—42no—
188C2×C8—42no—
189C2×C8—41no—
190C2×C8—41no—
191C2×C8—42no—
192C2×C8—81no—
193C2×C8—81no—
194C2×C8—81no—
195C2×C8—81no—
196C2×C8—81no—
197C4⋊C4—21no—
198C4⋊C4—21no—
199C4⋊C4—41no—
200C4⋊C4—41no—
201C4⋊C4—41no—
202C4⋊C4—41no—
203C4⋊C4—41no—
204C4⋊C4—41no—
205C4⋊C4—41no—
206C4⋊C4—41no—
207C4⋊C4—41no—
208C4⋊C4—41no—
209C4⋊C4—41no—
210C4⋊C4—41no—
211C4⋊C4—81no—
212C4⋊C4—81no—
213C4⋊C4—81no—
214C4⋊C4—82no—
215C4⋊C4—81no—
216C4⋊C4—81no—
217C22⋊C4—42no—
218C22⋊C4—41no—
219C22⋊C4—41no—
220C22⋊C4—41no—
221C22⋊C4—41no—
222C22⋊C4—41no—
223C22⋊C4—41no—
224C22⋊C4—41no—
225C22⋊C4—41no—
226C42—21no—
227C42—21no—
228C42—21no—
229C42—21no—
230C42—21no—
231C42—21no—
232C42—41no—
233C42—41no—
234C42—41no—
235C42—41no—
236C42—41no—
237C42C2×SD1611yescharacteristic
238C42#13511yescharacteristic
239C42AGL(1,ℤ/8)11yescharacteristic
240C42#13711yescharacteristic
241C23—41no—
242C23—41no—
243C23—41no—
244C23—41no—
245C23—41no—
246C23—41no—
247C23—41no—
248C23#38611yescharacteristic
249D4—42no—
250D4—81no—
251C2×C4—21no—
252C2×C4—21no—
253C2×C4—21no—
254C2×C4—21no—
255C2×C4—21no—
256C2×C4—21no—
257C2×C4—21no—
258C2×C4—21no—
259C2×C4—21no—
260C2×C4—21no—
261C2×C4—21no—
262C2×C4—21no—
263C2×C4—21no—
264C2×C4—21no—
265C2×C4—21no—
266C2×C4—41no—
267C2×C4—41no—
268C2×C4—41no—
269C2×C4—41no—
270C2×C4—41no—
271C2×C4—41no—
272C2×C4—41no—
273C2×C4—41no—
274C2×C4—41no—
275C2×C4—82no—
276C2×C4—82no—
277C2×C4—81no—
278C2×C4—81no—
279C2×C4—81no—
280C2×C4—81no—
281C2×C4—81no—
282C2×C4—81no—
283C2×C4#42111yescharacteristic
284C2×C4#42111yescharacteristic
285C2×C4C4×SD1611yescharacteristic
286C2×C4#44011yescharacteristic
287C2×C4#44311yescharacteristic
288C2×C4#44411yescharacteristic
289C2×C4#45011yescharacteristic
290C2×C4#45211yescharacteristic
291C2×C4#46111yescharacteristic
292C2×C4#46311yescharacteristic
293C2×C4#47111yescharacteristic
294C2×C4#47211yescharacteristic
295C8—42no—
296C8—42no—
297C8—42no—
298C8—82no—
299C8—81no—
300C8—82no—
301C8—161no—
302C22—81no—
303C22—81no—
304C22—81no—
305C22—81no—
306C22—81no—
307C22—81no—
308C22—81no—
309C22#187611yescharacteristic
310C22#187711yescharacteristic
311C22#188411yescharacteristic
312C22#189711yescharacteristic
313C22#190611yescharacteristic
314C22#190711yescharacteristic
315C22#191511yescharacteristic
316C4—21no—
317C4—21no—
318C4—21no—
319C4—21no—
320C4—21no—
321C4—21no—
322C4—21no—
323C4—21no—
324C4—21no—
325C4—21no—
326C4—21no—
327C4—21no—
328C4—41no—
329C4—81no—
330C4—81no—
331C4—81no—
332C4—161no—
333C4—161no—
334C2—161no—
335C2#905511yescharacteristic
336C2#921311yescharacteristic
337C2#923511yescharacteristic
338C2#926611yescharacteristic
339C2#926911yescharacteristic
340C2#939011yescharacteristic
341C2#943211yescharacteristic
subgroupG/H
center, socleC23#386
commutatorC2×C42C22×C4
Frattini, cosocleC43C23
prime radicalC22×D4C22⋊C4
LCSG ▷ C2×C42 ▷ C23 ▷ C1
UCSC1 ◁ C23 ◁ C43 ◁ G
LECSG ▷ C43 ▷ C23 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #9891C2semidirect
4. #7660C2semidirect
6. #7030C2semidirect
7. #7027C2semidirect

Maximal subgroups

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

subgroupindex
#295742
#212692
#98912
#76602
#76142
#70302
#70272