Finite Group Databasebeta

← #101327

Subgroups — #101327

Order 512.

409 conjugacy classes of subgroups; only the 88 normal subgroups are shown.

normal subgroups only

This group has 409 conjugacy classes of subgroups — too many to draw. Only its 88 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#23781C211yesmaximal, characteristic
2#22702C211yesmaximal, characteristic
3#7653C211yesmaximal, characteristic, complemented
4#7622C211yesmaximal, characteristic, complemented
5#7594C211yesmaximal, characteristic
6#7049C211yesmaximal, characteristic, complemented
7#7041C211yesmaximal, characteristic, complemented
8C2×#435—21no—
9C4×C4⋊Q8C2211yescharacteristic
10#2293C2211yescharacteristic
11#2099C411yescharacteristic
12#2093C411yescharacteristic
13#2087—21no—
14#2065—21no—
15#2058C411yescharacteristic
16#2023C411yescharacteristic
17#1940—21no—
18#1851—21no—
19#1850—21no—
20#1839C2211yescharacteristic
21#1831C2211yescharacteristic
22#1831C2211yescharacteristic
23#1798—21no—
24#1776—21no—
25#1767C2211yescharacteristic
26#1767C2211yescharacteristic
27#1762—21no—
28#1729—21no—
29#1272—21no—
30#1272—21no—
31#1271—21no—
32#1271—21no—
33#1271—21no—
34#1270—21no—
35C22×M16—21no—
36C23×C8—21no—
37C2×C4⋊Q8C2×C411yescharacteristic
38C2×C4×Q8—21no—
39C2×C4×D4D411yescharacteristic
40C2×#117—21no—
41#435—42no—
42#435—42no—
43C2×#107—21no—
44C2×#106—21no—
45C2×C4⋊C8—21no—
46C2×C4⋊C8—21no—
47C2×C4⋊C8—21no—
48C2×C4⋊C8—21no—
49C2×C4⋊C8—21no—
50C2×C4⋊C8—21no—
51C2×#102—21no—
52C2×#102—21no—
53C2×C22⋊C8—21no—
54C2×C22⋊C8—21no—
55C2×#97—21no—
56C2×#97—21no—
57#391—21no—
58#390C2×C411yescharacteristic
59#390D411yescharacteristic
60#389—21no—
61#389—21no—
62#389C2×C411yescharacteristic
63#389D411yescharacteristic
64#388—21no—
65#388—21no—
66#384C2×C411yescharacteristic
67C4×C4⋊C4—21no—
68C4×C4⋊C4—21no—
69C4×C4⋊C4D411yescharacteristic
70C4×C22⋊C4—21no—
71C43C2311yescharacteristic
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#336D411yescharacteristic
89#336D411yescharacteristic
90C22×Q8—21no—
91C22×D4C22⋊C411yescharacteristic
92C23×C4—21no—
93C2×M16—42no—
94C2×M16—42no—
95C2×M16—81no—
96C22×C8—21no—
97C22×C8—21no—
98C22×C8—21no—
99C22×C8—21no—
100C22×C8—21no—
101C22×C8—21no—
102C22×C8—42no—
103C22×C8—42no—
104C22×C8—41no—
105C22×C8—41no—
106C22×C8—81no—
107C4⋊Q8—42no—
108C4×Q8—82no—
109C4×D4—21no—
110C4×D4—21no—
111C4×D4SD1612yes—
112C4×D4Q1612yes—
113#117—81no—
114#117—81no—
115C2×C4⋊C4—21no—
116C2×C4⋊C4—21no—
117C2×C4⋊C4—21no—
118C2×C4⋊C4—21no—
119C2×C4⋊C4—21no—
120C2×C4⋊C4—21no—
121C2×C4⋊C4—21no—
122C2×C4⋊C4—41no—
123C2×C4⋊C4C22⋊C411yescharacteristic
124C2×C4⋊C4C22⋊C411yescharacteristic
125C2×C4⋊C4C22⋊C411yescharacteristic
126C2×C22⋊C4—21no—
127C2×C22⋊C4—21no—
128C2×C22⋊C4—21no—
129C2×C22⋊C4—21no—
130C2×C22⋊C4—21no—
131C2×C42—21no—
132C2×C42—41no—
133C2×C42C22×C411yescharacteristic
134C2×C42C2×D411yescharacteristic
135C2×C42C2×D411yescharacteristic
136C2×C42C2×D411yescharacteristic
137C2×C42C2×D411yescharacteristic
138C2×C42#4111yescharacteristic
139C2×C42#4111yescharacteristic
140#107—42no—
141#106—42no—
142C4⋊C8—42no—
143C4⋊C8—42no—
144C4⋊C8—41no—
145C4⋊C8—41no—
146C4⋊C8—81no—
147C4⋊C8—81no—
148C4⋊C8—81no—
149#102—42no—
150#102—42no—
151C22⋊C8—42no—
152C22⋊C8—42no—
153#97—41no—
154#97—41no—
155#97—41no—
156#97—41no—
157#95—21no—
158#95—21no—
159#95—21no—
160#95—41no—
161#95—41no—
162C24—21no—
163C2×Q8—22no—
164C2×Q8—41no—
165C2×Q8—81no—
166C2×Q8—81no—
167C2×D4—21no—
168C2×D4—41no—
169C2×D4—41no—
170C2×D4#10312yes—
171C22×C4—21no—
172C22×C4—22no—
173C22×C4—22no—
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×C4C2×C22⋊C411yescharacteristic
183C22×C4C4×D411yescharacteristic
184C22×C4C4×D411yescharacteristic
185C22×C4#12011yescharacteristic
186C22×C4#12111yescharacteristic
187C22×C4#12111yescharacteristic
188C22×C4#12311yescharacteristic
189M16—84no—
190C2×C8—22no—
191C2×C8—22no—
192C2×C8—41no—
193C2×C8—41no—
194C2×C8—41no—
195C2×C8—41no—
196C2×C8—41no—
197C2×C8—41no—
198C2×C8—41no—
199C2×C8—41no—
200C2×C8—41no—
201C2×C8—41no—
202C2×C8—41no—
203C2×C8—41no—
204C2×C8—41no—
205C2×C8—41no—
206C2×C8—41no—
207C2×C8—41no—
208C2×C8—84no—
209C2×C8—81no—
210C2×C8—81no—
211C2×C8—81no—
212C2×C8—81no—
213C2×C8—81no—
214C2×C8—81no—
215C4⋊C4—21no—
216C4⋊C4—22no—
217C4⋊C4—22no—
218C4⋊C4—41no—
219C4⋊C4—41no—
220C4⋊C4—41no—
221C4⋊C4—41no—
222C4⋊C4—41no—
223C4⋊C4—41no—
224C4⋊C4—81no—
225C4⋊C4—81no—
226C4⋊C4—81no—
227C4⋊C4—81no—
228C4⋊C4—82no—
229C4⋊C4#10312yes—
230C22⋊C4—22no—
231C22⋊C4—22no—
232C22⋊C4—41no—
233C22⋊C4—41no—
234C22⋊C4—81no—
235C22⋊C4—81no—
236C22⋊C4—81no—
237C42—21no—
238C42—21no—
239C42—41no—
240C42—41no—
241C42—41no—
242C42—41no—
243C42—41no—
244C42—41no—
245C42—41no—
246C42—82no—
247C42C2×SD1611yescharacteristic
248C42C2×Q1611yescharacteristic
249C42AGL(1,ℤ/8)11yescharacteristic
250C42#13711yescharacteristic
251C23—22no—
252C23—41no—
253C23—41no—
254C23—41no—
255C23—41no—
256C23—41no—
257C23—41no—
258C23#38611yescharacteristic
259Q8—82no—
260Q8—82no—
261D4—42no—
262D4—81no—
263C2×C4—21no—
264C2×C4—21no—
265C2×C4—21no—
266C2×C4—21no—
267C2×C4—21no—
268C2×C4—21no—
269C2×C4—21no—
270C2×C4—21no—
271C2×C4—22no—
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—42no—
282C2×C4—42no—
283C2×C4—42no—
284C2×C4—42no—
285C2×C4—41no—
286C2×C4—41no—
287C2×C4—41no—
288C2×C4—42no—
289C2×C4—41no—
290C2×C4—41no—
291C2×C4—41no—
292C2×C4—41no—
293C2×C4—41no—
294C2×C4—81no—
295C2×C4—81no—
296C2×C4—81no—
297C2×C4—81no—
298C2×C4—81no—
299C2×C4C2×#10311yescharacteristic
300C2×C4#41711yescharacteristic
301C2×C4#45011yescharacteristic
302C2×C4#45111yescharacteristic
303C2×C4#47111yescharacteristic
304C2×C4#47311yescharacteristic
305C2×C4#48311yescharacteristic
306C2×C4#48411yescharacteristic
307C8—42no—
308C8—42no—
309C8—81no—
310C8—81no—
311C8—81no—
312C8—81no—
313C8—161no—
314C8—161no—
315C22—42no—
316C22—42no—
317C22—42no—
318C22—81no—
319C22—81no—
320C22—81no—
321C22—81no—
322C22#187211yescharacteristic
323C22#188411yescharacteristic
324C22#189111yescharacteristic
325C22#190311yescharacteristic
326C22#190411yescharacteristic
327C22#190611yescharacteristic
328C22#190811yescharacteristic
329C4—21no—
330C4—21no—
331C4—21no—
332C4—21no—
333C4—21no—
334C4—21no—
335C4—21no—
336C4—21no—
337C4—41no—
338C4—41no—
339C4—41no—
340C4—42no—
341C4—81no—
342C4—82no—
343C4—81no—
344C4—161no—
345C2—82no—
346C2#903311yescharacteristic
347C2#916911yescharacteristic
348C2#917511yescharacteristic
349C2#941711yescharacteristic
350C2#943111yescharacteristic
351C2#952711yescharacteristic
352C2#953211yescharacteristic
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. #7653C2semidirect
4. #7622C2semidirect
6. #7049C2semidirect
7. #7041C2semidirect

Maximal subgroups

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

subgroupindex
#237812
#227022
#76532
#76222
#75942
#70492
#70412