Finite Group Databasebeta

← #101371

Subgroups — #101371

Order 512.

410 conjugacy classes of subgroups; only the 76 normal subgroups are shown.

normal subgroups only

This group has 410 conjugacy classes of subgroups — too many to draw. Only its 76 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#27587C211yesmaximal, characteristic
2#21905C211yesmaximal, characteristic
3#8722C211yesmaximal, characteristic, complemented
4#8710C211yesmaximal, characteristic, complemented
5#7612C211yesmaximal, characteristic
6#7038C211yesmaximal, characteristic, complemented
7#7021C211yesmaximal, characteristic, complemented
8#2297C2211yescharacteristic
9#2285C2211yescharacteristic
10#2096C411yescharacteristic
11#2093C411yescharacteristic
12#2072C411yescharacteristic
13#2071C411yescharacteristic
14#1928—21no—
15#1926—21no—
16#1924—22no—
17#1923—22no—
18#1854—21no—
19#1817C2211yescharacteristic
20#1798—21no—
21#1797C2211yescharacteristic
22#1788—21no—
23#1780—21no—
24#1749C2211yescharacteristic
25#1749C2211yescharacteristic
26C4×C22⋊C8C2211yescharacteristic
27#1295—22no—
28#1279—22no—
29#1272—21no—
30#1270—21no—
31C22×#41D411yescharacteristic
32C2×#117—21no—
33C2×#117D411yescharacteristic
34C22×C42C2311yescharacteristic
35C2×#107—21no—
36C2×#106—21no—
37C2×C4⋊C8—21no—
38C2×C4⋊C8—22no—
39C2×#103—21no—
40C2×#103—21no—
41C2×#102—21no—
42C2×#102—21no—
43C2×C22⋊C8—21no—
44C2×C22⋊C8—21no—
45C2×C22⋊C8—21no—
46C2×C22⋊C8—21no—
47C2×C22⋊C8—22no—
48C2×C22⋊C8—22no—
49C2×C22⋊C8—21no—
50C2×#97—21no—
51C2×#97D412yes—
52C2×C4×C8D412yes—
53#391C2×C411yescharacteristic
54#391D411yescharacteristic
55#390C2×C411yescharacteristic
56#390D411yescharacteristic
57#386—21no—
58#386—21no—
59#384C2×C411yescharacteristic
60#384C2×C411yescharacteristic
61#382—21no—
62#380—21no—
63#380—21no—
64C4×C4⋊C4—21no—
65C4×C4⋊C4—21no—
66C4×C22⋊C4—21no—
67C4×C22⋊C4—21no—
68#340—21no—
69#340—21no—
70#336—22no—
71#336—21no—
72#336—21no—
73#336—21no—
74#336—21no—
75#336—21no—
76#336C2×C411yescharacteristic
77#336C2×C411yescharacteristic
78C2×#41—41no—
79C2×#41—41no—
80C2×#41—42no—
81C2×#41—81no—
82C22×Q8C22⋊C411yescharacteristic
83C22×D4—21no—
84C22×D4C22⋊C411yescharacteristic
85C23×C4—21no—
86C23×C4C2×D411yescharacteristic
87C23×C4#4111yescharacteristic
88C23×C4#4111yescharacteristic
89C22×C8—21no—
90C22×C8—21no—
91C22×C8—22no—
92C22×C8—22no—
93C22×C8—41no—
94C22×C8—41no—
95#117—81no—
96#117—82no—
97C2×C4⋊C4—21no—
98C2×C4⋊C4—21no—
99C2×C4⋊C4—21no—
100C2×C4⋊C4—21no—
101C2×C4⋊C4—21no—
102C2×C4⋊C4—21no—
103C2×C4⋊C4C22⋊C411yescharacteristic
104C2×C4⋊C4C22⋊C411yescharacteristic
105C2×C22⋊C4—21no—
106C2×C22⋊C4—21no—
107C2×C22⋊C4—21no—
108C2×C22⋊C4—21no—
109C2×C22⋊C4—21no—
110C2×C22⋊C4—41no—
111C2×C42—21no—
112C2×C42—42no—
113C2×C42—41no—
114C2×C42C22×C411yescharacteristic
115C2×C42C2×D411yescharacteristic
116C2×C42C2×D411yescharacteristic
117C2×C42C2×D411yescharacteristic
118#107—81no—
119#106—81no—
120C4⋊C8—42no—
121C4⋊C8—82no—
122#103—42no—
123#103—81no—
124#102—42no—
125#102—81no—
126C22⋊C8—22no—
127C22⋊C8—41no—
128C22⋊C8—41no—
129C22⋊C8—41no—
130C22⋊C8—41no—
131C22⋊C8—41no—
132C22⋊C8—42no—
133C22⋊C8—82no—
134C22⋊C8—82no—
135C22⋊C8—81no—
136#97—42no—
137#97—42no—
138C4×C8—42no—
139#95—21no—
140#95—21no—
141#95—21no—
142#95—21no—
143#95—41no—
144#95—41no—
145C24—21no—
146C24#12311yescharacteristic
147#41—84no—
148#41—84no—
149C2×Q8—21no—
150C2×Q8—21no—
151C2×Q8—41no—
152C2×Q8—41no—
153C2×D4—21no—
154C2×D4—21no—
155C2×D4—41no—
156C2×D4—41no—
157C2×D4—42no—
158C2×D4—81no—
159C2×D4—81no—
160C22×C4—21no—
161C22×C4—21no—
162C22×C4—22no—
163C22×C4—21no—
164C22×C4—21no—
165C22×C4—21no—
166C22×C4—21no—
167C22×C4—21no—
168C22×C4—21no—
169C22×C4—21no—
170C22×C4—41no—
171C22×C4—41no—
172C22×C4—41no—
173C22×C4—41no—
174C22×C4—41no—
175C22×C4—42no—
176C22×C4—42no—
177C22×C4—42no—
178C22×C4—41no—
179C22×C4—41no—
180C22×C4C2×C22⋊C411yescharacteristic
181C22×C4C4×D411yescharacteristic
182C22×C4C4×D411yescharacteristic
183C22×C4#12011yescharacteristic
184C22×C4#12111yescharacteristic
185C22×C4#12111yescharacteristic
186C2×C8—42no—
187C2×C8—41no—
188C2×C8—41no—
189C2×C8—42no—
190C2×C8—42no—
191C2×C8—42no—
192C2×C8—42no—
193C2×C8—41no—
194C2×C8—41no—
195C2×C8—41no—
196C2×C8—41no—
197C2×C8—42no—
198C2×C8—42no—
199C2×C8—81no—
200C2×C8—81no—
201C2×C8—81no—
202C2×C8—81no—
203C2×C8—81no—
204C4⋊C4—41no—
205C4⋊C4—41no—
206C4⋊C4—41no—
207C4⋊C4—41no—
208C4⋊C4—42no—
209C4⋊C4—41no—
210C4⋊C4—41no—
211C4⋊C4—41no—
212C4⋊C4—41no—
213C4⋊C4—81no—
214C4⋊C4—81no—
215C22⋊C4—41no—
216C22⋊C4—41no—
217C22⋊C4—41no—
218C22⋊C4—41no—
219C22⋊C4—81no—
220C22⋊C4—81no—
221C22⋊C4—81no—
222C22⋊C4—82no—
223C42—22no—
224C42—22no—
225C42—41no—
226C42—41no—
227C42—42no—
228C42—82no—
229C23—21no—
230C23—21no—
231C23—21no—
232C23—21no—
233C23—21no—
234C23—21no—
235C23—21no—
236C23—41no—
237C23—41no—
238C23—41no—
239C23—41no—
240C23—41no—
241C23—41no—
242C23—41no—
243C23#38611yescharacteristic
244Q8—42no—
245Q8—81no—
246D4—42no—
247D4—81no—
248D4—84no—
249C2×C4—21no—
250C2×C4—21no—
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—42no—
264C2×C4—42no—
265C2×C4—42no—
266C2×C4—42no—
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—41no—
276C2×C4—81no—
277C2×C4—81no—
278C2×C4—84no—
279C2×C4—81no—
280C2×C4—81no—
281C2×C4—81no—
282C2×C4—81no—
283C2×C4—81no—
284C2×C4—81no—
285C2×C4#42111yescharacteristic
286C2×C4#42111yescharacteristic
287C2×C4#44311yescharacteristic
288C2×C4#44411yescharacteristic
289C2×C4#45812yes—
290C2×C4#47111yescharacteristic
291C2×C4#47311yescharacteristic
292C8—82no—
293C8—82no—
294C8—81no—
295C8—81no—
296C8—82no—
297C8—161no—
298C22—41no—
299C22—41no—
300C22—41no—
301C22—41no—
302C22—41no—
303C22—41no—
304C22—41no—
305C22—81no—
306C22—81no—
307C22—81no—
308C22—81no—
309C22—81no—
310C22—81no—
311C22—81no—
312C22#187611yescharacteristic
313C22#188111yescharacteristic
314C22#189011yescharacteristic
315C22#189711yescharacteristic
316C22#190311yescharacteristic
317C22#190811yescharacteristic
318C22#191111yescharacteristic
319C4—22no—
320C4—21no—
321C4—21no—
322C4—21no—
323C4—21no—
324C4—21no—
325C4—21no—
326C4—41no—
327C4—41no—
328C4—81no—
329C4—81no—
330C4—81no—
331C4—161no—
332C4—161no—
333C2—81no—
334C2—161no—
335C2#903811yescharacteristic
336C2#921411yescharacteristic
337C2#928311yescharacteristic
338C2#931311yescharacteristic
339C2#937311yescharacteristic
340C2#948111yescharacteristic
341C2#952111yescharacteristic
subgroupG/H
center, socleC23#386
commutatorC2×C42C22×C4
Frattini, cosocleC22×C42C23
prime radicalC22×#41D4
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
3. #8722C2semidirect
4. #8710C2semidirect
6. #7038C2semidirect
7. #7021C2semidirect

Maximal subgroups

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

subgroupindex
#275872
#219052
#87222
#87102
#76122
#70382
#70212