Finite Group Databasebeta

← #100298

Subgroups — #100298

Order 512.

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

normal subgroups only

This group has 415 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#33072C211yesmaximal, characteristic
2#22012C211yesmaximal, characteristic
3#9891C211yesmaximal, characteristic, complemented
4#8172C211yesmaximal, characteristic
5#7680C211yesmaximal, characteristic, complemented
6#7042C211yesmaximal, characteristic, complemented
7#7030C211yesmaximal, characteristic, complemented
8#2859C2211yescharacteristic
9C42×D4C2211yescharacteristic
10#1954—21no—
11#1948C411yescharacteristic
12#1947C411yescharacteristic
13#1939—21no—
14#1924—22no—
15#1921—21no—
16#1919—21no—
17#1864—21no—
18#1862—21no—
19#1831C2211yescharacteristic
20#1817—21no—
21#1802C411yescharacteristic
22#1802C411yescharacteristic
23#1797—21no—
24#1767C2211yescharacteristic
25#1767C2211yescharacteristic
26#1744C2211yescharacteristic
27C4×#97C2211yescharacteristic
28C2×D4⋊C8—21no—
29#1295—22no—
30#1279—21no—
31#1271—21no—
32#1271—21no—
33C2×#125—21no—
34C2×C4×D4C2×C411yescharacteristic
35C2×C4×D4D411yescharacteristic
36C2×C4×D4D411yescharacteristic
37C22×C42—21no—
38C2×#107—21no—
39C2×#106—21no—
40C2×C4⋊C8—21no—
41C2×C4⋊C8—21no—
42C2×C4⋊C8—22no—
43C2×C4⋊C8C2×C411yescharacteristic
44C2×C4⋊C8C2×C411yescharacteristic
45C2×#102—21no—
46C2×#102—21no—
47C2×C22⋊C8—22no—
48C2×C22⋊C8—22no—
49C2×#97—21no—
50C2×#97—21no—
51C2×#97—21no—
52C2×#97D412yes—
53C2×C4×C8—21no—
54C2×C4×C8—21no—
55#389—21no—
56#389—21no—
57#389—21no—
58#389—21no—
59#384C2×C411yescharacteristic
60#384C2×C411yescharacteristic
61#382—21no—
62C4×C4⋊C4C2×C411yescharacteristic
63C4×C4⋊C4D411yescharacteristic
64C4×C4⋊C4D411yescharacteristic
65C4×C22⋊C4—21no—
66C4×C22⋊C4—21no—
67C4×C22⋊C4—21no—
68C43C2311yescharacteristic
69#340—21no—
70#340—21no—
71#340—21no—
72#340—21no—
73#336—22no—
74#336—21no—
75#336—21no—
76#336—21no—
77#336—21no—
78#336—21no—
79#336D411yescharacteristic
80#336D411yescharacteristic
81D4⋊C8—24no—
82C22×D4C22⋊C411yescharacteristic
83C23×C4—21no—
84C23×C4—21no—
85C23×C4—21no—
86C22×C8—21no—
87C22×C8—21no—
88C22×C8—21no—
89C22×C8—22no—
90C22×C8—21no—
91C22×C8—41no—
92C22×C8—41no—
93#125—81no—
94#125—81no—
95C4×D4—22no—
96C4×D4—22no—
97C4×D4—41no—
98C4×D4—41no—
99C4×D4—41no—
100C4×D4—41no—
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—41no—
108C2×C4⋊C4—41no—
109C2×C4⋊C4C22⋊C411yescharacteristic
110C2×C4⋊C4C22⋊C411yescharacteristic
111C2×C4⋊C4C22⋊C411yescharacteristic
112C2×C22⋊C4—21no—
113C2×C22⋊C4—21no—
114C2×C22⋊C4—21no—
115C2×C42—21no—
116C2×C42—21no—
117C2×C42—21no—
118C2×C42—82no—
119C2×C42C22×C411yescharacteristic
120C2×C42C2×D411yescharacteristic
121C2×C42C2×D411yescharacteristic
122C2×C42C2×D411yescharacteristic
123C2×C42C2×D411yescharacteristic
124C2×C42#4111yescharacteristic
125C2×C42#4111yescharacteristic
126#107—42no—
127#106—42no—
128C4⋊C8—22no—
129C4⋊C8—41no—
130C4⋊C8—41no—
131C4⋊C8—41no—
132C4⋊C8—41no—
133C4⋊C8—41no—
134C4⋊C8—82no—
135#102—42no—
136#102—42no—
137C22⋊C8—82no—
138C22⋊C8—82no—
139#97—22no—
140#97—22no—
141#97—42no—
142#97—41no—
143#97—41no—
144#97—81no—
145C4×C8—22no—
146C4×C8—41no—
147C4×C8—41no—
148C4×C8—41no—
149#95—21no—
150#95—21no—
151#95—21no—
152#95—41no—
153#95—41no—
154C24—21no—
155C2×D4—21no—
156C2×D4—21no—
157C2×D4—21no—
158C2×D4—21no—
159C2×D4—21no—
160C2×D4#10112yes—
161C22×C4—21no—
162C22×C4—21no—
163C22×C4—21no—
164C22×C4—42no—
165C22×C4—42no—
166C22×C4—42no—
167C22×C4—42no—
168C22×C4—41no—
169C22×C4—41no—
170C22×C4—41no—
171C22×C4—41no—
172C22×C4—41no—
173C22×C4—41no—
174C22×C4—41no—
175C22×C4—41no—
176C22×C4—81no—
177C22×C4—81no—
178C22×C4C2×C22⋊C411yescharacteristic
179C22×C4C4×D411yescharacteristic
180C22×C4C4×D411yescharacteristic
181C22×C4#12011yescharacteristic
182C22×C4#12111yescharacteristic
183C22×C4#12111yescharacteristic
184C22×C4#12311yescharacteristic
185C2×C8—42no—
186C2×C8—42no—
187C2×C8—42no—
188C2×C8—41no—
189C2×C8—41no—
190C2×C8—41no—
191C2×C8—42no—
192C2×C8—42no—
193C2×C8—42no—
194C2×C8—41no—
195C2×C8—41no—
196C2×C8—41no—
197C2×C8—41no—
198C2×C8—41no—
199C2×C8—41no—
200C2×C8—41no—
201C2×C8—42no—
202C2×C8—41no—
203C2×C8—41no—
204C2×C8—81no—
205C2×C8—81no—
206C4⋊C4—21no—
207C4⋊C4—21no—
208C4⋊C4—21no—
209C4⋊C4—21no—
210C4⋊C4—21no—
211C4⋊C4—41no—
212C4⋊C4—41no—
213C4⋊C4—41no—
214C4⋊C4—41no—
215C4⋊C4—41no—
216C4⋊C4—41no—
217C4⋊C4—81no—
218C4⋊C4—81no—
219C4⋊C4—81no—
220C4⋊C4—82no—
221C4⋊C4—81no—
222C4⋊C4—81no—
223C4⋊C4#10412yes—
224C22⋊C4—42no—
225C22⋊C4—42no—
226C22⋊C4—42no—
227C42—21no—
228C42—21no—
229C42—21no—
230C42—21no—
231C42—21no—
232C42—21no—
233C42—21no—
234C42—21no—
235C42—21no—
236C42—21no—
237C42—21no—
238C42—21no—
239C42—41no—
240C42—42no—
241C42—81no—
242C42—81no—
243C23—22no—
244C23—41no—
245C23—41no—
246C23—41no—
247C23—41no—
248C23—41no—
249C23—41no—
250C23#38611yescharacteristic
251D4—22no—
252D4—22no—
253D4—41no—
254D4—41no—
255C2×C4—21no—
256C2×C4—21no—
257C2×C4—21no—
258C2×C4—21no—
259C2×C4—21no—
260C2×C4—21no—
261C2×C4—21no—
262C2×C4—22no—
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—41no—
272C2×C4—41no—
273C2×C4—41no—
274C2×C4—41no—
275C2×C4—41no—
276C2×C4—41no—
277C2×C4—41no—
278C2×C4—41no—
279C2×C4—82no—
280C2×C4—82no—
281C2×C4—82no—
282C2×C4—82no—
283C2×C4—82no—
284C2×C4—82no—
285C2×C4—81no—
286C2×C4—81no—
287C2×C4—81no—
288C2×C4—81no—
289C2×C4—81no—
290C2×C4—81no—
291C2×C4C2×#10111yescharacteristic
292C2×C4#41311yescharacteristic
293C2×C4C2×#10411yescharacteristic
294C2×C4#42111yescharacteristic
295C2×C4#42111yescharacteristic
296C2×C4#42111yescharacteristic
297C2×C4#44311yescharacteristic
298C2×C4#44411yescharacteristic
299C2×C4#45511yescharacteristic
300C2×C4#45611yescharacteristic
301C2×C4#47111yescharacteristic
302C2×C4#47211yescharacteristic
303C8—44no—
304C8—81no—
305C8—82no—
306C8—81no—
307C8—82no—
308C8—81no—
309C8—81no—
310C22—42no—
311C22—42no—
312C22—42no—
313C22—81no—
314C22—81no—
315C22—81no—
316C22—81no—
317C22#187111yescharacteristic
318C22#187611yescharacteristic
319C22#188211yescharacteristic
320C22#189711yescharacteristic
321C22#190111yescharacteristic
322C22#190411yescharacteristic
323C22#190711yescharacteristic
324C4—21no—
325C4—21no—
326C4—21no—
327C4—21no—
328C4—21no—
329C4—21no—
330C4—41no—
331C4—41no—
332C4—42no—
333C4—41no—
334C4—81no—
335C4—81no—
336C4—161no—
337C4—161no—
338C4#152911yescharacteristic
339C4#153111yescharacteristic
340C4#153411yescharacteristic
341C4#153511yescharacteristic
342C2—82no—
343C2#897211yescharacteristic
344C2#912611yescharacteristic
345C2#916311yescharacteristic
346C2#921311yescharacteristic
347C2#924511yescharacteristic
348C2#932711yescharacteristic
349C2#935511yescharacteristic
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
5. #7680C2semidirect
6. #7042C2semidirect
7. #7030C2semidirect

Maximal subgroups

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

subgroupindex
#330722
#220122
#98912
#81722
#76802
#70422
#70302