Finite Group Databasebeta

← #100332

Subgroups — #100332

Order 512.

423 conjugacy classes of subgroups.

The subgroup lattice has 343 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#33198C211yesmaximal, characteristic
2#22825C211yesmaximal, characteristic
3#9896C211yesmaximal, characteristic
4#9691C211yesmaximal, characteristic
5#8564C211yesmaximal, characteristic
6#7050C211yesmaximal, characteristic
7#7031C211yesmaximal, characteristic
8C2×#477C412yes—
9#2868C2211yescharacteristic
10C42×Q8C2211yescharacteristic
11#1987C412yes—
12#1984—21no—
13#1947—21no—
14#1947—21no—
15#1917—22no—
16#1842C2211yescharacteristic
17#1842C2211yescharacteristic
18#1842C2211yescharacteristic
19#1831—21no—
20#1803—21no—
21#1803—21no—
22#1770—21no—
23#1767—21no—
24C4×C4⋊C8C2211yescharacteristic
25C4×C4⋊C8C2211yescharacteristic
26C2×#333—22no—
27C2×Q8⋊C8—21no—
28#1296—21no—
29#1296—21no—
30#1279—21no—
31C2×C4⋊Q8—21no—
32C2×C4⋊Q8—21no—
33C2×C4⋊Q8C2×C412yes—
34C2×C4×Q8C2×C411yescharacteristic
35C2×C4×Q8D411yescharacteristic
36C2×C4×Q8D411yescharacteristic
37#477—44no—
38C2×#107—22no—
39C2×C4⋊C8—21no—
40C2×C4⋊C8—21no—
41C2×C4⋊C8—21no—
42C2×C4⋊C8—21no—
43C2×C4⋊C8—21no—
44C2×C4⋊C8—21no—
45C2×C4⋊C8—21no—
46C2×C4⋊C8—21no—
47C2×C4⋊C8C2×C412yes—
48C2×C4⋊C8D411yescharacteristic
49C2×C4⋊C8D411yescharacteristic
50C2×C4⋊C8D412yes—
51C2×#103—22no—
52C2×#97—22no—
53C2×C4×C8—21no—
54C2×C4×C8—21no—
55#384—21no—
56#384—21no—
57#384—21no—
58#384—21no—
59C4×C4⋊C4—21no—
60C4×C4⋊C4—21no—
61C4×C4⋊C4—21no—
62C4×C4⋊C4C2×C411yescharacteristic
63C4×C4⋊C4D411yescharacteristic
64C4×C4⋊C4D411yescharacteristic
65C43—21no—
66C43C2311yescharacteristic
67#340—22no—
68#340—22no—
69#336—22no—
70#336—22no—
71#336—22no—
72#336—21no—
73#336—21no—
74#333—44no—
75Q8⋊C8—42no—
76C22×Q8—22no—
77C22×Q8C22⋊C411yescharacteristic
78C22×C8—21no—
79C22×C8—22no—
80C22×C8—21no—
81C22×C8—22no—
82C22×C8—41no—
83C22×C8—41no—
84C4⋊Q8—42no—
85C4⋊Q8—42no—
86C4⋊Q8—42no—
87C4⋊Q8—42no—
88C4⋊Q8—81no—
89C4⋊Q8—81no—
90C4×Q8—22no—
91C4×Q8—21no—
92C4×Q8—21no—
93C4×Q8—21no—
94C4×Q8—21no—
95C4×Q8—21no—
96C4×Q8—21no—
97C4×Q8—21no—
98C4×Q8—21no—
99C4×Q8—41no—
100C2×C4⋊C4—21no—
101C2×C4⋊C4—21no—
102C2×C4⋊C4—21no—
103C2×C4⋊C4—22no—
104C2×C4⋊C4—22no—
105C2×C4⋊C4—41no—
106C2×C4⋊C4—41no—
107C2×C4⋊C4—41no—
108C2×C4⋊C4—41no—
109C2×C4⋊C4C22⋊C411yescharacteristic
110C2×C4⋊C4C22⋊C411yescharacteristic
111C2×C4⋊C4C22⋊C411yescharacteristic
112C2×C42—21no—
113C2×C42—21no—
114C2×C42—21no—
115C2×C42—21no—
116C2×C42—21no—
117C2×C42—21no—
118C2×C42C22×C411yescharacteristic
119C2×C42C2×D411yescharacteristic
120C2×C42C2×D411yescharacteristic
121C2×C42C2×D411yescharacteristic
122C2×C42C2×D411yescharacteristic
123C2×C42#4111yescharacteristic
124C2×C42#4111yescharacteristic
125#107—44no—
126C4⋊C8—22no—
127C4⋊C8—21no—
128C4⋊C8—21no—
129C4⋊C8—21no—
130C4⋊C8—42no—
131C4⋊C8—42no—
132C4⋊C8—41no—
133C4⋊C8—41no—
134C4⋊C8—41no—
135C4⋊C8—41no—
136C4⋊C8—41no—
137C4⋊C8—41no—
138C4⋊C8—41no—
139C4⋊C8—41no—
140C4⋊C8—41no—
141C4⋊C8—41no—
142C4⋊C8—41no—
143C4⋊C8—41no—
144C4⋊C8—41no—
145C4⋊C8—81no—
146C4⋊C8Q1612yes—
147#103—44no—
148#97—82no—
149C4×C8—41no—
150C4×C8—41no—
151C4×C8—41no—
152C4×C8—41no—
153#95—21no—
154#95—21no—
155#95—21no—
156C2×Q8—21no—
157C2×Q8—21no—
158C2×Q8—21no—
159C2×Q8—21no—
160C2×Q8—21no—
161C2×Q8—44no—
162C2×Q8—82no—
163C2×Q8—82no—
164C2×Q8#10012yes—
165C22×C4—21no—
166C22×C4—21no—
167C22×C4—21no—
168C22×C4—21no—
169C22×C4—42no—
170C22×C4C2×C22⋊C411yescharacteristic
171C22×C4C4×D411yescharacteristic
172C22×C4C4×D411yescharacteristic
173C22×C4#12011yescharacteristic
174C22×C4#12111yescharacteristic
175C22×C4#12111yescharacteristic
176C22×C4#12311yescharacteristic
177C2×C8—22no—
178C2×C8—41no—
179C2×C8—41no—
180C2×C8—42no—
181C2×C8—42no—
182C2×C8—42no—
183C2×C8—42no—
184C2×C8—41no—
185C2×C8—41no—
186C2×C8—42no—
187C2×C8—42no—
188C2×C8—42no—
189C2×C8—41no—
190C2×C8—81no—
191C2×C8—81no—
192C2×C8—81no—
193C2×C8—81no—
194C2×C8—81no—
195C4⋊C4—21no—
196C4⋊C4—21no—
197C4⋊C4—21no—
198C4⋊C4—21no—
199C4⋊C4—21no—
200C4⋊C4—21no—
201C4⋊C4—42no—
202C4⋊C4—41no—
203C4⋊C4—41no—
204C4⋊C4—41no—
205C4⋊C4—41no—
206C4⋊C4—42no—
207C4⋊C4—42no—
208C4⋊C4—82no—
209C4⋊C4—82no—
210C4⋊C4—82no—
211C4⋊C4—81no—
212C4⋊C4—81no—
213C4⋊C4—81no—
214C4⋊C4—81no—
215C42—21no—
216C42—21no—
217C42—21no—
218C42—21no—
219C42—21no—
220C42—21no—
221C42—41no—
222C42—42no—
223C42—41no—
224C42—41no—
225C42—41no—
226C42—41no—
227C42—41no—
228C42—41no—
229C42—41no—
230C42—41no—
231C42—81no—
232C42C2×Q1611yescharacteristic
233C42#13511yescharacteristic
234C42#13511yescharacteristic
235C42#13511yescharacteristic
236C42AGL(1,ℤ/8)11yescharacteristic
237C42#13711yescharacteristic
238C42#13711yescharacteristic
239C42#13711yescharacteristic
240C42#13711yescharacteristic
241C42#13711yescharacteristic
242C42#13711yescharacteristic
243C42#13711yescharacteristic
244C23#38611yescharacteristic
245Q8—22no—
246Q8—22no—
247Q8—41no—
248Q8—41no—
249Q8—164no—
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—22no—
259C2×C4—21no—
260C2×C4—41no—
261C2×C4—41no—
262C2×C4—41no—
263C2×C4—41no—
264C2×C4—41no—
265C2×C4—41no—
266C2×C4—41no—
267C2×C4—41no—
268C2×C4—41no—
269C2×C4—41no—
270C2×C4—41no—
271C2×C4—82no—
272C2×C4—82no—
273C2×C4—82no—
274C2×C4C2×#10011yescharacteristic
275C2×C4#41311yescharacteristic
276C2×C4#41611yescharacteristic
277C2×C4#41711yescharacteristic
278C2×C4#41911yescharacteristic
279C2×C4#41911yescharacteristic
280C2×C4C4×Q1611yescharacteristic
281C2×C4#44111yescharacteristic
282C2×C4#45111yescharacteristic
283C2×C4#45211yescharacteristic
284C2×C4#45211yescharacteristic
285C2×C4#45211yescharacteristic
286C2×C4#45311yescharacteristic
287C2×C4#45411yescharacteristic
288C2×C4#46211yescharacteristic
289C2×C4#46411yescharacteristic
290C2×C4#46811yescharacteristic
291C2×C4#46811yescharacteristic
292C2×C4#47011yescharacteristic
293C2×C4#47011yescharacteristic
294C2×C4#47111yescharacteristic
295C2×C4#47211yescharacteristic
296C2×C4#48511yescharacteristic
297C2×C4#48511yescharacteristic
298C8—42no—
299C8—82no—
300C8—82no—
301C8—82no—
302C8—81no—
303C8—161no—
304C22#187311yescharacteristic
305C22#187811yescharacteristic
306C22#188311yescharacteristic
307C22#189411yescharacteristic
308C22#190011yescharacteristic
309C22#190211yescharacteristic
310C22#190711yescharacteristic
311C4—21no—
312C4—21no—
313C4—21no—
314C4—21no—
315C4—21no—
316C4—21no—
317C4—41no—
318C4—42no—
319C4—81no—
320C4—81no—
321C4—81no—
322C4—162no—
323C4#155111yescharacteristic
324C4#155411yescharacteristic
325C4#155811yescharacteristic
326C4#155911yescharacteristic
327C4#163011yescharacteristic
328C4#163211yescharacteristic
329C4#163611yescharacteristic
330C4#163811yescharacteristic
331C4#169211yescharacteristic
332C4#169611yescharacteristic
333C4#169811yescharacteristic
334C4#170411yescharacteristic
335C2#897411yescharacteristic
336C2#913611yescharacteristic
337C2#916811yescharacteristic
338C2#922411yescharacteristic
339C2#924811yescharacteristic
340C2#932111yescharacteristic
341C2#936011yescharacteristic
subgroupG/H
center, socle, prime radicalC23#386
commutatorC2×C42C22×C4
Frattini, cosocleC43C23
LCSG ▷ C2×C42 ▷ C23 ▷ C1
UCSC1 ◁ C23 ◁ C43 ◁ G
LECSG ▷ C43 ▷ C23 ▷ C1

Maximal subgroups

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

subgroupindex
#331982
#228252
#98962
#96912
#85642
#70502
#70312