Finite Group Databasebeta

← #99798

Subgroups — #99798

Order 512.

446 conjugacy classes of subgroups.

The subgroup lattice has 362 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22828C211yesmaximal, characteristic
2#21506C211yesmaximal, characteristic
3#8171C211yesmaximal, characteristic
4#7588C211yesmaximal, characteristic
5#7570C211yesmaximal, characteristic
6#7027C211yesmaximal, characteristic
7#7022C211yesmaximal, characteristic
8C2×#462C411yescharacteristic
9C2×#441—21no—
10C4×C4⋊Q8C2211yescharacteristic
11C42×Q8C2211yescharacteristic
12#1984—21no—
13#1981C411yescharacteristic
14#1948C411yescharacteristic
15#1927—21no—
16#1916—22no—
17#1878—21no—
18#1839—21no—
19#1831C2211yescharacteristic
20#1803C411yescharacteristic
21C8×C4⋊C4—21no—
22#1767C2211yescharacteristic
23C4×C4⋊C8C2211yescharacteristic
24C4×C4⋊C8C2211yescharacteristic
25#1763—21no—
26#1744—21no—
27#1744C2211yescharacteristic
28C2×Q8⋊C8—21no—
29#1279—21no—
30#1271—21no—
31#1271—21no—
32#1271—22no—
33C22×Q16—21no—
34C2×C4⋊Q8C2×C411yescharacteristic
35C2×C4×Q8—21no—
36C2×C4×Q8C2×C411yescharacteristic
37C2×C4×Q8D411yescharacteristic
38C2×C4×Q8D411yescharacteristic
39#462—22no—
40#462C814yescomplemented
41#441—42no—
42#441—42no—
43C2×#106—21no—
44C2×C4⋊C8—21no—
45C2×C4⋊C8—21no—
46C2×C4⋊C8—21no—
47C2×C4⋊C8C2×C411yescharacteristic
48C2×C4⋊C8C2×C411yescharacteristic
49C2×#103—21no—
50C2×#103—21no—
51C2×#103—21no—
52C2×#103—21no—
53C2×#97—21no—
54C2×C4×C8—21no—
55C2×C4×C8—22no—
56C2×C4×C8—21no—
57C2×C4×C8—21no—
58#391—21no—
59#389—21no—
60#389—21no—
61#384C2×C411yescharacteristic
62C4×C4⋊C4—21no—
63C4×C4⋊C4—21no—
64C4×C4⋊C4—21no—
65C4×C4⋊C4—21no—
66C4×C4⋊C4—21no—
67C4×C4⋊C4C2×C411yescharacteristic
68C4×C4⋊C4D411yescharacteristic
69C4×C4⋊C4D411yescharacteristic
70C43—21no—
71C43C2311yescharacteristic
72#340—21no—
73#340—21no—
74#336—22no—
75#336—22no—
76#336—22no—
77#336—21no—
78#336—21no—
79#336—21no—
80#336—21no—
81#336—21no—
82#336D412yes—
83#336D411yescharacteristic
84#336D411yescharacteristic
85Q8⋊C8—24no—
86C22×Q8—21no—
87C22×Q8C22⋊C411yescharacteristic
88C2×Q16—24no—
89C2×Q16—42no—
90C2×Q16—42no—
91C22×C8—21no—
92C22×C8—21no—
93C22×C8—21no—
94C22×C8—22no—
95C22×C8—21no—
96C22×C8—41no—
97C22×C8—41no—
98C4⋊Q8—21no—
99C4⋊Q8—21no—
100C4⋊Q8—21no—
101C4⋊Q8C2×C812yes—
102C4×Q8—21no—
103C4×Q8—21no—
104C4×Q8—22no—
105C4×Q8—21no—
106C4×Q8—21no—
107C4×Q8—21no—
108C4×Q8—21no—
109C4×Q8—41no—
110C4×Q8—41no—
111C4×Q8—41no—
112C4×Q8—41no—
113C4×Q8—41no—
114C4×Q8C2×C812yes—
115C4×Q8M1612yes—
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—21no—
123C2×C4⋊C4—21no—
124C2×C4⋊C4—41no—
125C2×C4⋊C4C22⋊C411yescharacteristic
126C2×C4⋊C4C22⋊C411yescharacteristic
127C2×C4⋊C4C22⋊C411yescharacteristic
128C2×C42—21no—
129C2×C42—21no—
130C2×C42—21no—
131C2×C42—21no—
132C2×C42—21no—
133C2×C42—21no—
134C2×C42—41no—
135C2×C42C22×C411yescharacteristic
136C2×C42C2×D411yescharacteristic
137C2×C42C2×D411yescharacteristic
138C2×C42C2×D411yescharacteristic
139C2×C42C2×D411yescharacteristic
140C2×C42#4111yescharacteristic
141C2×C42#4111yescharacteristic
142#106—42no—
143C4⋊C8—22no—
144C4⋊C8—21no—
145C4⋊C8—21no—
146C4⋊C8—21no—
147C4⋊C8—41no—
148C4⋊C8—42no—
149C4⋊C8—41no—
150C4⋊C8—41no—
151C4⋊C8C2×C812yes—
152#103—24no—
153#103—42no—
154#103—42no—
155#103—42no—
156#97—41no—
157#97—41no—
158C4×C8—22no—
159C4×C8—41no—
160C4×C8—44no—
161C4×C8—41no—
162C4×C8—41no—
163C4×C8—81no—
164#95—21no—
165#95—21no—
166#95—21no—
167#95—41no—
168#95—41no—
169C2×Q8—21no—
170C2×Q8—22no—
171C2×Q8—21no—
172C2×Q8—21no—
173C2×Q8—21no—
174C2×Q8—21no—
175C2×Q8—41no—
176C2×Q8—41no—
177C2×Q8—41no—
178C2×Q8—41no—
179C2×Q8—41no—
180C2×Q8C22⋊C812yes—
181C22×C4—21no—
182C22×C4—21no—
183C22×C4—21no—
184C22×C4—21no—
185C22×C4—41no—
186C22×C4—41no—
187C22×C4C2×C22⋊C411yescharacteristic
188C22×C4C4×D411yescharacteristic
189C22×C4C4×D411yescharacteristic
190C22×C4#12011yescharacteristic
191C22×C4#12111yescharacteristic
192C22×C4#12111yescharacteristic
193C22×C4#12311yescharacteristic
194Q16—48no—
195C2×C8—22no—
196C2×C8—22no—
197C2×C8—41no—
198C2×C8—41no—
199C2×C8—41no—
200C2×C8—41no—
201C2×C8—42no—
202C2×C8—42no—
203C2×C8—42no—
204C2×C8—41no—
205C2×C8—41no—
206C2×C8—42no—
207C2×C8—41no—
208C2×C8—41no—
209C2×C8—41no—
210C2×C8—41no—
211C2×C8—42no—
212C2×C8—81no—
213C2×C8—81no—
214C2×C8—81no—
215C2×C8—81no—
216C4⋊C4—21no—
217C4⋊C4—21no—
218C4⋊C4—22no—
219C4⋊C4—22no—
220C4⋊C4—21no—
221C4⋊C4—21no—
222C4⋊C4—22no—
223C4⋊C4—22no—
224C4⋊C4—42no—
225C4⋊C4—41no—
226C4⋊C4—41no—
227C4⋊C4—41no—
228C4⋊C4—41no—
229C4⋊C4—41no—
230C4⋊C4—41no—
231C4⋊C4—81no—
232C4⋊C4—81no—
233C4⋊C4—81no—
234C4⋊C4—81no—
235C4⋊C4C22⋊C812yes—
236C4⋊C4#10412yes—
237C42—21no—
238C42—21no—
239C42—22no—
240C42—22no—
241C42—21no—
242C42—21no—
243C42—41no—
244C42—41no—
245C42—41no—
246C42—42no—
247C42—42no—
248C42—41no—
249C42—41no—
250C42—41no—
251C42—41no—
252C42—81no—
253C42—81no—
254C42—81no—
255C42C22×C811yescharacteristic
256C42C2×M1611yescharacteristic
257C42#13111yescharacteristic
258C42#13111yescharacteristic
259C42#13511yescharacteristic
260C42#13511yescharacteristic
261C42AGL(1,ℤ/8)11yescharacteristic
262C42AGL(1,ℤ/8)11yescharacteristic
263C23#38611yescharacteristic
264Q8—22no—
265Q8—22no—
266Q8—41no—
267Q8—41no—
268Q8—42no—
269Q8—42no—
270Q8—81no—
271Q8—81no—
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—21no—
282C2×C4—22no—
283C2×C4—22no—
284C2×C4—21no—
285C2×C4—22no—
286C2×C4—21no—
287C2×C4—21no—
288C2×C4—41no—
289C2×C4—41no—
290C2×C4—41no—
291C2×C4—41no—
292C2×C4—41no—
293C2×C4—41no—
294C2×C4—42no—
295C2×C4—41no—
296C2×C4—41no—
297C2×C4—41no—
298C2×C4—81no—
299C2×C4—81no—
300C2×C4—81no—
301C2×C4—81no—
302C2×C4—81no—
303C2×C4C2×C22⋊C811yescharacteristic
304C2×C4#40811yescharacteristic
305C2×C4#41611yescharacteristic
306C2×C4#41811yescharacteristic
307C2×C4C2×#10411yescharacteristic
308C2×C4#42111yescharacteristic
309C2×C4C8×D411yescharacteristic
310C2×C4C8×D411yescharacteristic
311C2×C4#43511yescharacteristic
312C2×C4#43511yescharacteristic
313C2×C4#44911yescharacteristic
314C2×C4#44911yescharacteristic
315C2×C4#47111yescharacteristic
316C2×C4#47311yescharacteristic
317C2×C4#48211yescharacteristic
318C2×C4#48211yescharacteristic
319C8—42no—
320C8—42no—
321C8—81no—
322C8—82no—
323C8—82no—
324C8—82no—
325C8—81no—
326C22#186911yescharacteristic
327C22#187111yescharacteristic
328C22#187511yescharacteristic
329C22#190211yescharacteristic
330C22#190511yescharacteristic
331C22#190611yescharacteristic
332C22#190811yescharacteristic
333C4—21no—
334C4—21no—
335C4—21no—
336C4—21no—
337C4—21no—
338C4—21no—
339C4—21no—
340C4—21no—
341C4—21no—
342C4—21no—
343C4—42no—
344C4—41no—
345C4—42no—
346C4—42no—
347C4—81no—
348C4—82no—
349C4—161no—
350C4#147611yescharacteristic
351C4#147811yescharacteristic
352C4#148211yescharacteristic
353C4#148411yescharacteristic
354C2#894011yescharacteristic
355C2#908811yescharacteristic
356C2#909611yescharacteristic
357C2#915911yescharacteristic
358C2#916611yescharacteristic
359C2#917211yescharacteristic
360C2#918211yescharacteristic
subgroupG/H
center, socle, prime radicalC23#386
commutatorC42C22×C8
Frattini, cosocleC43C23
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C43 ◁ G
LECSG ▷ C43 ▷ C23 ▷ C1

Complements

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

NH ≅ G/Ntype
40. #462C8semidirect

Maximal subgroups

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

subgroupindex
#228282
#215062
#81712
#75882
#75702
#70272
#70222