Finite Group Databasebeta

← #99788

Subgroups — #99788

Order 512.

448 conjugacy classes of subgroups.

The subgroup lattice has 352 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22734C211yesmaximal, characteristic
2#21616C211yesmaximal, characteristic
3#8171C211yesmaximal, characteristic
4#7590C211yesmaximal, characteristic
5#7570C211yesmaximal, characteristic
6#7027C211yesmaximal, characteristic
7#7022C211yesmaximal, characteristic
8C2×#477C411yescharacteristic
9C2×C8×Q8—21no—
10C4×C4⋊Q8C2211yescharacteristic
11C42×Q8C2211yescharacteristic
12#1987C411yescharacteristic
13#1948C411yescharacteristic
14#1939—21no—
15#1939—22no—
16#1918—21no—
17#1865—21no—
18#1831—21no—
19#1831C2211yescharacteristic
20#1803C411yescharacteristic
21#1776—21no—
22#1767C2211yescharacteristic
23C4×C4⋊C8—21no—
24C4×C4⋊C8C2211yescharacteristic
25C4×C4⋊C8C2211yescharacteristic
26#1763—21no—
27#1744C2211yescharacteristic
28C2×Q8⋊C8—21no—
29#1279—21no—
30#1272—22no—
31#1271—21no—
32#1271—21no—
33C2×C4⋊Q8C2×C411yescharacteristic
34C2×C4×Q8—21no—
35C2×C4×Q8C2×C411yescharacteristic
36C2×C4×Q8D411yescharacteristic
37C2×C4×Q8D411yescharacteristic
38#477—22no—
39#477C814yescomplemented
40C8×Q8—42no—
41C8×Q8—42no—
42C2×#107—21no—
43C2×#106—21no—
44C2×C4⋊C8—21no—
45C2×C4⋊C8—22no—
46C2×C4⋊C8—22no—
47C2×C4⋊C8—21no—
48C2×C4⋊C8—21no—
49C2×C4⋊C8C2×C411yescharacteristic
50C2×C4⋊C8C2×C411yescharacteristic
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#340—21no—
75#340—21no—
76#336—22no—
77#336—21no—
78#336—21no—
79#336—21no—
80#336—21no—
81#336—21no—
82#336D411yescharacteristic
83#336D412yes—
84#336D411yescharacteristic
85Q8⋊C8—24no—
86C22×Q8—21no—
87C22×Q8C22⋊C411yescharacteristic
88C22×C8—21no—
89C22×C8—21no—
90C22×C8—21no—
91C22×C8—21no—
92C22×C8—22no—
93C22×C8—41no—
94C22×C8—41no—
95C4⋊Q8—21no—
96C4⋊Q8—21no—
97C4⋊Q8—21no—
98C4⋊Q8C2×C812yes—
99C4×Q8—21no—
100C4×Q8—21no—
101C4×Q8—21no—
102C4×Q8—21no—
103C4×Q8—22no—
104C4×Q8—41no—
105C4×Q8—81no—
106C4×Q8—81no—
107C4×Q8C2×C812yes—
108C4×Q8M1612yes—
109C4×Q8SD1612yes—
110C4×Q8Q1612yes—
111C2×C4⋊C4—21no—
112C2×C4⋊C4—21no—
113C2×C4⋊C4—21no—
114C2×C4⋊C4—21no—
115C2×C4⋊C4—21no—
116C2×C4⋊C4—21no—
117C2×C4⋊C4—21no—
118C2×C4⋊C4—21no—
119C2×C4⋊C4—41no—
120C2×C4⋊C4C22⋊C411yescharacteristic
121C2×C4⋊C4C22⋊C411yescharacteristic
122C2×C4⋊C4C22⋊C411yescharacteristic
123C2×C42—21no—
124C2×C42—21no—
125C2×C42—21no—
126C2×C42—21no—
127C2×C42—21no—
128C2×C42—21no—
129C2×C42—41no—
130C2×C42C22×C411yescharacteristic
131C2×C42C2×D411yescharacteristic
132C2×C42C2×D411yescharacteristic
133C2×C42C2×D411yescharacteristic
134C2×C42C2×D411yescharacteristic
135C2×C42#4111yescharacteristic
136C2×C42#4111yescharacteristic
137#107—24no—
138#106—42no—
139C4⋊C8—22no—
140C4⋊C8—21no—
141C4⋊C8—21no—
142C4⋊C8—21no—
143C4⋊C8—41no—
144C4⋊C8—44no—
145C4⋊C8—42no—
146C4⋊C8—44no—
147C4⋊C8—41no—
148C4⋊C8—41no—
149C4⋊C8C2×C812yes—
150#103—24no—
151#103—42no—
152#97—41no—
153#97—41no—
154C4×C8—22no—
155C4×C8—41no—
156C4×C8—44no—
157C4×C8—41no—
158C4×C8—41no—
159C4×C8—81no—
160#95—21no—
161#95—21no—
162#95—21no—
163#95—41no—
164#95—41no—
165C2×Q8—21no—
166C2×Q8—21no—
167C2×Q8—21no—
168C2×Q8—22no—
169C2×Q8—41no—
170C2×Q8—81no—
171C2×Q8—81no—
172C2×Q8C22⋊C812yes—
173C2×Q8#10312yes—
174C2×Q8#10412yes—
175C22×C4—21no—
176C22×C4—21no—
177C22×C4—21no—
178C22×C4—21no—
179C22×C4—41no—
180C22×C4—41no—
181C22×C4C2×C22⋊C411yescharacteristic
182C22×C4C4×D411yescharacteristic
183C22×C4C4×D411yescharacteristic
184C22×C4#12011yescharacteristic
185C22×C4#12111yescharacteristic
186C22×C4#12111yescharacteristic
187C22×C4#12311yescharacteristic
188C2×C8—22no—
189C2×C8—22no—
190C2×C8—41no—
191C2×C8—41no—
192C2×C8—41no—
193C2×C8—41no—
194C2×C8—42no—
195C2×C8—42no—
196C2×C8—41no—
197C2×C8—42no—
198C2×C8—41no—
199C2×C8—42no—
200C2×C8—41no—
201C2×C8—41no—
202C2×C8—41no—
203C2×C8—42no—
204C2×C8—41no—
205C2×C8—81no—
206C2×C8—81no—
207C2×C8—81no—
208C2×C8—81no—
209C4⋊C4—21no—
210C4⋊C4—21no—
211C4⋊C4—22no—
212C4⋊C4—22no—
213C4⋊C4—22no—
214C4⋊C4—22no—
215C4⋊C4—22no—
216C4⋊C4—21no—
217C4⋊C4—21no—
218C4⋊C4—22no—
219C4⋊C4—42no—
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⋊C4C22⋊C812yes—
229C4⋊C4#10312yes—
230C42—21no—
231C42—21no—
232C42—22no—
233C42—22no—
234C42—22no—
235C42—22no—
236C42—21no—
237C42—21no—
238C42—41no—
239C42—41no—
240C42—41no—
241C42—42no—
242C42—42no—
243C42—41no—
244C42—41no—
245C42—81no—
246C42—82no—
247C42C22×C811yescharacteristic
248C42C2×M1611yescharacteristic
249C42#13111yescharacteristic
250C42#13111yescharacteristic
251C42C2×SD1611yescharacteristic
252C42C2×Q1611yescharacteristic
253C42AGL(1,ℤ/8)11yescharacteristic
254C42#13711yescharacteristic
255C23#38611yescharacteristic
256Q8—22no—
257Q8—22no—
258Q8—22no—
259Q8—82no—
260Q8—82no—
261Q8Q8⋊C814yes—
262C2×C4—22no—
263C2×C4—21no—
264C2×C4—22no—
265C2×C4—22no—
266C2×C4—21no—
267C2×C4—21no—
268C2×C4—21no—
269C2×C4—21no—
270C2×C4—21no—
271C2×C4—21no—
272C2×C4—21no—
273C2×C4—21no—
274C2×C4—21no—
275C2×C4—22no—
276C2×C4—21no—
277C2×C4—22no—
278C2×C4—21no—
279C2×C4—21no—
280C2×C4—41no—
281C2×C4—41no—
282C2×C4—41no—
283C2×C4—41no—
284C2×C4—42no—
285C2×C4—41no—
286C2×C4—41no—
287C2×C4—41no—
288C2×C4—81no—
289C2×C4—81no—
290C2×C4—81no—
291C2×C4—81no—
292C2×C4—81no—
293C2×C4C2×C22⋊C811yescharacteristic
294C2×C4#40811yescharacteristic
295C2×C4C2×#10311yescharacteristic
296C2×C4#41711yescharacteristic
297C2×C4C2×#10411yescharacteristic
298C2×C4#42111yescharacteristic
299C2×C4C8×D411yescharacteristic
300C2×C4C8×D411yescharacteristic
301C2×C4#43511yescharacteristic
302C2×C4#43511yescharacteristic
303C2×C4#45011yescharacteristic
304C2×C4#45111yescharacteristic
305C2×C4#47111yescharacteristic
306C2×C4#47211yescharacteristic
307C2×C4#48311yescharacteristic
308C2×C4#48411yescharacteristic
309C8—42no—
310C8—42no—
311C8—82no—
312C8—81no—
313C8—82no—
314C8—82no—
315C8—81no—
316C22#186911yescharacteristic
317C22#187111yescharacteristic
318C22#187211yescharacteristic
319C22#190211yescharacteristic
320C22#190511yescharacteristic
321C22#190611yescharacteristic
322C22#190711yescharacteristic
323C4—21no—
324C4—21no—
325C4—21no—
326C4—21no—
327C4—21no—
328C4—21no—
329C4—21no—
330C4—21no—
331C4—21no—
332C4—21no—
333C4—24no—
334C4—42no—
335C4—41no—
336C4—42no—
337C4—82no—
338C4—81no—
339C4—161no—
340C4C2×Q8⋊C811yescharacteristic
341C4#147711yescharacteristic
342C4#148111yescharacteristic
343C4#148311yescharacteristic
344C2#893711yescharacteristic
345C2#908811yescharacteristic
346C2#909511yescharacteristic
347C2#916011yescharacteristic
348C2#916411yescharacteristic
349C2#916711yescharacteristic
350C2#917611yescharacteristic
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
39. #477C8semidirect

Maximal subgroups

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

subgroupindex
#227342
#216162
#81712
#75902
#75702
#70272
#70222