Finite Group Databasebeta

← #290107

Subgroups — #290107

Order 512.

539 conjugacy classes of subgroups.

The subgroup lattice has 358 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#29290C211yesmaximal, characteristic
2#29218C211yesmaximal, characteristic
3#29206C211yesmaximal, characteristic
4#23247C211yesmaximal, characteristic
5#10656C212yesmaximal
6#10601C212yesmaximal
7#9898C211yesmaximal, characteristic
8#9616C212yesmaximal
9#9601C211yesmaximal, characteristic
10#9567C212yesmaximal
11C4×#333C211yesmaximal, characteristic
12#2868C2211yescharacteristic
13C4×C4⋊Q8C2211yescharacteristic
14C4×C4⋊Q8C2211yescharacteristic
15#2100—22no—
16#2087—22no—
17#2058—22no—
18#1990—21no—
19#1990C2212yes—
20#1987—21no—
21#1987—21no—
22#1987—21no—
23#1987C2212yes—
24#1984—22no—
25#1948C2212yes—
26#1948C2212yes—
27#1947—21no—
28#1947C2212yes—
29#1939—21no—
30#1939C2212yes—
31#1844—21no—
32#1842C2211yescharacteristic
33#1834C2212yes—
34C8×C4⋊C4—21no—
35C8×C4⋊C4C2212yes—
36C4×C4⋊C8C2211yescharacteristic
37C4×C4⋊C8C2211yescharacteristic
38C4×C4⋊C8C2211yescharacteristic
39C2×#335C2212yes—
40C2×#333C2211yescharacteristic
41C2×#333C2211yescharacteristic
42#1296C2211yescharacteristic
43#1296C2211yescharacteristic
44#1296C2212yes—
45#1296C2212yes—
46#1277C2212yes—
47C2×C4⋊Q8—21no—
48C2×C4⋊Q8—21no—
49C2×C4⋊Q8—21no—
50C2×C4⋊Q8D411yescharacteristic
51C2×C4⋊Q8C2311yescharacteristic
52C2×C4×Q8—21no—
53C2×C4×Q8—21no—
54C2×#107—22no—
55C2×#106—22no—
56C2×#106—22no—
57C2×C4⋊C8—21no—
58C2×C4⋊C8—22no—
59C2×C4⋊C8—21no—
60C2×C4⋊C8D411yescharacteristic
61C2×C4⋊C8D411yescharacteristic
62C2×C4⋊C8C2311yescharacteristic
63C2×C4⋊C8C2312yes—
64C2×C4⋊C8C2311yescharacteristic
65C2×C4⋊C8C2311yescharacteristic
66C2×C4⋊C8C2311yescharacteristic
67C2×C4×C8—21no—
68C2×C4×C8—21no—
69C2×C4×C8—22no—
70C2×C4×C8C2312yes—
71#391—21no—
72#391—21no—
73#389—21no—
74#389—21no—
75#389—21no—
76#389D412yes—
77#384—21no—
78#384—21no—
79#384—21no—
80#384D411yescharacteristic
81#384C2311yescharacteristic
82#384C2312yes—
83C4×C4⋊C4—21no—
84C4×C4⋊C4—21no—
85C4×C4⋊C4—21no—
86C4×C4⋊C4C2312yes—
87C43C2311yescharacteristic
88#340—22no—
89#340—22no—
90#340—22no—
91#340—22no—
92#340—22no—
93#340—22no—
94#340—22no—
95#340—22no—
96#340—22no—
97#336—22no—
98#336—22no—
99#336—21no—
100#336—21no—
101#336D412yes—
102#335—24no—
103#333—22no—
104#333Q814yescomplemented
105C22×Q8—21no—
106C22×Q8—21no—
107C22×Q8—41no—
108C22×C8—21no—
109C22×C8—21no—
110C22×C8—21no—
111C22×C8—21no—
112C22×C8—22no—
113C22×C8C2×D412yes—
114C22×C8C2×D412yes—
115C4⋊Q8—22no—
116C4⋊Q8—21no—
117C4⋊Q8—21no—
118C4⋊Q8—21no—
119C4⋊Q8—42no—
120C4⋊Q8—42no—
121C4⋊Q8—81no—
122C4⋊Q8—81no—
123C4⋊Q8—82no—
124C4⋊Q8Q1614yes—
125C4⋊Q8C2×Q812yes—
126C4×Q8—81no—
127C4×Q8—81no—
128C4×Q8—82no—
129C2×C4⋊C4—21no—
130C2×C4⋊C4—21no—
131C2×C4⋊C4—21no—
132C2×C4⋊C4—21no—
133C2×C4⋊C4—22no—
134C2×C4⋊C4—22no—
135C2×C4⋊C4—21no—
136C2×C4⋊C4—21no—
137C2×C4⋊C4—21no—
138C2×C4⋊C4—22no—
139C2×C4⋊C4—21no—
140C2×C4⋊C4—22no—
141C2×C4⋊C4—41no—
142C2×C4⋊C4—41no—
143C2×C4⋊C4—41no—
144C2×C4⋊C4—41no—
145C2×C4⋊C4C2×D412yes—
146C2×C4⋊C4C2×D412yes—
147C2×C42—22no—
148C2×C42—41no—
149C2×C42C2×D411yescharacteristic
150C2×C42C2×D411yescharacteristic
151C2×C42C2×D411yescharacteristic
152C2×C42C2×D411yescharacteristic
153C2×C42#4111yescharacteristic
154C2×C42#4111yescharacteristic
155C2×C42C2411yescharacteristic
156#107—82no—
157#106—82no—
158#106—82no—
159C4⋊C8—22no—
160C4⋊C8—22no—
161C4⋊C8—21no—
162C4⋊C8—22no—
163C4⋊C8—21no—
164C4⋊C8—21no—
165C4⋊C8—21no—
166C4⋊C8—21no—
167C4⋊C8—22no—
168C4⋊C8—42no—
169C4⋊C8—81no—
170C4⋊C8—82no—
171C4⋊C8C2×Q812yes—
172C4⋊C8C2×Q812yes—
173C4⋊C8#4112yes—
174C4×C8—22no—
175C4×C8—22no—
176C4×C8—42no—
177C4×C8—42no—
178C4×C8—82no—
179#95—22no—
180#95—22no—
181#95—41no—
182#95—41no—
183C2×Q8—24no—
184C2×Q8—22no—
185C2×Q8—41no—
186C2×Q8—81no—
187C2×Q8—81no—
188C2×Q8—82no—
189C2×Q8—82no—
190C2×Q8—81no—
191C2×Q8—81no—
192C2×Q8—82no—
193C22×C4—22no—
194C22×C4—22no—
195C22×C4—41no—
196C22×C4—41no—
197C22×C4—42no—
198C22×C4#12111yescharacteristic
199C22×C4#12111yescharacteristic
200C22×C4#12111yescharacteristic
201C22×C4#12111yescharacteristic
202C22×C4C22×D411yescharacteristic
203C22×C4C22×D411yescharacteristic
204C22×C4C2×#4111yescharacteristic
205C2×C8—22no—
206C2×C8—22no—
207C2×C8—22no—
208C2×C8—22no—
209C2×C8—22no—
210C2×C8—22no—
211C2×C8—22no—
212C2×C8—22no—
213C2×C8—22no—
214C2×C8—41no—
215C2×C8—41no—
216C2×C8—41no—
217C2×C8—41no—
218C2×C8—41no—
219C2×C8—41no—
220C2×C8—41no—
221C2×C8—42no—
222C2×C8—41no—
223C2×C8—42no—
224C2×C8—42no—
225C2×C8—41no—
226C4⋊C4—22no—
227C4⋊C4—24no—
228C4⋊C4—22no—
229C4⋊C4—22no—
230C4⋊C4—22no—
231C4⋊C4—22no—
232C4⋊C4—41no—
233C4⋊C4—41no—
234C4⋊C4—42no—
235C4⋊C4—42no—
236C4⋊C4—42no—
237C4⋊C4—42no—
238C4⋊C4—41no—
239C4⋊C4—42no—
240C4⋊C4—41no—
241C4⋊C4—41no—
242C4⋊C4—41no—
243C4⋊C4—81no—
244C4⋊C4—82no—
245C4⋊C4—81no—
246C4⋊C4—82no—
247C4⋊C4—82no—
248C4⋊C4—82no—
249C4⋊C4—82no—
250C4⋊C4—82no—
251C4⋊C4C2×Q1614yes—
252C42—21no—
253C42—21no—
254C42—21no—
255C42—21no—
256C42—22no—
257C42—41no—
258C42—42no—
259C42—42no—
260C42—82no—
261C42C2×Q1612yes—
262C42#13512yes—
263C42#13512yes—
264C42#13712yes—
265C42C22×Q811yescharacteristic
266C42C2×#4111yescharacteristic
267C422−1+411yescharacteristic
268C422−1+411yescharacteristic
269C23C2×#12111yescharacteristic
270Q8—84no—
271Q8—82no—
272Q8—82no—
273Q8—164no—
274C2×C4—21no—
275C2×C4—21no—
276C2×C4—21no—
277C2×C4—21no—
278C2×C4—21no—
279C2×C4—21no—
280C2×C4—21no—
281C2×C4—24no—
282C2×C4—21no—
283C2×C4—21no—
284C2×C4—42no—
285C2×C4—42no—
286C2×C4—42no—
287C2×C4—42no—
288C2×C4—42no—
289C2×C4—42no—
290C2×C4—81no—
291C2×C4—81no—
292C2×C4—82no—
293C2×C4—81no—
294C2×C4—82no—
295C2×C4—81no—
296C2×C4—81no—
297C2×C4—82no—
298C2×C4#46412yes—
299C2×C4#46412yes—
300C2×C4#46712yes—
301C2×C4#46712yes—
302C2×C4#47112yes—
303C2×C4#53311yescharacteristic
304C2×C4#54111yescharacteristic
305C2×C4#54711yescharacteristic
306C2×C4#54811yescharacteristic
307C2×C4D4×Q811yescharacteristic
308C2×C4D4×Q811yescharacteristic
309C2×C4C22×Q1611yescharacteristic
310C2×C4C2×#13511yescharacteristic
311C2×C4C2×#13511yescharacteristic
312C2×C4C2×#13711yescharacteristic
313C2×C4#57611yescharacteristic
314C2×C4#57711yescharacteristic
315C2×C4#57711yescharacteristic
316C2×C4#57711yescharacteristic
317C8—42no—
318C8—42no—
319C8—42no—
320C8—42no—
321C8—42no—
322C8—81no—
323C8—82no—
324C22#242911yescharacteristic
325C22C2×#46411yescharacteristic
326C22#304211yescharacteristic
327C22C2×#46711yescharacteristic
328C22#306211yescharacteristic
329C22C2×#47111yescharacteristic
330C22#306811yescharacteristic
331C4—22no—
332C4—21no—
333C4—21no—
334C4—22no—
335C4—41no—
336C4—44no—
337C4—82no—
338C4—82no—
339C4—162no—
340C4—161no—
341C4#168414yes—
342C4#334011yescharacteristic
343C4#335611yescharacteristic
344C4#336111yescharacteristic
345C4#336311yescharacteristic
346C4#338111yescharacteristic
347C4#338111yescharacteristic
348C4Q8×Q1611yescharacteristic
349C4#338711yescharacteristic
350C2C2×#168411yescharacteristic
351C2#1924211yescharacteristic
352C2#1939311yescharacteristic
353C2#1955811yescharacteristic
354C2#2545411yescharacteristic
355C2#2547111yescharacteristic
356C2#2549011yescharacteristic
subgroupG/H
center, socle, prime radicalC23C2×#121
commutator, Frattini, cosocleC2×C42C24
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C43 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
104. #333Q8semidirect

Maximal subgroups

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

subgroupindex
#292902
#292182
#292062
#232472
#106562
#106012
#98982
#96162
#96012
#95672
C4×#3332