Finite Group Databasebeta

← #290101

Subgroups — #290101

Order 512.

565 conjugacy classes of subgroups.

The subgroup lattice has 361 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#29287C211yesmaximal, characteristic
2#29208C211yesmaximal, characteristic
3#29206C211yesmaximal, characteristic
4#23149C211yesmaximal, characteristic
5#10652C212yesmaximal
6#10601C212yesmaximal
7#9896C211yesmaximal, characteristic
8#9615C212yesmaximal
9#9601C211yesmaximal, characteristic
10#9567C212yesmaximal
11C4×#333C211yesmaximal, characteristic
12C2×C8×Q8—21no—
13#2868C2211yescharacteristic
14C4×C4⋊Q8C2211yescharacteristic
15C4×C4⋊Q8C2211yescharacteristic
16#2088—22no—
17#2058—22no—
18#2057—22no—
19#1990—21no—
20#1990C2212yes—
21#1988—21no—
22#1987C2212yes—
23#1984—21no—
24#1948—21no—
25#1948C2212yes—
26#1948C2212yes—
27#1948C2212yes—
28#1947—21no—
29#1947—21no—
30#1939C2212yes—
31#1916—22no—
32#1845C2212yes—
33#1844—21no—
34#1842C2211yescharacteristic
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—
54C8×Q8—44no—
55C2×#107—22no—
56C2×#107—22no—
57C2×#106—22no—
58C2×#106—22no—
59C2×#106—22no—
60C2×C4⋊C8—22no—
61C2×C4⋊C8—21no—
62C2×C4⋊C8—21no—
63C2×C4⋊C8D411yescharacteristic
64C2×C4⋊C8D411yescharacteristic
65C2×C4⋊C8C2311yescharacteristic
66C2×C4⋊C8C2312yes—
67C2×C4⋊C8C2311yescharacteristic
68C2×C4⋊C8C2311yescharacteristic
69C2×C4⋊C8C2311yescharacteristic
70C2×C4×C8—22no—
71C2×C4×C8—21no—
72C2×C4×C8—21no—
73C2×C4×C8C2312yes—
74#391—21no—
75#391—21no—
76#389—21no—
77#389—21no—
78#389—21no—
79#389D412yes—
80#384—21no—
81#384—21no—
82#384—21no—
83#384D411yescharacteristic
84#384C2311yescharacteristic
85#384C2312yes—
86C4×C4⋊C4—21no—
87C4×C4⋊C4—21no—
88C4×C4⋊C4—21no—
89C4×C4⋊C4C2312yes—
90C43C2311yescharacteristic
91#340—22no—
92#340—22no—
93#340—22no—
94#340—22no—
95#340—22no—
96#340—22no—
97#340—22no—
98#336—22no—
99#336—22no—
100#336—21no—
101#336—21no—
102#336D412yes—
103#335—24no—
104#333—22no—
105#333Q814yescomplemented
106C22×Q8—21no—
107C22×Q8—21no—
108C22×Q8—41no—
109C22×C8—21no—
110C22×C8—21no—
111C22×C8—21no—
112C22×C8—21no—
113C22×C8—22no—
114C22×C8C2×D412yes—
115C22×C8C2×D412yes—
116C4⋊Q8—22no—
117C4⋊Q8—21no—
118C4⋊Q8—21no—
119C4⋊Q8—21no—
120C4⋊Q8—81no—
121C4⋊Q8—81no—
122C4⋊Q8—81no—
123C4⋊Q8—81no—
124C4⋊Q8—82no—
125C4⋊Q8Q1614yes—
126C4⋊Q8C2×Q812yes—
127C4×Q8—42no—
128C4×Q8—42no—
129C4×Q8—82no—
130C2×C4⋊C4—21no—
131C2×C4⋊C4—21no—
132C2×C4⋊C4—21no—
133C2×C4⋊C4—21no—
134C2×C4⋊C4—22no—
135C2×C4⋊C4—22no—
136C2×C4⋊C4—22no—
137C2×C4⋊C4—21no—
138C2×C4⋊C4—21no—
139C2×C4⋊C4—21no—
140C2×C4⋊C4—21no—
141C2×C4⋊C4—22no—
142C2×C4⋊C4—41no—
143C2×C4⋊C4—41no—
144C2×C4⋊C4—41no—
145C2×C4⋊C4—41no—
146C2×C4⋊C4C2×D412yes—
147C2×C4⋊C4C2×D412yes—
148C2×C42—22no—
149C2×C42—41no—
150C2×C42C2×D411yescharacteristic
151C2×C42C2×D411yescharacteristic
152C2×C42C2×D411yescharacteristic
153C2×C42C2×D411yescharacteristic
154C2×C42#4111yescharacteristic
155C2×C42#4111yescharacteristic
156C2×C42C2411yescharacteristic
157#107—82no—
158#107—82no—
159#106—82no—
160#106—82no—
161#106—82no—
162C4⋊C8—22no—
163C4⋊C8—21no—
164C4⋊C8—22no—
165C4⋊C8—21no—
166C4⋊C8—21no—
167C4⋊C8—21no—
168C4⋊C8—21no—
169C4⋊C8—22no—
170C4⋊C8—44no—
171C4⋊C8—42no—
172C4⋊C8—81no—
173C4⋊C8SD1614yes—
174C4⋊C8C2×Q812yes—
175C4⋊C8C2×Q812yes—
176C4⋊C8#4112yes—
177C4×C8—22no—
178C4×C8—22no—
179C4×C8—44no—
180C4×C8—42no—
181C4×C8—42no—
182#95—22no—
183#95—22no—
184#95—41no—
185#95—41no—
186C2×Q8—24no—
187C2×Q8—22no—
188C2×Q8—41no—
189C2×Q8—42no—
190C2×Q8—42no—
191C2×Q8—82no—
192C2×Q8—82no—
193C2×Q8—82no—
194C2×Q8—81no—
195C2×Q8—81no—
196C22×C4—22no—
197C22×C4—22no—
198C22×C4—41no—
199C22×C4—41no—
200C22×C4—42no—
201C22×C4#12111yescharacteristic
202C22×C4#12111yescharacteristic
203C22×C4#12111yescharacteristic
204C22×C4#12111yescharacteristic
205C22×C4C22×D411yescharacteristic
206C22×C4C22×D411yescharacteristic
207C22×C4C2×#4111yescharacteristic
208C2×C8—22no—
209C2×C8—22no—
210C2×C8—22no—
211C2×C8—22no—
212C2×C8—22no—
213C2×C8—22no—
214C2×C8—22no—
215C2×C8—22no—
216C2×C8—41no—
217C2×C8—41no—
218C2×C8—41no—
219C2×C8—41no—
220C2×C8—41no—
221C2×C8—41no—
222C2×C8—41no—
223C2×C8—42no—
224C2×C8—41no—
225C2×C8—42no—
226C2×C8—42no—
227C2×C8—41no—
228C2×C8C2×SD1614yes—
229C4⋊C4—22no—
230C4⋊C4—24no—
231C4⋊C4—22no—
232C4⋊C4—22no—
233C4⋊C4—22no—
234C4⋊C4—22no—
235C4⋊C4—41no—
236C4⋊C4—41no—
237C4⋊C4—44no—
238C4⋊C4—42no—
239C4⋊C4—41no—
240C4⋊C4—42no—
241C4⋊C4—42no—
242C4⋊C4—41no—
243C4⋊C4—41no—
244C4⋊C4—42no—
245C4⋊C4—42no—
246C4⋊C4—41no—
247C4⋊C4—81no—
248C4⋊C4—81no—
249C4⋊C4—82no—
250C4⋊C4—82no—
251C4⋊C4—82no—
252C4⋊C4—82no—
253C4⋊C4—82no—
254C4⋊C4C2×Q1614yes—
255C42—21no—
256C42—21no—
257C42—21no—
258C42—21no—
259C42—22no—
260C42—41no—
261C42—42no—
262C42—42no—
263C42—82no—
264C42C2×SD1612yes—
265C42C2×Q1612yes—
266C42#13512yes—
267C42#13712yes—
268C42C22×Q811yescharacteristic
269C42C2×#4111yescharacteristic
270C422−1+411yescharacteristic
271C422−1+411yescharacteristic
272C23C2×#12111yescharacteristic
273Q8—48no—
274Q8—82no—
275Q8—82no—
276Q8—164no—
277C2×C4—21no—
278C2×C4—21no—
279C2×C4—21no—
280C2×C4—21no—
281C2×C4—21no—
282C2×C4—21no—
283C2×C4—21no—
284C2×C4—24no—
285C2×C4—21no—
286C2×C4—21no—
287C2×C4—42no—
288C2×C4—42no—
289C2×C4—42no—
290C2×C4—42no—
291C2×C4—42no—
292C2×C4—42no—
293C2×C4—81no—
294C2×C4—81no—
295C2×C4—82no—
296C2×C4—81no—
297C2×C4—82no—
298C2×C4—81no—
299C2×C4—81no—
300C2×C4—82no—
301C2×C4#46112yes—
302C2×C4#46112yes—
303C2×C4#46712yes—
304C2×C4#46712yes—
305C2×C4#47212yes—
306C2×C4#53311yescharacteristic
307C2×C4#54111yescharacteristic
308C2×C4#54711yescharacteristic
309C2×C4#54811yescharacteristic
310C2×C4D4×Q811yescharacteristic
311C2×C4D4×Q811yescharacteristic
312C2×C4C22×SD1611yescharacteristic
313C2×C4C22×Q1611yescharacteristic
314C2×C4C2×#13511yescharacteristic
315C2×C4C2×#13711yescharacteristic
316C2×C4#57611yescharacteristic
317C2×C4#57711yescharacteristic
318C2×C4#57711yescharacteristic
319C2×C4#57711yescharacteristic
320C8—24no—
321C8—42no—
322C8—42no—
323C8—42no—
324C8—42no—
325C8—81no—
326C8—82no—
327C22#242911yescharacteristic
328C22C2×#46111yescharacteristic
329C22#304311yescharacteristic
330C22C2×#46711yescharacteristic
331C22#306211yescharacteristic
332C22C2×#47211yescharacteristic
333C22#306911yescharacteristic
334C4—22no—
335C4—21no—
336C4—21no—
337C4—22no—
338C4—41no—
339C4—44no—
340C4—82no—
341C4—82no—
342C4—161no—
343C4—162no—
344C4#167514yes—
345C4#333511yescharacteristic
346C4#335611yescharacteristic
347C4#336111yescharacteristic
348C4#336311yescharacteristic
349C4Q8×SD1611yescharacteristic
350C4#338111yescharacteristic
351C4Q8×Q1611yescharacteristic
352C4#338711yescharacteristic
353C2C2×#167511yescharacteristic
354C2#1923211yescharacteristic
355C2#1939911yescharacteristic
356C2#1956611yescharacteristic
357C2#2545311yescharacteristic
358C2#2547111yescharacteristic
359C2#2548911yescharacteristic
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
105. #333Q8semidirect

Maximal subgroups

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

subgroupindex
#292872
#292082
#292062
#231492
#106522
#106012
#98962
#96152
#96012
#95672
C4×#3332