Finite Group Databasebeta

← #290100

Subgroups — #290100

Order 512.

583 conjugacy classes of subgroups.

The subgroup lattice has 369 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#29206C211yesmaximal, characteristic
2#29206C211yesmaximal, characteristic
3#28283C211yesmaximal, characteristic
4#23149C211yesmaximal, characteristic
5#10654C212yesmaximal
6#10598C212yesmaximal
7#9896C211yesmaximal, characteristic
8#9612C212yesmaximal
9#9602C211yesmaximal, characteristic
10#9565C212yesmaximal
11C4×#332C211yesmaximal, characteristic
12C2×#475—21no—
13C2×C8×Q8—21no—
14#2868C2211yescharacteristic
15C4×C4⋊Q8C2211yescharacteristic
16C4×C4⋊Q8C2211yescharacteristic
17#2066—22no—
18#2058—22no—
19#2058—22no—
20#1990—21no—
21#1990C2212yes—
22#1984—21no—
23#1981C2212yes—
24#1948—21no—
25#1948—21no—
26#1948C2212yes—
27#1948C2212yes—
28#1948C2212yes—
29#1939C2212yes—
30#1916—22no—
31#1844—21no—
32#1844C2212yes—
33#1842C2211yescharacteristic
34#1803—21no—
35C8×C4⋊C4C2212yes—
36C4×C4⋊C8C2211yescharacteristic
37C4×C4⋊C8C2211yescharacteristic
38C4×C4⋊C8C2211yescharacteristic
39C2×#334C2212yes—
40C2×#332C2211yescharacteristic
41C2×#332C2211yescharacteristic
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—
54#475—44no—
55C8×Q8—44no—
56C2×#106—22no—
57C2×#106—22no—
58C2×#106—22no—
59C2×#106—22no—
60C2×#106—22no—
61C2×C4⋊C8—22no—
62C2×C4⋊C8—21no—
63C2×C4⋊C8—21no—
64C2×C4⋊C8D411yescharacteristic
65C2×C4⋊C8D411yescharacteristic
66C2×C4⋊C8C2311yescharacteristic
67C2×C4⋊C8C2312yes—
68C2×C4⋊C8C2311yescharacteristic
69C2×C4⋊C8C2311yescharacteristic
70C2×C4⋊C8C2311yescharacteristic
71C2×#103—22no—
72C2×#103—22no—
73C2×C4×C8—22no—
74C2×C4×C8—21no—
75C2×C4×C8—21no—
76C2×C4×C8C2312yes—
77#391—21no—
78#391—21no—
79#389—21no—
80#389—21no—
81#389—21no—
82#389D412yes—
83#384—21no—
84#384—21no—
85#384—21no—
86#384D411yescharacteristic
87#384C2311yescharacteristic
88#384C2312yes—
89C4×C4⋊C4—21no—
90C4×C4⋊C4—21no—
91C4×C4⋊C4—21no—
92C4×C4⋊C4C2312yes—
93C43C2311yescharacteristic
94#340—22no—
95#340—22no—
96#340—22no—
97#340—22no—
98#340—22no—
99#336—22no—
100#336—22no—
101#336—21no—
102#336—21no—
103#336D412yes—
104#334—24no—
105#332—22no—
106#332Q814yescomplemented
107C22×Q8—21no—
108C22×Q8—21no—
109C22×Q8—41no—
110C22×C8—21no—
111C22×C8—21no—
112C22×C8—21no—
113C22×C8—21no—
114C22×C8—22no—
115C22×C8C2×D412yes—
116C22×C8C2×D412yes—
117C4⋊Q8—22no—
118C4⋊Q8—21no—
119C4⋊Q8—21no—
120C4⋊Q8—21no—
121C4⋊Q8—81no—
122C4⋊Q8—81no—
123C4⋊Q8—81no—
124C4⋊Q8—81no—
125C4⋊Q8—82no—
126C4⋊Q8SD1614yes—
127C4⋊Q8C2×Q812yes—
128C4×Q8—42no—
129C4×Q8—42no—
130C4×Q8—42no—
131C4×Q8—42no—
132C2×C4⋊C4—21no—
133C2×C4⋊C4—21no—
134C2×C4⋊C4—21no—
135C2×C4⋊C4—21no—
136C2×C4⋊C4—22no—
137C2×C4⋊C4—21no—
138C2×C4⋊C4—22no—
139C2×C4⋊C4—22no—
140C2×C4⋊C4—21no—
141C2×C4⋊C4—21no—
142C2×C4⋊C4—22no—
143C2×C4⋊C4—21no—
144C2×C4⋊C4—41no—
145C2×C4⋊C4—41no—
146C2×C4⋊C4—41no—
147C2×C4⋊C4—41no—
148C2×C4⋊C4C2×D412yes—
149C2×C4⋊C4C2×D412yes—
150C2×C42—22no—
151C2×C42—41no—
152C2×C42C2×D411yescharacteristic
153C2×C42C2×D411yescharacteristic
154C2×C42C2×D411yescharacteristic
155C2×C42C2×D411yescharacteristic
156C2×C42#4111yescharacteristic
157C2×C42#4111yescharacteristic
158C2×C42C2411yescharacteristic
159#106—44no—
160#106—82no—
161#106—82no—
162#106—82no—
163#106—82no—
164C4⋊C8—22no—
165C4⋊C8—21no—
166C4⋊C8—22no—
167C4⋊C8—21no—
168C4⋊C8—21no—
169C4⋊C8—21no—
170C4⋊C8—21no—
171C4⋊C8—22no—
172C4⋊C8—44no—
173C4⋊C8—42no—
174C4⋊C8—81no—
175C4⋊C8SD1614yes—
176C4⋊C8C2×Q812yes—
177C4⋊C8C2×Q812yes—
178C4⋊C8#4112yes—
179#103—44no—
180#103—82no—
181C4×C8—22no—
182C4×C8—22no—
183C4×C8—44no—
184C4×C8—42no—
185C4×C8—42no—
186#95—22no—
187#95—22no—
188#95—41no—
189#95—41no—
190C2×Q8—24no—
191C2×Q8—22no—
192C2×Q8—41no—
193C2×Q8—42no—
194C2×Q8—42no—
195C2×Q8—41no—
196C2×Q8—41no—
197C2×Q8—41no—
198C2×Q8—41no—
199C2×Q8—82no—
200C2×Q8—82no—
201C2×Q8—82no—
202C22×C4—22no—
203C22×C4—22no—
204C22×C4—41no—
205C22×C4—41no—
206C22×C4—42no—
207C22×C4#12111yescharacteristic
208C22×C4#12111yescharacteristic
209C22×C4#12111yescharacteristic
210C22×C4#12111yescharacteristic
211C22×C4C22×D411yescharacteristic
212C22×C4C22×D411yescharacteristic
213C22×C4C2×#4111yescharacteristic
214C2×C8—22no—
215C2×C8—22no—
216C2×C8—22no—
217C2×C8—22no—
218C2×C8—22no—
219C2×C8—22no—
220C2×C8—22no—
221C2×C8—22no—
222C2×C8—41no—
223C2×C8—41no—
224C2×C8—41no—
225C2×C8—41no—
226C2×C8—41no—
227C2×C8—41no—
228C2×C8—41no—
229C2×C8—42no—
230C2×C8—41no—
231C2×C8—42no—
232C2×C8—42no—
233C2×C8—41no—
234C2×C8C2×SD1614yes—
235C4⋊C4—22no—
236C4⋊C4—24no—
237C4⋊C4—22no—
238C4⋊C4—22no—
239C4⋊C4—22no—
240C4⋊C4—22no—
241C4⋊C4—41no—
242C4⋊C4—41no—
243C4⋊C4—44no—
244C4⋊C4—42no—
245C4⋊C4—41no—
246C4⋊C4—41no—
247C4⋊C4—42no—
248C4⋊C4—41no—
249C4⋊C4—41no—
250C4⋊C4—42no—
251C4⋊C4—42no—
252C4⋊C4—42no—
253C4⋊C4—81no—
254C4⋊C4—81no—
255C4⋊C4—82no—
256C4⋊C4—82no—
257C4⋊C4—82no—
258C4⋊C4—82no—
259C4⋊C4—82no—
260C4⋊C4C2×SD1614yes—
261C42—21no—
262C42—21no—
263C42—21no—
264C42—21no—
265C42—22no—
266C42—41no—
267C42—42no—
268C42—42no—
269C42—82no—
270C42C2×SD1612yes—
271C42C2×SD1612yes—
272C42#13512yes—
273C42#13712yes—
274C42C22×Q811yescharacteristic
275C42C2×#4111yescharacteristic
276C422−1+411yescharacteristic
277C422−1+411yescharacteristic
278C23C2×#12111yescharacteristic
279Q8—48no—
280Q8—42no—
281Q8—42no—
282Q8—81no—
283Q8—81no—
284Q8—164no—
285C2×C4—21no—
286C2×C4—21no—
287C2×C4—21no—
288C2×C4—21no—
289C2×C4—21no—
290C2×C4—21no—
291C2×C4—21no—
292C2×C4—24no—
293C2×C4—21no—
294C2×C4—21no—
295C2×C4—42no—
296C2×C4—42no—
297C2×C4—42no—
298C2×C4—42no—
299C2×C4—42no—
300C2×C4—42no—
301C2×C4—81no—
302C2×C4—81no—
303C2×C4—82no—
304C2×C4—81no—
305C2×C4—81no—
306C2×C4—82no—
307C2×C4—82no—
308C2×C4—81no—
309C2×C4#46112yes—
310C2×C4#46112yes—
311C2×C4#46512yes—
312C2×C4#46512yes—
313C2×C4#47112yes—
314C2×C4#53311yescharacteristic
315C2×C4#54111yescharacteristic
316C2×C4#54711yescharacteristic
317C2×C4#54811yescharacteristic
318C2×C4D4×Q811yescharacteristic
319C2×C4D4×Q811yescharacteristic
320C2×C4C22×SD1611yescharacteristic
321C2×C4C22×SD1611yescharacteristic
322C2×C4C2×#13511yescharacteristic
323C2×C4C2×#13711yescharacteristic
324C2×C4#57611yescharacteristic
325C2×C4#57711yescharacteristic
326C2×C4#57711yescharacteristic
327C2×C4#57711yescharacteristic
328C8—24no—
329C8—42no—
330C8—42no—
331C8—42no—
332C8—42no—
333C8—81no—
334C8—82no—
335C22#242911yescharacteristic
336C22C2×#46111yescharacteristic
337C22#304311yescharacteristic
338C22C2×#46511yescharacteristic
339C22#306211yescharacteristic
340C22C2×#47111yescharacteristic
341C22#306911yescharacteristic
342C4—22no—
343C4—21no—
344C4—21no—
345C4—22no—
346C4—41no—
347C4—44no—
348C4—82no—
349C4—82no—
350C4—162no—
351C4—161no—
352C4#167114yes—
353C4#333511yescharacteristic
354C4#335611yescharacteristic
355C4#336011yescharacteristic
356C4#336211yescharacteristic
357C4Q8×SD1611yescharacteristic
358C4Q8×SD1611yescharacteristic
359C4#338011yescharacteristic
360C4#338711yescharacteristic
361C2C2×#167111yescharacteristic
362C2#1923211yescharacteristic
363C2#1939711yescharacteristic
364C2#1956411yescharacteristic
365C2#2545311yescharacteristic
366C2#2547011yescharacteristic
367C2#2549211yescharacteristic
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
106. #332Q8semidirect

Maximal subgroups

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

subgroupindex
#292062
#282832
#231492
#106542
#105982
#98962
#96122
#96022
#95652
C4×#3322