Finite Group Databasebeta

← #290099

Subgroups — #290099

Order 512.

593 conjugacy classes of subgroups.

The subgroup lattice has 370 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#29208C211yesmaximal, characteristic
2#28687C211yesmaximal, characteristic
3#28283C211yesmaximal, characteristic
4#23149C211yesmaximal, characteristic
5#10651C212yesmaximal
6#10602C212yesmaximal
7#9896C211yesmaximal, characteristic
8#9615C212yesmaximal
9#9601C211yesmaximal, characteristic
10#9565C212yesmaximal
11C4×#332C211yesmaximal, characteristic
12C2×#498—21no—
13C2×#475—21no—
14C2×C8×Q8—21no—
15#2868C2211yescharacteristic
16C4×C4⋊Q8C2211yescharacteristic
17C4×C4⋊Q8C2211yescharacteristic
18#2084—22no—
19#2066—22no—
20#2057—22no—
21#1990—21no—
22#1990C2212yes—
23#1984—21no—
24#1981C2212yes—
25#1948C2212yes—
26#1948C2212yes—
27#1948C2212yes—
28#1947—21no—
29#1947—21no—
30#1939C2212yes—
31#1916—22no—
32#1845C2212yes—
33#1842C2211yescharacteristic
34#1803—21no—
35C8×C4⋊C4C2212yes—
36C4×C4⋊C8C2211yescharacteristic
37C4×C4⋊C8C2211yescharacteristic
38C4×C4⋊C8C2211yescharacteristic
39C2×#335C2212yes—
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#498—44no—
55#475—44no—
56C8×Q8—44no—
57C2×#107—22no—
58C2×#107—22no—
59C2×#106—22no—
60C2×#106—22no—
61C2×#106—22no—
62C2×C4⋊C8—22no—
63C2×C4⋊C8—21no—
64C2×C4⋊C8—21no—
65C2×C4⋊C8D411yescharacteristic
66C2×C4⋊C8D411yescharacteristic
67C2×C4⋊C8C2311yescharacteristic
68C2×C4⋊C8C2312yes—
69C2×C4⋊C8C2311yescharacteristic
70C2×C4⋊C8C2311yescharacteristic
71C2×C4⋊C8C2311yescharacteristic
72C2×#103—22no—
73C2×#103—22no—
74C2×C4×C8—22no—
75C2×C4×C8—21no—
76C2×C4×C8—21no—
77C2×C4×C8C2312yes—
78#391—21no—
79#391—21no—
80#389—21no—
81#389—21no—
82#389—21no—
83#389D412yes—
84#384—21no—
85#384—21no—
86#384—21no—
87#384D411yescharacteristic
88#384C2311yescharacteristic
89#384C2312yes—
90C4×C4⋊C4—21no—
91C4×C4⋊C4—21no—
92C4×C4⋊C4—21no—
93C4×C4⋊C4C2312yes—
94C43C2311yescharacteristic
95#340—22no—
96#340—22no—
97#340—22no—
98#340—22no—
99#340—22no—
100#336—22no—
101#336—22no—
102#336—21no—
103#336—21no—
104#336D412yes—
105#335—24no—
106#332—22no—
107#332Q814yescomplemented
108C22×Q8—21no—
109C22×Q8—21no—
110C22×Q8—41no—
111C22×C8—21no—
112C22×C8—21no—
113C22×C8—21no—
114C22×C8—21no—
115C22×C8—22no—
116C22×C8C2×D412yes—
117C22×C8C2×D412yes—
118C4⋊Q8—22no—
119C4⋊Q8—21no—
120C4⋊Q8—21no—
121C4⋊Q8—21no—
122C4⋊Q8—42no—
123C4⋊Q8—42no—
124C4⋊Q8—81no—
125C4⋊Q8—81no—
126C4⋊Q8—82no—
127C4⋊Q8D814yes—
128C4⋊Q8C2×Q812yes—
129C4×Q8—42no—
130C4×Q8—42no—
131C4×Q8—42no—
132C4×Q8—42no—
133C2×C4⋊C4—21no—
134C2×C4⋊C4—21no—
135C2×C4⋊C4—21no—
136C2×C4⋊C4—21no—
137C2×C4⋊C4—22no—
138C2×C4⋊C4—21no—
139C2×C4⋊C4—22no—
140C2×C4⋊C4—22no—
141C2×C4⋊C4—21no—
142C2×C4⋊C4—21no—
143C2×C4⋊C4—22no—
144C2×C4⋊C4—21no—
145C2×C4⋊C4—41no—
146C2×C4⋊C4—41no—
147C2×C4⋊C4—41no—
148C2×C4⋊C4—41no—
149C2×C4⋊C4C2×D412yes—
150C2×C4⋊C4C2×D412yes—
151C2×C42—22no—
152C2×C42—41no—
153C2×C42C2×D411yescharacteristic
154C2×C42C2×D411yescharacteristic
155C2×C42C2×D411yescharacteristic
156C2×C42C2×D411yescharacteristic
157C2×C42#4111yescharacteristic
158C2×C42#4111yescharacteristic
159C2×C42C2411yescharacteristic
160#107—82no—
161#107—82no—
162#106—44no—
163#106—44no—
164#106—44no—
165C4⋊C8—22no—
166C4⋊C8—21no—
167C4⋊C8—22no—
168C4⋊C8—21no—
169C4⋊C8—21no—
170C4⋊C8—21no—
171C4⋊C8—21no—
172C4⋊C8—22no—
173C4⋊C8—44no—
174C4⋊C8—42no—
175C4⋊C8—81no—
176C4⋊C8SD1614yes—
177C4⋊C8C2×Q812yes—
178C4⋊C8C2×Q812yes—
179C4⋊C8#4112yes—
180#103—44no—
181#103—82no—
182C4×C8—22no—
183C4×C8—22no—
184C4×C8—44no—
185C4×C8—42no—
186C4×C8—42no—
187#95—22no—
188#95—22no—
189#95—41no—
190#95—41no—
191C2×Q8—24no—
192C2×Q8—22no—
193C2×Q8—41no—
194C2×Q8—42no—
195C2×Q8—42no—
196C2×Q8—41no—
197C2×Q8—41no—
198C2×Q8—41no—
199C2×Q8—41no—
200C2×Q8—82no—
201C2×Q8—82no—
202C2×Q8—82no—
203C22×C4—22no—
204C22×C4—22no—
205C22×C4—41no—
206C22×C4—41no—
207C22×C4—42no—
208C22×C4#12111yescharacteristic
209C22×C4#12111yescharacteristic
210C22×C4#12111yescharacteristic
211C22×C4#12111yescharacteristic
212C22×C4C22×D411yescharacteristic
213C22×C4C22×D411yescharacteristic
214C22×C4C2×#4111yescharacteristic
215C2×C8—22no—
216C2×C8—22no—
217C2×C8—22no—
218C2×C8—22no—
219C2×C8—22no—
220C2×C8—22no—
221C2×C8—22no—
222C2×C8—22no—
223C2×C8—41no—
224C2×C8—41no—
225C2×C8—41no—
226C2×C8—41no—
227C2×C8—41no—
228C2×C8—41no—
229C2×C8—41no—
230C2×C8—42no—
231C2×C8—42no—
232C2×C8—41no—
233C2×C8—42no—
234C2×C8—41no—
235C2×C8C2×SD1614yes—
236C4⋊C4—22no—
237C4⋊C4—24no—
238C4⋊C4—22no—
239C4⋊C4—22no—
240C4⋊C4—22no—
241C4⋊C4—22no—
242C4⋊C4—41no—
243C4⋊C4—41no—
244C4⋊C4—44no—
245C4⋊C4—42no—
246C4⋊C4—41no—
247C4⋊C4—42no—
248C4⋊C4—41no—
249C4⋊C4—42no—
250C4⋊C4—41no—
251C4⋊C4—41no—
252C4⋊C4—42no—
253C4⋊C4—42no—
254C4⋊C4—81no—
255C4⋊C4—81no—
256C4⋊C4—82no—
257C4⋊C4—82no—
258C4⋊C4—82no—
259C4⋊C4—82no—
260C4⋊C4—82no—
261C4⋊C4C2×D814yes—
262C42—21no—
263C42—21no—
264C42—21no—
265C42—21no—
266C42—22no—
267C42—41no—
268C42—42no—
269C42—42no—
270C42—82no—
271C42C2×D812yes—
272C42C2×SD1612yes—
273C42#13512yes—
274C42#13712yes—
275C42C22×Q811yescharacteristic
276C42C2×#4111yescharacteristic
277C422−1+411yescharacteristic
278C422−1+411yescharacteristic
279C23C2×#12111yescharacteristic
280Q8—48no—
281Q8—42no—
282Q8—42no—
283Q8—81no—
284Q8—81no—
285Q8—164no—
286C2×C4—21no—
287C2×C4—21no—
288C2×C4—21no—
289C2×C4—21no—
290C2×C4—21no—
291C2×C4—21no—
292C2×C4—21no—
293C2×C4—24no—
294C2×C4—21no—
295C2×C4—21no—
296C2×C4—42no—
297C2×C4—42no—
298C2×C4—42no—
299C2×C4—42no—
300C2×C4—42no—
301C2×C4—42no—
302C2×C4—81no—
303C2×C4—81no—
304C2×C4—82no—
305C2×C4—81no—
306C2×C4—81no—
307C2×C4—82no—
308C2×C4—82no—
309C2×C4—81no—
310C2×C4#46112yes—
311C2×C4#46112yes—
312C2×C4#46612yes—
313C2×C4#46612yes—
314C2×C4#47312yes—
315C2×C4#53311yescharacteristic
316C2×C4#54111yescharacteristic
317C2×C4#54711yescharacteristic
318C2×C4#54811yescharacteristic
319C2×C4D4×Q811yescharacteristic
320C2×C4D4×Q811yescharacteristic
321C2×C4C22×D811yescharacteristic
322C2×C4C22×SD1611yescharacteristic
323C2×C4C2×#13511yescharacteristic
324C2×C4C2×#13711yescharacteristic
325C2×C4#57711yescharacteristic
326C2×C4#57711yescharacteristic
327C2×C4#57711yescharacteristic
328C2×C4#57811yescharacteristic
329C8—24no—
330C8—42no—
331C8—42no—
332C8—42no—
333C8—42no—
334C8—81no—
335C8—82no—
336C22#242911yescharacteristic
337C22C2×#46111yescharacteristic
338C22#304311yescharacteristic
339C22C2×#46611yescharacteristic
340C22#306311yescharacteristic
341C22C2×#47311yescharacteristic
342C22#306811yescharacteristic
343C4—22no—
344C4—21no—
345C4—21no—
346C4—22no—
347C4—41no—
348C4—44no—
349C4—82no—
350C4—82no—
351C4—161no—
352C4—162no—
353C4#167314yes—
354C4#333511yescharacteristic
355C4#335611yescharacteristic
356C4#335811yescharacteristic
357C4#336411yescharacteristic
358C4Q8×D811yescharacteristic
359C4Q8×SD1611yescharacteristic
360C4#338111yescharacteristic
361C4#338711yescharacteristic
362C2C2×#167311yescharacteristic
363C2#1923311yescharacteristic
364C2#1939811yescharacteristic
365C2#1956711yescharacteristic
366C2#2545311yescharacteristic
367C2#2546911yescharacteristic
368C2#2549111yescharacteristic
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
107. #332Q8semidirect

Maximal subgroups

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

subgroupindex
#292082
#286872
#282832
#231492
#106512
#106022
#98962
#96152
#96012
#95652
C4×#3322