Finite Group Databasebeta

← #286690

Subgroups — #286690

Order 512.

583 conjugacy classes of subgroups.

The subgroup lattice has 376 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#29290C211yesmaximal, characteristic
2#28748C211yesmaximal, characteristic
3#28687C211yesmaximal, characteristic
4#23247C211yesmaximal, characteristic
5#10590C212yesmaximal
6#10544C212yesmaximal
7#9729C212yesmaximal
8#9692C211yesmaximal, characteristic
9#9602C211yesmaximal, characteristic
10#9566C212yesmaximal
11C4×#333C211yesmaximal, characteristic
12C2×#498C2212yes—
13C2×#487—21no—
14#2868C2211yescharacteristic
15C4×C4⋊Q8C2211yescharacteristic
16C4×C4⋊Q8C2211yescharacteristic
17#2087—22no—
18#2084—22no—
19#2043—22no—
20#1990C2212yes—
21#1987—21no—
22#1987C2212yes—
23#1984—22no—
24#1981C2212yes—
25#1981C2212yes—
26#1956—21no—
27#1948—21no—
28#1948—21no—
29#1947—21no—
30#1947C2212yes—
31#1939—21no—
32#1939C2212yes—
33#1842C2211yescharacteristic
34C8×C4⋊C4—21no—
35C8×C4⋊C4C2212yes—
36C4×C4⋊C8C2211yescharacteristic
37C4×C4⋊C8C2211yescharacteristic
38C4×C4⋊C8C2211yescharacteristic
39C2×#334C2212yes—
40C2×#333C2211yescharacteristic
41C2×#333C2211yescharacteristic
42C2×#332C2212yes—
43C2×#332C2212yes—
44#1296C2211yescharacteristic
45#1296C2211yescharacteristic
46#1277C2212yes—
47C2×C4⋊Q8—21no—
48C2×C4⋊Q8—21no—
49C2×C4⋊Q8D411yescharacteristic
50C2×C4⋊Q8C2312yes—
51C2×C4⋊Q8C2311yescharacteristic
52C2×C4×Q8—21no—
53C2×C4×Q8—21no—
54#498—24no—
55#498—42no—
56#487—44no—
57C2×#107—22no—
58C2×#106—22no—
59C2×#106—22no—
60C2×C4⋊C8—21no—
61C2×C4⋊C8—22no—
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×#103—22no—
71C2×#103—22no—
72C2×#103—22no—
73C2×#103—22no—
74C2×C4×C8—21no—
75C2×C4×C8—21no—
76C2×C4×C8—22no—
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#384—21no—
88#384D411yescharacteristic
89#384C2311yescharacteristic
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#334—42no—
106#333—22no—
107#333—22no—
108#332—24no—
109#332—42no—
110C22×Q8—21no—
111C22×Q8—22no—
112C22×Q8—21no—
113C22×C8—21no—
114C22×C8—21no—
115C22×C8—21no—
116C22×C8—21no—
117C22×C8—22no—
118C22×C8C2×D412yes—
119C22×C8C2×D412yes—
120C4⋊Q8—22no—
121C4⋊Q8—21no—
122C4⋊Q8—22no—
123C4⋊Q8—22no—
124C4⋊Q8—21no—
125C4⋊Q8—22no—
126C4⋊Q8—21no—
127C4⋊Q8—22no—
128C4⋊Q8—42no—
129C4⋊Q8—42no—
130C4⋊Q8—81no—
131C4⋊Q8—81no—
132C4⋊Q8SD1614yes—
133C4⋊Q8#4112yes—
134C4×Q8—81no—
135C4×Q8—81no—
136C4×Q8—82no—
137C2×C4⋊C4—21no—
138C2×C4⋊C4—21no—
139C2×C4⋊C4—21no—
140C2×C4⋊C4—21no—
141C2×C4⋊C4—22no—
142C2×C4⋊C4—21no—
143C2×C4⋊C4—21no—
144C2×C4⋊C4—22no—
145C2×C4⋊C4—21no—
146C2×C4⋊C4—22no—
147C2×C4⋊C4—21no—
148C2×C4⋊C4—41no—
149C2×C4⋊C4—41no—
150C2×C4⋊C4—41no—
151C2×C4⋊C4—41no—
152C2×C4⋊C4—41no—
153C2×C4⋊C4C2×D412yes—
154C2×C4⋊C4C2×D412yes—
155C2×C42—22no—
156C2×C42—41no—
157C2×C42C2×D411yescharacteristic
158C2×C42C2×D411yescharacteristic
159C2×C42C2×D411yescharacteristic
160C2×C42C2×D411yescharacteristic
161C2×C42#4111yescharacteristic
162C2×C42#4111yescharacteristic
163C2×C42C2411yescharacteristic
164#107—82no—
165#106—44no—
166#106—44no—
167C4⋊C8—22no—
168C4⋊C8—22no—
169C4⋊C8—21no—
170C4⋊C8—21no—
171C4⋊C8—21no—
172C4⋊C8—21no—
173C4⋊C8—22no—
174C4⋊C8—21no—
175C4⋊C8—21no—
176C4⋊C8—21no—
177C4⋊C8—42no—
178C4⋊C8—81no—
179C4⋊C8—82no—
180C4⋊C8Q1614yes—
181C4⋊C8#4112yes—
182#103—44no—
183#103—44no—
184#103—82no—
185#103—82no—
186C4×C8—22no—
187C4×C8—22no—
188C4×C8—42no—
189C4×C8—42no—
190C4×C8—82no—
191#95—22no—
192#95—22no—
193#95—41no—
194#95—41no—
195C2×Q8—24no—
196C2×Q8—22no—
197C2×Q8—41no—
198C2×Q8—42no—
199C2×Q8—42no—
200C2×Q8—42no—
201C2×Q8—42no—
202C2×Q8—42no—
203C2×Q8—42no—
204C2×Q8—81no—
205C2×Q8—81no—
206C2×Q8—81no—
207C2×Q8—81no—
208C22×C4—22no—
209C22×C4—22no—
210C22×C4—41no—
211C22×C4—41no—
212C22×C4—42no—
213C22×C4#12111yescharacteristic
214C22×C4#12111yescharacteristic
215C22×C4#12111yescharacteristic
216C22×C4#12111yescharacteristic
217C22×C4C22×D411yescharacteristic
218C22×C4C22×D411yescharacteristic
219C22×C4C2×#4111yescharacteristic
220C2×C8—22no—
221C2×C8—22no—
222C2×C8—22no—
223C2×C8—22no—
224C2×C8—22no—
225C2×C8—22no—
226C2×C8—41no—
227C2×C8—41no—
228C2×C8—41no—
229C2×C8—41no—
230C2×C8—41no—
231C2×C8—42no—
232C2×C8—41no—
233C2×C8—41no—
234C2×C8—42no—
235C2×C8—41no—
236C2×C8—41no—
237C2×C8—42no—
238C2×C8—41no—
239C2×C8—41no—
240C2×C8C2×Q1614yes—
241C4⋊C4—22no—
242C4⋊C4—24no—
243C4⋊C4—22no—
244C4⋊C4—22no—
245C4⋊C4—22no—
246C4⋊C4—22no—
247C4⋊C4—41no—
248C4⋊C4—41no—
249C4⋊C4—42no—
250C4⋊C4—42no—
251C4⋊C4—42no—
252C4⋊C4—42no—
253C4⋊C4—42no—
254C4⋊C4—41no—
255C4⋊C4—41no—
256C4⋊C4—41no—
257C4⋊C4—41no—
258C4⋊C4—81no—
259C4⋊C4—82no—
260C4⋊C4—81no—
261C4⋊C4—82no—
262C4⋊C4—82no—
263C4⋊C4—82no—
264C4⋊C4—82no—
265C4⋊C4—82no—
266C4⋊C4C2×SD1614yes—
267C42—21no—
268C42—21no—
269C42—21no—
270C42—21no—
271C42—22no—
272C42—41no—
273C42—42no—
274C42—42no—
275C42—82no—
276C42C2×SD1612yes—
277C42C2×Q1612yes—
278C42#13512yes—
279C42#13712yes—
280C42C2×#4111yescharacteristic
281C42C2×#4111yescharacteristic
282C42H(5,𝔽2)11yescharacteristic
283C422−1+411yescharacteristic
284C23C2×#12111yescharacteristic
285Q8—84no—
286Q8—82no—
287Q8—82no—
288Q8—82no—
289Q8—82no—
290Q8—82no—
291Q8—82no—
292C2×C4—21no—
293C2×C4—21no—
294C2×C4—21no—
295C2×C4—21no—
296C2×C4—21no—
297C2×C4—21no—
298C2×C4—21no—
299C2×C4—24no—
300C2×C4—21no—
301C2×C4—21no—
302C2×C4—42no—
303C2×C4—42no—
304C2×C4—42no—
305C2×C4—42no—
306C2×C4—42no—
307C2×C4—42no—
308C2×C4—81no—
309C2×C4—81no—
310C2×C4—82no—
311C2×C4—81no—
312C2×C4—82no—
313C2×C4—81no—
314C2×C4—82no—
315C2×C4—81no—
316C2×C4#46212yes—
317C2×C4#46212yes—
318C2×C4#46512yes—
319C2×C4#46512yes—
320C2×C4#47112yes—
321C2×C4#52911yescharacteristic
322C2×C4#54211yescharacteristic
323C2×C4#54611yescharacteristic
324C2×C4#54811yescharacteristic
325C2×C4#54811yescharacteristic
326C2×C4#55011yescharacteristic
327C2×C4C22×SD1611yescharacteristic
328C2×C4C22×Q1611yescharacteristic
329C2×C4C2×#13511yescharacteristic
330C2×C4C2×#13711yescharacteristic
331C2×C4#57611yescharacteristic
332C2×C4#57711yescharacteristic
333C2×C4#57711yescharacteristic
334C2×C4#57711yescharacteristic
335C8—24no—
336C8—42no—
337C8—42no—
338C8—81no—
339C8—81no—
340C8—81no—
341C8—82no—
342C22#242611yescharacteristic
343C22C2×#46211yescharacteristic
344C22#304211yescharacteristic
345C22C2×#46511yescharacteristic
346C22#306211yescharacteristic
347C22C2×#47111yescharacteristic
348C22#306911yescharacteristic
349C4—22no—
350C4—22no—
351C4—21no—
352C4—21no—
353C4—41no—
354C4—44no—
355C4—82no—
356C4—82no—
357C4—161no—
358C4—162no—
359C4#167214yes—
360C4#329911yescharacteristic
361C4#332911yescharacteristic
362C4#336211yescharacteristic
363C4#336311yescharacteristic
364C4#336511yescharacteristic
365C4#337211yescharacteristic
366C4#339211yescharacteristic
367C4#339611yescharacteristic
368C2C2×#167211yescharacteristic
369C2#1924011yescharacteristic
370C2#1939011yescharacteristic
371C2#1955811yescharacteristic
372C2#2527811yescharacteristic
373C2#2529511yescharacteristic
374C2#2531311yescharacteristic
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

Maximal subgroups

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

subgroupindex
#292902
#287482
#286872
#232472
#105902
#105442
#97292
#96922
#96022
#95662
C4×#3332