Finite Group Databasebeta

← #320247

Subgroups — #320247

Order 512.

724 conjugacy classes of subgroups.

The subgroup lattice has 362 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#33289C211yesmaximal, characteristic
2#29218C211yesmaximal, characteristic
3#29206C211yesmaximal, characteristic
4#28687C211yesmaximal, characteristic
5#28440C211yesmaximal, characteristic
6C4×#487C211yesmaximal, characteristic
7#10686C212yesmaximal
8#10318C212yesmaximal
9#9839C212yesmaximal
10#9828C212yesmaximal
11#8261C211yesmaximal, characteristic
12C2×#498C2212yes—
13C2×#494C2212yes—
14C2×#487C2211yescharacteristic
15C2×#462—22no—
16#2868C2211yescharacteristic
17#2868C2211yescharacteristic
18C4×C4⋊Q8C2211yescharacteristic
19C4×C4⋊Q8C2211yescharacteristic
20#2100—22no—
21#2084—22no—
22#2058—22no—
23#2016—22no—
24#1990—22no—
25#1990C2212yes—
26#1990C2212yes—
27#1990C2212yes—
28#1990C2212yes—
29#1990C2212yes—
30#1990C2211yescharacteristic
31#1987—22no—
32#1956C2212yes—
33#1956C2211yescharacteristic
34#1948—22no—
35#1948—22no—
36#1933—22no—
37#1844C2212yes—
38#1834—22no—
39#1834C2212yes—
40#1834C2211yescharacteristic
41C4×C4⋊C8C2211yescharacteristic
42C4×C4⋊C8C2211yescharacteristic
43C4×#103—22no—
44C42×C8C2211yescharacteristic
45C2×#322C2212yes—
46#1273C2212yes—
47C22×Q16—24no—
48C2×C4⋊Q8—21no—
49C2×C4⋊Q8—21no—
50C2×C4⋊Q8—21no—
51C2×C4⋊Q8—21no—
52C2×C4⋊Q8—21no—
53C2×C4⋊Q8C2312yes—
54C2×C4⋊Q8C2311yescharacteristic
55C2×C4⋊Q8C2311yescharacteristic
56C2×C4×Q8—21no—
57C2×C4×Q8—21no—
58#498—24no—
59#498—42no—
60#494—24no—
61#494—42no—
62#487—22no—
63#487—22no—
64#462—48no—
65C2×#106—24no—
66C2×#106—24no—
67C2×C4⋊C8D412yes—
68C2×C4⋊C8D412yes—
69C2×#103—22no—
70C2×#103—22no—
71C2×#103—24no—
72C2×C4×C8D412yes—
73C2×C4×C8D412yes—
74C2×C4×C8C2311yescharacteristic
75C2×C4×C8C2312yes—
76C2×C4×C8C2312yes—
77C2×C4×C8C2311yescharacteristic
78#391—21no—
79#391—21no—
80#389—21no—
81#389—21no—
82#389—21no—
83#389—21no—
84#384—21no—
85#384—21no—
86#384—21no—
87#384—21no—
88#384—21no—
89#384—21no—
90#384—21no—
91#384C2311yescharacteristic
92#384C2311yescharacteristic
93#384C2312yes—
94C4×C4⋊C4—21no—
95C4×C4⋊C4—21no—
96C4×C4⋊C4—21no—
97C4×C4⋊C4—21no—
98C43C2311yescharacteristic
99#340—24no—
100#340—24no—
101#340—24no—
102#340—24no—
103#340—24no—
104#340—22no—
105#340—24no—
106#340—22no—
107#336D412yes—
108#336D412yes—
109#322—42no—
110C22×Q8—21no—
111C22×Q8—21no—
112C22×Q8—22no—
113C22×Q8—41no—
114C2×Q16—48no—
115C2×Q16—48no—
116C2×Q16—84no—
117C22×C8C2×D414yes—
118C22×C8C2×D412yes—
119C22×C8C2×D412yes—
120C22×C8C2×D414yes—
121C4⋊Q8—21no—
122C4⋊Q8—21no—
123C4⋊Q8—22no—
124C4⋊Q8—22no—
125C4⋊Q8—21no—
126C4⋊Q8—21no—
127C4⋊Q8—22no—
128C4⋊Q8—21no—
129C4⋊Q8—21no—
130C4⋊Q8—22no—
131C4⋊Q8—44no—
132C4⋊Q8—44no—
133C4⋊Q8—44no—
134C4⋊Q8—82no—
135C4⋊Q8—82no—
136C4⋊Q8#4112yes—
137C4⋊Q8#4112yes—
138C4×Q8—44no—
139C4×Q8—44no—
140C2×C4⋊C4—21no—
141C2×C4⋊C4—21no—
142C2×C4⋊C4—21no—
143C2×C4⋊C4—21no—
144C2×C4⋊C4—21no—
145C2×C4⋊C4—21no—
146C2×C4⋊C4—22no—
147C2×C4⋊C4—22no—
148C2×C4⋊C4—22no—
149C2×C4⋊C4—22no—
150C2×C4⋊C4—22no—
151C2×C4⋊C4—21no—
152C2×C4⋊C4—21no—
153C2×C4⋊C4—21no—
154C2×C4⋊C4—21no—
155C2×C4⋊C4—41no—
156C2×C4⋊C4—41no—
157C2×C4⋊C4—41no—
158C2×C4⋊C4—41no—
159C2×C4⋊C4—41no—
160C2×C4⋊C4—41no—
161C2×C4⋊C4—41no—
162C2×C4⋊C4—41no—
163C2×C4⋊C4—41no—
164C2×C42—41no—
165C2×C42—41no—
166C2×C42C2×D411yescharacteristic
167C2×C42C2×D411yescharacteristic
168C2×C42C2×D411yescharacteristic
169C2×C42C2×D411yescharacteristic
170C2×C42C2×D411yescharacteristic
171C2×C42C2×D411yescharacteristic
172C2×C42C2411yescharacteristic
173#106—48no—
174#106—84no—
175C4⋊C8—24no—
176C4⋊C8SD1618yes—
177#103—28no—
178#103—44no—
179#103—48no—
180C4×C8—24no—
181C4×C8—21no—
182C4×C8—21no—
183C4×C8—22no—
184C4×C8—22no—
185C4×C8—21no—
186C4×C8—22no—
187C4×C8—22no—
188C4×C8—21no—
189C4×C8—42no—
190#95—41no—
191#95—41no—
192#95—41no—
193#95—41no—
194C2×Q8—22no—
195C2×Q8—22no—
196C2×Q8—41no—
197C2×Q8—41no—
198C2×Q8—42no—
199C2×Q8—42no—
200C2×Q8—42no—
201C2×Q8—44no—
202C2×Q8—42no—
203C2×Q8—42no—
204C2×Q8—42no—
205C2×Q8—44no—
206C2×Q8—82no—
207C2×Q8—82no—
208C2×Q8—82no—
209C22×C4—41no—
210C22×C4—41no—
211C22×C4—41no—
212C22×C4—41no—
213C22×C4—42no—
214C22×C4—42no—
215C22×C4#12711yescharacteristic
216C22×C4#12711yescharacteristic
217C22×C4#12711yescharacteristic
218C22×C4#12711yescharacteristic
219C22×C4C22×D411yescharacteristic
220C22×C4C22×D411yescharacteristic
221C22×C4C22×D411yescharacteristic
222Q16—816no—
223C2×C8—24no—
224C2×C8—24no—
225C2×C8—22no—
226C2×C8—22no—
227C2×C8—24no—
228C2×C8—22no—
229C2×C8—22no—
230C2×C8—24no—
231C2×C8—22no—
232C2×C8—22no—
233C2×C8—24no—
234C2×C8C2×SD1618yes—
235C4⋊C4—22no—
236C4⋊C4—22no—
237C4⋊C4—22no—
238C4⋊C4—22no—
239C4⋊C4—41no—
240C4⋊C4—41no—
241C4⋊C4—41no—
242C4⋊C4—41no—
243C4⋊C4—42no—
244C4⋊C4—42no—
245C4⋊C4—42no—
246C4⋊C4—42no—
247C4⋊C4—42no—
248C4⋊C4—42no—
249C4⋊C4—44no—
250C4⋊C4—42no—
251C4⋊C4—44no—
252C4⋊C4—42no—
253C4⋊C4—44no—
254C4⋊C4—42no—
255C4⋊C4—42no—
256C4⋊C4—82no—
257C4⋊C4—82no—
258C4⋊C4—82no—
259C4⋊C4—82no—
260C4⋊C4—82no—
261C4⋊C4—82no—
262C4⋊C4—82no—
263C4⋊C4—82no—
264C4⋊C4—82no—
265C42—22no—
266C42—22no—
267C42—22no—
268C42—82no—
269C42—82no—
270C42C2×SD1614yes—
271C42#13714yes—
272C42#13714yes—
273C42C2×#4111yescharacteristic
274C42C2×#4111yescharacteristic
275C42H(5,𝔽2)11yescharacteristic
276C422−1+411yescharacteristic
277C23C2×#12711yescharacteristic
278Q8—44no—
279Q8—44no—
280Q8—82no—
281Q8—82no—
282Q8—84no—
283Q8—84no—
284Q8—164no—
285C2×C4—21no—
286C2×C4—21no—
287C2×C4—21no—
288C2×C4—21no—
289C2×C4—21no—
290C2×C4—21no—
291C2×C4—42no—
292C2×C4—42no—
293C2×C4—81no—
294C2×C4—81no—
295C2×C4—81no—
296C2×C4—81no—
297C2×C4—81no—
298C2×C4—82no—
299C2×C4—81no—
300C2×C4—82no—
301C2×C4—82no—
302C2×C4—81no—
303C2×C4—82no—
304C2×C4—81no—
305C2×C4—82no—
306C2×C4—81no—
307C2×C4—81no—
308C2×C4—82no—
309C2×C4#49212yes—
310C2×C4#49212yes—
311C2×C4#49712yes—
312C2×C4#49712yes—
313C2×C4#49712yes—
314C2×C4#49712yes—
315C2×C4#52511yescharacteristic
316C2×C4#53711yescharacteristic
317C2×C4#54611yescharacteristic
318C2×C4#54611yescharacteristic
319C2×C4#54811yescharacteristic
320C2×C4#54811yescharacteristic
321C2×C4C22×SD1612yes—
322C2×C4C2×#13712yes—
323C2×C4C2×#13712yes—
324C2×C4#57512yes—
325C2×C4#57612yes—
326C2×C4#57812yes—
327C8—28no—
328C8—44no—
329C8—42no—
330C8—42no—
331C22#254011yescharacteristic
332C22C2×#49211yescharacteristic
333C22C2×#49711yescharacteristic
334C22C2×#49711yescharacteristic
335C22#315311yescharacteristic
336C22#315511yescharacteristic
337C22#315611yescharacteristic
338C4—22no—
339C4—22no—
340C4—22no—
341C4—82no—
342C4—82no—
343C4—161no—
344C4—162no—
345C4—162no—
346C4—161no—
347C4#171614yes—
348C4#311612yes—
349C4#325212yes—
350C4#329512yes—
351C4#332812yes—
352C4#336212yes—
353C4#337412yes—
354C2C2×#171611yescharacteristic
355C2#1962411yescharacteristic
356C2#1968211yescharacteristic
357C2#1970411yescharacteristic
358C2#2838411yescharacteristic
359C2#2840411yescharacteristic
360C2#2841711yescharacteristic
subgroupG/H
center, socle, prime radicalC23C2×#127
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
#332892
#292182
#292062
#286872
#284402
C4×#4872
#106862
#103182
#98392
#98282
#82612