Finite Group Databasebeta

← #413780

Subgroups — #413780

Order 512.

760 conjugacy classes of subgroups.

The subgroup lattice has 363 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23747C212yesmaximal
2C2×#1942C211yesmaximal, characteristic
3#20437C211yesmaximal, characteristic
4#11303C214yesmaximal
5#11204C212yesmaximal
6#10870C212yesmaximal
7#10870C212yesmaximal
8#10864C211yesmaximal, characteristic
9C22×#108C2212yes—
10C22×#107C2211yescharacteristic
11C22×#107C2211yescharacteristic
12C2×#423C2211yescharacteristic
13C2×#405C2212yes—
14C22×C4×C8C2211yescharacteristic
15C2×#389C2212yes—
16#2285C2211yescharacteristic
17#2101—24no—
18#2100—22no—
19#2099—22no—
20#1942—21no—
21#1942—21no—
22#1942—21no—
23#1942—21no—
24#1770—21no—
25#1770—21no—
26C8×C4⋊C4—21no—
27C8×C4⋊C4—21no—
28C2×#343C2214yes—
29C2×#343C2214yes—
30C2×#341C2212yes—
31#1736C2214yes—
32#1736C2214yes—
33C2×#340C2212yes—
34#1729C2212yes—
35#1728C2212yes—
36C22×M16C2314yes—
37C22×M16C2312yes—
38C23×C8C2311yescharacteristic
39C23×C8C2311yescharacteristic
40C2×#117C2312yes—
41C22×C4⋊C4—22no—
42C22×C4⋊C4C2312yes—
43C22×C4⋊C4C2312yes—
44C22×C42C2311yescharacteristic
45C2×#108—24no—
46C2×#108—24no—
47C2×#108—24no—
48C2×#107—21no—
49C2×#107—21no—
50C2×#107—22no—
51C2×#107—21no—
52C2×#107—21no—
53C2×#107—24no—
54C2×#107Q814yes—
55#423—24no—
56C2×C4⋊C8—21no—
57C2×C4⋊C8—21no—
58#405—24no—
59#405—24no—
60C2×#97—22no—
61C2×C4×C8—24no—
62C2×C4×C8—22no—
63C2×C4×C8D411yescharacteristic
64C2×C4×C8D411yescharacteristic
65C2×C4×C8D411yescharacteristic
66C2×C4×C8D411yescharacteristic
67#389—22no—
68#389—22no—
69#389—22no—
70#389—22no—
71#384D411yescharacteristic
72#384D411yescharacteristic
73#382D412yes—
74#380—21no—
75#380—21no—
76C4×C4⋊C4—21no—
77C4×C4⋊C4—21no—
78C2×#95—22no—
79#343—28no—
80#343—44no—
81#341—42no—
82#340—22no—
83#340—22no—
84#340—24no—
85#340—24no—
86#340—24no—
87#340—24no—
88#336—22no—
89#336—22no—
90#336—22no—
91#336—22no—
92C23×C4—22no—
93C23×C4—22no—
94C23×C4#4111yescharacteristic
95C23×C4#4111yescharacteristic
96C23×C4C2411yescharacteristic
97C2×M16—22no—
98C2×M16—22no—
99C2×M16—22no—
100C2×M16—22no—
101C2×M16—22no—
102C2×M16—24no—
103C2×M16—24no—
104C2×M16—24no—
105C2×M16—24no—
106C2×M16—24no—
107C2×M16—24no—
108C2×M16#4114yes—
109C22×C8—22no—
110C22×C8—22no—
111C22×C8—21no—
112C22×C8—22no—
113C22×C8—21no—
114C22×C8—24no—
115C22×C8C2×D411yescharacteristic
116C22×C8C2×D411yescharacteristic
117C22×C8C2×D411yescharacteristic
118C22×C8C2×D411yescharacteristic
119C22×C8C2×Q812yes—
120C22×C8#4112yes—
121C22×C8#4114yes—
122C22×C8#4114yes—
123#117—22no—
124#117—22no—
125#117—22no—
126#117—22no—
127C2×C4⋊C4—22no—
128C2×C4⋊C4—22no—
129C2×C4⋊C4—22no—
130C2×C4⋊C4—22no—
131C2×C4⋊C4—21no—
132C2×C4⋊C4—21no—
133C2×C4⋊C4—21no—
134C2×C4⋊C4—21no—
135C2×C4⋊C4—22no—
136C2×C4⋊C4—22no—
137C2×C4⋊C4—22no—
138C2×C4⋊C4—24no—
139C2×C4⋊C4—24no—
140C2×C4⋊C4—42no—
141C2×C4⋊C4—42no—
142C2×C4⋊C4—84no—
143C2×C4⋊C4C2×D412yes—
144C2×C4⋊C4C2×D412yes—
145C2×C4⋊C4C2×Q814yes—
146C2×C22⋊C4—22no—
147C2×C42—22no—
148C2×C42—22no—
149C2×C42—22no—
150C2×C42C2×D411yescharacteristic
151C2×C42C2×D411yescharacteristic
152C2×C42C2×D411yescharacteristic
153C2×C42C2×D411yescharacteristic
154#108—48no—
155#107—24no—
156#107—24no—
157#107—42no—
158#107—42no—
159C4⋊C8—24no—
160C4⋊C8—24no—
161#97—44no—
162C4×C8—22no—
163C4×C8—22no—
164C4×C8—22no—
165C4×C8—22no—
166C4×C8—44no—
167#95—22no—
168#95—22no—
169#95—42no—
170#95—42no—
171C24C2×#4111yescharacteristic
172C22×C4—22no—
173C22×C4—22no—
174C22×C4—22no—
175C22×C4—22no—
176C22×C4—22no—
177C22×C4—22no—
178C22×C4—24no—
179C22×C4—41no—
180C22×C4—41no—
181C22×C4—42no—
182C22×C4—42no—
183C22×C4—42no—
184C22×C4—44no—
185C22×C4—82no—
186C22×C4—82no—
187C22×C4—82no—
188C22×C4—82no—
189C22×C4#12111yescharacteristic
190C22×C4#12111yescharacteristic
191C22×C4#12111yescharacteristic
192C22×C4#12111yescharacteristic
193C22×C4C22×D411yescharacteristic
194C22×C4C22×D411yescharacteristic
195C22×C4C22×Q811yescharacteristic
196C22×C4C2×#4111yescharacteristic
197C22×C4C2×#4112yes—
198C22×C4C2×#4112yes—
199C22×C4C2×#4112yes—
200C22×C4H(5,𝔽2)11yescharacteristic
201C22×C42−1+412yes—
202C22×C42−1+411yescharacteristic
203M16—24no—
204M16—24no—
205M16—42no—
206M16—42no—
207M16—44no—
208M16—44no—
209M16—44no—
210M16—44no—
211C2×C8—21no—
212C2×C8—21no—
213C2×C8—24no—
214C2×C8—21no—
215C2×C8—21no—
216C2×C8—21no—
217C2×C8—21no—
218C2×C8—42no—
219C2×C8—42no—
220C2×C8—42no—
221C2×C8—42no—
222C2×C8—44no—
223C2×C8—44no—
224C2×C8—44no—
225C2×C8C22⋊Q812yes—
226C2×C8C22⋊Q812yes—
227C2×C8#12312yes—
228C2×C8#12312yes—
229C2×C8#12412yes—
230C2×C8#12412yes—
231C2×C8#12518yes—
232C2×C8#12618yes—
233C4⋊C4—24no—
234C4⋊C4—22no—
235C4⋊C4—22no—
236C4⋊C4—22no—
237C4⋊C4—22no—
238C4⋊C4—24no—
239C4⋊C4—42no—
240C4⋊C4—42no—
241C4⋊C4—42no—
242C4⋊C4—42no—
243C4⋊C4—42no—
244C4⋊C4—42no—
245C4⋊C4—42no—
246C4⋊C4—42no—
247C4⋊C4—44no—
248C4⋊C4—84no—
249C4⋊C4—84no—
250C22⋊C4—42no—
251C22⋊C4—42no—
252C42—21no—
253C42—21no—
254C42—21no—
255C42—21no—
256C42—22no—
257C42—22no—
258C42—42no—
259C42—42no—
260C23—21no—
261C23—21no—
262C23—21no—
263C23—21no—
264C23C2×#12111yescharacteristic
265C23#52912yes—
266C23#55411yescharacteristic
267C23#55612yes—
268C23#55611yescharacteristic
269C2×C4—21no—
270C2×C4—21no—
271C2×C4—21no—
272C2×C4—21no—
273C2×C4—21no—
274C2×C4—21no—
275C2×C4—21no—
276C2×C4—21no—
277C2×C4—21no—
278C2×C4—21no—
279C2×C4—44no—
280C2×C4—44no—
281C2×C4—44no—
282C2×C4—42no—
283C2×C4—42no—
284C2×C4—42no—
285C2×C4—42no—
286C2×C4—42no—
287C2×C4—42no—
288C2×C4—84no—
289C2×C4—82no—
290C2×C4—82no—
291C2×C4—84no—
292C2×C4—82no—
293C2×C4—82no—
294C2×C4—82no—
295C2×C4—164no—
296C2×C4#47212yes—
297C2×C4#47312yes—
298C2×C4C2×C22⋊Q811yescharacteristic
299C2×C4C2×#12311yescharacteristic
300C2×C4#52511yescharacteristic
301C2×C4C2×#12411yescharacteristic
302C2×C4C2×#12512yes—
303C2×C4C2×#12612yes—
304C2×C4#53212yes—
305C2×C4#53611yescharacteristic
306C2×C4#53711yescharacteristic
307C2×C4#54112yes—
308C2×C4#54212yes—
309C2×C4#54712yes—
310C2×C4D4×Q811yescharacteristic
311C2×C4#55011yescharacteristic
312C2×C4#55212yes—
313C2×C4#55312yes—
314C2×C4#55612yes—
315C2×C4#55712yes—
316C8—44no—
317C8—41no—
318C8—41no—
319C8—84no—
320C8#40014yes—
321C8#40014yes—
322C22—22no—
323C22—22no—
324C22—22no—
325C22—22no—
326C22—21no—
327C22—21no—
328C22—21no—
329C22—21no—
330C22#250912yes—
331C22#251412yes—
332C22#253811yescharacteristic
333C22#254711yescharacteristic
334C22#285411yescharacteristic
335C22C2×#47211yescharacteristic
336C22C2×#47311yescharacteristic
337C22#306711yescharacteristic
338C22#306711yescharacteristic
339C4—22no—
340C4—22no—
341C4—41no—
342C4—41no—
343C4—82no—
344C4—82no—
345C4—84no—
346C4—162no—
347C4C2×#40011yescharacteristic
348C4#250612yes—
349C4#251212yes—
350C4#256211yescharacteristic
351C4#258111yescharacteristic
352C4#262511yescharacteristic
353C2—22no—
354C2—22no—
355C2#1515611yescharacteristic
356C2#2829511yescharacteristic
357C2#2829711yescharacteristic
358C2#2873111yescharacteristic
359C2#2873411yescharacteristic
360C2#3312311yescharacteristic
361C2#3312411yescharacteristic
subgroupG/H
center, socleC23C2×#121
commutator, Frattini, cosocleC23×C4C24
prime radicalC24C2×#41
LCSG ▷ C23×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Maximal subgroups

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

subgroupindex
#237472
C2×#19422
#204372
#113032
#112042
#108702
#108642