Finite Group Databasebeta

← #413569

Subgroups — #413569

Order 512.

832 conjugacy classes of subgroups.

The subgroup lattice has 368 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23657C212yesmaximal
2C2×#1952C212yesmaximal
3C2×#1942C211yesmaximal, characteristic
4#20437C211yesmaximal, characteristic
5#11290C214yesmaximal
6#11204C212yesmaximal
7#10870C212yesmaximal
8#10864C211yesmaximal, characteristic
9C22×#108C2212yes—
10C2×#428C2214yes—
11C22×#107C2211yescharacteristic
12C22×#107C2211yescharacteristic
13C2×#423C2211yescharacteristic
14C2×#405C2212yes—
15C22×C4×C8C2211yescharacteristic
16C2×#389C2212yes—
17#2285C2211yescharacteristic
18#2085—24no—
19#2084—22no—
20#2083—22no—
21#1952—28no—
22#1942—21no—
23#1942—21no—
24#1942—21no—
25#1942—21no—
26#1770—21no—
27#1770—21no—
28C8×C4⋊C4—21no—
29C8×C4⋊C4—21no—
30C2×#343C2214yes—
31C2×#343C2214yes—
32C2×#341C2212yes—
33#1736C2214yes—
34C2×#340C2212yes—
35#1729C2212yes—
36#1728C2212yes—
37C22×M16C2314yes—
38C22×M16C2312yes—
39C23×C8C2311yescharacteristic
40C23×C8C2311yescharacteristic
41C2×#117C2312yes—
42C22×C4⋊C4—22no—
43C22×C4⋊C4C2312yes—
44C22×C4⋊C4C2312yes—
45C22×C42C2311yescharacteristic
46C2×#108—24no—
47C2×#108—24no—
48C2×#108Q818yes—
49#428—28no—
50#428—44no—
51C2×#107—21no—
52C2×#107—21no—
53C2×#107—22no—
54C2×#107—21no—
55C2×#107—21no—
56C2×#107—24no—
57C2×#107—24no—
58C2×#107D414yes—
59C2×#106—24no—
60#423—24no—
61C2×C4⋊C8—21no—
62C2×C4⋊C8—21no—
63#405—24no—
64#405—24no—
65C2×#97—22no—
66C2×C4×C8—24no—
67C2×C4×C8—22no—
68C2×C4×C8D411yescharacteristic
69C2×C4×C8D411yescharacteristic
70C2×C4×C8D411yescharacteristic
71C2×C4×C8D411yescharacteristic
72#389—22no—
73#389—22no—
74#389—22no—
75#389—22no—
76#384D411yescharacteristic
77#384D411yescharacteristic
78#382D412yes—
79#380—21no—
80#380—21no—
81C4×C4⋊C4—21no—
82C4×C4⋊C4—21no—
83C2×#95—22no—
84#343—28no—
85#343—28no—
86#341—42no—
87#340—22no—
88#340—22no—
89#340—24no—
90#340—24no—
91#336—22no—
92#336—22no—
93#336—22no—
94#336—22no—
95C23×C4—22no—
96C23×C4—22no—
97C23×C4#4111yescharacteristic
98C23×C4#4111yescharacteristic
99C23×C4C2411yescharacteristic
100C2×M16—22no—
101C2×M16—24no—
102C2×M16—22no—
103C2×M16—24no—
104C2×M16—24no—
105C2×M16—22no—
106C2×M16—24no—
107C2×M16—24no—
108C2×M16—22no—
109C2×M16—22no—
110C2×M16C2×Q818yes—
111C2×M16#4114yes—
112C22×C8—24no—
113C22×C8—22no—
114C22×C8—22no—
115C22×C8—21no—
116C22×C8—22no—
117C22×C8—21no—
118C22×C8C2×D411yescharacteristic
119C22×C8C2×D411yescharacteristic
120C22×C8C2×D412yes—
121C22×C8C2×D411yescharacteristic
122C22×C8C2×D411yescharacteristic
123C22×C8C2×Q814yes—
124C22×C8#4112yes—
125C22×C8#4114yes—
126#117—22no—
127#117—22no—
128#117—22no—
129#117—22no—
130C2×C4⋊C4—22no—
131C2×C4⋊C4—22no—
132C2×C4⋊C4—22no—
133C2×C4⋊C4—22no—
134C2×C4⋊C4—21no—
135C2×C4⋊C4—21no—
136C2×C4⋊C4—21no—
137C2×C4⋊C4—21no—
138C2×C4⋊C4—22no—
139C2×C4⋊C4—22no—
140C2×C4⋊C4—22no—
141C2×C4⋊C4—24no—
142C2×C4⋊C4—24no—
143C2×C4⋊C4—42no—
144C2×C4⋊C4—42no—
145C2×C4⋊C4—84no—
146C2×C4⋊C4C2×D414yes—
147C2×C4⋊C4C2×D412yes—
148C2×C4⋊C4C2×D412yes—
149C2×C22⋊C4—22no—
150C2×C42—22no—
151C2×C42—22no—
152C2×C42—22no—
153C2×C42C2×D411yescharacteristic
154C2×C42C2×D411yescharacteristic
155C2×C42C2×D411yescharacteristic
156C2×C42C2×D411yescharacteristic
157#108—216no—
158#107—24no—
159#107—24no—
160#107—48no—
161#107—42no—
162#107—42no—
163#106—48no—
164C4⋊C8—24no—
165C4⋊C8—24no—
166#97—44no—
167C4×C8—22no—
168C4×C8—22no—
169C4×C8—22no—
170C4×C8—22no—
171C4×C8—44no—
172#95—22no—
173#95—22no—
174#95—42no—
175#95—42no—
176C24C2×#4111yescharacteristic
177C22×C4—22no—
178C22×C4—22no—
179C22×C4—22no—
180C22×C4—22no—
181C22×C4—22no—
182C22×C4—22no—
183C22×C4—24no—
184C22×C4—41no—
185C22×C4—41no—
186C22×C4—42no—
187C22×C4—42no—
188C22×C4—42no—
189C22×C4—44no—
190C22×C4—82no—
191C22×C4—82no—
192C22×C4—82no—
193C22×C4—82no—
194C22×C4#12111yescharacteristic
195C22×C4#12111yescharacteristic
196C22×C4#12111yescharacteristic
197C22×C4#12111yescharacteristic
198C22×C4C22×D411yescharacteristic
199C22×C4C22×D411yescharacteristic
200C22×C4C22×D411yescharacteristic
201C22×C4C22×Q812yes—
202C22×C4C2×#4111yescharacteristic
203C22×C4C2×#4112yes—
204C22×C4C2×#4112yes—
205C22×C4H(5,𝔽2)11yescharacteristic
206C22×C42−1+412yes—
207C22×C42−1+411yescharacteristic
208M16—28no—
209M16—24no—
210M16—24no—
211M16—28no—
212M16—44no—
213M16—44no—
214M16—42no—
215M16—42no—
216C2×C8—21no—
217C2×C8—21no—
218C2×C8—24no—
219C2×C8—21no—
220C2×C8—21no—
221C2×C8—21no—
222C2×C8—21no—
223C2×C8—28no—
224C2×C8—42no—
225C2×C8—44no—
226C2×C8—42no—
227C2×C8—42no—
228C2×C8—42no—
229C2×C8—44no—
230C2×C8#12112yes—
231C2×C8#12112yes—
232C2×C8C22⋊Q818yes—
233C2×C8#12312yes—
234C2×C8#12312yes—
235C2×C8#12518yes—
236C2×C8C4⋊Q812yes—
237C2×C8C4⋊Q812yes—
238C4⋊C4—24no—
239C4⋊C4—22no—
240C4⋊C4—24no—
241C4⋊C4—22no—
242C4⋊C4—22no—
243C4⋊C4—22no—
244C4⋊C4—42no—
245C4⋊C4—42no—
246C4⋊C4—42no—
247C4⋊C4—42no—
248C4⋊C4—42no—
249C4⋊C4—42no—
250C4⋊C4—42no—
251C4⋊C4—42no—
252C4⋊C4—44no—
253C4⋊C4—84no—
254C4⋊C4—84no—
255C22⋊C4—42no—
256C22⋊C4—42no—
257C42—21no—
258C42—21no—
259C42—21no—
260C42—21no—
261C42—22no—
262C42—22no—
263C42—42no—
264C42—42no—
265C23—21no—
266C23—21no—
267C23—21no—
268C23—21no—
269C23C2×#12111yescharacteristic
270C23#52912yes—
271C23#54711yescharacteristic
272C23#55511yescharacteristic
273C23#55712yes—
274C2×C4—21no—
275C2×C4—21no—
276C2×C4—21no—
277C2×C4—21no—
278C2×C4—21no—
279C2×C4—21no—
280C2×C4—21no—
281C2×C4—21no—
282C2×C4—21no—
283C2×C4—21no—
284C2×C4—44no—
285C2×C4—44no—
286C2×C4—42no—
287C2×C4—42no—
288C2×C4—44no—
289C2×C4—42no—
290C2×C4—42no—
291C2×C4—42no—
292C2×C4—42no—
293C2×C4—84no—
294C2×C4—82no—
295C2×C4—82no—
296C2×C4—84no—
297C2×C4—82no—
298C2×C4—82no—
299C2×C4—82no—
300C2×C4—164no—
301C2×C4#47212yes—
302C2×C4#47312yes—
303C2×C4C2×#12111yescharacteristic
304C2×C4C2×C22⋊Q812yes—
305C2×C4C2×#12311yescharacteristic
306C2×C4#52511yescharacteristic
307C2×C4C2×#12512yes—
308C2×C4C2×C4⋊Q811yescharacteristic
309C2×C4#53212yes—
310C2×C4#53611yescharacteristic
311C2×C4#53711yescharacteristic
312C2×C4#54112yes—
313C2×C4#54212yes—
314C2×C4D4211yescharacteristic
315C2×C4#54711yescharacteristic
316C2×C4#54812yes—
317C2×C4D4×Q812yes—
318C2×C4#55212yes—
319C2×C4#55412yes—
320C2×C4#55712yes—
321C8—41no—
322C8—41no—
323C8—48no—
324C8—44no—
325C8#39814yes—
326C8#39814yes—
327C22—22no—
328C22—22no—
329C22—21no—
330C22—21no—
331C22—21no—
332C22—21no—
333C22—22no—
334C22—22no—
335C22#245312yes—
336C22#250812yes—
337C22#253711yescharacteristic
338C22#254611yescharacteristic
339C22#285411yescharacteristic
340C22C2×#47211yescharacteristic
341C22C2×#47311yescharacteristic
342C22#306711yescharacteristic
343C22#306711yescharacteristic
344C4—22no—
345C4—22no—
346C4—41no—
347C4—41no—
348C4—82no—
349C4—84no—
350C4—82no—
351C4—162no—
352C4C2×#39811yescharacteristic
353C4#245212yes—
354C4#250612yes—
355C4#253611yescharacteristic
356C4#258411yescharacteristic
357C4#262511yescharacteristic
358C2—22no—
359C2—22no—
360C2#1514911yescharacteristic
361C2#2824011yescharacteristic
362C2#2824111yescharacteristic
363C2#2869911yescharacteristic
364C2#2870211yescharacteristic
365C2#3312311yescharacteristic
366C2#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
#236572
C2×#19522
C2×#19422
#204372
#112902
#112042
#108702
#108642