Finite Group Databasebeta

← #411142

Subgroups — #411142

Order 512.

782 conjugacy classes of subgroups.

The subgroup lattice has 336 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23657C212yesmaximal
2C2×#1942C211yesmaximal, characteristic
3#20437C211yesmaximal, characteristic
4#11365C212yesmaximal
5#11331C212yesmaximal
6#11204C212yesmaximal
7#10896C214yesmaximal
8#10822C211yesmaximal, characteristic
9C2×#428C2214yes—
10C22×#107C2211yescharacteristic
11C22×#107C2211yescharacteristic
12C2×#423C2211yescharacteristic
13C2×#405C2212yes—
14C22×C4×C8C2211yescharacteristic
15C2×#389C2212yes—
16#2285C2211yescharacteristic
17#2085—24no—
18#2084—22no—
19#2083—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—
31C2×#341C2212yes—
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
45#428—28no—
46#428—44no—
47C2×#107—21no—
48C2×#107—21no—
49C2×#107—24no—
50C2×#107—21no—
51C2×#107—21no—
52C2×#107—24no—
53C2×#107—24no—
54C2×#106—24no—
55#423—24no—
56C2×C4⋊C8—21no—
57C2×C4⋊C8—21no—
58#405—44no—
59C2×#97—22no—
60C2×C4×C8—24no—
61C2×C4×C8—22no—
62C2×C4×C8D411yescharacteristic
63C2×C4×C8D411yescharacteristic
64C2×C4×C8D411yescharacteristic
65C2×C4×C8D411yescharacteristic
66#389—22no—
67#389—22no—
68#389—22no—
69#389—22no—
70#384D411yescharacteristic
71#384D411yescharacteristic
72#382D412yes—
73#380—21no—
74#380—21no—
75C4×C4⋊C4—21no—
76C4×C4⋊C4—21no—
77C2×#95—22no—
78#343—28no—
79#343—44no—
80#341—24no—
81#341Q818yes—
82#340—22no—
83#340—22no—
84#340—24no—
85#340—24no—
86#336—22no—
87#336—22no—
88#336—22no—
89#336—22no—
90C23×C4—22no—
91C23×C4—22no—
92C23×C4#4111yescharacteristic
93C23×C4#4111yescharacteristic
94C23×C4C2411yescharacteristic
95C2×M16—22no—
96C2×M16—24no—
97C2×M16—24no—
98C2×M16—24no—
99C2×M16—24no—
100C2×M16—24no—
101C2×M16—24no—
102C2×M16—24no—
103C2×M16C2×Q818yes—
104C2×M16#4114yes—
105C22×C8—22no—
106C22×C8—22no—
107C22×C8—22no—
108C22×C8—22no—
109C22×C8—24no—
110C22×C8—22no—
111C22×C8C2×D411yescharacteristic
112C22×C8C2×D411yescharacteristic
113C22×C8C2×D411yescharacteristic
114C22×C8C2×D411yescharacteristic
115C22×C8C2×Q814yes—
116C22×C8#4114yes—
117#117—24no—
118#117—24no—
119C2×C4⋊C4—22no—
120C2×C4⋊C4—22no—
121C2×C4⋊C4—22no—
122C2×C4⋊C4—22no—
123C2×C4⋊C4—21no—
124C2×C4⋊C4—21no—
125C2×C4⋊C4—21no—
126C2×C4⋊C4—21no—
127C2×C4⋊C4—22no—
128C2×C4⋊C4—22no—
129C2×C4⋊C4—22no—
130C2×C4⋊C4—24no—
131C2×C4⋊C4—24no—
132C2×C4⋊C4—42no—
133C2×C4⋊C4—42no—
134C2×C4⋊C4—84no—
135C2×C4⋊C4C2×D412yes—
136C2×C4⋊C4C2×D412yes—
137C2×C4⋊C4#4114yes—
138C2×C22⋊C4—22no—
139C2×C42—22no—
140C2×C42—22no—
141C2×C42—22no—
142C2×C42C2×D411yescharacteristic
143C2×C42C2×D411yescharacteristic
144C2×C42C2×D411yescharacteristic
145C2×C42C2×D411yescharacteristic
146#107—24no—
147#107—24no—
148#107—48no—
149#107—42no—
150#107—42no—
151#106—48no—
152C4⋊C8—24no—
153C4⋊C8—24no—
154#97—44no—
155C4×C8—22no—
156C4×C8—22no—
157C4×C8—22no—
158C4×C8—22no—
159C4×C8—44no—
160#95—22no—
161#95—22no—
162#95—42no—
163#95—42no—
164C24C2×#4111yescharacteristic
165C22×C4—22no—
166C22×C4—22no—
167C22×C4—22no—
168C22×C4—22no—
169C22×C4—22no—
170C22×C4—22no—
171C22×C4—24no—
172C22×C4—41no—
173C22×C4—41no—
174C22×C4—42no—
175C22×C4—42no—
176C22×C4—42no—
177C22×C4—44no—
178C22×C4—82no—
179C22×C4—82no—
180C22×C4—82no—
181C22×C4—82no—
182C22×C4#12111yescharacteristic
183C22×C4#12111yescharacteristic
184C22×C4#12111yescharacteristic
185C22×C4#12111yescharacteristic
186C22×C4C22×D411yescharacteristic
187C22×C4C22×D411yescharacteristic
188C22×C4C22×Q812yes—
189C22×C4C2×#4112yes—
190C22×C4C2×#4112yes—
191C22×C4C2×#4112yes—
192C22×C4H(5,𝔽2)12yes—
193C22×C42−1+412yes—
194M16—28no—
195M16—28no—
196M16—28no—
197M16—44no—
198M16—44no—
199M16—44no—
200C2×C8—21no—
201C2×C8—21no—
202C2×C8—21no—
203C2×C8—21no—
204C2×C8—21no—
205C2×C8—21no—
206C2×C8—21no—
207C2×C8—21no—
208C2×C8—28no—
209C2×C8—28no—
210C2×C8—24no—
211C2×C8—28no—
212C2×C8—44no—
213C2×C8—42no—
214C2×C8—42no—
215C2×C8—44no—
216C2×C8#12312yes—
217C2×C8#12312yes—
218C2×C8C4⋊Q812yes—
219C2×C8C4⋊Q812yes—
220C4⋊C4—24no—
221C4⋊C4—24no—
222C4⋊C4—24no—
223C4⋊C4—24no—
224C4⋊C4—42no—
225C4⋊C4—42no—
226C4⋊C4—42no—
227C4⋊C4—42no—
228C4⋊C4—42no—
229C4⋊C4—42no—
230C4⋊C4—42no—
231C4⋊C4—42no—
232C4⋊C4—44no—
233C4⋊C4—84no—
234C4⋊C4—84no—
235C22⋊C4—44no—
236C42—21no—
237C42—21no—
238C42—21no—
239C42—21no—
240C42—22no—
241C42—22no—
242C42—42no—
243C42—42no—
244C23—21no—
245C23—21no—
246C23—21no—
247C23—21no—
248C23C2×#12111yescharacteristic
249C23#52912yes—
250C23#55312yes—
251C23#55712yes—
252C2×C4—21no—
253C2×C4—21no—
254C2×C4—21no—
255C2×C4—21no—
256C2×C4—21no—
257C2×C4—21no—
258C2×C4—21no—
259C2×C4—21no—
260C2×C4—21no—
261C2×C4—21no—
262C2×C4—44no—
263C2×C4—44no—
264C2×C4—42no—
265C2×C4—42no—
266C2×C4—44no—
267C2×C4—42no—
268C2×C4—42no—
269C2×C4—42no—
270C2×C4—42no—
271C2×C4—84no—
272C2×C4—82no—
273C2×C4—82no—
274C2×C4—84no—
275C2×C4—82no—
276C2×C4—82no—
277C2×C4—82no—
278C2×C4—164no—
279C2×C4#47212yes—
280C2×C4#47312yes—
281C2×C4C2×#12311yescharacteristic
282C2×C4#52511yescharacteristic
283C2×C4#52511yescharacteristic
284C2×C4C2×C4⋊Q811yescharacteristic
285C2×C4#53212yes—
286C2×C4#53314yes—
287C2×C4#53511yescharacteristic
288C2×C4#53611yescharacteristic
289C2×C4#53914yes—
290C2×C4#54612yes—
291C2×C4D4×Q812yes—
292C2×C4#55214yes—
293C2×C4#55414yes—
294C8—22no—
295C8—22no—
296C8—22no—
297C8—22no—
298C8—44no—
299C8—48no—
300C22—22no—
301C22—22no—
302C22—22no—
303C22—22no—
304C22—22no—
305C22—22no—
306C22#253511yescharacteristic
307C22#254611yescharacteristic
308C22#259014yes—
309C22#285411yescharacteristic
310C22C2×#47211yescharacteristic
311C22C2×#47311yescharacteristic
312C22#306711yescharacteristic
313C22#306711yescharacteristic
314C4—22no—
315C4—22no—
316C4—41no—
317C4—41no—
318C4—82no—
319C4—84no—
320C4—82no—
321C4—162no—
322C4#239911yescharacteristic
323C4#248411yescharacteristic
324C4#250211yescharacteristic
325C4#255614yes—
326C4#263311yescharacteristic
327C2—24no—
328C2#1510611yescharacteristic
329C2#2817411yescharacteristic
330C2#2817611yescharacteristic
331C2#2869911yescharacteristic
332C2#2870111yescharacteristic
333C2#3312311yescharacteristic
334C2#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×#19422
#204372
#113652
#113312
#112042
#108962
#108222