Finite Group Databasebeta

← #417104

Subgroups — #417104

Order 512.

770 conjugacy classes of subgroups.

The subgroup lattice has 361 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21144C211yesmaximal, characteristic
2C2×#1847C211yesmaximal, characteristic
3#20765C211yesmaximal, characteristic
4#20764C211yesmaximal, characteristic
5#11366C212yesmaximal
6#11365C212yesmaximal
7#11239C212yesmaximal
8#11238C212yesmaximal
9#10858C211yesmaximal, characteristic
10#10857C211yesmaximal, characteristic
11#10705C211yesmaximal, characteristic
12C22×#107C2211yescharacteristic
13C22×#107C2211yescharacteristic
14C22×#106C2211yescharacteristic
15C22×#106C2211yescharacteristic
16C2×#424C2212yes—
17C2×#423C2211yescharacteristic
18C2×C4×M16C2211yescharacteristic
19C2×C4×M16C2211yescharacteristic
20C22×#97C2211yescharacteristic
21#2285C2211yescharacteristic
22C2×#384C2211yescharacteristic
23C2×#382C2211yescharacteristic
24#1940—24no—
25#1847—24no—
26#1836—22no—
27#1836—22no—
28#1835—22no—
29#1834—22no—
30C2×#343C2212yes—
31C2×#343C2212yes—
32C2×#343C2212yes—
33C2×#343C2212yes—
34C2×#341C2212yes—
35C2×#341C2212yes—
36#1736C2212yes—
37#1736C2212yes—
38#1736C2212yes—
39#1736C2212yes—
40#1729C2212yes—
41C22×M16C2312yes—
42C22×M16C2312yes—
43C22×M16C2312yes—
44C23×C8C2311yescharacteristic
45C23×C8C2311yescharacteristic
46C2×#117C2312yes—
47C22×C4⋊C4—22no—
48C22×C4⋊C4C2311yescharacteristic
49C22×C4⋊C4C2311yescharacteristic
50C22×C4⋊C4C2311yescharacteristic
51C22×C4⋊C4C2311yescharacteristic
52C22×C42C2311yescharacteristic
53C2×#107—22no—
54C2×#107—22no—
55C2×#107—24no—
56C2×#107—42no—
57C2×#106—22no—
58C2×#106—22no—
59C2×#106—24no—
60C2×#106—42no—
61#424—24no—
62#424—24no—
63#423—22no—
64#423—22no—
65C2×C4⋊C8—24no—
66C2×C4⋊C8—22no—
67C4×M16—22no—
68C4×M16—22no—
69C4×M16Q814yes—
70C4×M16Q814yes—
71C2×#97—21no—
72C2×#97—21no—
73C2×#97—22no—
74C2×#97—22no—
75C2×#97D414yes—
76C2×C4×C8—21no—
77C2×C4×C8—21no—
78#384—21no—
79#384—21no—
80#384—22no—
81#384D412yes—
82#382—21no—
83#382—21no—
84#382—22no—
85#382D412yes—
86#380—22no—
87C4×C4⋊C4—22no—
88C2×#95—22no—
89#343—42no—
90#343—42no—
91#343—42no—
92#343—42no—
93#341—24no—
94#341—24no—
95#340—24no—
96#340—24no—
97#340—24no—
98#340—24no—
99#336—24no—
100C23×C4—21no—
101C23×C4—21no—
102C23×C4—21no—
103C23×C4—21no—
104C23×C4#4112yes—
105C23×C4C2411yescharacteristic
106C2×M16—22no—
107C2×M16—22no—
108C2×M16—22no—
109C2×M16—22no—
110C2×M16—22no—
111C2×M16—22no—
112C2×M16—22no—
113C2×M16—22no—
114C2×M16—22no—
115C2×M16—22no—
116C2×M16—22no—
117C2×M16—22no—
118C2×M16—22no—
119C2×M16—22no—
120C2×M16—22no—
121C2×M16—22no—
122C2×M16C2×Q814yes—
123C2×M16C2×Q814yes—
124C22×C8—21no—
125C22×C8—21no—
126C22×C8—21no—
127C22×C8—21no—
128C22×C8—22no—
129C22×C8—22no—
130C22×C8—22no—
131C22×C8—22no—
132C22×C8—22no—
133C22×C8C2×D412yes—
134C22×C8C2×D412yes—
135C22×C8#4114yes—
136C22×C8#4114yes—
137#117—22no—
138#117—22no—
139#117—22no—
140#117—22no—
141C2×C4⋊C4—21no—
142C2×C4⋊C4—21no—
143C2×C4⋊C4—21no—
144C2×C4⋊C4—21no—
145C2×C4⋊C4—21no—
146C2×C4⋊C4—21no—
147C2×C4⋊C4—21no—
148C2×C4⋊C4—21no—
149C2×C4⋊C4—22no—
150C2×C4⋊C4—22no—
151C2×C4⋊C4—22no—
152C2×C4⋊C4—22no—
153C2×C4⋊C4—22no—
154C2×C4⋊C4—22no—
155C2×C4⋊C4—22no—
156C2×C4⋊C4—22no—
157C2×C4⋊C4—22no—
158C2×C4⋊C4—22no—
159C2×C4⋊C4—42no—
160C2×C4⋊C4—42no—
161C2×C4⋊C4—84no—
162C2×C4⋊C4C2×D414yes—
163C2×C22⋊C4—22no—
164C2×C42—22no—
165C2×C42C2×D411yescharacteristic
166C2×C42C2×D411yescharacteristic
167C2×C42C2×D412yes—
168C2×C42C2×Q812yes—
169C2×C42C2×Q812yes—
170C2×C42#4112yes—
171C2×C42#4112yes—
172#107—44no—
173#107—44no—
174#106—44no—
175#106—44no—
176C4⋊C8—48no—
177C4⋊C8—44no—
178#97—24no—
179#97—24no—
180#97—28no—
181C4×C8—24no—
182C4×C8—24no—
183#95—24no—
184#95—42no—
185#95—42no—
186C24C2×#4111yescharacteristic
187C22×C4—24no—
188C22×C4—42no—
189C22×C4—42no—
190C22×C4—41no—
191C22×C4—41no—
192C22×C4—41no—
193C22×C4—41no—
194C22×C4—42no—
195C22×C4—42no—
196C22×C4—42no—
197C22×C4—42no—
198C22×C4—82no—
199C22×C4—82no—
200C22×C4—82no—
201C22×C4—82no—
202C22×C4#12112yes—
203C22×C4#12112yes—
204C22×C4#12514yes—
205C22×C4#12514yes—
206C22×C4C22×D411yescharacteristic
207C22×C4C22×D411yescharacteristic
208C22×C4C22×Q811yescharacteristic
209C22×C4C22×Q811yescharacteristic
210C22×C4C2×#4111yescharacteristic
211C22×C4C2×#4111yescharacteristic
212C22×C4C2×#4112yes—
213C22×C4C2×#4112yes—
214C22×C4H(5,𝔽2)12yes—
215C22×C42−1+412yes—
216M16—24no—
217M16—24no—
218M16—24no—
219M16—24no—
220M16—42no—
221M16—42no—
222M16—42no—
223M16—42no—
224M16—42no—
225M16—42no—
226M16—42no—
227M16—42no—
228C2×C8—22no—
229C2×C8—22no—
230C2×C8—22no—
231C2×C8—22no—
232C2×C8—24no—
233C2×C8—24no—
234C2×C8—24no—
235C2×C8—24no—
236C2×C8—24no—
237C2×C8—24no—
238C2×C8—42no—
239C2×C8—42no—
240C2×C8—42no—
241C2×C8—42no—
242C2×C8—42no—
243C2×C8—42no—
244C2×C8—42no—
245C2×C8#12414yes—
246C2×C8#12414yes—
247C4⋊C4—24no—
248C4⋊C4—24no—
249C4⋊C4—41no—
250C4⋊C4—41no—
251C4⋊C4—41no—
252C4⋊C4—41no—
253C4⋊C4—41no—
254C4⋊C4—41no—
255C4⋊C4—41no—
256C4⋊C4—41no—
257C4⋊C4—42no—
258C4⋊C4—42no—
259C4⋊C4—42no—
260C4⋊C4—42no—
261C4⋊C4—82no—
262C4⋊C4—82no—
263C4⋊C4—84no—
264C4⋊C4—84no—
265C22⋊C4—42no—
266C22⋊C4—42no—
267C42—42no—
268C42—42no—
269C42C22⋊Q812yes—
270C42C22⋊Q812yes—
271C42C22⋊Q812yes—
272C42C22⋊Q812yes—
273C42#12314yes—
274C42C4⋊Q814yes—
275C23—22no—
276C23—22no—
277C23C2×#12111yescharacteristic
278C23C2×#12511yescharacteristic
279C23C2×#12511yescharacteristic
280C23#55612yes—
281C23#55912yes—
282C2×C4—22no—
283C2×C4—22no—
284C2×C4—22no—
285C2×C4—22no—
286C2×C4—44no—
287C2×C4—44no—
288C2×C4—44no—
289C2×C4—44no—
290C2×C4—44no—
291C2×C4—81no—
292C2×C4—81no—
293C2×C4—81no—
294C2×C4—81no—
295C2×C4—82no—
296C2×C4—82no—
297C2×C4—82no—
298C2×C4—82no—
299C2×C4—162no—
300C2×C4—162no—
301C2×C4—162no—
302C2×C4—162no—
303C2×C4#39814yes—
304C2×C4#39814yes—
305C2×C4C2×C22⋊Q811yescharacteristic
306C2×C4C2×C22⋊Q811yescharacteristic
307C2×C4C2×#12311yescharacteristic
308C2×C4C2×#12411yescharacteristic
309C2×C4C2×#12411yescharacteristic
310C2×C4C2×C4⋊Q811yescharacteristic
311C2×C4#53312yes—
312C2×C4#53312yes—
313C2×C4#53511yescharacteristic
314C2×C4#53611yescharacteristic
315C2×C4#54412yes—
316C2×C4#54412yes—
317C2×C4#54712yes—
318C2×C4#55012yes—
319C2×C4#55212yes—
320C2×C4#55212yes—
321C2×C4#55312yes—
322C2×C4#55312yes—
323C8—24no—
324C8—24no—
325C8—44no—
326C8—44no—
327C8—82no—
328C22—24no—
329C22—22no—
330C22—22no—
331C22—22no—
332C22—22no—
333C22C2×#39811yescharacteristic
334C22#254711yescharacteristic
335C22#254811yescharacteristic
336C22#257012yes—
337C22#257012yes—
338C22#306812yes—
339C22#306912yes—
340C4—24no—
341C4—24no—
342C4—84no—
343C4—161no—
344C4—161no—
345C4—161no—
346C4—161no—
347C4#244112yes—
348C4#244112yes—
349C4#250311yescharacteristic
350C4#250411yescharacteristic
351C4#266011yescharacteristic
352C4#286211yescharacteristic
353C2—22no—
354C2—22no—
355C2#1524211yescharacteristic
356C2#2367512yes—
357C2#2873512yes—
358C2#2876711yescharacteristic
359C2#2876811yescharacteristic
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
#211442
C2×#18472
#207652
#207642
#113662
#113652
#112392
#112382
#108582
#108572
#107052