Finite Group Databasebeta

← #414281

Subgroups — #414281

Order 512.

770 conjugacy classes of subgroups.

The subgroup lattice has 378 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23670C212yesmaximal
2C2×#1943C211yesmaximal, characteristic
3#20436C211yesmaximal, characteristic
4#11330C211yesmaximal, characteristic
5#11270C211yesmaximal, characteristic
6#11239C211yesmaximal, characteristic
7#11238C211yesmaximal, characteristic
8#11204C212yesmaximal
9#10896C212yesmaximal
10#10845C211yesmaximal, characteristic
11#10827C212yesmaximal
12C2×#428C2212yes—
13C22×#107C2211yescharacteristic
14C22×#106C2211yescharacteristic
15C2×#424C2211yescharacteristic
16C2×#423C2212yes—
17C2×#405C2211yescharacteristic
18C2×#405C2211yescharacteristic
19C22×#97C2211yescharacteristic
20#2296C2211yescharacteristic
21C2×#389C2212yes—
22#2088—22no—
23#2088—22no—
24#2087—22no—
25#2086—22no—
26#1943—22no—
27#1943—22no—
28#1771—22no—
29#1770—22no—
30C2×#343C2212yes—
31C2×#343C2212yes—
32C2×#343C2212yes—
33C2×#343C2212yes—
34C2×#341C2211yescharacteristic
35C2×#341C2211yescharacteristic
36C2×#341C2212yes—
37#1736C2212yes—
38#1736C2212yes—
39#1736C2212yes—
40C2×#340C2212yes—
41#1729C2211yescharacteristic
42#1728C2211yescharacteristic
43C22×M16C2312yes—
44C22×M16C2311yescharacteristic
45C22×M16C2311yescharacteristic
46C22×M16C2312yes—
47C23×C8C2311yescharacteristic
48C23×C8C2311yescharacteristic
49C2×#117C2311yescharacteristic
50C2×#117C2311yescharacteristic
51C22×C4⋊C4—22no—
52C22×C4⋊C4C2312yes—
53C22×C4⋊C4C2312yes—
54C22×C42C2311yescharacteristic
55#428—24no—
56#428—42no—
57C2×#107—22no—
58C2×#107—24no—
59C2×#107—22no—
60C2×#106—22no—
61C2×#106—22no—
62C2×#106—42no—
63#424—22no—
64#424Q814yes—
65#423—24no—
66#423—42no—
67C2×C4⋊C8—22no—
68C2×C4⋊C8—22no—
69C2×C4⋊C8—22no—
70#405—22no—
71#405—42no—
72#405Q814yes—
73C2×#97—21no—
74C2×#97—21no—
75C2×#97—42no—
76C2×#97D412yes—
77C2×#97D412yes—
78C2×C4×C8—21no—
79C2×C4×C8—21no—
80#389—22no—
81#389—22no—
82#389—22no—
83#389—22no—
84#389D412yes—
85#389D412yes—
86#385—22no—
87C4×C4⋊C4—22no—
88C2×#95—22no—
89#343—24no—
90#343—24no—
91#343—24no—
92#343—42no—
93#341—24no—
94#341—22no—
95#341—22no—
96#340—22no—
97#340—22no—
98#340—22no—
99#340—22no—
100#340—22no—
101#340—22no—
102#340—22no—
103#340—22no—
104#336—22no—
105#336—22no—
106C23×C4—22no—
107C23×C4—22no—
108C23×C4#4112yes—
109C23×C4C2411yescharacteristic
110C2×M16—21no—
111C2×M16—21no—
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×M16—22no—
123C2×M16—22no—
124C2×M16—22no—
125C2×M16—22no—
126C2×M16—22no—
127C2×M16C2×Q812yes—
128C2×M16C2×Q812yes—
129C2×M16#4114yes—
130C22×C8—22no—
131C22×C8—22no—
132C22×C8—22no—
133C22×C8—22no—
134C22×C8—21no—
135C22×C8—21no—
136C22×C8—22no—
137C22×C8—22no—
138C22×C8C2×D412yes—
139C22×C8C2×D411yescharacteristic
140C22×C8C2×D411yescharacteristic
141C22×C8C2×Q812yes—
142C22×C8C2×Q812yes—
143C22×C8#4112yes—
144C22×C8#4112yes—
145#117—21no—
146#117—21no—
147#117—22no—
148#117—22no—
149#117C2×Q812yes—
150#117C2×Q812yes—
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—22no—
160C2×C4⋊C4—22no—
161C2×C4⋊C4—22no—
162C2×C4⋊C4—22no—
163C2×C4⋊C4—22no—
164C2×C4⋊C4—42no—
165C2×C4⋊C4—42no—
166C2×C4⋊C4—84no—
167C2×C4⋊C4C2×D412yes—
168C2×C4⋊C4C2×D412yes—
169C2×C4⋊C4#4114yes—
170C2×C22⋊C4—21no—
171C2×C22⋊C4—21no—
172C2×C42—21no—
173C2×C42—22no—
174C2×C42—22no—
175C2×C42—21no—
176C2×C42C2×D412yes—
177C2×C42C2×D411yescharacteristic
178C2×C42C2×D411yescharacteristic
179#107—28no—
180#107—44no—
181#106—44no—
182#106—44no—
183C4⋊C8—28no—
184C4⋊C8—44no—
185C4⋊C8—44no—
186#97—24no—
187#97—42no—
188#97—42no—
189#97—42no—
190C4×C8—24no—
191C4×C8—42no—
192#95—22no—
193#95—22no—
194#95—42no—
195#95—42no—
196C24C2×#4111yescharacteristic
197C22×C4—22no—
198C22×C4—22no—
199C22×C4—22no—
200C22×C4—22no—
201C22×C4—24no—
202C22×C4—42no—
203C22×C4—42no—
204C22×C4—42no—
205C22×C4—42no—
206C22×C4—42no—
207C22×C4—42no—
208C22×C4—42no—
209C22×C4—82no—
210C22×C4—82no—
211C22×C4—82no—
212C22×C4—82no—
213C22×C4#12112yes—
214C22×C4#12112yes—
215C22×C4C22×D411yescharacteristic
216C22×C4C22×D411yescharacteristic
217C22×C4C22×Q811yescharacteristic
218C22×C4C22×Q811yescharacteristic
219C22×C4C2×#4111yescharacteristic
220C22×C4C2×#4111yescharacteristic
221C22×C4C2×#4112yes—
222C22×C4C2×#4112yes—
223C22×C4H(5,𝔽2)12yes—
224C22×C42−1+412yes—
225M16—24no—
226M16—24no—
227M16—24no—
228M16—24no—
229M16—42no—
230M16—42no—
231M16—42no—
232M16—42no—
233M16—42no—
234M16—42no—
235M16—42no—
236M16—42no—
237C2×C8—22no—
238C2×C8—21no—
239C2×C8—21no—
240C2×C8—22no—
241C2×C8—21no—
242C2×C8—21no—
243C2×C8—24no—
244C2×C8—24no—
245C2×C8—24no—
246C2×C8—24no—
247C2×C8—22no—
248C2×C8—22no—
249C2×C8—24no—
250C2×C8—41no—
251C2×C8—41no—
252C2×C8—41no—
253C2×C8—42no—
254C2×C8—41no—
255C2×C8—42no—
256C2×C8—42no—
257C2×C8—42no—
258C2×C8—42no—
259C2×C8C22⋊Q812yes—
260C2×C8C22⋊Q812yes—
261C2×C8C22⋊Q814yes—
262C4⋊C4—22no—
263C4⋊C4—24no—
264C4⋊C4—22no—
265C4⋊C4—22no—
266C4⋊C4—24no—
267C4⋊C4—42no—
268C4⋊C4—42no—
269C4⋊C4—42no—
270C4⋊C4—42no—
271C4⋊C4—42no—
272C4⋊C4—42no—
273C4⋊C4—82no—
274C4⋊C4—82no—
275C4⋊C4—84no—
276C4⋊C4—84no—
277C4⋊C4C4⋊Q814yes—
278C22⋊C4—22no—
279C22⋊C4—22no—
280C22⋊C4—42no—
281C42—22no—
282C42—22no—
283C42—22no—
284C42—22no—
285C42—22no—
286C42—22no—
287C42—41no—
288C42—41no—
289C23—22no—
290C23—22no—
291C23C2×#12111yescharacteristic
292C23#53311yescharacteristic
293C23#53311yescharacteristic
294C23#55212yes—
295C23#55312yes—
296C2×C4—24no—
297C2×C4—22no—
298C2×C4—22no—
299C2×C4—22no—
300C2×C4—22no—
301C2×C4—22no—
302C2×C4—22no—
303C2×C4—42no—
304C2×C4—42no—
305C2×C4—44no—
306C2×C4—42no—
307C2×C4—42no—
308C2×C4—42no—
309C2×C4—82no—
310C2×C4—82no—
311C2×C4—84no—
312C2×C4—82no—
313C2×C4—82no—
314C2×C4—84no—
315C2×C4—82no—
316C2×C4—82no—
317C2×C4—82no—
318C2×C4—162no—
319C2×C4—162no—
320C2×C4C2×C22⋊Q811yescharacteristic
321C2×C4C2×C22⋊Q811yescharacteristic
322C2×C4#52511yescharacteristic
323C2×C4#52912yes—
324C2×C4C2×C4⋊Q811yescharacteristic
325C2×C4#53211yescharacteristic
326C2×C4#53211yescharacteristic
327C2×C4#53312yes—
328C2×C4#53411yescharacteristic
329C2×C4#53611yescharacteristic
330C2×C4#53912yes—
331C2×C4#54412yes—
332C2×C4#54612yes—
333C2×C4#54812yes—
334C2×C4#55212yes—
335C2×C4#55312yes—
336C2×C4#55412yes—
337C2×C4#55712yes—
338C8—24no—
339C8—22no—
340C8—22no—
341C8—42no—
342C8—42no—
343C8—44no—
344C8—82no—
345C22—22no—
346C22—22no—
347C22—22no—
348C22—22no—
349C22—22no—
350C22—22no—
351C22#253611yescharacteristic
352C22#254311yescharacteristic
353C22#259012yes—
354C22#260612yes—
355C22#285711yescharacteristic
356C22#306812yes—
357C22#306912yes—
358C4—42no—
359C4—42no—
360C4—44no—
361C4—84no—
362C4—82no—
363C4—162no—
364C4#239611yescharacteristic
365C4#255012yes—
366C4#255812yes—
367C4#256511yescharacteristic
368C4#257911yescharacteristic
369C4#262711yescharacteristic
370C2—22no—
371C2—22no—
372C2#1516611yescharacteristic
373C2#2822011yescharacteristic
374C2#2822211yescharacteristic
375C2#2859612yes—
376C2#3322312yes—
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
#236702
C2×#19432
#204362
#113302
#112702
#112392
#112382
#112042
#108962
#108452
#108272