Finite Group Databasebeta

← #416849

Subgroups — #416849

Order 512.

816 conjugacy classes of subgroups.

The subgroup lattice has 399 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21171C211yesmaximal, characteristic
2#21171C211yesmaximal, characteristic
3C2×#1834C211yesmaximal, characteristic
4C4×#424C211yesmaximal, characteristic
5#11290C214yesmaximal
6#11216C214yesmaximal
7#10857C211yesmaximal, characteristic
8#10857C211yesmaximal, characteristic
9#10844C211yesmaximal, characteristic
10C2×#428C2212yes—
11C2×#428C2212yes—
12C2×#424C2212yes—
13C2×#423C2211yescharacteristic
14C2×#423C2211yescharacteristic
15C2×C4×M16C2211yescharacteristic
16C22×C4×C8C2211yescharacteristic
17C2×#384C2211yescharacteristic
18C2×#384C2211yescharacteristic
19C4×#117C2211yescharacteristic
20#1948—21no—
21#1948—21no—
22#1948—21no—
23#1948—21no—
24#1947—21no—
25#1947—21no—
26#1947—21no—
27#1947—21no—
28#1834—22no—
29#1834—22no—
30#1768—22no—
31C4×C4⋊C8—22no—
32C2×#343C2214yes—
33C2×#343C2214yes—
34C2×#341C2214yes—
35#1736C2212yes—
36#1736C2212yes—
37C2×#340C2214yes—
38#1729C2212yes—
39C22×M16C2312yes—
40C22×M16C2312yes—
41C22×M16C2312yes—
42C23×C8C2312yes—
43C2×#117C2311yescharacteristic
44C2×#117C2311yescharacteristic
45C22×C4⋊C4—21no—
46C22×C4⋊C4—21no—
47C22×C4⋊C4—21no—
48C22×C4⋊C4—21no—
49C22×C4⋊C4C2312yes—
50C22×C4⋊C4C2312yes—
51C22×C42C2311yescharacteristic
52#428—28no—
53#428—44no—
54C2×#107—22no—
55C2×#107—22no—
56C2×#106—22no—
57C2×#106—22no—
58#424—24no—
59#424—24no—
60#423—22no—
61#423—22no—
62#423Q814yes—
63#423Q814yes—
64C2×C4⋊C8—24no—
65C2×C4⋊C8—21no—
66C2×C4⋊C8—21no—
67C2×C4⋊C8—21no—
68C2×C4⋊C8—21no—
69C4×M16—22no—
70C4×M16—22no—
71C2×#97—21no—
72C2×C4×C8—22no—
73C2×C4×C8—22no—
74C2×C4×C8—21no—
75C2×C4×C8D412yes—
76C2×C4×C8D412yes—
77#384—21no—
78#384—21no—
79#384—21no—
80#384—21no—
81#384—21no—
82#384—21no—
83#384—21no—
84#384—21no—
85C4×C4⋊C4D411yescharacteristic
86C4×C4⋊C4D411yescharacteristic
87C4×C4⋊C4D411yescharacteristic
88C4×C4⋊C4D411yescharacteristic
89C4×C22⋊C4—21no—
90C4×C22⋊C4—21no—
91#376—21no—
92C43—21no—
93#343—28no—
94#343—44no—
95#341—44no—
96#340—28no—
97#340—22no—
98#340—22no—
99#340—22no—
100#340—22no—
101#336—24no—
102C23×C4—22no—
103C23×C4—22no—
104C23×C4#4111yescharacteristic
105C23×C4#4111yescharacteristic
106C23×C4C2411yescharacteristic
107C2×M16—22no—
108C2×M16—22no—
109C2×M16—22no—
110C2×M16—24no—
111C2×M16—22no—
112C2×M16—22no—
113C2×M16—24no—
114C2×M16—24no—
115C2×M16—24no—
116C2×M16—22no—
117C2×M16—22no—
118C2×M16C2×Q814yes—
119C2×M16C2×Q814yes—
120C2×M16#4114yes—
121C22×C8—22no—
122C22×C8—22no—
123C22×C8—22no—
124C22×C8—22no—
125C22×C8—22no—
126C22×C8—22no—
127C22×C8—22no—
128C22×C8—22no—
129C22×C8—22no—
130C22×C8C2×D414yes—
131#117—22no—
132#117—22no—
133#117—22no—
134#117#4114yes—
135C2×C4⋊C4—22no—
136C2×C4⋊C4—22no—
137C2×C4⋊C4—22no—
138C2×C4⋊C4—22no—
139C2×C4⋊C4—24no—
140C2×C4⋊C4—24no—
141C2×C4⋊C4—24no—
142C2×C4⋊C4—24no—
143C2×C4⋊C4—41no—
144C2×C4⋊C4—41no—
145C2×C4⋊C4—41no—
146C2×C4⋊C4—41no—
147C2×C4⋊C4—41no—
148C2×C4⋊C4—41no—
149C2×C4⋊C4—41no—
150C2×C4⋊C4—41no—
151C2×C4⋊C4—82no—
152C2×C4⋊C4—82no—
153C2×C4⋊C4—82no—
154C2×C4⋊C4—82no—
155C2×C4⋊C4C2×D411yescharacteristic
156C2×C4⋊C4C2×D411yescharacteristic
157C2×C4⋊C4C2×D411yescharacteristic
158C2×C4⋊C4C2×D411yescharacteristic
159C2×C22⋊C4—21no—
160C2×C22⋊C4—21no—
161C2×C42—21no—
162C2×C42—21no—
163C2×C42—21no—
164C2×C42—21no—
165C2×C42—21no—
166C2×C42—21no—
167C2×C42C2×D411yescharacteristic
168C2×C42C2×D411yescharacteristic
169C2×C42C2×D411yescharacteristic
170C2×C42C2×D411yescharacteristic
171C2×C42C2×Q812yes—
172C2×C42C2×Q812yes—
173C2×C42#4112yes—
174C2×C42#4112yes—
175#107—44no—
176#107—44no—
177#106—44no—
178#106—44no—
179C4⋊C8—216no—
180C4⋊C8—24no—
181C4⋊C8—24no—
182C4⋊C8—24no—
183C4⋊C8—24no—
184#97—42no—
185C4×C8—24no—
186C4×C8—24no—
187C4×C8—42no—
188#95—21no—
189#95—21no—
190#95—21no—
191#95—21no—
192C24C2×#4111yescharacteristic
193C22×C4—22no—
194C22×C4—22no—
195C22×C4—22no—
196C22×C4—22no—
197C22×C4—21no—
198C22×C4—21no—
199C22×C4—21no—
200C22×C4—21no—
201C22×C4—42no—
202C22×C4—42no—
203C22×C4—44no—
204C22×C4—44no—
205C22×C4—82no—
206C22×C4—82no—
207C22×C4—82no—
208C22×C4—82no—
209C22×C4#12111yescharacteristic
210C22×C4#12111yescharacteristic
211C22×C4#12111yescharacteristic
212C22×C4#12111yescharacteristic
213C22×C4#12512yes—
214C22×C4#12512yes—
215C22×C4#12512yes—
216C22×C4#12512yes—
217C22×C4C22×D411yescharacteristic
218C22×C4C22×D411yescharacteristic
219C22×C4C22×Q811yescharacteristic
220C22×C4C22×Q811yescharacteristic
221C22×C4C2×#4111yescharacteristic
222C22×C4C2×#4111yescharacteristic
223C22×C4C2×#4112yes—
224C22×C4C2×#4112yes—
225C22×C4H(5,𝔽2)12yes—
226C22×C42−1+412yes—
227M16—24no—
228M16—24no—
229M16—28no—
230M16—28no—
231M16—44no—
232M16—44no—
233M16—42no—
234M16—42no—
235C2×C8—24no—
236C2×C8—24no—
237C2×C8—24no—
238C2×C8—24no—
239C2×C8—24no—
240C2×C8—24no—
241C2×C8—42no—
242C2×C8—42no—
243C2×C8—42no—
244C2×C8—42no—
245C2×C8—42no—
246C2×C8—42no—
247C2×C8—42no—
248C2×C8—42no—
249C2×C8—42no—
250C2×C8—42no—
251C4⋊C4—21no—
252C4⋊C4—21no—
253C4⋊C4—21no—
254C4⋊C4—21no—
255C4⋊C4—21no—
256C4⋊C4—21no—
257C4⋊C4—44no—
258C4⋊C4—42no—
259C4⋊C4—42no—
260C4⋊C4—42no—
261C4⋊C4—42no—
262C4⋊C4—44no—
263C4⋊C4—82no—
264C4⋊C4—82no—
265C4⋊C4—82no—
266C4⋊C4—82no—
267C4⋊C4—82no—
268C4⋊C4—82no—
269C4⋊C4—82no—
270C4⋊C4—82no—
271C4⋊C4#12412yes—
272C4⋊C4#12412yes—
273C22⋊C4—24no—
274C22⋊C4—42no—
275C42—24no—
276C42—42no—
277C42—42no—
278C42—42no—
279C42—42no—
280C42—42no—
281C42C22⋊Q812yes—
282C42C22⋊Q812yes—
283C42C22⋊Q812yes—
284C42C22⋊Q812yes—
285C42#12312yes—
286C42#12312yes—
287C42C4⋊Q812yes—
288C42C4⋊Q812yes—
289C23—21no—
290C23—21no—
291C23—21no—
292C23—21no—
293C23C2×#12111yescharacteristic
294C23C2×#12511yescharacteristic
295C23C2×#12511yescharacteristic
296C23#52912yes—
297C23#54212yes—
298C2×C4—21no—
299C2×C4—21no—
300C2×C4—21no—
301C2×C4—21no—
302C2×C4—24no—
303C2×C4—24no—
304C2×C4—24no—
305C2×C4—24no—
306C2×C4—22no—
307C2×C4—22no—
308C2×C4—41no—
309C2×C4—41no—
310C2×C4—41no—
311C2×C4—41no—
312C2×C4—41no—
313C2×C4—41no—
314C2×C4—41no—
315C2×C4—41no—
316C2×C4—41no—
317C2×C4—41no—
318C2×C4—82no—
319C2×C4—82no—
320C2×C4—82no—
321C2×C4—82no—
322C2×C4—82no—
323C2×C4—82no—
324C2×C4—164no—
325C2×C4—164no—
326C2×C4#39812yes—
327C2×C4#39812yes—
328C2×C4#39812yes—
329C2×C4#39812yes—
330C2×C4#47112yes—
331C2×C4#47112yes—
332C2×C4#47212yes—
333C2×C4#47312yes—
334C2×C4C2×C22⋊Q811yescharacteristic
335C2×C4C2×C22⋊Q811yescharacteristic
336C2×C4C2×#12311yescharacteristic
337C2×C4#52512yes—
338C2×C4C2×#12411yescharacteristic
339C2×C4C2×C4⋊Q811yescharacteristic
340C2×C4#53211yescharacteristic
341C2×C4#53511yescharacteristic
342C2×C4#53611yescharacteristic
343C2×C4#53712yes—
344C2×C4#55212yes—
345C2×C4#55312yes—
346C2×C4#55412yes—
347C2×C4#55412yes—
348C2×C4#55512yes—
349C2×C4#55612yes—
350C2×C4#55712yes—
351C2×C4#55712yes—
352C8—44no—
353C8—44no—
354C8—44no—
355C8—44no—
356C22—22no—
357C22—22no—
358C22—21no—
359C22—21no—
360C22—21no—
361C22—21no—
362C22—22no—
363C22—22no—
364C22C2×#39811yescharacteristic
365C22#250812yes—
366C22#250912yes—
367C22#262011yescharacteristic
368C22#285011yescharacteristic
369C22C2×#47111yescharacteristic
370C22C2×#47111yescharacteristic
371C22C2×#47211yescharacteristic
372C22C2×#47311yescharacteristic
373C4—22no—
374C4—22no—
375C4—42no—
376C4—42no—
377C4—81no—
378C4—81no—
379C4—162no—
380C4—162no—
381C4#208914yes—
382C4#209014yes—
383C4#248512yes—
384C4#256311yescharacteristic
385C4#256511yescharacteristic
386C4#256611yescharacteristic
387C4#256811yescharacteristic
388C4#258412yes—
389C2—22no—
390C2—22no—
391C2#1523411yescharacteristic
392C2C2×#208911yescharacteristic
393C2C2×#209011yescharacteristic
394C2#2985811yescharacteristic
395C2#2985911yescharacteristic
396C2#3306011yescharacteristic
397C2#3306111yescharacteristic
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
#211712
C2×#18342
C4×#4242
#112902
#112162
#108572
#108442