Finite Group Databasebeta

← #415348

Subgroups — #415348

Order 512.

1122 conjugacy classes of subgroups.

The subgroup lattice has 416 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35167C211yesmaximal, characteristic
2#21325C212yesmaximal
3#14160C211yesmaximal, characteristic
4#11392C212yesmaximal
5#11392C212yesmaximal
6#11296C212yesmaximal
7#11296C212yesmaximal
8#10851C212yesmaximal
9C4×#343C211yesmaximal, characteristic
10C2×#544C2212yes—
11C22×#125C2211yescharacteristic
12C2×C4×M16C2212yes—
13#2857—22no—
14#2659—22no—
15#2552—22no—
16#2551—22no—
17#2550—24no—
18#2316C2212yes—
19#2285C2212yes—
20C2×C4×C4⋊C4C2211yescharacteristic
21C2×C4×C4⋊C4C2211yescharacteristic
22#1992—22no—
23#1992—22no—
24#1991—24no—
25C2×#343C2212yes—
26C2×#343C2212yes—
27#1736C2214yes—
28#1736C2214yes—
29C2×#339C2214yes—
30C2×#339C2214yes—
31#1441—22no—
32#1436—22no—
33C4⋊#95C2211yescharacteristic
34C4⋊#95C2211yescharacteristic
35C4⋊#95C2211yescharacteristic
36C4⋊#95C2211yescharacteristic
37C22×M16C2314yes—
38#544—24no—
39#544—24no—
40#544—44no—
41C2×C4⋊Q8—22no—
42C2×C4⋊Q8—24no—
43C2×#125—24no—
44C2×#125—22no—
45C2×#125—22no—
46C2×#125—22no—
47C2×#125—24no—
48C2×#125Q814yescomplemented
49C2×C22⋊Q8—24no—
50C2×#117C2314yes—
51C22×C4⋊C4—21no—
52C22×C4⋊C4—21no—
53C22×C4⋊C4—21no—
54C22×C4⋊C4C2311yescharacteristic
55C22×C4⋊C4C2311yescharacteristic
56C22×C4⋊C4C2311yescharacteristic
57C22×C4⋊C4C2311yescharacteristic
58C22×C42D412yes—
59C22×C42C2311yescharacteristic
60C22×C42C2311yescharacteristic
61C22×C42C2311yescharacteristic
62C4×M16—24no—
63C4×M16—24no—
64C2×#97—22no—
65C2×C4×C8—22no—
66#398—28no—
67#395—28no—
68#391—24no—
69#389—24no—
70#389—24no—
71#389—24no—
72#389—24no—
73#384—22no—
74#384D412yes—
75#384D412yes—
76#382D414yes—
77#380—24no—
78#380—22no—
79#380—22no—
80C4×C4⋊C4—22no—
81C4×C4⋊C4—22no—
82C4×C4⋊C4—22no—
83C4×C4⋊C4—22no—
84C4×C4⋊C4—24no—
85C2×#95—22no—
86C2×#95—22no—
87C2×#95—21no—
88C2×#95—22no—
89C2×#95—22no—
90C2×#95D412yes—
91#343—24no—
92#343Q818yescomplemented
93#340—24no—
94#340—24no—
95#340—24no—
96#340—24no—
97#339—44no—
98#339—44no—
99C22×Q8—24no—
100C23×C4—21no—
101C23×C4—21no—
102C23×C4—21no—
103C23×C4—21no—
104C23×C4C2×D412yes—
105C23×C4C2×D412yes—
106C23×C4C2×D411yescharacteristic
107C23×C4C2×D411yescharacteristic
108C23×C4C2411yescharacteristic
109C2×M16—24no—
110C2×M16—24no—
111C2×M16—24no—
112C2×M16—24no—
113C2×M16—24no—
114C2×M16C2×Q818yes—
115C22×C8—24no—
116C4⋊Q8—48no—
117C4⋊Q8—48no—
118C4⋊Q8—84no—
119#125—28no—
120#125—28no—
121#125—48no—
122#125—48no—
123#125—44no—
124#125—44no—
125C22⋊Q8—48no—
126C22⋊Q8—48no—
127#117—24no—
128#117—24no—
129#117—24no—
130#117—24no—
131C2×C4⋊C4—22no—
132C2×C4⋊C4—22no—
133C2×C4⋊C4—22no—
134C2×C4⋊C4—22no—
135C2×C4⋊C4—22no—
136C2×C4⋊C4—22no—
137C2×C4⋊C4—22no—
138C2×C4⋊C4—22no—
139C2×C4⋊C4—24no—
140C2×C4⋊C4—28no—
141C2×C4⋊C4—22no—
142C2×C4⋊C4—22no—
143C2×C4⋊C4—24no—
144C2×C4⋊C4—24no—
145C2×C4⋊C4—21no—
146C2×C4⋊C4—21no—
147C2×C4⋊C4—24no—
148C2×C4⋊C4—24no—
149C2×C4⋊C4—22no—
150C2×C4⋊C4—22no—
151C2×C4⋊C4—42no—
152C2×C4⋊C4—42no—
153C2×C4⋊C4—41no—
154C2×C4⋊C4—42no—
155C2×C4⋊C4—41no—
156C2×C4⋊C4—42no—
157C2×C4⋊C4—42no—
158C2×C4⋊C4—42no—
159C2×C4⋊C4C2×D414yes—
160C2×C4⋊C4C2×D414yes—
161C2×C4⋊C4C2×Q812yes—
162C2×C4⋊C4C2×Q812yes—
163C2×C4⋊C4C2×Q814yes—
164C2×C4⋊C4#4114yes—
165C2×C22⋊C4—24no—
166C2×C42—22no—
167C2×C42—24no—
168C2×C42—22no—
169C2×C42—22no—
170C2×C42—22no—
171C2×C42—22no—
172C2×C42—22no—
173C2×C42—22no—
174C2×C42—22no—
175C2×C42—82no—
176C2×C42C2×D412yes—
177C2×C42C2×D412yes—
178C2×C42C2×Q812yes—
179C2×C42#4114yes—
180C2×C42#4112yes—
181#97—44no—
182C4×C8—44no—
183#95—28no—
184#95—24no—
185#95—24no—
186#95—24no—
187#95—44no—
188#95—44no—
189#95—42no—
190#95—82no—
191#95—82no—
192C24C22×D411yescharacteristic
193C2×Q8—48no—
194C2×Q8—84no—
195C2×Q8—84no—
196C22×C4—22no—
197C22×C4—22no—
198C22×C4—22no—
199C22×C4—22no—
200C22×C4—22no—
201C22×C4—22no—
202C22×C4—24no—
203C22×C4—24no—
204C22×C4—22no—
205C22×C4—22no—
206C22×C4—41no—
207C22×C4—41no—
208C22×C4—41no—
209C22×C4—41no—
210C22×C4—42no—
211C22×C4—42no—
212C22×C4—42no—
213C22×C4—42no—
214C22×C4—42no—
215C22×C4—42no—
216C22×C4—41no—
217C22×C4—41no—
218C22×C4—42no—
219C22×C4—41no—
220C22×C4—41no—
221C22×C4—41no—
222C22×C4—41no—
223C22×C4—42no—
224C22×C4—41no—
225C22×C4—41no—
226C22×C4—41no—
227C22×C4—41no—
228C22×C4—41no—
229C22×C4—41no—
230C22×C4—41no—
231C22×C4—41no—
232C22×C4—41no—
233C22×C4—41no—
234C22×C4#12011yescharacteristic
235C22×C4#12012yes—
236C22×C4#12011yescharacteristic
237C22×C4C22⋊Q812yes—
238C22×C4C22⋊Q812yes—
239C22×C4#12414yes—
240C22×C4#12412yes—
241C22×C4#12412yes—
242C22×C4C4⋊Q814yes—
243C22×C4C22×D412yes—
244C22×C4C22×Q811yescharacteristic
245C22×C4C22×Q812yes—
246C22×C4C2×#4112yes—
247C22×C4C2×#4111yescharacteristic
248C22×C4C2×#4112yes—
249C22×C4H(5,𝔽2)12yes—
250C22×C42−1+412yes—
251M16—28no—
252M16—28no—
253M16—44no—
254M16—44no—
255C2×C8—28no—
256C2×C8—44no—
257C2×C8—44no—
258C4⋊C4—28no—
259C4⋊C4—28no—
260C4⋊C4—24no—
261C4⋊C4—24no—
262C4⋊C4—24no—
263C4⋊C4—24no—
264C4⋊C4—24no—
265C4⋊C4—24no—
266C4⋊C4—44no—
267C4⋊C4—48no—
268C4⋊C4—416no—
269C4⋊C4—44no—
270C4⋊C4—48no—
271C4⋊C4—42no—
272C4⋊C4—42no—
273C4⋊C4—48no—
274C4⋊C4—44no—
275C4⋊C4—44no—
276C4⋊C4—82no—
277C4⋊C4—82no—
278C4⋊C4—84no—
279C4⋊C4—84no—
280C22⋊C4—44no—
281C22⋊C4—44no—
282C42—44no—
283C42—44no—
284C42—44no—
285C42—44no—
286C42—44no—
287C42—82no—
288C42C22⋊Q814yes—
289C42C22⋊Q814yes—
290C42#12414yes—
291C42#12414yes—
292C23—21no—
293C23—21no—
294C23—21no—
295C23—21no—
296C23C2×#12011yescharacteristic
297C23C2×C22⋊Q811yescharacteristic
298C23C2×#12411yescharacteristic
299C23C2×#12411yescharacteristic
300C23C2×C4⋊Q811yescharacteristic
301C23#53511yescharacteristic
302C23#53611yescharacteristic
303Q8—168no—
304C2×C4—22no—
305C2×C4—22no—
306C2×C4—21no—
307C2×C4—21no—
308C2×C4—21no—
309C2×C4—21no—
310C2×C4—24no—
311C2×C4—24no—
312C2×C4—42no—
313C2×C4—42no—
314C2×C4—42no—
315C2×C4—42no—
316C2×C4—42no—
317C2×C4—42no—
318C2×C4—42no—
319C2×C4—42no—
320C2×C4—44no—
321C2×C4—44no—
322C2×C4—42no—
323C2×C4—42no—
324C2×C4—42no—
325C2×C4—42no—
326C2×C4—42no—
327C2×C4—42no—
328C2×C4—44no—
329C2×C4—44no—
330C2×C4—42no—
331C2×C4—42no—
332C2×C4—44no—
333C2×C4—44no—
334C2×C4—44no—
335C2×C4—44no—
336C2×C4—44no—
337C2×C4—44no—
338C2×C4—82no—
339C2×C4—82no—
340C2×C4—81no—
341C2×C4—81no—
342C2×C4—81no—
343C2×C4—81no—
344C2×C4—81no—
345C2×C4—81no—
346C2×C4—82no—
347C2×C4—82no—
348C2×C4—81no—
349C2×C4—81no—
350C2×C4—81no—
351C2×C4—81no—
352C2×C4—81no—
353C2×C4—81no—
354C2×C4#39314yes—
355C2×C4#39312yes—
356C2×C4#39312yes—
357C2×C4C2×C22⋊Q812yes—
358C2×C4C2×#12412yes—
359C2×C4#53314yes—
360C2×C4#54214yes—
361C2×C4#54614yes—
362C2×C4D4×Q814yes—
363C2×C4#55412yes—
364C2×C4#55612yes—
365C2×C4#55712yes—
366C2×C4Q8212yes—
367C8—48no—
368C22—21no—
369C22—21no—
370C22—21no—
371C22—21no—
372C22—21no—
373C22—21no—
374C22—21no—
375C22—21no—
376C22—21no—
377C22—21no—
378C22—21no—
379C22—21no—
380C22C2×#39311yescharacteristic
381C22#243411yescharacteristic
382C22#243511yescharacteristic
383C22#256711yescharacteristic
384C22#256811yescharacteristic
385C22#264011yescharacteristic
386C22#286211yescharacteristic
387C22#301811yescharacteristic
388C22#301811yescharacteristic
389C22#301911yescharacteristic
390C22#302011yescharacteristic
391C4—24no—
392C4—22no—
393C4—22no—
394C4—44no—
395C4—44no—
396C4—82no—
397C4—82no—
398C4—84no—
399C4—84no—
400C4—82no—
401C4—82no—
402C4#245114yes—
403C4#249214yes—
404C2—21no—
405C2—21no—
406C2—21no—
407C2—21no—
408C2#1519111yescharacteristic
409C2#2346111yescharacteristic
410C2#2346211yescharacteristic
411C2#2582711yescharacteristic
412C2#2582811yescharacteristic
413C2#2587311yescharacteristic
414C2#2587411yescharacteristic
subgroupG/H
center, socleC23C2×#120
commutator, Frattini, cosocleC23×C4C24
prime radicalC24C22×D4
LCSG ▷ C23×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
48. C2×#125Q8semidirect
92. #343Q8semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
#351672
#213252
#141602
#113922
#112962
#108512
C4×#3432