Finite Group Databasebeta

← #382817

Subgroups — #382817

Order 512.

1193 conjugacy classes of subgroups.

The subgroup lattice has 406 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34266C211yesmaximal, characteristic
2#20695C211yesmaximal, characteristic
3#20640C211yesmaximal, characteristic
4#11223C212yesmaximal, complemented
5#11175C212yesmaximal
6#11048C212yesmaximal, complemented
7#10974C212yesmaximal
8#10830C212yesmaximal, complemented
9#10763C212yesmaximal, complemented
10C2×#536C2211yescharacteristic
11C2×#520C2211yescharacteristic
12C22×C4×Q8C2211yescharacteristic
13#2905C2212yes—
14C2×#413C2212yes—
15C22×#101C2214yes—
16#2879C2212yes—
17#2367—22no—
18#2365—22no—
19#2362—22no—
20#2360—22no—
21#2322C414yes—
22#2310C2211yescharacteristic
23C4×#123C414yes—
24#2297C2211yescharacteristic
25#2296C2211yescharacteristic
26#2274C2211yescharacteristic
27#1822—22no—
28#1821—22no—
29#1801—22no—
30#1800—22no—
31C2×#344C2214yes—
32C2×#344C2214yes—
33#1742C2212yes—
34C2×#343C2212yes—
35C2×#343C2212yes—
36#1728C2214yes—
37C22×#41C2311yescharacteristic
38C23×Q8C2311yescharacteristic
39C22×M16C2314yes—
40C22×M16C2314yes—
41#536—24no—
42#536—24no—
43C2×C4⋊Q8—21no—
44C2×#124—21no—
45C2×#123C2×C414yes—
46C2×C22⋊Q8—22no—
47#520—44no—
48C2×C4×Q8—22no—
49C2×C4×Q8—22no—
50C2×C4×Q8—44no—
51C2×C4×Q8D412yes—
52C2×C4×D4C2×C414yes—
53C2×C4×D4D412yes—
54C2×#117D412yes—
55C2×#117C2311yescharacteristic
56C2×#117C2311yescharacteristic
57C2×#117C2311yescharacteristic
58C22×C4⋊C4C2×C412yes—
59C22×C4⋊C4D412yes—
60C22×C4⋊C4C2311yescharacteristic
61C22×C42C2×C412yes—
62C22×C42C2311yescharacteristic
63#424—44no—
64#423—24no—
65#413—44no—
66C2×#101—22no—
67C2×#101—28no—
68C2×#101—44no—
69C2×#101—44no—
70C2×#100—22no—
71#408—44no—
72#407—24no—
73#391—21no—
74#391—21no—
75#391—21no—
76#391—22no—
77#390—21no—
78#389—24no—
79#389—21no—
80#389—21no—
81#388—24no—
82#388—24no—
83#387—24no—
84#387—24no—
85#386C2×C412yes—
86#386C2×C412yes—
87#385—24no—
88#385C2×C412yes—
89#385C2×C412yes—
90#381C2×C414yes—
91C4×C4⋊C4—22no—
92C4×C4⋊C4—22no—
93C4×C4⋊C4—24no—
94C4×C4⋊C4—22no—
95C4×C22⋊C4—24no—
96C4×C22⋊C4—22no—
97C4×C22⋊C4C2×C414yes—
98#344—28no—
99#344—44no—
100#343—24no—
101#343—22no—
102#343—24no—
103#343—22no—
104#336—44no—
105C2×#41—22no—
106C2×#41—22no—
107C2×#41—24no—
108C2×#41—22no—
109C2×#41—22no—
110C22×Q8—21no—
111C22×Q8—21no—
112C22×Q8—22no—
113C22×Q8—24no—
114C22×Q8—22no—
115C22×Q8—22no—
116C22×Q8C2×D411yescharacteristic
117C22×Q8#4114yes—
118C22×D4C22×C412yes—
119C22×D4C2×D411yescharacteristic
120C23×C4C22⋊C414yes—
121C23×C4C22×C412yes—
122C23×C4C22×C411yescharacteristic
123C23×C4C22×C411yescharacteristic
124C23×C4C22×C412yes—
125C23×C4C2×D412yes—
126C23×C4C2411yescharacteristic
127C2×M16—24no—
128C2×M16—24no—
129C2×M16—24no—
130C2×M16—24no—
131C2×M16—24no—
132C2×M16—24no—
133C2×M16—24no—
134C2×M16—24no—
135C2×M16—24no—
136C2×M16—24no—
137C2×M16—24no—
138C2×M16—24no—
139C22×C8—24no—
140C22×C8—24no—
141C4⋊Q8—28no—
142#124—28no—
143#123—28no—
144#123—28no—
145C22⋊Q8—216no—
146C4×Q8—44no—
147C4×Q8—48no—
148C4×Q8—48no—
149C4×D4—44no—
150C4×D4—48no—
151#117—22no—
152#117—24no—
153#117—22no—
154#117—22no—
155#117—44no—
156#117#4114yes—
157C2×C4⋊C4—24no—
158C2×C4⋊C4—21no—
159C2×C4⋊C4—24no—
160C2×C4⋊C4—24no—
161C2×C4⋊C4—22no—
162C2×C4⋊C4—21no—
163C2×C4⋊C4—22no—
164C2×C4⋊C4—44no—
165C2×C4⋊C4—44no—
166C2×C4⋊C4C22⋊C414yes—
167C2×C4⋊C4C2×D411yescharacteristic
168C2×C4⋊C4C2×D412yes—
169C2×C4⋊C4C2×D412yes—
170C2×C4⋊C4C2×D411yescharacteristic
171C2×C22⋊C4—24no—
172C2×C22⋊C4—22no—
173C2×C22⋊C4—22no—
174C2×C22⋊C4C22⋊C414yes—
175C2×C22⋊C4C22×C412yes—
176C2×C22⋊C4C22×C412yes—
177C2×C22⋊C4C22×C412yes—
178C2×C42—21no—
179C2×C42—24no—
180C2×C42—21no—
181C2×C42—21no—
182C2×C42—21no—
183C2×C42—24no—
184C2×C42—24no—
185C2×C42—44no—
186C2×C42C22⋊C414yes—
187C2×C42C2×D412yes—
188C4⋊C8—48no—
189#101—416no—
190#101—44no—
191#100—44no—
192C22⋊C8—48no—
193#95—22no—
194#95—22no—
195#95—24no—
196#95—22no—
197#95—24no—
198#95—24no—
199#95—22no—
200C24C2×C22⋊C412yes—
201C24C23×C411yescharacteristic
202#41—44no—
203#41—44no—
204#41—44no—
205#41—44no—
206C2×Q8—24no—
207C2×Q8—24no—
208C2×Q8—22no—
209C2×Q8—22no—
210C2×Q8—21no—
211C2×Q8—21no—
212C2×Q8—28no—
213C2×Q8—28no—
214C2×Q8—42no—
215C2×Q8—42no—
216C2×Q8—41no—
217C2×Q8—41no—
218C2×Q8—42no—
219C2×Q8—44no—
220C2×Q8—44no—
221C2×D4—22no—
222C2×D4—22no—
223C2×D4—24no—
224C2×D4—24no—
225C2×D4—22no—
226C2×D4—22no—
227C2×D4—21no—
228C2×D4—21no—
229C22×C4—21no—
230C22×C4—22no—
231C22×C4—22no—
232C22×C4—22no—
233C22×C4—22no—
234C22×C4—24no—
235C22×C4—24no—
236C22×C4—24no—
237C22×C4—24no—
238C22×C4—24no—
239C22×C4—24no—
240C22×C4—22no—
241C22×C4—21no—
242C22×C4—22no—
243C22×C4—42no—
244C22×C4—44no—
245C22×C4—81no—
246C22×C4—82no—
247C22×C4—84no—
248C22×C4C2×C22⋊C412yes—
249C22×C4C2×C22⋊C412yes—
250C22×C4C2×C22⋊C412yes—
251C22×C4C2×C22⋊C414yes—
252C22×C4#11714yes—
253C22×C4C22×D411yescharacteristic
254C22×C4C22×D411yescharacteristic
255C22×C4C2×#4112yes—
256C22×C4C2×#4112yes—
257C22×C4H(5,𝔽2)14yes—
258C22×C4H(5,𝔽2)12yes—
259C22×C42−1+412yes—
260M16—44no—
261M16—44no—
262M16—44no—
263M16—44no—
264M16—44no—
265M16—44no—
266M16—44no—
267M16—44no—
268C2×C8—44no—
269C2×C8—44no—
270C2×C8—44no—
271C2×C8—44no—
272C2×C8—44no—
273C2×C8—44no—
274C4⋊C4—216no—
275C4⋊C4—24no—
276C4⋊C4—24no—
277C4⋊C4—24no—
278C4⋊C4—22no—
279C4⋊C4—22no—
280C4⋊C4—44no—
281C4⋊C4—44no—
282C4⋊C4—48no—
283C4⋊C4—41no—
284C4⋊C4—41no—
285C4⋊C4—82no—
286C22⋊C4—216no—
287C22⋊C4—22no—
288C22⋊C4—24no—
289C22⋊C4—24no—
290C22⋊C4—44no—
291C22⋊C4—44no—
292C22⋊C4—82no—
293C22⋊C4#11714yes—
294C42—24no—
295C42—24no—
296C42—44no—
297C42—48no—
298C42—42no—
299C42—42no—
300C42—42no—
301C42—84no—
302C23—22no—
303C23—22no—
304C23—22no—
305C23—41no—
306C23—41no—
307C23—42no—
308C23—42no—
309C23C22×C22⋊C411yescharacteristic
310C23C2×#11711yescharacteristic
311C23C2×#11711yescharacteristic
312C23#51812yes—
313C23#52012yes—
314Q8—48no—
315Q8—44no—
316Q8—42no—
317Q8—44no—
318Q8—42no—
319D4—44no—
320D4—44no—
321D4—42no—
322D4—42no—
323C2×C4—22no—
324C2×C4—24no—
325C2×C4—22no—
326C2×C4—22no—
327C2×C4—24no—
328C2×C4—22no—
329C2×C4—22no—
330C2×C4—28no—
331C2×C4—24no—
332C2×C4—24no—
333C2×C4—22no—
334C2×C4—42no—
335C2×C4—42no—
336C2×C4—41no—
337C2×C4—42no—
338C2×C4—41no—
339C2×C4—42no—
340C2×C4—42no—
341C2×C4—44no—
342C2×C4—48no—
343C2×C4—44no—
344C2×C4—42no—
345C2×C4—42no—
346C2×C4—42no—
347C2×C4—42no—
348C2×C4—41no—
349C2×C4—41no—
350C2×C4—44no—
351C2×C4—41no—
352C2×C4—42no—
353C2×C4—84no—
354C2×C4—164no—
355C2×C4#52512yes—
356C2×C4#52512yes—
357C2×C4#53412yes—
358C2×C4#53412yes—
359C2×C4#53511yescharacteristic
360C2×C4#53611yescharacteristic
361C2×C4#53712yes—
362C2×C4#53814yes—
363C2×C4#53914yes—
364C2×C4#54014yes—
365C2×C4#54214yes—
366C8—84no—
367C8—84no—
368C22—21no—
369C22—22no—
370C22—21no—
371C22—22no—
372C22—41no—
373C22—41no—
374C22—42no—
375C22—44no—
376C22—42no—
377C22—42no—
378C22—42no—
379C22—82no—
380C22#228011yescharacteristic
381C22#230811yescharacteristic
382C22#231111yescharacteristic
383C22#231812yes—
384C22#231912yes—
385C4—44no—
386C4—42no—
387C4—44no—
388C4—44no—
389C4—42no—
390C4—42no—
391C4—42no—
392C4—82no—
393C4—81no—
394C4#263014yes—
395C4#263512yes—
396C4#263812yes—
397C2—21no—
398C2—21no—
399C2—22no—
400C2—22no—
401C2—82no—
402C2#1434411yescharacteristic
403C2#2177911yescharacteristic
404C2#2178011yescharacteristic
subgroupG/H
center, socleC22#2308
commutatorC24C23×C4
Frattini, cosocleC23×C4C24
prime radicalC22×#41C23
LCSG ▷ C24 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C23×C4 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #11223C2semidirect
6. #11048C2semidirect
8. #10830C2semidirect
9. #10763C2semidirect

Maximal subgroups

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

subgroupindex
#342662
#206952
#206402
#112232
#111752
#110482
#109742
#108302
#107632