Finite Group Databasebeta

← #214889

Subgroups — #214889

Order 512.

833 conjugacy classes of subgroups.

The subgroup lattice has 400 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35711C211yesmaximal, characteristic
2#20730C211yesmaximal, characteristic
3#17371C211yesmaximal, characteristic
4#9488C212yesmaximal, complemented
5#9404C212yesmaximal, complemented
6#9194C212yesmaximal
7#9034C212yesmaximal
8#8767C212yesmaximal, complemented
9#8396C212yesmaximal, complemented
10C2×Q82C2211yescharacteristic
11C2×#519C2211yescharacteristic
12#2983C2212yes—
13#2921C2212yes—
14#2628C412yes—
15#2628C2211yescharacteristic
16#2581C412yes—
17#2581C2211yescharacteristic
18#2568C412yes—
19#2568C2211yescharacteristic
20#2451—21no—
21#2451—22no—
22#2439—21no—
23#2439—22no—
24#2396C412yes—
25#2396C2211yescharacteristic
26#2351—21no—
27#2351C2211yescharacteristic
28Q8×C22⋊C4—21no—
29#1822—21no—
30#1801C2212yes—
31#1801C2212yes—
32#1784—21no—
33C4×#101—21no—
34C4×#101—21no—
35#1549C2214yes—
36#1487C2214yes—
37#1467—22no—
38#1464—22no—
39#1461C2212yes—
40#1458C2212yes—
41#1303C2214yes—
42#1286C2214yes—
43C23×Q8D411yescharacteristic
44Q82—44no—
45C2×C4⋊Q8—22no—
46C2×C4⋊Q8—21no—
47C2×C4⋊Q8C2×C412yes—
48C2×C4⋊Q8C2311yescharacteristic
49#519—44no—
50C2×C4×Q8—21no—
51C2×C4×Q8—21no—
52C2×C4×Q8C2×C412yes—
53C2×C4×Q8C2×C412yes—
54C2×C4×Q8D411yescharacteristic
55C2×C4×Q8D411yescharacteristic
56C2×C4×Q8D412yes—
57C2×C4×Q8C2311yescharacteristic
58C2×C4×Q8C2311yescharacteristic
59C2×C4×Q8C2311yescharacteristic
60C2×#117C2×C412yes—
61C2×#117D411yescharacteristic
62C2×#117D412yes—
63C2×#117C2311yescharacteristic
64#446—44no—
65#433—44no—
66#432—24no—
67#431C2314yes—
68#423C2314yes—
69C2×#101—22no—
70C2×#101—22no—
71#407—22no—
72C4×M16—22no—
73#400C2×C411yescharacteristic
74#400C2×C412yes—
75#400C2×C412yes—
76#400C2×C412yes—
77#400C2×C412yes—
78#399C2×C411yescharacteristic
79#399C2×C412yes—
80#398—22no—
81#398—21no—
82#396—22no—
83#396—21no—
84#395—22no—
85#395—21no—
86#395C2×C411yescharacteristic
87#395C2×C412yes—
88#393—22no—
89#393—21no—
90#393C2×C411yescharacteristic
91#391—22no—
92#391—22no—
93#391—21no—
94#391—21no—
95#391—22no—
96#391—22no—
97#391—22no—
98#389—22no—
99#389—21no—
100#389—21no—
101#389—22no—
102#389—22no—
103#389—21no—
104#387—22no—
105#387—22no—
106#387—22no—
107#382—22no—
108#382—21no—
109#382C2×C412yes—
110#382C2311yescharacteristic
111#380—22no—
112#380—21no—
113C4×C4⋊C4—22no—
114C4×C4⋊C4—21no—
115C4×C4⋊C4C2×C412yes—
116C4×C4⋊C4C2311yescharacteristic
117C4×C22⋊C4—22no—
118C4×C22⋊C4—21no—
119#343—22no—
120#343—22no—
121#332—44no—
122Q8⋊C8—44no—
123#324—22no—
124#324—24no—
125#323—22no—
126C22×Q8—24no—
127C22×Q8—22no—
128C22×Q8—21no—
129C22×Q8—42no—
130C22×Q8—42no—
131C22×Q8C22⋊C412yes—
132C22×Q8C22×C411yescharacteristic
133C22×Q8C2×D411yescharacteristic
134C22×Q8C2×D411yescharacteristic
135C23×C4C22⋊C412yes—
136C23×C4C2×D411yescharacteristic
137C2×M16—24no—
138C4⋊Q8—24no—
139C4⋊Q8—44no—
140C4⋊Q8—48no—
141C4⋊Q8—44no—
142C4×Q8—24no—
143C4×Q8—24no—
144C4×Q8—24no—
145C4×Q8—44no—
146C4×Q8—44no—
147C4×Q8—44no—
148C4×Q8—42no—
149C4×Q8—42no—
150C4×Q8—44no—
151C4×Q8—44no—
152#117—24no—
153#117—44no—
154#117—44no—
155#117—42no—
156C2×C4⋊C4—22no—
157C2×C4⋊C4—22no—
158C2×C4⋊C4—22no—
159C2×C4⋊C4—22no—
160C2×C4⋊C4—21no—
161C2×C4⋊C4—21no—
162C2×C4⋊C4—21no—
163C2×C4⋊C4—21no—
164C2×C4⋊C4—22no—
165C2×C4⋊C4—21no—
166C2×C4⋊C4—22no—
167C2×C4⋊C4—21no—
168C2×C4⋊C4C22⋊C412yes—
169C2×C4⋊C4C22⋊C412yes—
170C2×C4⋊C4C22×C412yes—
171C2×C4⋊C4C22×C411yescharacteristic
172C2×C4⋊C4C22×C412yes—
173C2×C4⋊C4C22×C411yescharacteristic
174C2×C4⋊C4C2×D412yes—
175C2×C4⋊C4C2×D411yescharacteristic
176C2×C4⋊C4C2×D411yescharacteristic
177C2×C4⋊C4C2×D411yescharacteristic
178C2×C22⋊C4—21no—
179C2×C22⋊C4—22no—
180C2×C22⋊C4—21no—
181C2×C22⋊C4C22⋊C412yes—
182C2×C22⋊C4C22×C411yescharacteristic
183C2×C22⋊C4C2×D411yescharacteristic
184C2×C42—22no—
185C2×C42—22no—
186C2×C42—21no—
187C2×C42—21no—
188C2×C42C22⋊C412yes—
189C2×C42C22⋊C412yes—
190C2×C42C22⋊C412yes—
191C2×C42C22×C412yes—
192C2×C42C2×D412yes—
193C2×C42C2×D411yescharacteristic
194C2×C42C2411yescharacteristic
195C4⋊C8—24no—
196C4⋊C8—24no—
197C4⋊C8—24no—
198C4⋊C8—44no—
199#101—44no—
200#101—44no—
201C22⋊C8—24no—
202C22⋊C8—44no—
203#97—42no—
204#97—44no—
205C4×C8—24no—
206C4×C8—42no—
207#95—21no—
208#95—22no—
209#95—21no—
210#95—22no—
211#95—21no—
212#95—22no—
213#95—21no—
214#95—22no—
215#95—22no—
216#95—22no—
217#95—22no—
218#95—22no—
219#95—22no—
220#95—22no—
221#95—21no—
222#95C22×C411yescharacteristic
223#95C22×C412yes—
224#95C22×C411yescharacteristic
225C24C2×C22⋊C411yescharacteristic
226C2×Q8—22no—
227C2×Q8—22no—
228C2×Q8—22no—
229C2×Q8—22no—
230C2×Q8—22no—
231C2×Q8—24no—
232C2×Q8—22no—
233C2×Q8—42no—
234C2×Q8—42no—
235C2×Q8—42no—
236C2×Q8—44no—
237C2×Q8—416no—
238C2×Q8—42no—
239C2×Q8—82no—
240C2×Q8—82no—
241C2×Q8—82no—
242C2×Q8#13714yes—
243C22×C4—21no—
244C22×C4—22no—
245C22×C4—22no—
246C22×C4—22no—
247C22×C4—24no—
248C22×C4—21no—
249C22×C4—21no—
250C22×C4—22no—
251C22×C4—22no—
252C22×C4—21no—
253C22×C4—42no—
254C22×C4—42no—
255C22×C4C2×C22⋊C411yescharacteristic
256C22×C4C2×C22⋊C412yes—
257C22×C4C2×C22⋊C412yes—
258C22×C4C2×C22⋊C412yes—
259C22×C4C2×C22⋊C411yescharacteristic
260C22×C4C2×C22⋊C412yes—
261C22×C4C2×C22⋊C411yescharacteristic
262C22×C4C23×C411yescharacteristic
263C22×C4C22×D411yescharacteristic
264C22×C4C22×D411yescharacteristic
265M16—44no—
266M16—44no—
267C2×C8—44no—
268C2×C8—44no—
269C4⋊C4—22no—
270C4⋊C4—24no—
271C4⋊C4—22no—
272C4⋊C4—22no—
273C4⋊C4—24no—
274C4⋊C4—22no—
275C4⋊C4—22no—
276C4⋊C4—24no—
277C4⋊C4—42no—
278C4⋊C4—42no—
279C4⋊C4—44no—
280C4⋊C4—42no—
281C4⋊C4—44no—
282C4⋊C4—44no—
283C4⋊C4—44no—
284C4⋊C4—42no—
285C4⋊C4—42no—
286C4⋊C4—42no—
287C4⋊C4—81no—
288C4⋊C4—82no—
289C4⋊C4—81no—
290C4⋊C4—82no—
291C22⋊C4—22no—
292C22⋊C4—22no—
293C22⋊C4—42no—
294C22⋊C4—42no—
295C22⋊C4—82no—
296C22⋊C4—81no—
297C42—24no—
298C42—22no—
299C42—24no—
300C42—42no—
301C42—42no—
302C42—42no—
303C42—44no—
304C42—44no—
305C42—42no—
306C42—42no—
307C42H(5,𝔽2)14yes—
308C23—21no—
309C23—21no—
310C23—21no—
311C23—41no—
312C23—41no—
313C23C22×C22⋊C411yescharacteristic
314Q8—28no—
315Q8—24no—
316Q8—42no—
317Q8—48no—
318Q8—44no—
319Q8—162no—
320Q8—164no—
321C2×C4—22no—
322C2×C4—22no—
323C2×C4—22no—
324C2×C4—21no—
325C2×C4—21no—
326C2×C4—22no—
327C2×C4—22no—
328C2×C4—22no—
329C2×C4—21no—
330C2×C4—21no—
331C2×C4—22no—
332C2×C4—21no—
333C2×C4—22no—
334C2×C4—41no—
335C2×C4—41no—
336C2×C4—48no—
337C2×C4—41no—
338C2×C4—41no—
339C2×C4—44no—
340C2×C4—42no—
341C2×C4—41no—
342C2×C4—41no—
343C2×C4—42no—
344C2×C4—41no—
345C2×C4—41no—
346C2×C4—42no—
347C2×C4—42no—
348C2×C4—41no—
349C2×C4—41no—
350C2×C4—42no—
351C2×C4—41no—
352C2×C4—42no—
353C2×C4—41no—
354C2×C4—42no—
355C2×C4—41no—
356C2×C4—41no—
357C2×C4—42no—
358C2×C4—41no—
359C2×C4#41914yes—
360C2×C4#51812yes—
361C2×C4#53412yes—
362C2×C4#53412yes—
363C2×C4C2×#13712yes—
364C2×C4#57512yes—
365C8—84no—
366C8—84no—
367C22—21no—
368C22—21no—
369C22—42no—
370C22—41no—
371C22—41no—
372C22—41no—
373C22—81no—
374C22#230811yescharacteristic
375C22C2×#41911yescharacteristic
376C22#289611yescharacteristic
377C4—24no—
378C4—22no—
379C4—22no—
380C4—42no—
381C4—41no—
382C4—41no—
383C4—42no—
384C4—42no—
385C4—41no—
386C4—42no—
387C4—81no—
388C4—82no—
389C4—81no—
390C4—82no—
391C4—81no—
392C4#319614yes—
393C2—21no—
394C2—21no—
395C2—81no—
396C2#2176811yescharacteristic
397C2#2178011yescharacteristic
398C2#2178511yescharacteristic
subgroupG/H
center, socleC22#2308
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
prime radicalC24C2×C22⋊C4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C2×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #9488C2semidirect
5. #9404C2semidirect
8. #8767C2semidirect
9. #8396C2semidirect

Maximal subgroups

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

subgroupindex
#357112
#207302
#173712
#94882
#94042
#91942
#90342
#87672
#83962