Finite Group Databasebeta

← #215582

Subgroups — #215582

Order 512.

888 conjugacy classes of subgroups.

The subgroup lattice has 421 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35281C211yesmaximal, characteristic
2#21926C211yesmaximal, characteristic
3#20786C211yesmaximal, characteristic
4#9709C212yesmaximal, complemented
5#9702C212yesmaximal, complemented
6#9702C211yesmaximal, characteristic, complemented
7#9701C211yesmaximal, characteristic, complemented
8#9693C211yesmaximal, characteristic, complemented
9#9691C211yesmaximal, characteristic, complemented
10#8343C212yesmaximal
11#8342C212yesmaximal
12C2×#533C2211yescharacteristic
13C22×C4⋊Q8C2211yescharacteristic
14#2868C2211yescharacteristic
15#2868C2211yescharacteristic
16#2862C2211yescharacteristic
17#2862C2211yescharacteristic
18#2660—21no—
19#2660—21no—
20#2567—22no—
21#2546C414yes—
22#2502—22no—
23#2434C414yes—
24#2317—21no—
25#2317—21no—
26C4×#117C2211yescharacteristic
27#1924—22no—
28#1917—22no—
29#1842C2211yescharacteristic
30#1840—21no—
31#1831C2212yes—
32#1831C2211yescharacteristic
33#1818—21no—
34#1797C2212yes—
35#1797C2212yes—
36#1767C2211yescharacteristic
37#1767C2212yes—
38C4×C4⋊C8C2211yescharacteristic
39#1750—21no—
40#1746—21no—
41C2×#333C2212yes—
42C2×#333C2212yes—
43C2×#333C2212yes—
44C2×#333C2212yes—
45C2×#333C2212yes—
46C2×#333C2212yes—
47C2×#332C2212yes—
48C2×#332C2212yes—
49C23×Q8C2×C412yes—
50#533—42no—
51#533—42no—
52C2×C4⋊Q8—21no—
53C2×C4⋊Q8—21no—
54C2×C4⋊Q8—42no—
55C2×C4⋊Q8—42no—
56C2×C4⋊Q8C2×C412yes—
57C2×C4⋊Q8C2×C414yes—
58C2×C4⋊Q8C2×C414yes—
59C2×C4⋊Q8C2311yescharacteristic
60C2×C4⋊Q8C2311yescharacteristic
61C2×C4⋊Q8C2311yescharacteristic
62C2×C4⋊Q8C2311yescharacteristic
63C2×#125—21no—
64C2×C22⋊Q8—21no—
65C2×C22⋊Q8C2×C412yes—
66C2×C4×Q8—21no—
67C2×C4×Q8C2×C412yes—
68C2×#117—21no—
69C2×#117D411yescharacteristic
70C2×#117D411yescharacteristic
71C22×C4⋊C4—21no—
72C22×C4⋊C4—21no—
73C22×C42C2311yescharacteristic
74C2×C4⋊C8—22no—
75C2×C4⋊C8C2312yes—
76C2×C4⋊C8C2312yes—
77C2×C4⋊C8C2312yes—
78C2×C4⋊C8C2312yes—
79C2×C22⋊C8—22no—
80C2×C22⋊C8—22no—
81C2×#97—21no—
82C2×#97—22no—
83C2×C4×C8—21no—
84#398—24no—
85#398—24no—
86#395—24no—
87#395—24no—
88#391—22no—
89#391—22no—
90#391C2×C414yes—
91#391C2×C414yes—
92#389—24no—
93#389—24no—
94#385—24no—
95#385—24no—
96#384—22no—
97#384—22no—
98#384—21no—
99#384—21no—
100#384—21no—
101#384—21no—
102#382—21no—
103#382—21no—
104#382—21no—
105#382—21no—
106C4×C4⋊C4C2×C412yes—
107C4×C4⋊C4D412yes—
108C4×C22⋊C4C2×C412yes—
109C4×C22⋊C4D412yes—
110#376D411yescharacteristic
111#376D411yescharacteristic
112C43C2311yescharacteristic
113C43C2311yescharacteristic
114#336—22no—
115#336—22no—
116#336—22no—
117#336—21no—
118#336—21no—
119#336—22no—
120#336—22no—
121#336—22no—
122#333—42no—
123#333—42no—
124#333—42no—
125#333—42no—
126#333—42no—
127#333—42no—
128#332—42no—
129#332—42no—
130C22×Q8—22no—
131C22×Q8—21no—
132C22×Q8—44no—
133C22×Q8—84no—
134C22×Q8C22×C412yes—
135C22×Q8C22×C412yes—
136C22×Q8C22×C412yes—
137C23×C4—22no—
138C23×C4C22×C411yescharacteristic
139C23×C4C2×D411yescharacteristic
140C23×C4C2×D411yescharacteristic
141C22×C8—22no—
142C22×C8—22no—
143C22×C8—22no—
144C22×C8—22no—
145C4⋊Q8—22no—
146C4⋊Q8—22no—
147C4⋊Q8—22no—
148C4⋊Q8—22no—
149C4⋊Q8—22no—
150C4⋊Q8—22no—
151C4⋊Q8—22no—
152C4⋊Q8—22no—
153C4⋊Q8—44no—
154C4⋊Q8—42no—
155C4⋊Q8—42no—
156C4⋊Q8—44no—
157C4⋊Q8—84no—
158C4⋊Q8—84no—
159#125—44no—
160C22⋊Q8—44no—
161C22⋊Q8—42no—
162C22⋊Q8—42no—
163C4×Q8—44no—
164C4×Q8—42no—
165C4×Q8—42no—
166#117—24no—
167#117—22no—
168#117—22no—
169#117—44no—
170C2×C4⋊C4—22no—
171C2×C4⋊C4—22no—
172C2×C4⋊C4—22no—
173C2×C4⋊C4—22no—
174C2×C4⋊C4—21no—
175C2×C4⋊C4—21no—
176C2×C4⋊C4—21no—
177C2×C4⋊C4—21no—
178C2×C4⋊C4—21no—
179C2×C4⋊C4—21no—
180C2×C4⋊C4—21no—
181C2×C4⋊C4—24no—
182C2×C4⋊C4—24no—
183C2×C4⋊C4—24no—
184C2×C4⋊C4—24no—
185C2×C4⋊C4—44no—
186C2×C4⋊C4—42no—
187C2×C4⋊C4—42no—
188C2×C4⋊C4C22⋊C412yes—
189C2×C4⋊C4C22⋊C412yes—
190C2×C4⋊C4C22×C412yes—
191C2×C22⋊C4—22no—
192C2×C22⋊C4C22⋊C412yes—
193C2×C22⋊C4C22⋊C412yes—
194C2×C42—22no—
195C2×C42—22no—
196C2×C42—22no—
197C2×C42C22×C411yescharacteristic
198C2×C42C22×C412yes—
199C2×C42C22×C412yes—
200C2×C42C2×D411yescharacteristic
201C2×C42C2×D411yescharacteristic
202C2×C42C2×D411yescharacteristic
203C2×C42C2×D411yescharacteristic
204C2×C42C2×D411yescharacteristic
205C2×C42C2×D412yes—
206C2×C42C2×D412yes—
207C2×C42C2×D411yescharacteristic
208C2×C42C2411yescharacteristic
209C4⋊C8—22no—
210C4⋊C8—22no—
211C4⋊C8—22no—
212C4⋊C8—22no—
213C4⋊C8—22no—
214C4⋊C8—22no—
215C4⋊C8—22no—
216C4⋊C8—22no—
217C4⋊C8—82no—
218C22⋊C8—44no—
219C22⋊C8—82no—
220#97—42no—
221#97—82no—
222C4×C8—42no—
223#95—24no—
224#95—24no—
225#95—24no—
226#95—24no—
227#95C22⋊C412yes—
228#95C22⋊C412yes—
229#95C22⋊C412yes—
230#95C22⋊C412yes—
231C24C2×C22⋊C411yescharacteristic
232C2×Q8—24no—
233C2×Q8—24no—
234C2×Q8—24no—
235C2×Q8—44no—
236C2×Q8—44no—
237C2×Q8—44no—
238C2×Q8—44no—
239C2×Q8—41no—
240C2×Q8—41no—
241C2×Q8—42no—
242C2×Q8—42no—
243C2×Q8—84no—
244C2×Q8—816no—
245C22×C4—22no—
246C22×C4—21no—
247C22×C4—21no—
248C22×C4—22no—
249C22×C4—21no—
250C22×C4—21no—
251C22×C4—22no—
252C22×C4—22no—
253C22×C4—22no—
254C22×C4—22no—
255C22×C4—22no—
256C22×C4—22no—
257C22×C4—42no—
258C22×C4—44no—
259C22×C4—84no—
260C22×C4C2×C22⋊C412yes—
261C22×C4C2×C22⋊C412yes—
262C22×C4C2×C22⋊C412yes—
263C22×C4C2×C22⋊C411yescharacteristic
264C22×C4C2×C22⋊C411yescharacteristic
265C22×C4C2×C22⋊C412yes—
266C22×C4C2×C22⋊C411yescharacteristic
267C22×C4#13714yes—
268C22×C4C23×C411yescharacteristic
269C22×C4C22×D411yescharacteristic
270C22×C4C22×D411yescharacteristic
271C2×C8—42no—
272C2×C8—42no—
273C2×C8—42no—
274C2×C8—42no—
275C2×C8—42no—
276C2×C8—42no—
277C2×C8—42no—
278C2×C8—42no—
279C2×C8—42no—
280C2×C8—42no—
281C2×C8—42no—
282C2×C8—42no—
283C4⋊C4—24no—
284C4⋊C4—24no—
285C4⋊C4—24no—
286C4⋊C4—44no—
287C4⋊C4—44no—
288C4⋊C4—44no—
289C4⋊C4—44no—
290C4⋊C4—42no—
291C4⋊C4—42no—
292C4⋊C4—41no—
293C4⋊C4—41no—
294C4⋊C4—41no—
295C4⋊C4—41no—
296C4⋊C4—42no—
297C4⋊C4—42no—
298C4⋊C4—41no—
299C4⋊C4—41no—
300C4⋊C4—84no—
301C4⋊C4—84no—
302C4⋊C4—84no—
303C4⋊C4—84no—
304C22⋊C4—24no—
305C22⋊C4—24no—
306C22⋊C4—44no—
307C42—22no—
308C42—21no—
309C42—21no—
310C42—22no—
311C42—22no—
312C42—22no—
313C42—21no—
314C42—21no—
315C42—22no—
316C42—22no—
317C42—22no—
318C42—22no—
319C42—42no—
320C42—42no—
321C42—44no—
322C42#13712yes—
323C42#13712yes—
324C42H(5,𝔽2)12yes—
325C42H(5,𝔽2)12yes—
326C23—21no—
327C23—22no—
328C23—21no—
329C23—22no—
330C23#41912yes—
331C23C22×C22⋊C411yescharacteristic
332Q8—84no—
333Q8—84no—
334Q8—84no—
335Q8—88no—
336Q8—81no—
337Q8—81no—
338Q8—82no—
339C2×C4—21no—
340C2×C4—22no—
341C2×C4—22no—
342C2×C4—21no—
343C2×C4—22no—
344C2×C4—24no—
345C2×C4—24no—
346C2×C4—21no—
347C2×C4—21no—
348C2×C4—22no—
349C2×C4—22no—
350C2×C4—22no—
351C2×C4—22no—
352C2×C4—22no—
353C2×C4—22no—
354C2×C4—22no—
355C2×C4—22no—
356C2×C4—22no—
357C2×C4—22no—
358C2×C4—22no—
359C2×C4—44no—
360C2×C4—41no—
361C2×C4—41no—
362C2×C4—42no—
363C2×C4—42no—
364C2×C4—42no—
365C2×C4—42no—
366C2×C4—42no—
367C2×C4—44no—
368C2×C4—44no—
369C2×C4—44no—
370C2×C4—44no—
371C2×C4—88no—
372C2×C4#41914yes—
373C2×C4#41912yes—
374C2×C4#51812yes—
375C2×C4#53512yes—
376C2×C4#53511yescharacteristic
377C2×C4#53511yescharacteristic
378C2×C4C2×#13712yes—
379C2×C4C2×#13711yescharacteristic
380C2×C4C2×#13711yescharacteristic
381C2×C4#57712yes—
382C2×C4#57812yes—
383C8—82no—
384C8—82no—
385C8—82no—
386C8—82no—
387C22—22no—
388C22—24no—
389C22—41no—
390C22—41no—
391C22—42no—
392C22#230911yescharacteristic
393C22#288812yes—
394C22C2×#41911yescharacteristic
395C22C2×#41911yescharacteristic
396C22#289812yes—
397C4—22no—
398C4—22no—
399C4—24no—
400C4—41no—
401C4—41no—
402C4—42no—
403C4—42no—
404C4—42no—
405C4—82no—
406C4—82no—
407C4—82no—
408C4—82no—
409C4#321812yes—
410C4#321812yes—
411C4#322111yescharacteristic
412C4#322311yescharacteristic
413C4#322512yes—
414C2—42no—
415C2#1776012yes—
416C2#1776012yes—
417C2#2180111yescharacteristic
418C2#2181111yescharacteristic
419C2#2181411yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
prime radicalC24C2×C22⋊C4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C4×#117 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #9709C2semidirect
5. #9702C2semidirect
6. #9702C2semidirect
7. #9701C2semidirect
8. #9693C2semidirect
9. #9691C2semidirect

Maximal subgroups

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

subgroupindex
#352812
#219262
#207862
#97092
#97022
#97012
#96932
#96912
#83432
#83422