Finite Group Databasebeta

← #215824

Subgroups — #215824

Order 512.

900 conjugacy classes of subgroups.

The subgroup lattice has 420 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34938C211yesmaximal, characteristic
2#21430C211yesmaximal, characteristic
3#20812C211yesmaximal, characteristic
4#9874C212yesmaximal, complemented
5#9870C212yesmaximal, complemented
6#9869C212yesmaximal, complemented
7#9865C212yesmaximal, complemented
8#8709C212yesmaximal
9#8706C212yesmaximal
10C2×#533—21no—
11C2×#433—21no—
12#2552C412yes—
13#2551C412yes—
14#2550—22no—
15#2549—22no—
16#2440C412yes—
17#2440C412yes—
18#2399—21no—
19#2399—21no—
20C4×C4⋊Q8C2212yes—
21C4×C4⋊Q8C2212yes—
22#2296—21no—
23#2285C2211yescharacteristic
24C4×#117C2211yescharacteristic
25C2×C4×C4⋊C4C2211yescharacteristic
26#1923—22no—
27#1916—22no—
28#1842C2212yes—
29#1832—21no—
30#1798—21no—
31#1797C2212yes—
32#1768—21no—
33#1767C2211yescharacteristic
34C4×C4⋊C8C2211yescharacteristic
35C4×C4⋊C8C2212yes—
36C4×C22⋊C8C2212yes—
37#1744C2211yescharacteristic
38C42×C8C2211yescharacteristic
39#1279C2212yes—
40#1279C2212yes—
41#1279C2212yes—
42#1279C2212yes—
43#1279C2212yes—
44#1279C2212yes—
45#1279C2212yes—
46#1279C2212yes—
47#533—44no—
48#533—44no—
49C2×C4⋊Q8—22no—
50C2×C4⋊Q8C2×C412yes—
51C2×C4⋊Q8C2×C412yes—
52C2×#125—22no—
53C2×C22⋊Q8—22no—
54C2×C22⋊Q8—22no—
55C2×C4×Q8—22no—
56C2×C4×Q8—22no—
57C2×#117—21no—
58C2×#117D411yescharacteristic
59C2×#117D411yescharacteristic
60C22×C4⋊C4C2×C412yes—
61C22×C42—21no—
62C22×C42C2311yescharacteristic
63#433—44no—
64C2×C4⋊C8—22no—
65C2×C4⋊C8—22no—
66C2×C4⋊C8—22no—
67C2×C4⋊C8C2312yes—
68C2×C4⋊C8C2312yes—
69C2×C22⋊C8—22no—
70C2×C22⋊C8—22no—
71C2×#97—21no—
72C2×C4×C8—21no—
73C2×C4×C8—22no—
74C2×C4×C8C2312yes—
75C2×C4×C8C2312yes—
76#398—22no—
77#398—22no—
78#398—22no—
79#398—22no—
80#395—22no—
81#395—22no—
82#395—22no—
83#395—22no—
84#391—22no—
85#391—22no—
86#389—22no—
87#389—22no—
88#389—22no—
89#389—22no—
90#389C2×C412yes—
91#389C2×C412yes—
92#389C2×C412yes—
93#389C2×C412yes—
94#385—22no—
95#385—22no—
96#384C2×C412yes—
97#384C2×C412yes—
98#384C2×C412yes—
99#382—21no—
100#380—21no—
101#380C2×C411yescharacteristic
102#380C2×C411yescharacteristic
103C4×C4⋊C4—22no—
104C4×C4⋊C4—22no—
105C4×C4⋊C4—21no—
106C4×C4⋊C4—21no—
107C4×C4⋊C4—21no—
108C4×C4⋊C4C2×C411yescharacteristic
109C4×C4⋊C4C2×C411yescharacteristic
110C4×C4⋊C4C2×C412yes—
111C4×C4⋊C4D412yes—
112C4×C4⋊C4C2312yes—
113C4×C4⋊C4C2312yes—
114C4×C22⋊C4C2×C412yes—
115C4×C22⋊C4D412yes—
116#376D411yescharacteristic
117#376D411yescharacteristic
118C2×#95—21no—
119C43C2311yescharacteristic
120C43C2311yescharacteristic
121#336—22no—
122#336—21no—
123#336—21no—
124#336—22no—
125#336—22no—
126#336—22no—
127C22×Q8—22no—
128C22×Q8—22no—
129C23×C4—22no—
130C23×C4C22×C411yescharacteristic
131C23×C4C2×D411yescharacteristic
132C23×C4C2×D411yescharacteristic
133C22×C8—22no—
134C22×C8—22no—
135C22×C8—22no—
136C22×C8—22no—
137C4⋊Q8—24no—
138C4⋊Q8—24no—
139C4⋊Q8—42no—
140C4⋊Q8—42no—
141C4⋊Q8—48no—
142#125—48no—
143C22⋊Q8—48no—
144C22⋊Q8—48no—
145C4×Q8—44no—
146C4×Q8—44no—
147C4×Q8—44no—
148C4×Q8—44no—
149#117—24no—
150#117—42no—
151#117—42no—
152#117D814yes—
153#117SD1614yes—
154C2×C4⋊C4—24no—
155C2×C4⋊C4—22no—
156C2×C4⋊C4—22no—
157C2×C4⋊C4—22no—
158C2×C4⋊C4—22no—
159C2×C4⋊C4—22no—
160C2×C4⋊C4—22no—
161C2×C4⋊C4—22no—
162C2×C4⋊C4—22no—
163C2×C4⋊C4—22no—
164C2×C4⋊C4—22no—
165C2×C4⋊C4—22no—
166C2×C4⋊C4—22no—
167C2×C4⋊C4—22no—
168C2×C4⋊C4—42no—
169C2×C4⋊C4C22⋊C412yes—
170C2×C4⋊C4C22⋊C412yes—
171C2×C4⋊C4C22×C412yes—
172C2×C4⋊C4C22×C412yes—
173C2×C4⋊C4C22×C412yes—
174C2×C4⋊C4C22×C412yes—
175C2×C22⋊C4—21no—
176C2×C22⋊C4—21no—
177C2×C22⋊C4C22⋊C412yes—
178C2×C22⋊C4C22⋊C412yes—
179C2×C42—22no—
180C2×C42—22no—
181C2×C42—21no—
182C2×C42—21no—
183C2×C42—22no—
184C2×C42—22no—
185C2×C42—42no—
186C2×C42—42no—
187C2×C42C22×C412yes—
188C2×C42C22×C412yes—
189C2×C42C22×C411yescharacteristic
190C2×C42C2×D411yescharacteristic
191C2×C42C2×D411yescharacteristic
192C2×C42C2×D411yescharacteristic
193C2×C42C2×D412yes—
194C2×C42C2×D412yes—
195C2×C42C2×D411yescharacteristic
196C2×C42C2×D411yescharacteristic
197C2×C42C2×D411yescharacteristic
198C2×C42C2411yescharacteristic
199C4⋊C8—22no—
200C4⋊C8—22no—
201C4⋊C8—22no—
202C4⋊C8—22no—
203C4⋊C8—44no—
204C4⋊C8—44no—
205C4⋊C8—44no—
206C22⋊C8—44no—
207C22⋊C8—44no—
208#97—42no—
209C4×C8—22no—
210C4×C8—22no—
211C4×C8—22no—
212C4×C8—22no—
213C4×C8—44no—
214C4×C8—42no—
215#95—21no—
216#95—21no—
217#95—21no—
218#95—21no—
219#95—22no—
220#95—22no—
221#95—22no—
222#95—22no—
223#95—22no—
224#95—22no—
225#95C22⋊C412yes—
226#95C22⋊C412yes—
227#95C22⋊C412yes—
228#95C22⋊C412yes—
229C24C2×C22⋊C411yescharacteristic
230C2×Q8—44no—
231C2×Q8—44no—
232C2×Q8—44no—
233C2×Q8—44no—
234C2×Q8—82no—
235C2×Q8—82no—
236C22×C4—22no—
237C22×C4—22no—
238C22×C4—22no—
239C22×C4—21no—
240C22×C4—21no—
241C22×C4—22no—
242C22×C4—22no—
243C22×C4—22no—
244C22×C4—22no—
245C22×C4—22no—
246C22×C4—22no—
247C22×C4—42no—
248C22×C4—44no—
249C22×C4—44no—
250C22×C4—82no—
251C22×C4C2×C22⋊C411yescharacteristic
252C22×C4C2×C22⋊C411yescharacteristic
253C22×C4C2×C22⋊C412yes—
254C22×C4C2×C22⋊C412yes—
255C22×C4C2×C22⋊C412yes—
256C22×C4C2×C22⋊C411yescharacteristic
257C22×C4C2×C22⋊C412yes—
258C22×C4C2×D812yes—
259C22×C4C2×SD1612yes—
260C22×C4C23×C411yescharacteristic
261C22×C4C22×D411yescharacteristic
262C22×C4C22×D411yescharacteristic
263C2×C8—24no—
264C2×C8—42no—
265C2×C8—42no—
266C2×C8—42no—
267C2×C8—42no—
268C2×C8—42no—
269C2×C8—42no—
270C2×C8—42no—
271C2×C8—42no—
272C2×C8—42no—
273C2×C8—42no—
274C2×C8—42no—
275C4⋊C4—22no—
276C4⋊C4—24no—
277C4⋊C4—22no—
278C4⋊C4—22no—
279C4⋊C4—22no—
280C4⋊C4—22no—
281C4⋊C4—22no—
282C4⋊C4—22no—
283C4⋊C4—44no—
284C4⋊C4—44no—
285C4⋊C4—44no—
286C4⋊C4—44no—
287C4⋊C4—44no—
288C4⋊C4—44no—
289C4⋊C4—44no—
290C4⋊C4—44no—
291C4⋊C4—44no—
292C4⋊C4—44no—
293C4⋊C4—82no—
294C4⋊C4—82no—
295C4⋊C4—82no—
296C4⋊C4#10218yes—
297C4⋊C4#13114yes—
298C22⋊C4—24no—
299C22⋊C4—42no—
300C22⋊C4—42no—
301C22⋊C4#10218yes—
302C42—21no—
303C42—21no—
304C42—22no—
305C42—22no—
306C42—22no—
307C42—22no—
308C42—22no—
309C42—22no—
310C42—22no—
311C42—42no—
312C42—42no—
313C42—42no—
314C42—42no—
315C42—42no—
316C42—42no—
317C42—42no—
318C42—42no—
319C42C2×D812yes—
320C42C2×D812yes—
321C42C2×SD1612yes—
322C42C2×SD1612yes—
323C42#13714yes—
324C422−1+412yes—
325C422−1+412yes—
326C23—21no—
327C23—22no—
328C23—21no—
329C23—22no—
330C23C2×#10212yes—
331C23C22×C22⋊C411yescharacteristic
332Q8—88no—
333Q8—88no—
334C2×C4—21no—
335C2×C4—22no—
336C2×C4—22no—
337C2×C4—22no—
338C2×C4—21no—
339C2×C4—22no—
340C2×C4—22no—
341C2×C4—22no—
342C2×C4—22no—
343C2×C4—24no—
344C2×C4—24no—
345C2×C4—21no—
346C2×C4—21no—
347C2×C4—22no—
348C2×C4—24no—
349C2×C4—22no—
350C2×C4—22no—
351C2×C4—22no—
352C2×C4—22no—
353C2×C4—42no—
354C2×C4—42no—
355C2×C4—42no—
356C2×C4—41no—
357C2×C4—41no—
358C2×C4—42no—
359C2×C4—42no—
360C2×C4—42no—
361C2×C4—42no—
362C2×C4—42no—
363C2×C4—42no—
364C2×C4—42no—
365C2×C4—42no—
366C2×C4—42no—
367C2×C4—82no—
368C2×C4—88no—
369C2×C4C2×#10212yes—
370C2×C4C2×#10214yes—
371C2×C4C2×#10214yes—
372C2×C4#41914yes—
373C2×C4#51911yescharacteristic
374C2×C4#51911yescharacteristic
375C2×C4#53612yes—
376C2×C4#53612yes—
377C2×C4C2×#13112yes—
378C2×C4#56812yes—
379C2×C4C22×D811yescharacteristic
380C2×C4C22×SD1611yescharacteristic
381C2×C4C2×#13712yes—
382C8—44no—
383C8—82no—
384C8—82no—
385C8—82no—
386C22—22no—
387C22—24no—
388C22—41no—
389C22—41no—
390C22—42no—
391C22#231011yescharacteristic
392C22#288012yes—
393C22C22×#10211yescharacteristic
394C22C2×#41911yescharacteristic
395C22#290112yes—
396C4—22no—
397C4—22no—
398C4—24no—
399C4—24no—
400C4—24no—
401C4—41no—
402C4—41no—
403C4—44no—
404C4—42no—
405C4—82no—
406C4—82no—
407C4—82no—
408C4#298512yes—
409C4#298611yescharacteristic
410C4#298711yescharacteristic
411C4#322612yes—
412C4#322912yes—
413C2—42no—
414C2#1750712yes—
415C2#1751412yes—
416C2#2182411yescharacteristic
417C2#2183311yescharacteristic
418C2#2184611yescharacteristic
subgroupG/H
centerC22×C4C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
socleC23C22×C22⋊C4
prime radicalC24C2×C22⋊C4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C4 ◁ C4×#117 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #9874C2semidirect
5. #9870C2semidirect
6. #9869C2semidirect
7. #9865C2semidirect

Maximal subgroups

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

subgroupindex
#349382
#214302
#208122
#98742
#98702
#98692
#98652
#87092
#87062