Finite Group Databasebeta

← #215831

Subgroups — #215831

Order 512.

760 conjugacy classes of subgroups.

The subgroup lattice has 398 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34948C211yesmaximal, characteristic
2#21431C211yesmaximal, characteristic
3#20812C211yesmaximal, characteristic
4#9876C212yesmaximal, complemented
5#9872C212yesmaximal, complemented
6#9867C212yesmaximal, complemented
7#9866C212yesmaximal, complemented
8#8709C212yesmaximal
9#8707C212yesmaximal
10C2×#433—21no—
11#2571C412yes—
12#2570—22no—
13#2569—22no—
14#2552C412yes—
15#2442C412yes—
16#2442C412yes—
17#2402—21no—
18#2402—21no—
19C4×#125C2212yes—
20C4×#125C2212yes—
21#2296—21no—
22#2285—21no—
23#2285C2211yescharacteristic
24C4×#117C2211yescharacteristic
25C2×C4×C4⋊C4C2211yescharacteristic
26#1924—22no—
27#1917—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
38C4×#97C2211yescharacteristic
39#1279C2212yes—
40#1279C2212yes—
41#1279C2212yes—
42#1279C2212yes—
43#1279C2212yes—
44#1279C2212yes—
45#1279C2212yes—
46#1279C2212yes—
47C2×#125C2×C412yes—
48C2×#125C2×C412yes—
49C2×#117—21no—
50C2×#117D411yescharacteristic
51C2×#117D411yescharacteristic
52C22×C4⋊C4C2×C412yes—
53C22×C42—21no—
54C22×C42C2311yescharacteristic
55#433—44no—
56C2×C4⋊C8—22no—
57C2×C4⋊C8—22no—
58C2×C4⋊C8—22no—
59C2×C4⋊C8C2312yes—
60C2×C4⋊C8C2312yes—
61C2×C22⋊C8—22no—
62C2×C22⋊C8—22no—
63C2×#97—21no—
64C2×#97—22no—
65C2×C4×C8—21no—
66C2×C4×C8C2312yes—
67C2×C4×C8C2312yes—
68#400—22no—
69#400—22no—
70#400—22no—
71#400—22no—
72#398—22no—
73#398—22no—
74#398—22no—
75#398—22no—
76#389—22no—
77#389—22no—
78#389C2×C412yes—
79#389C2×C412yes—
80#389C2×C412yes—
81#389C2×C412yes—
82#387—22no—
83#387—22no—
84#387—22no—
85#387—22no—
86#385—22no—
87#385—22no—
88#384—22no—
89#384C2×C412yes—
90#382—22no—
91#382—21no—
92#382C2×C412yes—
93#382C2×C412yes—
94#380—21no—
95#380—22no—
96#380—22no—
97#380C2×C411yescharacteristic
98#380C2×C411yescharacteristic
99C4×C4⋊C4—22no—
100C4×C4⋊C4—22no—
101C4×C4⋊C4—22no—
102C4×C4⋊C4—22no—
103C4×C4⋊C4—21no—
104C4×C4⋊C4—21no—
105C4×C4⋊C4—21no—
106C4×C4⋊C4C2×C411yescharacteristic
107C4×C4⋊C4C2×C411yescharacteristic
108C4×C4⋊C4C2×C412yes—
109C4×C4⋊C4D412yes—
110C4×C4⋊C4C2312yes—
111C4×C4⋊C4C2312yes—
112C4×C22⋊C4C2×C412yes—
113C4×C22⋊C4D412yes—
114#376D411yescharacteristic
115#376D411yescharacteristic
116C2×#95—21no—
117C43C2311yescharacteristic
118C43C2311yescharacteristic
119#336—22no—
120#336—21no—
121#336—21no—
122#336—22no—
123#336—22no—
124#336—22no—
125C23×C4—22no—
126C23×C4C22×C411yescharacteristic
127C23×C4C2×D411yescharacteristic
128C23×C4C2×D411yescharacteristic
129C22×C8—22no—
130C22×C8—22no—
131C22×C8—22no—
132C22×C8—22no—
133#125—24no—
134#125—24no—
135#125—42no—
136#125—42no—
137#117—22no—
138#117—22no—
139#117—42no—
140#117—42no—
141#117—42no—
142C2×C4⋊C4—24no—
143C2×C4⋊C4—22no—
144C2×C4⋊C4—22no—
145C2×C4⋊C4—22no—
146C2×C4⋊C4—22no—
147C2×C4⋊C4—22no—
148C2×C4⋊C4—22no—
149C2×C4⋊C4—22no—
150C2×C4⋊C4—22no—
151C2×C4⋊C4—22no—
152C2×C4⋊C4—22no—
153C2×C4⋊C4—22no—
154C2×C4⋊C4—42no—
155C2×C4⋊C4C22⋊C412yes—
156C2×C4⋊C4C22⋊C412yes—
157C2×C4⋊C4C22×C412yes—
158C2×C4⋊C4C22×C412yes—
159C2×C4⋊C4C22×C412yes—
160C2×C4⋊C4C22×C412yes—
161C2×C22⋊C4—21no—
162C2×C22⋊C4—21no—
163C2×C22⋊C4C22⋊C412yes—
164C2×C22⋊C4C22⋊C412yes—
165C2×C42—22no—
166C2×C42—22no—
167C2×C42—21no—
168C2×C42—21no—
169C2×C42—22no—
170C2×C42—22no—
171C2×C42—44no—
172C2×C42C22×C412yes—
173C2×C42C22×C412yes—
174C2×C42C22×C411yescharacteristic
175C2×C42C2×D411yescharacteristic
176C2×C42C2×D411yescharacteristic
177C2×C42C2×D411yescharacteristic
178C2×C42C2×D411yescharacteristic
179C2×C42C2×D411yescharacteristic
180C2×C42C2×D411yescharacteristic
181C2×C42C2×D412yes—
182C2×C42C2×D412yes—
183C2×C42C2411yescharacteristic
184C4⋊C8—22no—
185C4⋊C8—22no—
186C4⋊C8—22no—
187C4⋊C8—22no—
188C4⋊C8—44no—
189C4⋊C8—44no—
190C4⋊C8—44no—
191C22⋊C8—44no—
192C22⋊C8—44no—
193#97—42no—
194#97—44no—
195C4×C8—22no—
196C4×C8—22no—
197C4×C8—22no—
198C4×C8—22no—
199C4×C8—42no—
200#95—21no—
201#95—21no—
202#95—22no—
203#95—22no—
204#95—22no—
205#95—22no—
206#95—21no—
207#95—21no—
208#95—22no—
209#95—22no—
210#95—22no—
211#95—22no—
212#95—22no—
213#95—22no—
214#95C22⋊C412yes—
215#95C22⋊C412yes—
216#95C22⋊C412yes—
217#95C22⋊C412yes—
218C24C2×C22⋊C411yescharacteristic
219C22×C4—22no—
220C22×C4—22no—
221C22×C4—22no—
222C22×C4—21no—
223C22×C4—21no—
224C22×C4—22no—
225C22×C4—22no—
226C22×C4—22no—
227C22×C4—22no—
228C22×C4—22no—
229C22×C4—22no—
230C22×C4—42no—
231C22×C4—44no—
232C22×C4—44no—
233C22×C4—82no—
234C22×C4C2×C22⋊C411yescharacteristic
235C22×C4C2×C22⋊C411yescharacteristic
236C22×C4C2×C22⋊C412yes—
237C22×C4C2×C22⋊C412yes—
238C22×C4C2×C22⋊C412yes—
239C22×C4C2×C22⋊C412yes—
240C22×C4C2×C22⋊C411yescharacteristic
241C22×C4AGL(1,ℤ/8)12yes—
242C22×C4#13712yes—
243C22×C4C23×C411yescharacteristic
244C22×C4C22×D411yescharacteristic
245C22×C4C22×D411yescharacteristic
246C2×C8—24no—
247C2×C8—42no—
248C2×C8—42no—
249C2×C8—42no—
250C2×C8—42no—
251C2×C8—42no—
252C2×C8—42no—
253C2×C8—42no—
254C2×C8—42no—
255C2×C8—42no—
256C2×C8—42no—
257C2×C8—42no—
258C4⋊C4—22no—
259C4⋊C4—24no—
260C4⋊C4—22no—
261C4⋊C4—22no—
262C4⋊C4—22no—
263C4⋊C4—22no—
264C4⋊C4—22no—
265C4⋊C4—22no—
266C4⋊C4—42no—
267C4⋊C4—44no—
268C4⋊C4—44no—
269C4⋊C4—44no—
270C4⋊C4—44no—
271C4⋊C4—44no—
272C4⋊C4—44no—
273C4⋊C4—82no—
274C4⋊C4—82no—
275C4⋊C4—82no—
276C4⋊C4—82no—
277C4⋊C4—82no—
278C4⋊C4#13114yes—
279C22⋊C4—24no—
280C22⋊C4—42no—
281C22⋊C4—42no—
282C22⋊C4—42no—
283C42—21no—
284C42—21no—
285C42—22no—
286C42—22no—
287C42—22no—
288C42—22no—
289C42—22no—
290C42—22no—
291C42—21no—
292C42—21no—
293C42—21no—
294C42—21no—
295C42—22no—
296C42—22no—
297C42—42no—
298C42—42no—
299C42—42no—
300C42—42no—
301C42—42no—
302C42—42no—
303C42—44no—
304C422−1+412yes—
305C422−1+412yes—
306C23—21no—
307C23—22no—
308C23—21no—
309C23—22no—
310C23#41712yes—
311C23C22×C22⋊C411yescharacteristic
312C2×C4—21no—
313C2×C4—21no—
314C2×C4—22no—
315C2×C4—24no—
316C2×C4—24no—
317C2×C4—21no—
318C2×C4—21no—
319C2×C4—22no—
320C2×C4—22no—
321C2×C4—22no—
322C2×C4—22no—
323C2×C4—24no—
324C2×C4—22no—
325C2×C4—22no—
326C2×C4—22no—
327C2×C4—22no—
328C2×C4—22no—
329C2×C4—22no—
330C2×C4—22no—
331C2×C4—22no—
332C2×C4—22no—
333C2×C4—22no—
334C2×C4—42no—
335C2×C4—42no—
336C2×C4—42no—
337C2×C4—41no—
338C2×C4—41no—
339C2×C4—42no—
340C2×C4—42no—
341C2×C4—42no—
342C2×C4—42no—
343C2×C4—42no—
344C2×C4—42no—
345C2×C4—42no—
346C2×C4—42no—
347C2×C4—42no—
348C2×C4—82no—
349C2×C4—88no—
350C2×C4#41712yes—
351C2×C4#51911yescharacteristic
352C2×C4#51911yescharacteristic
353C2×C4#53612yes—
354C2×C4#53612yes—
355C2×C4C2×#13112yes—
356C2×C4#56812yes—
357C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
358C2×C4C2×#13711yescharacteristic
359C2×C4#57512yes—
360C8—44no—
361C8—82no—
362C8—82no—
363C8—82no—
364C22—22no—
365C22—24no—
366C22—41no—
367C22—41no—
368C22—42no—
369C22#231011yescharacteristic
370C22#288012yes—
371C22C2×#41711yescharacteristic
372C22#289611yescharacteristic
373C22#290112yes—
374C4—22no—
375C4—22no—
376C4—41no—
377C4—41no—
378C4—42no—
379C4—42no—
380C4—44no—
381C4—42no—
382C4—42no—
383C4—82no—
384C4—82no—
385C4—82no—
386C4#298512yes—
387C4#298611yescharacteristic
388C4#298711yescharacteristic
389C4#323412yes—
390C4#323612yes—
391C2—42no—
392C2#1751312yes—
393C2#1751512yes—
394C2#2182411yescharacteristic
395C2#2183811yescharacteristic
396C2#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. #9876C2semidirect
5. #9872C2semidirect
6. #9867C2semidirect
7. #9866C2semidirect

Maximal subgroups

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

subgroupindex
#349482
#214312
#208122
#98762
#98722
#98672
#98662
#87092
#87072