Finite Group Databasebeta

← #202522

Subgroups — #202522

Order 512.

645 conjugacy classes of subgroups.

The subgroup lattice has 402 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21425C211yesmaximal, characteristic
2#17797C211yesmaximal, characteristic
3#17479C211yesmaximal, characteristic
4C2×#1497C211yesmaximal, characteristic
5#17324C211yesmaximal, characteristic
6#9551C214yesmaximal, complemented
7#9549C214yesmaximal, complemented
8#7933C211yesmaximal, characteristic
9#7882C211yesmaximal, characteristic
10C2×#533C2211yescharacteristic
11#2920C2211yescharacteristic
12#2920C2211yescharacteristic
13C2×#432C2211yescharacteristic
14C2×#431C2211yescharacteristic
15C2×#431C2211yescharacteristic
16C2×C4×M16C2211yescharacteristic
17#1997C411yescharacteristic
18#1997C411yescharacteristic
19#1966C411yescharacteristic
20#1966C411yescharacteristic
21#1901C414yes—
22C4×#101C2214yes—
23#1569—22no—
24#1568—22no—
25#1565C2214yes—
26#1497—22no—
27#1497—22no—
28#1497—22no—
29#1493—22no—
30#1479C2214yes—
31C2×Q8⋊C8C2212yes—
32C2×Q8⋊C8C2212yes—
33#1454—22no—
34#1453—22no—
35#1451C2212yes—
36C8×M16C2212yes—
37#1283C2214yes—
38#1278C2212yes—
39#1278C2212yes—
40C22×M16C2×C412yes—
41#533—21no—
42#533—21no—
43#533—21no—
44#533—21no—
45#533—21no—
46#533—21no—
47#533—21no—
48#533—21no—
49C2×C4⋊Q8C2×C411yescharacteristic
50C2×C4⋊Q8C2×C411yescharacteristic
51C2×#125—21no—
52C2×C22⋊Q8—21no—
53C2×C22⋊Q8C2×C412yes—
54C2×C4×Q8—21no—
55C2×C4×Q8C2312yes—
56C2×#117—21no—
57C22×C42C2311yescharacteristic
58#432—21no—
59#432—21no—
60#432—21no—
61#432—21no—
62#432—21no—
63#432—21no—
64#432—21no—
65#432D412yes—
66#431—21no—
67#431—21no—
68#431—21no—
69#431—21no—
70#431—21no—
71#431—21no—
72#431—21no—
73#431—21no—
74#431D412yes—
75#431D412yes—
76#431C2312yes—
77#431C2312yes—
78#423C2×C411yescharacteristic
79#423C2×C411yescharacteristic
80#423D412yes—
81C2×C4⋊C8—21no—
82C2×C4⋊C8—21no—
83C2×C4⋊C8C2312yes—
84C2×#101C2×C414yes—
85C2×#101C2×C414yes—
86#407—21no—
87#407C2×C412yes—
88C2×C22⋊C8—21no—
89C2×C22⋊C8—21no—
90C2×C22⋊C8—21no—
91C4×M16—21no—
92C4×M16—21no—
93C4×M16—21no—
94C4×M16C2×C411yescharacteristic
95C4×M16C2×C411yescharacteristic
96C4×M16C2312yes—
97C4×M16C2312yes—
98C2×#97—21no—
99C2×#97—21no—
100C2×C4×C8—21no—
101C2×C4×C8—21no—
102C2×C4×C8C2311yescharacteristic
103C2×C4×C8C2311yescharacteristic
104#343C2×C414yes—
105#343C2×C414yes—
106#335—24no—
107#334—24no—
108Q8⋊C8—28no—
109Q8⋊C8—24no—
110Q8⋊C8—24no—
111#322—22no—
112#322—24no—
113C82—22no—
114C22×Q8—21no—
115C22×Q8C22×C412yes—
116C23×C4C22×C411yescharacteristic
117C23×C4#4111yescharacteristic
118C23×C4#4111yescharacteristic
119C2×M16—22no—
120C2×M16—22no—
121C2×M16—24no—
122C2×M16—42no—
123C2×M16C22×C412yes—
124C2×M16C22×C412yes—
125C2×M16C22×C412yes—
126C2×M16C22×C412yes—
127C22×C8—22no—
128C22×C8—21no—
129C22×C8—21no—
130C22×C8—22no—
131C22×C8—41no—
132C4⋊Q8—21no—
133C4⋊Q8—21no—
134C4⋊Q8—21no—
135C4⋊Q8—21no—
136C4⋊Q8—21no—
137C4⋊Q8—21no—
138C4⋊Q8—21no—
139C4⋊Q8—21no—
140#125—41no—
141#125—41no—
142#125—41no—
143#125—41no—
144C22⋊Q8—22no—
145C22⋊Q8—22no—
146C22⋊Q8—22no—
147C22⋊Q8—22no—
148C22⋊Q8—41no—
149C22⋊Q8—41no—
150C22⋊Q8—41no—
151C22⋊Q8—41no—
152C4×Q8—22no—
153C4×Q8—22no—
154C4×Q8—22no—
155C4×Q8—22no—
156C4×Q8—41no—
157C4×Q8—41no—
158C4×Q8—41no—
159C4×Q8—41no—
160#117—41no—
161#117—41no—
162#117—41no—
163#117—41no—
164C2×C4⋊C4—21no—
165C2×C4⋊C4—21no—
166C2×C4⋊C4—21no—
167C2×C4⋊C4—22no—
168C2×C4⋊C4—22no—
169C2×C4⋊C4C22×C412yes—
170C2×C22⋊C4—22no—
171C2×C42—22no—
172C2×C42C22×C411yescharacteristic
173C2×C42C2×D411yescharacteristic
174C2×C42C2×D411yescharacteristic
175C2×C42C2×D411yescharacteristic
176C2×C42C2×D411yescharacteristic
177C2×C42#4111yescharacteristic
178C2×C42#4111yescharacteristic
179C2×C42#4111yescharacteristic
180C2×C42#4111yescharacteristic
181C2×C42#4111yescharacteristic
182C2×C42#4111yescharacteristic
183C2×C42C2411yescharacteristic
184C4⋊C8—21no—
185C4⋊C8—21no—
186C4⋊C8—21no—
187C4⋊C8—21no—
188C4⋊C8—22no—
189C4⋊C8—22no—
190C4⋊C8—22no—
191C4⋊C8—22no—
192C4⋊C8—22no—
193C4⋊C8—22no—
194C4⋊C8—41no—
195C4⋊C8C2×D414yes—
196#101—28no—
197#101—44no—
198C22⋊C8—22no—
199C22⋊C8—22no—
200C22⋊C8—22no—
201C22⋊C8—22no—
202C22⋊C8—22no—
203C22⋊C8—22no—
204C22⋊C8—22no—
205C22⋊C8—41no—
206C22⋊C8—41no—
207C22⋊C8—41no—
208#97—21no—
209#97—21no—
210#97—22no—
211#97—22no—
212#97—22no—
213#97—41no—
214#97—41no—
215C4×C8—21no—
216C4×C8—21no—
217C4×C8—22no—
218C4×C8—22no—
219C4×C8—22no—
220C4×C8—22no—
221C4×C8—21no—
222C4×C8—21no—
223C4×C8—41no—
224C4×C8C2×D412yes—
225C4×C8C2×D412yes—
226C24#11711yescharacteristic
227C2×Q8—22no—
228C2×Q8—22no—
229C2×Q8—24no—
230C2×Q8—24no—
231C2×Q8—41no—
232C2×Q8—41no—
233C2×Q8—82no—
234C22×C4—22no—
235C22×C4—22no—
236C22×C4—22no—
237C22×C4—22no—
238C22×C4—42no—
239C22×C4—81no—
240C22×C4#11711yescharacteristic
241C22×C4#11711yescharacteristic
242C22×C4#11711yescharacteristic
243C22×C4C4×D411yescharacteristic
244C22×C4C4×D411yescharacteristic
245C22×C4C4×D411yescharacteristic
246C22×C4C4×D411yescharacteristic
247C22×C4#12111yescharacteristic
248C22×C4#12111yescharacteristic
249C22×C4#12311yescharacteristic
250C22×C4#12311yescharacteristic
251C22×C4#12411yescharacteristic
252C22×C4#12411yescharacteristic
253C22×C4#12611yescharacteristic
254C22×C4#12611yescharacteristic
255C22×C4C23×C411yescharacteristic
256C22×C4C2×#4111yescharacteristic
257C22×C4C2×#4111yescharacteristic
258M16—22no—
259M16—24no—
260M16—22no—
261M16—22no—
262M16—24no—
263M16—44no—
264M16—82no—
265M16#13114yes—
266C2×C8—22no—
267C2×C8—22no—
268C2×C8—22no—
269C2×C8—22no—
270C2×C8—24no—
271C2×C8—22no—
272C2×C8—22no—
273C2×C8—44no—
274C2×C8—42no—
275C2×C8—42no—
276C2×C8—81no—
277C2×C8—81no—
278C2×C8—82no—
279C2×C8—82no—
280C4⋊C4—22no—
281C4⋊C4—22no—
282C4⋊C4—42no—
283C4⋊C4—42no—
284C4⋊C4—41no—
285C4⋊C4—41no—
286C4⋊C4—41no—
287C4⋊C4—41no—
288C4⋊C4—42no—
289C4⋊C4—42no—
290C4⋊C4—41no—
291C4⋊C4—41no—
292C22⋊C4—42no—
293C22⋊C4—42no—
294C42—42no—
295C42—42no—
296C42C4×D411yescharacteristic
297C42C4×D411yescharacteristic
298C42C4×D411yescharacteristic
299C42C4×D411yescharacteristic
300C42#12111yescharacteristic
301C42#12111yescharacteristic
302C42#12311yescharacteristic
303C42#12311yescharacteristic
304C42#12411yescharacteristic
305C42#12411yescharacteristic
306C42#12611yescharacteristic
307C42#12611yescharacteristic
308C42C22×D411yescharacteristic
309C42C22×D411yescharacteristic
310C42C2×#4111yescharacteristic
311C42C2×#4111yescharacteristic
312C23—22no—
313C23—41no—
314C23—41no—
315C23#38811yescharacteristic
316C23#38811yescharacteristic
317C23C2×#11711yescharacteristic
318Q8—42no—
319Q8—42no—
320Q8—44no—
321Q8—82no—
322Q8—82no—
323C2×C4—22no—
324C2×C4—21no—
325C2×C4—21no—
326C2×C4—21no—
327C2×C4—21no—
328C2×C4—22no—
329C2×C4—22no—
330C2×C4—22no—
331C2×C4—22no—
332C2×C4—22no—
333C2×C4—41no—
334C2×C4—41no—
335C2×C4—41no—
336C2×C4—41no—
337C2×C4—41no—
338C2×C4—41no—
339C2×C4—41no—
340C2×C4—41no—
341C2×C4—41no—
342C2×C4—44no—
343C2×C4—42no—
344C2×C4—42no—
345C2×C4—82no—
346C2×C4#38811yescharacteristic
347C2×C4#38811yescharacteristic
348C2×C4#38811yescharacteristic
349C2×C4#38811yescharacteristic
350C2×C4#38811yescharacteristic
351C2×C4#38811yescharacteristic
352C2×C4C2×C4×D411yescharacteristic
353C2×C4C2×C4×D411yescharacteristic
354C2×C4C2×#12111yescharacteristic
355C2×C4C2×#12311yescharacteristic
356C2×C4C2×#12411yescharacteristic
357C2×C4C2×#12611yescharacteristic
358C2×C4C2×#13112yes—
359C2×C4#56812yes—
360C8—24no—
361C8—42no—
362C8—42no—
363C8—42no—
364C8—42no—
365C8—42no—
366C8—44no—
367C8—82no—
368C8—82no—
369C22—21no—
370C22—21no—
371C22—21no—
372C22—21no—
373C22—21no—
374C22—21no—
375C22—21no—
376C22—21no—
377C22—82no—
378C22C2×#38811yescharacteristic
379C22#292512yes—
380C4—21no—
381C4—21no—
382C4—21no—
383C4—21no—
384C4—21no—
385C4—21no—
386C4—21no—
387C4—21no—
388C4—21no—
389C4—82no—
390C4—82no—
391C4#293111yescharacteristic
392C4#293211yescharacteristic
393C4#293312yes—
394C4#306912yes—
395C2—22no—
396C2—41no—
397C2—41no—
398C2#2105111yescharacteristic
399C2#2107711yescharacteristic
400C2#2108011yescharacteristic
subgroupG/H
centerC2×C4C2×#121
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
socleC22C2×#388
prime radicalC24#117
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C2×C4 ◁ C22×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
6. #9551C2semidirect
7. #9549C2semidirect

Maximal subgroups

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

subgroupindex
#214252
#177972
#174792
C2×#14972
#173242
#95512
#95492
#79332
#78822