Finite Group Databasebeta

← #203239

Subgroups — #203239

Order 512.

611 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#22013C211yesmaximal, characteristic
2#21120C211yesmaximal, characteristic
3#20569C211yesmaximal, characteristic, complemented
4#9542C211yesmaximal, characteristic
5#9541C211yesmaximal, characteristic
6#9536C211yesmaximal, characteristic, complemented
7#9533C211yesmaximal, characteristic, complemented
8#8948C211yesmaximal, characteristic
9#8948C211yesmaximal, characteristic
10#8846C211yesmaximal, characteristic, complemented
11#8844C211yesmaximal, characteristic, complemented
12#8279C211yesmaximal, characteristic, complemented
13#8183C211yesmaximal, characteristic
14#8175C211yesmaximal, characteristic, complemented
15#8135C211yesmaximal, characteristic, complemented
16C2×C8×Q8C2211yescharacteristic
17C2×C8×Q8C2211yescharacteristic
18C2×C4×SD16C2211yescharacteristic
19#2293C2211yescharacteristic
20C42×Q8C2211yescharacteristic
21#1954C2211yescharacteristic
22#1954C2211yescharacteristic
23#1939C2211yescharacteristic
24#1939C2211yescharacteristic
25#1925C412yes—
26#1924—21no—
27#1924—21no—
28#1920C412yes—
29#1880C412yes—
30#1863C412yes—
31#1818—21no—
32#1798—21no—
33C4×#107C2211yescharacteristic
34C8×C4⋊C4—21no—
35C8×C4⋊C4—21no—
36C8×C4⋊C4C2211yescharacteristic
37C8×C4⋊C4C2211yescharacteristic
38#1767C2211yescharacteristic
39C4×C4⋊C8—21no—
40C4×C4⋊C8C2211yescharacteristic
41C4×#103C2211yescharacteristic
42C4×#102C2211yescharacteristic
43#1744—21no—
44#1744C2211yescharacteristic
45#1744C2211yescharacteristic
46#1744C2211yescharacteristic
47C2×#335C2211yescharacteristic
48C2×#334C2211yescharacteristic
49C2×Q8⋊C8C2211yescharacteristic
50C2×Q8⋊C8C2211yescharacteristic
51C2×D4⋊C8C2211yescharacteristic
52C2×D4⋊C8C2211yescharacteristic
53#1279C2211yescharacteristic
54#1279C2211yescharacteristic
55#1279C2211yescharacteristic
56#1279C2211yescharacteristic
57#1277C2211yescharacteristic
58#1277C2211yescharacteristic
59#1273C2211yescharacteristic
60#1273C2211yescharacteristic
61#1273C2211yescharacteristic
62#1273C2211yescharacteristic
63C22×SD16C2×C411yescharacteristic
64C2×C4×Q8C2×C412yes—
65C2×C4×Q8C2311yescharacteristic
66C2×C4×D4C2311yescharacteristic
67C2×#117—21no—
68C8×Q8—22no—
69C8×Q8—22no—
70C8×Q8—22no—
71C8×Q8—22no—
72C4×SD16—22no—
73C4×SD16—22no—
74C2×#107C2×C412yes—
75C2×#106C2×C411yescharacteristic
76C2×C4⋊C8—21no—
77C2×C4⋊C8—21no—
78C2×C4⋊C8—21no—
79C2×C4⋊C8—21no—
80C2×C4⋊C8C2311yescharacteristic
81C2×C4⋊C8C2311yescharacteristic
82C2×C4⋊C8C2311yescharacteristic
83C2×C4⋊C8C2311yescharacteristic
84C2×#103C2×C411yescharacteristic
85C2×#103C2×C412yes—
86C2×#102C2×C412yes—
87C2×#102C2×C411yescharacteristic
88C2×C22⋊C8—21no—
89C2×C22⋊C8—21no—
90C2×C22⋊C8—21no—
91C2×C22⋊C8—21no—
92C2×#97—21no—
93C2×C4×C8—21no—
94C2×C4×C8—21no—
95C2×C4×C8—21no—
96C2×C4×C8—21no—
97C2×C4×C8—21no—
98C2×C4×C8C2311yescharacteristic
99C2×C4×C8C2311yescharacteristic
100C2×C4×C8C2311yescharacteristic
101C2×C4×C8C2311yescharacteristic
102C2×C4×C8C2311yescharacteristic
103C2×C4×C8C2311yescharacteristic
104#390C2×C412yes—
105#389C2×C412yes—
106#388—22no—
107C4×C4⋊C4—21no—
108C4×C4⋊C4—22no—
109C4×C4⋊C4C2×C412yes—
110C4×C4⋊C4C2311yescharacteristic
111C4×C4⋊C4C2311yescharacteristic
112C4×C22⋊C4—21no—
113C43—21no—
114C43C2311yescharacteristic
115#340C2×C412yes—
116#340C2×C412yes—
117#340C2×C412yes—
118#336—21no—
119#336—21no—
120#336—21no—
121#336—21no—
122#336—21no—
123#336—21no—
124#336—22no—
125#336C2×C412yes—
126#336C2×C412yes—
127#336D411yescharacteristic
128#336D411yescharacteristic
129#336D411yescharacteristic
130#336D411yescharacteristic
131#335—22no—
132#334—22no—
133Q8⋊C8—22no—
134Q8⋊C8—22no—
135D4⋊C8—22no—
136D4⋊C8—22no—
137C22×Q8C22×C411yescharacteristic
138C22×D4C22×C411yescharacteristic
139C23×C4—21no—
140C2×SD16—22no—
141C2×SD16—24no—
142C22×C8—21no—
143C22×C8—21no—
144C22×C8—21no—
145C22×C8—21no—
146C22×C8—21no—
147C22×C8—21no—
148C22×C8C22×C411yescharacteristic
149C22×C8C22×C412yes—
150C22×C8C22×C411yescharacteristic
151C22×C8C2×D411yescharacteristic
152C22×C8C2×D411yescharacteristic
153C22×C8C2×D411yescharacteristic
154C22×C8C2×D411yescharacteristic
155C4×Q8—24no—
156C4×Q8—24no—
157C4×Q8—21no—
158C4×Q8—21no—
159C4×Q8#4112yes—
160C4×Q8#4112yes—
161C4×D4—21no—
162C4×D4—21no—
163C4×D4—21no—
164C4×D4—21no—
165#117—81no—
166#117—81no—
167C2×C4⋊C4—22no—
168C2×C4⋊C4—21no—
169C2×C4⋊C4—22no—
170C2×C4⋊C4C22×C411yescharacteristic
171C2×C4⋊C4C22×C412yes—
172C2×C4⋊C4C22×C411yescharacteristic
173C2×C4⋊C4C22×C412yes—
174C2×C22⋊C4—21no—
175C2×C22⋊C4—22no—
176C2×C22⋊C4—22no—
177C2×C42—22no—
178C2×C42—21no—
179C2×C42—21no—
180C2×C42—21no—
181C2×C42—22no—
182C2×C42C22×C412yes—
183C2×C42C2×D411yescharacteristic
184C2×C42C2×D411yescharacteristic
185C2×C42#4112yes—
186C2×C42C2411yescharacteristic
187#107—24no—
188#106—22no—
189C4⋊C8—24no—
190C4⋊C8—24no—
191C4⋊C8—21no—
192C4⋊C8—21no—
193C4⋊C8—21no—
194C4⋊C8—21no—
195C4⋊C8—21no—
196C4⋊C8—21no—
197C4⋊C8—81no—
198C4⋊C8—81no—
199C4⋊C8#4112yes—
200C4⋊C8#4112yes—
201#103—22no—
202#103—24no—
203#102—22no—
204#102—24no—
205C22⋊C8—81no—
206C22⋊C8—81no—
207C22⋊C8—81no—
208C22⋊C8—81no—
209#97—81no—
210C4×C8—24no—
211C4×C8—24no—
212C4×C8—21no—
213C4×C8—21no—
214C4×C8—21no—
215C4×C8—21no—
216C4×C8—21no—
217C4×C8—21no—
218C4×C8—21no—
219C4×C8—21no—
220C4×C8—21no—
221C4×C8—21no—
222C4×C8—21no—
223C4×C8—21no—
224C4×C8—81no—
225C4×C8—81no—
226C4×C8—81no—
227#95—22no—
228#95—22no—
229#95—21no—
230#95—21no—
231C24—21no—
232C2×Q8—21no—
233C2×Q8—22no—
234C2×Q8#11712yes—
235C2×Q8#13114yes—
236C2×D4—21no—
237C2×D4—21no—
238C2×D4—22no—
239C2×D4—22no—
240C22×C4—21no—
241C22×C4—22no—
242C22×C4—21no—
243C22×C4—22no—
244C22×C4—21no—
245C22×C4—41no—
246C22×C4—41no—
247C22×C4—41no—
248C22×C4—41no—
249C22×C4—41no—
250C22×C4—41no—
251C22×C4C4×D412yes—
252C22×C4C4×D412yes—
253C22×C4C23×C411yescharacteristic
254C22×C4C22×D411yescharacteristic
255C22×C4C2×#4111yescharacteristic
256SD16—44no—
257C2×C8—21no—
258C2×C8—21no—
259C2×C8—22no—
260C2×C8—21no—
261C2×C8—21no—
262C2×C8—21no—
263C2×C8—21no—
264C2×C8—22no—
265C2×C8—22no—
266C2×C8—22no—
267C2×C8—22no—
268C2×C8—22no—
269C2×C8—22no—
270C2×C8—21no—
271C2×C8—21no—
272C2×C8—21no—
273C2×C8—21no—
274C2×C8—21no—
275C2×C8—21no—
276C2×C8—21no—
277C2×C8—21no—
278C2×C8—41no—
279C2×C8—41no—
280C2×C8—41no—
281C2×C8—41no—
282C2×C8—42no—
283C2×C8—42no—
284C2×C8—41no—
285C2×C8—41no—
286C2×C8—41no—
287C2×C8—41no—
288C2×C8—41no—
289C2×C8—41no—
290C2×C8—41no—
291C2×C8—41no—
292C4⋊C4—28no—
293C4⋊C4—21no—
294C4⋊C4—22no—
295C4⋊C4—22no—
296C4⋊C4—22no—
297C4⋊C4—21no—
298C4⋊C4—21no—
299C4⋊C4—22no—
300C4⋊C4—22no—
301C4⋊C4—82no—
302C4⋊C4#11712yes—
303C4⋊C4#13114yes—
304C22⋊C4—41no—
305C22⋊C4—41no—
306C22⋊C4—82no—
307C22⋊C4—82no—
308C42—28no—
309C42—22no—
310C42—22no—
311C42—22no—
312C42—22no—
313C42—21no—
314C42—21no—
315C42—21no—
316C42—21no—
317C42—42no—
318C42—41no—
319C42—41no—
320C42—41no—
321C42—41no—
322C42—82no—
323C42C2×#4111yescharacteristic
324C42C2×#4111yescharacteristic
325C42C2×#4111yescharacteristic
326C42C2×#4111yescharacteristic
327C23—42no—
328C23—41no—
329C23—41no—
330C23—42no—
331C23—41no—
332C23C2×C4×D411yescharacteristic
333Q8—22no—
334Q8—24no—
335Q8#43314yes—
336D4—41no—
337D4—41no—
338D4—42no—
339C2×C4—24no—
340C2×C4—22no—
341C2×C4—22no—
342C2×C4—22no—
343C2×C4—22no—
344C2×C4—24no—
345C2×C4—22no—
346C2×C4—22no—
347C2×C4—22no—
348C2×C4—22no—
349C2×C4—22no—
350C2×C4—42no—
351C2×C4—42no—
352C2×C4—42no—
353C2×C4—42no—
354C2×C4—42no—
355C2×C4—41no—
356C2×C4—41no—
357C2×C4—42no—
358C2×C4—41no—
359C2×C4—81no—
360C2×C4—81no—
361C2×C4—81no—
362C2×C4—81no—
363C2×C4C2×#11711yescharacteristic
364C2×C4C4×#4111yescharacteristic
365C2×C4#52511yescharacteristic
366C2×C4#52511yescharacteristic
367C2×C4#52911yescharacteristic
368C2×C4#52911yescharacteristic
369C2×C4C2×#13112yes—
370C2×C4C2×#13112yes—
371C2×C4#57611yescharacteristic
372C2×C4#57711yescharacteristic
373C2×C4#57711yescharacteristic
374C2×C4#57811yescharacteristic
375C8—41no—
376C8—42no—
377C8—41no—
378C8—41no—
379C8—42no—
380C8—41no—
381C8—41no—
382C8—41no—
383C8—42no—
384C8—81no—
385C8—81no—
386C8—81no—
387C8—81no—
388C22—82no—
389C22—81no—
390C22—81no—
391C22—81no—
392C22—82no—
393C22#229011yescharacteristic
394C22#292912yes—
395C22#294611yescharacteristic
396C22#294711yescharacteristic
397C22#295712yes—
398C4—24no—
399C4—21no—
400C4—21no—
401C4—42no—
402C4—44no—
403C4—42no—
404C4—42no—
405C4—82no—
406C4—81no—
407C4C2×#43311yescharacteristic
408C4#292211yescharacteristic
409C4#296612yes—
410C4#311911yescharacteristic
411C4#312111yescharacteristic
412C4#312411yescharacteristic
413C4#312611yescharacteristic
414C2—161no—
415C2#1791912yes—
416C2#1792012yes—
417C2#2110311yescharacteristic
418C2#2112811yescharacteristic
419C2#2114011yescharacteristic
subgroupG/H
centerC22×C4C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
socleC23C2×C4×D4
prime radicalC22×D4C22×C4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C4 ◁ #1744 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #20569C2semidirect
6. #9536C2semidirect
7. #9533C2semidirect
10. #8846C2semidirect
11. #8844C2semidirect
12. #8279C2semidirect
14. #8175C2semidirect
15. #8135C2semidirect

Maximal subgroups

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

subgroupindex
#220132
#211202
#205692
#95422
#95412
#95362
#95332
#89482
#88462
#88442
#82792
#81832
#81752
#81352