Finite Group Databasebeta

← #203237

Subgroups — #203237

Order 512.

669 conjugacy classes of subgroups.

The subgroup lattice has 449 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22013C211yesmaximal, characteristic, complemented
2#21118C211yesmaximal, characteristic
3#20555C211yesmaximal, characteristic, complemented
4#9542C211yesmaximal, characteristic, complemented
5#9541C211yesmaximal, characteristic, complemented
6#9535C211yesmaximal, characteristic, complemented
7#9534C211yesmaximal, characteristic, complemented
8#8947C211yesmaximal, characteristic
9#8947C211yesmaximal, characteristic
10#8845C211yesmaximal, characteristic, complemented
11#8843C211yesmaximal, characteristic, complemented
12#8279C211yesmaximal, characteristic, complemented
13#8183C211yesmaximal, characteristic, complemented
14#8174C211yesmaximal, characteristic, complemented
15#8135C211yesmaximal, characteristic, complemented
16C2×C4×D8C2211yescharacteristic
17C2×C8×D4C2211yescharacteristic
18C2×C8×D4C2211yescharacteristic
19#2293C2211yescharacteristic
20C42×D4C2211yescharacteristic
21#1954C2211yescharacteristic
22#1954C2211yescharacteristic
23#1939C2211yescharacteristic
24#1939C2211yescharacteristic
25#1926C412yes—
26#1924—21no—
27#1924—21no—
28#1921C412yes—
29#1879C412yes—
30#1864C412yes—
31#1818—21no—
32#1798—21no—
33C4×#106C2211yescharacteristic
34C8×C4⋊C4C2211yescharacteristic
35C8×C4⋊C4C2211yescharacteristic
36#1767C2211yescharacteristic
37C4×C4⋊C8C2211yescharacteristic
38C4×#102C2211yescharacteristic
39C4×#102C2211yescharacteristic
40C8×C22⋊C4—21no—
41C8×C22⋊C4—21no—
42C4×C22⋊C8—21no—
43#1744C2211yescharacteristic
44#1744C2211yescharacteristic
45#1744C2211yescharacteristic
46C2×#336—21no—
47C2×#335C2211yescharacteristic
48C2×#334C2211yescharacteristic
49C2×D4⋊C8C2211yescharacteristic
50C2×D4⋊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×D8C2×C411yescharacteristic
64C23×C8—21no—
65C23×C8—21no—
66C2×C4×D4C2×C412yes—
67C2×C4×D4C2311yescharacteristic
68C2×C4×D4C2311yescharacteristic
69C2×#117—21no—
70C22×C42—21no—
71C4×D8—22no—
72C4×D8—22no—
73C8×D4—22no—
74C8×D4—22no—
75C8×D4—22no—
76C8×D4—22no—
77C2×#107C2×C411yescharacteristic
78C2×#106C2×C412yes—
79C2×C4⋊C8—21no—
80C2×C4⋊C8—21no—
81C2×C4⋊C8C2311yescharacteristic
82C2×C4⋊C8C2311yescharacteristic
83C2×C4⋊C8C2311yescharacteristic
84C2×C4⋊C8C2311yescharacteristic
85C2×#102C2×C411yescharacteristic
86C2×#102C2×C412yes—
87C2×#102C2×C412yes—
88C2×#102C2×C411yescharacteristic
89C2×C22⋊C8—21no—
90C2×C22⋊C8—21no—
91C2×C22⋊C8—21no—
92C2×C22⋊C8—21no—
93C2×C22⋊C8—21no—
94C2×C22⋊C8—21no—
95C2×#97—21no—
96C2×C4×C8—21no—
97C2×C4×C8—21no—
98C2×C4×C8—21no—
99C2×C4×C8C2311yescharacteristic
100C2×C4×C8C2311yescharacteristic
101C2×C4×C8C2311yescharacteristic
102C2×C4×C8C2311yescharacteristic
103C2×C4×C8C2311yescharacteristic
104C2×C4×C8C2311yescharacteristic
105#390C2×C412yes—
106#389C2×C412yes—
107#388—22no—
108C4×C4⋊C4C2×C412yes—
109C4×C4⋊C4C2311yescharacteristic
110C4×C4⋊C4C2311yescharacteristic
111C4×C22⋊C4—21no—
112C4×C22⋊C4—21no—
113C4×C22⋊C4—22no—
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—
133D4⋊C8—22no—
134D4⋊C8—22no—
135D4⋊C8—22no—
136D4⋊C8—22no—
137C22×D4C22×C411yescharacteristic
138C22×D4C22×C411yescharacteristic
139C23×C4—21no—
140C23×C4—22no—
141C23×C4—21no—
142C2×D8—22no—
143C2×D8—24no—
144C22×C8—21no—
145C22×C8—21no—
146C22×C8—24no—
147C22×C8—24no—
148C22×C8—21no—
149C22×C8—21no—
150C22×C8—21no—
151C22×C8—21no—
152C22×C8—81no—
153C22×C8—81no—
154C22×C8—81no—
155C22×C8—81no—
156C22×C8C22×C411yescharacteristic
157C22×C8C22×C412yes—
158C22×C8C22×C411yescharacteristic
159C22×C8C2×D411yescharacteristic
160C22×C8C2×D411yescharacteristic
161C22×C8C2×D411yescharacteristic
162C22×C8C2×D411yescharacteristic
163C4×D4—24no—
164C4×D4—24no—
165C4×D4—21no—
166C4×D4—21no—
167C4×D4—21no—
168C4×D4—21no—
169C4×D4—21no—
170C4×D4—21no—
171C4×D4#4112yes—
172C4×D4#4112yes—
173#117—81no—
174#117—81no—
175C2×C4⋊C4—22no—
176C2×C4⋊C4C22×C411yescharacteristic
177C2×C4⋊C4C22×C412yes—
178C2×C4⋊C4C22×C411yescharacteristic
179C2×C4⋊C4C22×C412yes—
180C2×C22⋊C4—22no—
181C2×C22⋊C4—21no—
182C2×C22⋊C4—21no—
183C2×C22⋊C4—22no—
184C2×C22⋊C4—22no—
185C2×C42—21no—
186C2×C42—21no—
187C2×C42—22no—
188C2×C42—81no—
189C2×C42—81no—
190C2×C42C22×C412yes—
191C2×C42C2×D411yescharacteristic
192C2×C42C2×D411yescharacteristic
193C2×C42#4112yes—
194C2×C42C2411yescharacteristic
195#107—22no—
196#106—24no—
197C4⋊C8—21no—
198C4⋊C8—21no—
199C4⋊C8—21no—
200C4⋊C8—21no—
201C4⋊C8—21no—
202C4⋊C8—21no—
203C4⋊C8—81no—
204C4⋊C8—81no—
205C4⋊C8#4112yes—
206C4⋊C8#4112yes—
207#102—22no—
208#102—24no—
209#102—22no—
210#102—24no—
211C22⋊C8—24no—
212C22⋊C8—24no—
213C22⋊C8—81no—
214C22⋊C8—81no—
215C22⋊C8—81no—
216C22⋊C8—81no—
217#97—81no—
218C4×C8—21no—
219C4×C8—21no—
220C4×C8—21no—
221C4×C8—21no—
222C4×C8—21no—
223C4×C8—21no—
224C4×C8—21no—
225C4×C8—21no—
226C4×C8—21no—
227C4×C8—21no—
228C4×C8—21no—
229C4×C8—21no—
230C4×C8—81no—
231C4×C8—81no—
232C4×C8—81no—
233#95—22no—
234#95—22no—
235#95—21no—
236#95—21no—
237C24—21no—
238C24—21no—
239C2×D4—21no—
240C2×D4—21no—
241C2×D4—21no—
242C2×D4—22no—
243C2×D4—22no—
244C2×D4—22no—
245C2×D4#11712yes—
246C2×D4#13114yes—
247C22×C4—22no—
248C22×C4—21no—
249C22×C4—28no—
250C22×C4—22no—
251C22×C4—22no—
252C22×C4—22no—
253C22×C4—21no—
254C22×C4—41no—
255C22×C4—41no—
256C22×C4—41no—
257C22×C4—41no—
258C22×C4—41no—
259C22×C4—41no—
260C22×C4—41no—
261C22×C4—41no—
262C22×C4—41no—
263C22×C4—41no—
264C22×C4—82no—
265C22×C4—82no—
266C22×C4C4×D412yes—
267C22×C4C4×D412yes—
268C22×C4C23×C411yescharacteristic
269C22×C4C22×D411yescharacteristic
270C22×C4C2×#4111yescharacteristic
271D8—44no—
272C2×C8—21no—
273C2×C8—21no—
274C2×C8—22no—
275C2×C8—21no—
276C2×C8—21no—
277C2×C8—21no—
278C2×C8—21no—
279C2×C8—22no—
280C2×C8—22no—
281C2×C8—22no—
282C2×C8—22no—
283C2×C8—22no—
284C2×C8—22no—
285C2×C8—21no—
286C2×C8—21no—
287C2×C8—21no—
288C2×C8—21no—
289C2×C8—21no—
290C2×C8—21no—
291C2×C8—21no—
292C2×C8—21no—
293C2×C8—41no—
294C2×C8—41no—
295C2×C8—41no—
296C2×C8—41no—
297C2×C8—42no—
298C2×C8—42no—
299C2×C8—41no—
300C2×C8—41no—
301C2×C8—41no—
302C2×C8—41no—
303C2×C8—41no—
304C2×C8—41no—
305C2×C8—41no—
306C2×C8—41no—
307C2×C8—84no—
308C2×C8—84no—
309C4⋊C4—21no—
310C4⋊C4—22no—
311C4⋊C4—21no—
312C4⋊C4—21no—
313C4⋊C4—22no—
314C4⋊C4—22no—
315C4⋊C4—82no—
316C4⋊C4#11712yes—
317C4⋊C4#13114yes—
318C22⋊C4—28no—
319C22⋊C4—22no—
320C22⋊C4—22no—
321C22⋊C4—41no—
322C22⋊C4—41no—
323C22⋊C4—82no—
324C22⋊C4—82no—
325C42—22no—
326C42—22no—
327C42—21no—
328C42—21no—
329C42—21no—
330C42—21no—
331C42—42no—
332C42—41no—
333C42—41no—
334C42—41no—
335C42—41no—
336C42—82no—
337C42C2×#4111yescharacteristic
338C42C2×#4111yescharacteristic
339C42C2×#4111yescharacteristic
340C42C2×#4111yescharacteristic
341C23—24no—
342C23—22no—
343C23—41no—
344C23—41no—
345C23—42no—
346C23—42no—
347C23—41no—
348C23—41no—
349C23—42no—
350C23—41no—
351C23C2×C4×D411yescharacteristic
352D4—22no—
353D4—24no—
354D4—41no—
355D4—41no—
356D4—42no—
357D4#43314yes—
358C2×C4—22no—
359C2×C4—22no—
360C2×C4—22no—
361C2×C4—24no—
362C2×C4—22no—
363C2×C4—22no—
364C2×C4—22no—
365C2×C4—22no—
366C2×C4—22no—
367C2×C4—42no—
368C2×C4—41no—
369C2×C4—41no—
370C2×C4—42no—
371C2×C4—42no—
372C2×C4—42no—
373C2×C4—42no—
374C2×C4—42no—
375C2×C4—42no—
376C2×C4—42no—
377C2×C4—41no—
378C2×C4—41no—
379C2×C4—42no—
380C2×C4—41no—
381C2×C4—88no—
382C2×C4—81no—
383C2×C4—81no—
384C2×C4—81no—
385C2×C4—81no—
386C2×C4C2×#11711yescharacteristic
387C2×C4C4×#4111yescharacteristic
388C2×C4#52511yescharacteristic
389C2×C4#52511yescharacteristic
390C2×C4#52911yescharacteristic
391C2×C4#52911yescharacteristic
392C2×C4C2×#13112yes—
393C2×C4C2×#13112yes—
394C2×C4#57611yescharacteristic
395C2×C4#57611yescharacteristic
396C2×C4#57711yescharacteristic
397C2×C4#57711yescharacteristic
398C8—41no—
399C8—42no—
400C8—41no—
401C8—41no—
402C8—42no—
403C8—41no—
404C8—41no—
405C8—41no—
406C8—42no—
407C8—81no—
408C8—81no—
409C8—81no—
410C8—81no—
411C22—24no—
412C22—44no—
413C22—42no—
414C22—42no—
415C22—44no—
416C22—82no—
417C22—81no—
418C22—81no—
419C22—81no—
420C22—82no—
421C22#229011yescharacteristic
422C22#292912yes—
423C22#294611yescharacteristic
424C22#294611yescharacteristic
425C22#295712yes—
426C4—21no—
427C4—21no—
428C4—42no—
429C4—42no—
430C4—42no—
431C4—44no—
432C4—82no—
433C4—81no—
434C4C2×#43311yescharacteristic
435C4#292211yescharacteristic
436C4#296612yes—
437C4#311811yescharacteristic
438C4#311911yescharacteristic
439C4#312211yescharacteristic
440C4#312511yescharacteristic
441C2—44no—
442C2—161no—
443C2#1791812yes—
444C2#1792012yes—
445C2#2110311yescharacteristic
446C2#2112611yescharacteristic
447C2#2114011yescharacteristic
subgroupG/H
centerC22×C4C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
socleC23C2×C4×D4
prime radicalC22×D8C2×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
1. #22013C2semidirect
3. #20555C2semidirect
4. #9542C2semidirect
5. #9541C2semidirect
6. #9535C2semidirect
7. #9534C2semidirect
10. #8845C2semidirect
11. #8843C2semidirect
12. #8279C2semidirect
13. #8183C2semidirect
14. #8174C2semidirect
15. #8135C2semidirect

Maximal subgroups

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

subgroupindex
#220132
#211182
#205552
#95422
#95412
#95352
#95342
#89472
#88452
#88432
#82792
#81832
#81742
#81352