Finite Group Databasebeta

← #220108

Subgroups — #220108

Order 512.

987 conjugacy classes of subgroups.

The subgroup lattice has 484 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36134C211yesmaximal, characteristic
2#22799C211yesmaximal, characteristic
3#22650C211yesmaximal, characteristic
4C2×#1559C212yesmaximal, complemented
5#9691C211yesmaximal, characteristic, complemented
6#9691C211yesmaximal, characteristic, complemented
7#9136C212yesmaximal, complemented
8#9135C212yesmaximal, complemented
9#8586C211yesmaximal, characteristic, complemented
10#8343C212yesmaximal, complemented
11C4×#332C211yesmaximal, characteristic, complemented
12C2×#533C2212yes—
13C2×#436—21no—
14C2×#432C2212yes—
15C2×#423C2212yes—
16#2868C2211yescharacteristic
17#2857C2212yes—
18#2552—21no—
19#2542C412yes—
20#2538—21no—
21#2441—22no—
22#2429C414yes—
23#2423—22no—
24#2365C412yes—
25#2354—21no—
26Q8×C4⋊C4—21no—
27C4×C4⋊Q8C2211yescharacteristic
28C42×D4C2211yescharacteristic
29#1954—21no—
30#1939—21no—
31#1924—22no—
32#1851—22no—
33#1842C2211yescharacteristic
34#1842C2211yescharacteristic
35#1797C2212yes—
36#1797C2212yes—
37#1770—21no—
38C4×C4⋊C8C2211yescharacteristic
39C4×C4⋊C8C2211yescharacteristic
40#1559—28no—
41C2×#333C2212yes—
42C2×#333C2212yes—
43C2×#333C2212yes—
44C2×#333C2212yes—
45C2×#332C2211yescharacteristic
46C2×#332C2212yes—
47C2×#332C2211yescharacteristic
48#1296C2211yescharacteristic
49#1296C2211yescharacteristic
50#1296C2212yes—
51C22×M16—22no—
52#533—24no—
53#533—24no—
54#533—42no—
55#533—42no—
56C2×C4⋊Q8C2×C412yes—
57C2×C4⋊Q8C2×C412yes—
58C2×C4⋊Q8C2312yes—
59C2×C4⋊Q8C2311yescharacteristic
60C2×#125—22no—
61C2×C22⋊Q8—22no—
62C2×C22⋊Q8C2×C414yes—
63C2×C22⋊Q8C2×C414yes—
64C2×C4×Q8—21no—
65C2×C4×Q8—22no—
66C2×C4×Q8C2×C414yes—
67C2×C4×D4C2×C411yescharacteristic
68C2×C4×D4D411yescharacteristic
69C2×C4×D4D411yescharacteristic
70C2×#117—22no—
71C22×C42C2312yes—
72#436—44no—
73#432—24no—
74#432—42no—
75#423—24no—
76#423—42no—
77C2×C4⋊C8—21no—
78C2×C4⋊C8—21no—
79C2×C4⋊C8—22no—
80C2×C4⋊C8—21no—
81C2×C4⋊C8—21no—
82C2×C4⋊C8C2311yescharacteristic
83C2×C4⋊C8C2311yescharacteristic
84C2×C4⋊C8C2312yes—
85C2×C4⋊C8C2312yes—
86C2×C4⋊C8C2311yescharacteristic
87C2×C4⋊C8C2311yescharacteristic
88C2×C22⋊C8—22no—
89C2×C22⋊C8—22no—
90C2×C22⋊C8—22no—
91C2×#97—22no—
92C2×C4×C8—21no—
93C2×C4×C8—21no—
94#399—24no—
95#398—24no—
96#396—24no—
97#391—21no—
98#391C2×C414yes—
99#390—22no—
100#389—22no—
101#389—21no—
102#389—22no—
103#389—22no—
104#389—24no—
105#389C2×C412yes—
106#388—24no—
107#387—24no—
108#384—22no—
109#384—21no—
110#384—21no—
111#384—21no—
112#384—21no—
113#384C2311yescharacteristic
114#380—22no—
115#380—22no—
116C4×C4⋊C4—22no—
117C4×C4⋊C4—21no—
118C4×C4⋊C4C2×C412yes—
119C4×C4⋊C4C2×C411yescharacteristic
120C4×C4⋊C4D411yescharacteristic
121C4×C4⋊C4D411yescharacteristic
122C4×C22⋊C4C2×C412yes—
123C4×C22⋊C4D412yes—
124C4×C22⋊C4D412yes—
125C43C2311yescharacteristic
126#336—22no—
127#336—22no—
128#336—21no—
129#336—21no—
130#333—24no—
131#333—24no—
132#333—42no—
133#333—42no—
134#332—24no—
135#332—22no—
136#332D414yescomplemented
137C22×Q8—21no—
138C22×Q8C22×C414yes—
139C22×D4C22⋊C411yescharacteristic
140C23×C4C22×C412yes—
141C23×C4C2×D412yes—
142C23×C4C2×D412yes—
143C2×M16—44no—
144C2×M16—44no—
145C2×M16—82no—
146C22×C8—21no—
147C22×C8—21no—
148C22×C8—22no—
149C22×C8—21no—
150C22×C8—21no—
151C22×C8—22no—
152C4⋊Q8—22no—
153C4⋊Q8—22no—
154C4⋊Q8—21no—
155C4⋊Q8—21no—
156C4⋊Q8—22no—
157C4⋊Q8—22no—
158C4⋊Q8—44no—
159C4⋊Q8—82no—
160C4⋊Q8C2×D412yes—
161C4⋊Q8#4112yes—
162#125—44no—
163#125—44no—
164C22⋊Q8—28no—
165C22⋊Q8—44no—
166C22⋊Q8—44no—
167C22⋊Q8—44no—
168C22⋊Q8—84no—
169C4×Q8—28no—
170C4×Q8—44no—
171C4×Q8—44no—
172C4×Q8—44no—
173C4×Q8—42no—
174C4×Q8—42no—
175C4×D4—22no—
176C4×D4—22no—
177C4×D4—22no—
178C4×D4—22no—
179C4×D4—42no—
180#117—44no—
181#117—44no—
182C2×C4⋊C4—21no—
183C2×C4⋊C4—24no—
184C2×C4⋊C4—22no—
185C2×C4⋊C4—24no—
186C2×C4⋊C4—22no—
187C2×C4⋊C4—22no—
188C2×C4⋊C4—22no—
189C2×C4⋊C4—22no—
190C2×C4⋊C4—21no—
191C2×C4⋊C4—21no—
192C2×C4⋊C4—24no—
193C2×C4⋊C4—22no—
194C2×C4⋊C4—22no—
195C2×C4⋊C4C22⋊C411yescharacteristic
196C2×C4⋊C4C22⋊C411yescharacteristic
197C2×C4⋊C4C22⋊C411yescharacteristic
198C2×C4⋊C4C22×C412yes—
199C2×C4⋊C4C22×C412yes—
200C2×C22⋊C4—24no—
201C2×C22⋊C4—24no—
202C2×C22⋊C4C22⋊C412yes—
203C2×C22⋊C4C22⋊C412yes—
204C2×C22⋊C4C22⋊C412yes—
205C2×C42—22no—
206C2×C42—22no—
207C2×C42—22no—
208C2×C42—22no—
209C2×C42—24no—
210C2×C42—22no—
211C2×C42C22×C411yescharacteristic
212C2×C42C22×C412yes—
213C2×C42C22×C411yescharacteristic
214C2×C42C2×D412yes—
215C2×C42C2×D412yes—
216C2×C42C2×D411yescharacteristic
217C2×C42C2×D411yescharacteristic
218C2×C42C2×D411yescharacteristic
219C2×C42C2×D411yescharacteristic
220C2×C42C2411yescharacteristic
221C4⋊C8—22no—
222C4⋊C8—22no—
223C4⋊C8—21no—
224C4⋊C8—21no—
225C4⋊C8—21no—
226C4⋊C8—21no—
227C4⋊C8—21no—
228C4⋊C8—21no—
229C4⋊C8—22no—
230C4⋊C8—22no—
231C4⋊C8—44no—
232C4⋊C8—42no—
233C4⋊C8—42no—
234C4⋊C8—81no—
235C4⋊C8—81no—
236C4⋊C8C2×D412yes—
237C4⋊C8C2×D412yes—
238C22⋊C8—44no—
239C22⋊C8—44no—
240C22⋊C8—82no—
241#97—44no—
242C4×C8—42no—
243C4×C8—42no—
244#95—24no—
245#95—22no—
246#95—22no—
247#95—24no—
248#95C22⋊C412yes—
249#95C22⋊C412yes—
250#95C22⋊C412yes—
251C24C2×C22⋊C412yes—
252C2×Q8—22no—
253C2×Q8—22no—
254C2×Q8—24no—
255C2×Q8—24no—
256C2×Q8—44no—
257C2×Q8—44no—
258C2×Q8—41no—
259C2×Q8—41no—
260C2×Q8—41no—
261C2×Q8—41no—
262C2×D4—22no—
263C2×D4—22no—
264C2×D4—22no—
265C22×C4—22no—
266C22×C4—22no—
267C22×C4—22no—
268C22×C4—22no—
269C22×C4—22no—
270C22×C4—22no—
271C22×C4—22no—
272C22×C4—22no—
273C22×C4—22no—
274C22×C4—22no—
275C22×C4—22no—
276C22×C4—22no—
277C22×C4—22no—
278C22×C4—22no—
279C22×C4—22no—
280C22×C4—22no—
281C22×C4—24no—
282C22×C4—42no—
283C22×C4—42no—
284C22×C4C2×C22⋊C412yes—
285C22×C4C2×C22⋊C411yescharacteristic
286C22×C4C2×C22⋊C412yes—
287C22×C4C2×C22⋊C412yes—
288C22×C4C2×C22⋊C411yescharacteristic
289C22×C4C2×C22⋊C411yescharacteristic
290C22×C4C2×C22⋊C411yescharacteristic
291C22×C4C23×C411yescharacteristic
292C22×C4C22×D411yescharacteristic
293C22×C4C22×D411yescharacteristic
294M16—88no—
295C2×C8—22no—
296C2×C8—22no—
297C2×C8—41no—
298C2×C8—41no—
299C2×C8—41no—
300C2×C8—41no—
301C2×C8—42no—
302C2×C8—42no—
303C2×C8—41no—
304C2×C8—41no—
305C2×C8—41no—
306C2×C8—42no—
307C2×C8—41no—
308C2×C8—42no—
309C2×C8—41no—
310C2×C8—42no—
311C2×C8—42no—
312C2×C8—41no—
313C4⋊C4—22no—
314C4⋊C4—22no—
315C4⋊C4—22no—
316C4⋊C4—22no—
317C4⋊C4—22no—
318C4⋊C4—22no—
319C4⋊C4—48no—
320C4⋊C4—48no—
321C4⋊C4—42no—
322C4⋊C4—44no—
323C4⋊C4—42no—
324C4⋊C4—42no—
325C4⋊C4—42no—
326C4⋊C4—42no—
327C4⋊C4—41no—
328C4⋊C4—41no—
329C4⋊C4—41no—
330C4⋊C4—41no—
331C4⋊C4—44no—
332C4⋊C4—44no—
333C4⋊C4—84no—
334C4⋊C4—82no—
335C4⋊C4—82no—
336C4⋊C4C4×D414yes—
337C22⋊C4—24no—
338C22⋊C4—24no—
339C22⋊C4—42no—
340C22⋊C4—48no—
341C22⋊C4—84no—
342C42—22no—
343C42—22no—
344C42—22no—
345C42—22no—
346C42—22no—
347C42—22no—
348C42—21no—
349C42—21no—
350C42—21no—
351C42—21no—
352C42—22no—
353C42—41no—
354C42—48no—
355C42—44no—
356C42AGL(1,ℤ/8)12yes—
357C42#13712yes—
358C42#13712yes—
359C42#13712yes—
360C42C22×D411yescharacteristic
361C42C2×#4111yescharacteristic
362C42H(5,𝔽2)11yescharacteristic
363C42H(5,𝔽2)11yescharacteristic
364C23—22no—
365C23—22no—
366C23—22no—
367C23—22no—
368C23—22no—
369C23—22no—
370C23—22no—
371C23C22×C22⋊C411yescharacteristic
372Q8—42no—
373Q8—42no—
374Q8—42no—
375Q8—42no—
376Q8—84no—
377Q8—84no—
378D4—44no—
379C2×C4—21no—
380C2×C4—21no—
381C2×C4—21no—
382C2×C4—21no—
383C2×C4—21no—
384C2×C4—21no—
385C2×C4—22no—
386C2×C4—22no—
387C2×C4—22no—
388C2×C4—24no—
389C2×C4—21no—
390C2×C4—22no—
391C2×C4—22no—
392C2×C4—22no—
393C2×C4—22no—
394C2×C4—21no—
395C2×C4—21no—
396C2×C4—22no—
397C2×C4—22no—
398C2×C4—42no—
399C2×C4—42no—
400C2×C4—42no—
401C2×C4—42no—
402C2×C4—42no—
403C2×C4—42no—
404C2×C4—42no—
405C2×C4—42no—
406C2×C4—42no—
407C2×C4—42no—
408C2×C4—44no—
409C2×C4—44no—
410C2×C4—42no—
411C2×C4—44no—
412C2×C4—42no—
413C2×C4—42no—
414C2×C4—82no—
415C2×C4—82no—
416C2×C4#41312yes—
417C2×C4#41712yes—
418C2×C4#41712yes—
419C2×C4#41912yes—
420C2×C4#41912yes—
421C2×C4C2×C4×D411yescharacteristic
422C2×C4#51811yescharacteristic
423C2×C4D4211yescharacteristic
424C2×C4D4211yescharacteristic
425C2×C4#54611yescharacteristic
426C2×C4#54611yescharacteristic
427C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
428C2×C4C2×#13711yescharacteristic
429C2×C4C2×#13711yescharacteristic
430C2×C4C2×#13711yescharacteristic
431C2×C4#57711yescharacteristic
432C2×C4#57711yescharacteristic
433C2×C4#57811yescharacteristic
434C2×C4#57811yescharacteristic
435C8—42no—
436C8—42no—
437C8—81no—
438C8—81no—
439C8—82no—
440C8—82no—
441C22—42no—
442C22—42no—
443C22—42no—
444C22—42no—
445C22—42no—
446C22—42no—
447C22—42no—
448C22D4×C22⋊C411yescharacteristic
449C22C2×#41311yescharacteristic
450C22#288811yescharacteristic
451C22C2×#41911yescharacteristic
452C22C2×#41711yescharacteristic
453C22#289811yescharacteristic
454C22#289811yescharacteristic
455C4—22no—
456C4—22no—
457C4—21no—
458C4—21no—
459C4—41no—
460C4—42no—
461C4—42no—
462C4—42no—
463C4—44no—
464C4—82no—
465C4—84no—
466C4#155914yescomplemented
467C4#327211yescharacteristic
468C4#327611yescharacteristic
469C4#327611yescharacteristic
470C4#327811yescharacteristic
471C4#332811yescharacteristic
472C4#332911yescharacteristic
473C4#333011yescharacteristic
474C4#333111yescharacteristic
475C2—82no—
476C2C2×#155911yescharacteristic
477C2#1776011yescharacteristic
478C2#1776411yescharacteristic
479C2#1777811yescharacteristic
480C2#2241211yescharacteristic
481C2#2243011yescharacteristic
482C2#2243111yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
prime radicalC22×D4C22⋊C4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C42×D4 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
4. C2×#1559C2semidirect
5. #9691C2semidirect
6. #9691C2semidirect
7. #9136C2semidirect
8. #9135C2semidirect
9. #8586C2semidirect
10. #8343C2semidirect
11. C4×#332C2semidirect
136. #332D4semidirect
466. C4#1559semidirect

Maximal subgroups

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

subgroupindex
#361342
#227992
#226502
C2×#15592
#96912
#91362
#91352
#85862
#83432
C4×#3322