Finite Group Databasebeta

← #220049

Subgroups — #220049

Order 512.

967 conjugacy classes of subgroups.

The subgroup lattice has 480 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36134C211yesmaximal, characteristic
2#22799C211yesmaximal, characteristic
3#22795C211yesmaximal, characteristic
4C2×#1554C212yesmaximal, complemented
5#9691C211yesmaximal, characteristic, complemented
6#9691C211yesmaximal, characteristic, complemented
7#9136C212yesmaximal, complemented
8#9128C212yesmaximal, complemented
9#8586C211yesmaximal, characteristic, complemented
10#8343C212yesmaximal, complemented
11C4×#332C211yesmaximal, characteristic, complemented
12C2×#533C2212yes—
13C2×#432C2212yes—
14C2×#431C2212yes—
15#2868C2211yescharacteristic
16#2857C2212yes—
17#2552—21no—
18#2542C412yes—
19#2538—21no—
20#2441—22no—
21#2429C414yes—
22#2423—22no—
23#2365C412yes—
24#2354—21no—
25Q8×C4⋊C4—21no—
26C4×C4⋊Q8C2211yescharacteristic
27C42×D4C2211yescharacteristic
28#1954—21no—
29#1954—21no—
30#1924—22no—
31#1923—22no—
32#1842C2211yescharacteristic
33#1842C2211yescharacteristic
34#1797C2212yes—
35#1797C2212yes—
36#1770—21no—
37C8×C4⋊C4—21no—
38C4×C4⋊C8C2211yescharacteristic
39C4×C4⋊C8C2211yescharacteristic
40#1554—28no—
41C2×#333C2212yes—
42C2×#333C2212yes—
43C2×#333C2212yes—
44C2×#333C2212yes—
45C2×#332C2211yescharacteristic
46C2×#332C2212yes—
47C2×#332C2211yescharacteristic
48#1296C2211yescharacteristic
49#1296C2211yescharacteristic
50#1296C2212yes—
51#533—24no—
52#533—24no—
53#533—42no—
54#533—42no—
55C2×C4⋊Q8C2×C412yes—
56C2×C4⋊Q8C2×C412yes—
57C2×C4⋊Q8C2312yes—
58C2×C4⋊Q8C2311yescharacteristic
59C2×#125—22no—
60C2×C22⋊Q8—22no—
61C2×C22⋊Q8C2×C414yes—
62C2×C22⋊Q8C2×C414yes—
63C2×C4×Q8—21no—
64C2×C4×Q8—22no—
65C2×C4×Q8C2×C414yes—
66C2×C4×D4C2×C411yescharacteristic
67C2×C4×D4D411yescharacteristic
68C2×C4×D4D411yescharacteristic
69C2×#117—22no—
70C22×C42C2312yes—
71#432—24no—
72#432—42no—
73#431—24no—
74#431—42no—
75C2×C4⋊C8—21no—
76C2×C4⋊C8—21no—
77C2×C4⋊C8—21no—
78C2×C4⋊C8—21no—
79C2×C4⋊C8C2311yescharacteristic
80C2×C4⋊C8C2311yescharacteristic
81C2×C4⋊C8C2312yes—
82C2×C4⋊C8C2312yes—
83C2×C4⋊C8C2311yescharacteristic
84C2×C4⋊C8C2311yescharacteristic
85C2×C22⋊C8—22no—
86C2×C22⋊C8—22no—
87C2×C22⋊C8—22no—
88C2×C22⋊C8—22no—
89C2×#97—22no—
90C2×C4×C8—22no—
91C2×C4×C8—21no—
92C2×C4×C8—21no—
93#399—24no—
94#398—24no—
95#396—24no—
96#391—21no—
97#391C2×C414yes—
98#390—22no—
99#389—22no—
100#389—21no—
101#389—22no—
102#389—22no—
103#389—24no—
104#389C2×C412yes—
105#388—24no—
106#387—24no—
107#384—22no—
108#384—21no—
109#384—21no—
110#384—21no—
111#384—21no—
112#384C2311yescharacteristic
113#380—22no—
114#380—22no—
115C4×C4⋊C4—22no—
116C4×C4⋊C4—21no—
117C4×C4⋊C4C2×C412yes—
118C4×C4⋊C4C2×C411yescharacteristic
119C4×C4⋊C4D411yescharacteristic
120C4×C4⋊C4D411yescharacteristic
121C4×C22⋊C4C2×C412yes—
122C4×C22⋊C4D412yes—
123C4×C22⋊C4D412yes—
124C43C2311yescharacteristic
125#336—22no—
126#336—22no—
127#336—21no—
128#336—21no—
129#333—24no—
130#333—24no—
131#333—42no—
132#333—42no—
133#332—24no—
134#332—22no—
135#332D414yescomplemented
136C22×Q8—21no—
137C22×Q8C22×C414yes—
138C22×D4C22⋊C411yescharacteristic
139C23×C4C22×C412yes—
140C23×C4C2×D412yes—
141C23×C4C2×D412yes—
142C22×C8—21no—
143C22×C8—21no—
144C22×C8—21no—
145C22×C8—21no—
146C22×C8—22no—
147C22×C8—22no—
148C4⋊Q8—22no—
149C4⋊Q8—22no—
150C4⋊Q8—21no—
151C4⋊Q8—21no—
152C4⋊Q8—22no—
153C4⋊Q8—22no—
154C4⋊Q8—44no—
155C4⋊Q8—82no—
156C4⋊Q8C2×D412yes—
157C4⋊Q8#4112yes—
158#125—44no—
159#125—44no—
160C22⋊Q8—28no—
161C22⋊Q8—44no—
162C22⋊Q8—44no—
163C22⋊Q8—44no—
164C22⋊Q8—84no—
165C4×Q8—28no—
166C4×Q8—44no—
167C4×Q8—44no—
168C4×Q8—44no—
169C4×Q8—42no—
170C4×Q8—42no—
171C4×D4—22no—
172C4×D4—22no—
173C4×D4—22no—
174C4×D4—22no—
175C4×D4—42no—
176#117—44no—
177#117—44no—
178C2×C4⋊C4—21no—
179C2×C4⋊C4—24no—
180C2×C4⋊C4—22no—
181C2×C4⋊C4—24no—
182C2×C4⋊C4—22no—
183C2×C4⋊C4—22no—
184C2×C4⋊C4—22no—
185C2×C4⋊C4—22no—
186C2×C4⋊C4—21no—
187C2×C4⋊C4—21no—
188C2×C4⋊C4—24no—
189C2×C4⋊C4—22no—
190C2×C4⋊C4—22no—
191C2×C4⋊C4C22⋊C411yescharacteristic
192C2×C4⋊C4C22⋊C411yescharacteristic
193C2×C4⋊C4C22⋊C411yescharacteristic
194C2×C4⋊C4C22×C412yes—
195C2×C4⋊C4C22×C412yes—
196C2×C22⋊C4—24no—
197C2×C22⋊C4—24no—
198C2×C22⋊C4C22⋊C412yes—
199C2×C22⋊C4C22⋊C412yes—
200C2×C22⋊C4C22⋊C412yes—
201C2×C42—22no—
202C2×C42—22no—
203C2×C42—22no—
204C2×C42—22no—
205C2×C42—24no—
206C2×C42—22no—
207C2×C42C22×C411yescharacteristic
208C2×C42C22×C412yes—
209C2×C42C22×C411yescharacteristic
210C2×C42C2×D412yes—
211C2×C42C2×D412yes—
212C2×C42C2×D411yescharacteristic
213C2×C42C2×D411yescharacteristic
214C2×C42C2×D411yescharacteristic
215C2×C42C2×D411yescharacteristic
216C2×C42C2411yescharacteristic
217C4⋊C8—22no—
218C4⋊C8—22no—
219C4⋊C8—21no—
220C4⋊C8—21no—
221C4⋊C8—21no—
222C4⋊C8—21no—
223C4⋊C8—21no—
224C4⋊C8—21no—
225C4⋊C8—22no—
226C4⋊C8—22no—
227C4⋊C8—42no—
228C4⋊C8—42no—
229C4⋊C8—81no—
230C4⋊C8—81no—
231C4⋊C8C2×D412yes—
232C4⋊C8C2×D412yes—
233C22⋊C8—44no—
234C22⋊C8—44no—
235C22⋊C8—82no—
236C22⋊C8—82no—
237#97—44no—
238C4×C8—44no—
239C4×C8—42no—
240C4×C8—42no—
241#95—24no—
242#95—22no—
243#95—22no—
244#95—24no—
245#95C22⋊C412yes—
246#95C22⋊C412yes—
247#95C22⋊C412yes—
248C24C2×C22⋊C412yes—
249C2×Q8—22no—
250C2×Q8—22no—
251C2×Q8—24no—
252C2×Q8—24no—
253C2×Q8—44no—
254C2×Q8—44no—
255C2×Q8—41no—
256C2×Q8—41no—
257C2×Q8—41no—
258C2×Q8—41no—
259C2×D4—22no—
260C2×D4—22no—
261C2×D4—22no—
262C22×C4—22no—
263C22×C4—22no—
264C22×C4—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—24no—
279C22×C4—42no—
280C22×C4—42no—
281C22×C4C2×C22⋊C412yes—
282C22×C4C2×C22⋊C411yescharacteristic
283C22×C4C2×C22⋊C412yes—
284C22×C4C2×C22⋊C412yes—
285C22×C4C2×C22⋊C411yescharacteristic
286C22×C4C2×C22⋊C411yescharacteristic
287C22×C4C2×C22⋊C411yescharacteristic
288C22×C4C23×C411yescharacteristic
289C22×C4C22×D411yescharacteristic
290C22×C4C22×D411yescharacteristic
291C2×C8—22no—
292C2×C8—22no—
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—42no—
303C2×C8—41no—
304C2×C8—42no—
305C2×C8—41no—
306C2×C8—41no—
307C2×C8—42no—
308C2×C8—42no—
309C4⋊C4—22no—
310C4⋊C4—22no—
311C4⋊C4—22no—
312C4⋊C4—22no—
313C4⋊C4—22no—
314C4⋊C4—22no—
315C4⋊C4—48no—
316C4⋊C4—48no—
317C4⋊C4—42no—
318C4⋊C4—44no—
319C4⋊C4—42no—
320C4⋊C4—42no—
321C4⋊C4—42no—
322C4⋊C4—42no—
323C4⋊C4—41no—
324C4⋊C4—41no—
325C4⋊C4—41no—
326C4⋊C4—41no—
327C4⋊C4—44no—
328C4⋊C4—44no—
329C4⋊C4—84no—
330C4⋊C4—82no—
331C4⋊C4—82no—
332C4⋊C4C4×D414yes—
333C22⋊C4—24no—
334C22⋊C4—24no—
335C22⋊C4—42no—
336C22⋊C4—48no—
337C22⋊C4—84no—
338C42—22no—
339C42—22no—
340C42—22no—
341C42—22no—
342C42—22no—
343C42—22no—
344C42—21no—
345C42—21no—
346C42—21no—
347C42—21no—
348C42—22no—
349C42—41no—
350C42—48no—
351C42—44no—
352C42#13512yes—
353C42#13512yes—
354C42#13712yes—
355C42#13712yes—
356C42C22×D411yescharacteristic
357C42C2×#4111yescharacteristic
358C42H(5,𝔽2)11yescharacteristic
359C42H(5,𝔽2)11yescharacteristic
360C23—22no—
361C23—22no—
362C23—22no—
363C23—22no—
364C23—22no—
365C23—22no—
366C23—22no—
367C23C22×C22⋊C411yescharacteristic
368Q8—42no—
369Q8—42no—
370Q8—42no—
371Q8—42no—
372Q8—84no—
373Q8—84no—
374D4—44no—
375C2×C4—21no—
376C2×C4—21no—
377C2×C4—21no—
378C2×C4—21no—
379C2×C4—21no—
380C2×C4—21no—
381C2×C4—22no—
382C2×C4—22no—
383C2×C4—22no—
384C2×C4—24no—
385C2×C4—21no—
386C2×C4—22no—
387C2×C4—22no—
388C2×C4—22no—
389C2×C4—22no—
390C2×C4—21no—
391C2×C4—21no—
392C2×C4—22no—
393C2×C4—22no—
394C2×C4—42no—
395C2×C4—42no—
396C2×C4—42no—
397C2×C4—42no—
398C2×C4—42no—
399C2×C4—42no—
400C2×C4—42no—
401C2×C4—42no—
402C2×C4—42no—
403C2×C4—42no—
404C2×C4—44no—
405C2×C4—44no—
406C2×C4—42no—
407C2×C4—44no—
408C2×C4—42no—
409C2×C4—42no—
410C2×C4—82no—
411C2×C4—82no—
412C2×C4#41312yes—
413C2×C4#41612yes—
414C2×C4#41612yes—
415C2×C4#41912yes—
416C2×C4#41912yes—
417C2×C4C2×C4×D411yescharacteristic
418C2×C4#51811yescharacteristic
419C2×C4D4211yescharacteristic
420C2×C4D4211yescharacteristic
421C2×C4#54611yescharacteristic
422C2×C4#54611yescharacteristic
423C2×C4C2×#13511yescharacteristic
424C2×C4C2×#13511yescharacteristic
425C2×C4C2×#13711yescharacteristic
426C2×C4C2×#13711yescharacteristic
427C2×C4#57711yescharacteristic
428C2×C4#57711yescharacteristic
429C2×C4#57811yescharacteristic
430C2×C4#57811yescharacteristic
431C8—42no—
432C8—42no—
433C8—81no—
434C8—82no—
435C8—82no—
436C8—81no—
437C22—42no—
438C22—42no—
439C22—42no—
440C22—42no—
441C22—42no—
442C22—42no—
443C22—42no—
444C22D4×C22⋊C411yescharacteristic
445C22C2×#41311yescharacteristic
446C22#288811yescharacteristic
447C22C2×#41611yescharacteristic
448C22C2×#41911yescharacteristic
449C22#289811yescharacteristic
450C22#289811yescharacteristic
451C4—22no—
452C4—22no—
453C4—21no—
454C4—21no—
455C4—41no—
456C4—42no—
457C4—42no—
458C4—44no—
459C4—42no—
460C4—82no—
461C4—84no—
462C4#155414yescomplemented
463C4#327611yescharacteristic
464C4#327811yescharacteristic
465C4#328311yescharacteristic
466C4#328411yescharacteristic
467C4#330211yescharacteristic
468C4#330311yescharacteristic
469C4#332811yescharacteristic
470C4#332911yescharacteristic
471C2—82no—
472C2C2×#155411yescharacteristic
473C2#1772611yescharacteristic
474C2#1776011yescharacteristic
475C2#1777811yescharacteristic
476C2#2241211yescharacteristic
477C2#2241711yescharacteristic
478C2#2243011yescharacteristic
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×#1554C2semidirect
5. #9691C2semidirect
6. #9691C2semidirect
7. #9136C2semidirect
8. #9128C2semidirect
9. #8586C2semidirect
10. #8343C2semidirect
11. C4×#332C2semidirect
135. #332D4semidirect
462. C4#1554semidirect

Maximal subgroups

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

subgroupindex
#361342
#227992
#227952
C2×#15542
#96912
#91362
#91282
#85862
#83432
C4×#3322