Finite Group Databasebeta

← #196305

Subgroups — #196305

Order 512.

1054 conjugacy classes of subgroups.

The subgroup lattice has 518 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#60250C211yesmaximal, characteristic
2#35322C211yesmaximal, characteristic
3#17479C211yesmaximal, characteristic
4#17464C211yesmaximal, characteristic
5#17385C211yesmaximal, characteristic
6#9104C214yesmaximal, complemented
7#9100C214yesmaximal, complemented
8#7856C211yesmaximal, characteristic
9#7854C211yesmaximal, characteristic
10C2×#533C2211yescharacteristic
11C2×#533C2211yescharacteristic
12#2986—22no—
13#2985—22no—
14#2984—22no—
15#2983C2212yes—
16#2982C2212yes—
17#2962—22no—
18#2960—22no—
19#2924—22no—
20#2920C2211yescharacteristic
21#2920C2211yescharacteristic
22C2×#431C2211yescharacteristic
23C2×#431C2211yescharacteristic
24#2868C411yescharacteristic
25#2862C411yescharacteristic
26#2862C411yescharacteristic
27#2862C411yescharacteristic
28#2621C414yes—
29#2565—22no—
30#2565—22no—
31#2499—22no—
32#2499—22no—
33#2327C2214yes—
34C4×#117C2211yescharacteristic
35#1497—22no—
36#1493—22no—
37#1485—22no—
38#1481—22no—
39#1479C2214yes—
40#1479C2212yes—
41#1479C2212yes—
42#1472—22no—
43#1470—22no—
44#1461C2214yes—
45#1274C2212yes—
46#1274C2214yes—
47#1274C2212yes—
48#533—21no—
49#533—21no—
50#533—21no—
51#533—21no—
52#533—21no—
53#533—21no—
54#533—21no—
55#533—21no—
56#533—21no—
57#533—21no—
58#533—21no—
59#533—21no—
60#533—21no—
61#533—21no—
62#533—21no—
63#533—21no—
64C2×C4⋊Q8C2×C411yescharacteristic
65C2×C4⋊Q8C2×C414yes—
66C2×C4⋊Q8C2×C411yescharacteristic
67C2×C4⋊Q8C2×C411yescharacteristic
68C2×C4⋊Q8C2×C411yescharacteristic
69C2×#125—21no—
70C2×#125D412yes—
71C2×C22⋊Q8—21no—
72C2×C22⋊Q8C2×C412yes—
73C2×C22⋊Q8C2×C412yes—
74C2×C22⋊Q8D412yes—
75C2×C4×Q8—21no—
76C2×C4×Q8D412yes—
77C2×C4×Q8C2312yes—
78C2×C4×Q8C2312yes—
79C2×#117—21no—
80C2×#117C2×C412yes—
81C2×#117D412yes—
82C22×C42C2311yescharacteristic
83#446—24no—
84#446—28no—
85#446—24no—
86#446—24no—
87C8×Q8—24no—
88C8×Q8—24no—
89C8×Q8—24no—
90#433—28no—
91#432—21no—
92#432—21no—
93#432—24no—
94#432—21no—
95#432—21no—
96#431—21no—
97#431—21no—
98#431—24no—
99#431—21no—
100#431—21no—
101#431—21no—
102#431—21no—
103#431D412yes—
104#431D412yes—
105#431C2312yes—
106#431C2312yes—
107#431C2311yescharacteristic
108#431C2311yescharacteristic
109#431C2312yes—
110#424—24no—
111#423—21no—
112#423—21no—
113C2×C4⋊C8—21no—
114C2×C4⋊C8—21no—
115#407—21no—
116#407—21no—
117C2×C22⋊C8—21no—
118C2×C22⋊C8—21no—
119#405—24no—
120C4×M16—21no—
121C4×M16D412yes—
122C2×C4×C8—21no—
123C2×C4×C8D412yes—
124#398—28no—
125#397—28no—
126#391—22no—
127#391—22no—
128#391—22no—
129#391—22no—
130#391C2×C414yes—
131#391C2×C414yes—
132#387—28no—
133#387—28no—
134#384—22no—
135#384—22no—
136#382—22no—
137#382—22no—
138#382C2×C414yes—
139C4×C4⋊C4—21no—
140C4×C4⋊C4—21no—
141C4×C22⋊C4—21no—
142C4×C22⋊C4—21no—
143#376C2312yes—
144C43C2×C411yescharacteristic
145C43C2×C411yescharacteristic
146Q8⋊C8—28no—
147Q8⋊C8—24no—
148Q8⋊C8—24no—
149#324—28no—
150C22×Q8—21no—
151C22×Q8C22⋊C412yes—
152C22×Q8C22×C412yes—
153C22×Q8C22×C412yes—
154C23×C4C22×C411yescharacteristic
155C23×C4C2×D411yescharacteristic
156C23×C4C2×D411yescharacteristic
157C2×M16—22no—
158C2×M16—41no—
159C22×C8—22no—
160C22×C8—41no—
161C4⋊Q8—21no—
162C4⋊Q8—21no—
163C4⋊Q8—21no—
164C4⋊Q8—21no—
165C4⋊Q8—21no—
166C4⋊Q8—21no—
167C4⋊Q8—21no—
168C4⋊Q8—21no—
169C4⋊Q8—21no—
170C4⋊Q8—21no—
171C4⋊Q8—21no—
172C4⋊Q8—21no—
173C4⋊Q8—21no—
174C4⋊Q8—21no—
175C4⋊Q8—21no—
176C4⋊Q8—21no—
177C4⋊Q8—28no—
178C4⋊Q8—44no—
179#125—22no—
180#125—22no—
181#125—22no—
182#125—22no—
183#125—41no—
184#125—41no—
185#125—41no—
186#125—41no—
187C22⋊Q8—22no—
188C22⋊Q8—22no—
189C22⋊Q8—22no—
190C22⋊Q8—22no—
191C22⋊Q8—22no—
192C22⋊Q8—22no—
193C22⋊Q8—22no—
194C22⋊Q8—22no—
195C22⋊Q8—22no—
196C22⋊Q8—22no—
197C22⋊Q8—22no—
198C22⋊Q8—22no—
199C22⋊Q8—41no—
200C22⋊Q8—41no—
201C22⋊Q8—41no—
202C22⋊Q8—41no—
203C4×Q8—22no—
204C4×Q8—22no—
205C4×Q8—22no—
206C4×Q8—22no—
207C4×Q8—22no—
208C4×Q8—22no—
209C4×Q8—22no—
210C4×Q8—22no—
211C4×Q8—22no—
212C4×Q8—22no—
213C4×Q8—22no—
214C4×Q8—22no—
215C4×Q8—41no—
216C4×Q8—41no—
217C4×Q8—41no—
218C4×Q8—41no—
219#117—22no—
220#117—22no—
221#117—22no—
222#117—22no—
223#117—24no—
224#117—42no—
225#117—41no—
226#117—41no—
227#117—41no—
228#117—41no—
229C2×C4⋊C4—21no—
230C2×C4⋊C4—21no—
231C2×C4⋊C4—22no—
232C2×C4⋊C4—21no—
233C2×C4⋊C4—22no—
234C2×C4⋊C4—28no—
235C2×C4⋊C4—22no—
236C2×C4⋊C4C22⋊C412yes—
237C2×C4⋊C4C22⋊C412yes—
238C2×C4⋊C4C22⋊C414yes—
239C2×C4⋊C4C22⋊C412yes—
240C2×C4⋊C4C22×C412yes—
241C2×C4⋊C4C22×C412yes—
242C2×C4⋊C4C2×D414yes—
243C2×C22⋊C4—22no—
244C2×C22⋊C4—22no—
245C2×C22⋊C4C22⋊C414yes—
246C2×C42—21no—
247C2×C42—21no—
248C2×C42—21no—
249C2×C42—21no—
250C2×C42—22no—
251C2×C42C22×C412yes—
252C2×C42C22×C412yes—
253C2×C42C22×C411yescharacteristic
254C2×C42C2×D414yes—
255C2×C42C2×D411yescharacteristic
256C2×C42C2×D411yescharacteristic
257C2×C42C2×D411yescharacteristic
258C2×C42C2×D411yescharacteristic
259C2×C42C2×D411yescharacteristic
260C2×C42C2×D411yescharacteristic
261C2×C42#4111yescharacteristic
262C2×C42#4111yescharacteristic
263C2×C42#4111yescharacteristic
264C2×C42#4111yescharacteristic
265C2×C42C2411yescharacteristic
266C4⋊C8—28no—
267C4⋊C8—28no—
268C4⋊C8—22no—
269C4⋊C8—22no—
270C4⋊C8—21no—
271C4⋊C8—21no—
272C4⋊C8—22no—
273C4⋊C8—22no—
274C4⋊C8—22no—
275C4⋊C8—41no—
276C4⋊C8—41no—
277C22⋊C8—22no—
278C22⋊C8—22no—
279C22⋊C8—21no—
280C22⋊C8—21no—
281C22⋊C8—22no—
282C22⋊C8—22no—
283C22⋊C8—22no—
284C22⋊C8—41no—
285C22⋊C8—41no—
286#97—28no—
287#97—22no—
288#97—41no—
289C4×C8—28no—
290C4×C8—22no—
291C4×C8—22no—
292C4×C8—22no—
293C4×C8—41no—
294C4×C8C2×D412yes—
295C4×C8C2×D412yes—
296C4×C8C2×D414yes—
297#95—28no—
298#95—28no—
299#95—28no—
300#95—22no—
301#95—22no—
302C24C2×C22⋊C411yescharacteristic
303C2×Q8—22no—
304C2×Q8—22no—
305C2×Q8—24no—
306C2×Q8—24no—
307C2×Q8—22no—
308C2×Q8—22no—
309C2×Q8—22no—
310C2×Q8—22no—
311C2×Q8—24no—
312C2×Q8—24no—
313C2×Q8—44no—
314C2×Q8—44no—
315C2×Q8—41no—
316C2×Q8—41no—
317C22×C4—22no—
318C22×C4—22no—
319C22×C4—22no—
320C22×C4—22no—
321C22×C4—22no—
322C22×C4—22no—
323C22×C4—42no—
324C22×C4—42no—
325C22×C4C2×C22⋊C414yes—
326C22×C4C2×C22⋊C411yescharacteristic
327C22×C4C2×C22⋊C411yescharacteristic
328C22×C4C2×C22⋊C411yescharacteristic
329C22×C4C2×C22⋊C414yes—
330C22×C4C4×D411yescharacteristic
331C22×C4C4×D411yescharacteristic
332C22×C4C4×D411yescharacteristic
333C22×C4C4×D411yescharacteristic
334C22×C4#12011yescharacteristic
335C22×C4#12011yescharacteristic
336C22×C4#12111yescharacteristic
337C22×C4#12111yescharacteristic
338C22×C4#12111yescharacteristic
339C22×C4#12111yescharacteristic
340C22×C4#12311yescharacteristic
341C22×C4#12311yescharacteristic
342C22×C4C23×C411yescharacteristic
343C22×C4C22×D411yescharacteristic
344C22×C4C22×D411yescharacteristic
345M16—44no—
346M16—82no—
347C2×C8—22no—
348C2×C8—24no—
349C2×C8—22no—
350C2×C8—44no—
351C2×C8—42no—
352C2×C8—42no—
353C2×C8—82no—
354C4⋊C4—22no—
355C4⋊C4—22no—
356C4⋊C4—24no—
357C4⋊C4—24no—
358C4⋊C4—22no—
359C4⋊C4—22no—
360C4⋊C4—22no—
361C4⋊C4—22no—
362C4⋊C4—24no—
363C4⋊C4—24no—
364C4⋊C4—22no—
365C4⋊C4—22no—
366C4⋊C4—22no—
367C4⋊C4—22no—
368C4⋊C4—44no—
369C4⋊C4—416no—
370C4⋊C4—42no—
371C4⋊C4—42no—
372C4⋊C4—41no—
373C4⋊C4—41no—
374C4⋊C4—41no—
375C4⋊C4—41no—
376C4⋊C4—42no—
377C4⋊C4—42no—
378C4⋊C4—41no—
379C4⋊C4—41no—
380C22⋊C4—24no—
381C22⋊C4—24no—
382C22⋊C4—44no—
383C22⋊C4—42no—
384C22⋊C4—42no—
385C42—24no—
386C42—24no—
387C42—22no—
388C42—22no—
389C42—22no—
390C42—42no—
391C42—42no—
392C42—42no—
393C42—42no—
394C42—42no—
395C42—42no—
396C42C4×D411yescharacteristic
397C42C4×D411yescharacteristic
398C42C4×D411yescharacteristic
399C42C4×D411yescharacteristic
400C42#12011yescharacteristic
401C42#12011yescharacteristic
402C42#12111yescharacteristic
403C42#12111yescharacteristic
404C42#12111yescharacteristic
405C42#12111yescharacteristic
406C42#12311yescharacteristic
407C42#12311yescharacteristic
408C42#13114yes—
409C42C22×D411yescharacteristic
410C42C22×D411yescharacteristic
411C42C2×#4111yescharacteristic
412C42C2×#4111yescharacteristic
413C23—22no—
414C23—41no—
415C23—41no—
416C23#38611yescharacteristic
417C23#38611yescharacteristic
418C23C22×C22⋊C411yescharacteristic
419Q8—42no—
420Q8—42no—
421Q8—44no—
422Q8—44no—
423Q8—42no—
424Q8—42no—
425Q8—44no—
426Q8—82no—
427Q8—82no—
428Q8—82no—
429C2×C4—22no—
430C2×C4—21no—
431C2×C4—21no—
432C2×C4—21no—
433C2×C4—21no—
434C2×C4—22no—
435C2×C4—22no—
436C2×C4—22no—
437C2×C4—22no—
438C2×C4—22no—
439C2×C4—22no—
440C2×C4—22no—
441C2×C4—22no—
442C2×C4—24no—
443C2×C4—28no—
444C2×C4—24no—
445C2×C4—24no—
446C2×C4—41no—
447C2×C4—41no—
448C2×C4—41no—
449C2×C4—41no—
450C2×C4—41no—
451C2×C4—41no—
452C2×C4—42no—
453C2×C4—42no—
454C2×C4—44no—
455C2×C4—42no—
456C2×C4—42no—
457C2×C4—44no—
458C2×C4—42no—
459C2×C4—44no—
460C2×C4—42no—
461C2×C4—42no—
462C2×C4#38611yescharacteristic
463C2×C4#38611yescharacteristic
464C2×C4#38611yescharacteristic
465C2×C4#38611yescharacteristic
466C2×C4#38611yescharacteristic
467C2×C4#38611yescharacteristic
468C2×C4C2×C4×D411yescharacteristic
469C2×C4C2×C4×D411yescharacteristic
470C2×C4C2×#12011yescharacteristic
471C2×C4C2×#12111yescharacteristic
472C2×C4C2×#12111yescharacteristic
473C2×C4C2×#12311yescharacteristic
474C2×C4C2×#13112yes—
475C2×C4#56812yes—
476C8—42no—
477C8—42no—
478C8—44no—
479C8—82no—
480C8—82no—
481C22—21no—
482C22—21no—
483C22—21no—
484C22—21no—
485C22—21no—
486C22—21no—
487C22—21no—
488C22—21no—
489C22—82no—
490C22C2×#38611yescharacteristic
491C22#288012yes—
492C4—21no—
493C4—21no—
494C4—21no—
495C4—21no—
496C4—21no—
497C4—21no—
498C4—21no—
499C4—21no—
500C4—44no—
501C4—44no—
502C4—44no—
503C4—82no—
504C4—82no—
505C4—82no—
506C4#190912yes—
507C4#191212yes—
508C4#293111yescharacteristic
509C4#293211yescharacteristic
510C4#293512yes—
511C2—22no—
512C2—41no—
513C2—41no—
514C2#2090011yescharacteristic
515C2C2×#190911yescharacteristic
516C2C2×#191211yescharacteristic
subgroupG/H
center, socleC22C2×#386
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
prime radicalC24C2×C22⋊C4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C22×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
6. #9104C2semidirect
7. #9100C2semidirect

Maximal subgroups

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

subgroupindex
#602502
#353222
#174792
#174642
#173852
#91042
#91002
#78562
#78542