Finite Group Databasebeta

← #224059

Subgroups — #224059

Order 512.

826 conjugacy classes of subgroups.

The subgroup lattice has 528 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2511C211yesmaximal, characteristic
2#22629C211yesmaximal, characteristic
3#22625C211yesmaximal, characteristic
4#9881C212yesmaximal
5#9880C212yesmaximal
6#9878C211yesmaximal, characteristic
7#9878C211yesmaximal, characteristic
8#8562C212yesmaximal
9#8558C211yesmaximal, characteristic
10#8154C212yesmaximal
11C4×#324C211yesmaximal, characteristic
12#2511—22no—
13#2511—22no—
14#2511—21no—
15#2511—21no—
16#2511—21no—
17#2511—21no—
18C2×#400C412yes—
19C2×#400C412yes—
20C2×#391C411yescharacteristic
21C2×#391C411yescharacteristic
22C2×#391C2211yescharacteristic
23C2×#389C411yescharacteristic
24C2×#387C2212yes—
25C2×#387C2212yes—
26C2×#384C411yescharacteristic
27C2×C4×C4⋊C4C2211yescharacteristic
28C2×C4×C4⋊C4C2211yescharacteristic
29#1939—21no—
30#1939—21no—
31#1917—22no—
32#1916—22no—
33#1797C2211yescharacteristic
34#1797C2212yes—
35#1797C2211yescharacteristic
36#1797C2212yes—
37#1770—21no—
38C8×C4⋊C4—21no—
39#1749C2212yes—
40C4×C22⋊C8C2211yescharacteristic
41C4×C22⋊C8C2212yes—
42C4×C22⋊C8C2211yescharacteristic
43C2×#324C2212yes—
44C2×#324C2211yescharacteristic
45C2×#324C2211yescharacteristic
46#1280C2212yes—
47#1280C2212yes—
48#1280C2212yes—
49#1280C2212yes—
50#1280C2212yes—
51#1280C2211yescharacteristic
52#1280C2211yescharacteristic
53C23×Q8C2×C411yescharacteristic
54C22×C4⋊C4C2×C411yescharacteristic
55C22×C4⋊C4C2×C412yes—
56C22×C4⋊C4C2×C411yescharacteristic
57C22×C4⋊C4C2×C411yescharacteristic
58C22×C4⋊C4C2×C412yes—
59C22×C4⋊C4C2×C412yes—
60C22×C4⋊C4C2×C411yescharacteristic
61C22×C4⋊C4C2311yescharacteristic
62C22×C42C2×C411yescharacteristic
63C22×C42C2×C411yescharacteristic
64C22×C42C2311yescharacteristic
65C22×C42C2312yes—
66C2×C4⋊C8—21no—
67C2×C4⋊C8—21no—
68C2×C4⋊C8—22no—
69C2×C4⋊C8—22no—
70C2×C22⋊C8C2312yes—
71C2×C22⋊C8C2311yescharacteristic
72C2×C22⋊C8C2311yescharacteristic
73C2×C22⋊C8C2312yes—
74C2×C22⋊C8C2311yescharacteristic
75C2×C22⋊C8C2311yescharacteristic
76C2×#97—22no—
77C2×C4×C8—21no—
78C2×C4×C8—21no—
79C2×C4×C8—22no—
80#400—22no—
81#400—22no—
82#400—22no—
83#400—22no—
84#400—22no—
85#400—22no—
86#400—22no—
87#400—22no—
88#391—22no—
89#391—21no—
90#391—21no—
91#391—21no—
92#391—21no—
93#391—22no—
94#391—21no—
95#391—21no—
96#391—21no—
97#391—21no—
98#389—22no—
99#389—21no—
100#389—21no—
101#387—22no—
102#387—22no—
103#387—22no—
104#387—22no—
105#387—22no—
106#387—22no—
107#387—22no—
108#387—22no—
109#384—21no—
110#384—21no—
111#384—22no—
112C4×C4⋊C4—21no—
113C4×C4⋊C4—21no—
114C4×C4⋊C4—21no—
115C4×C4⋊C4—21no—
116C4×C4⋊C4D412yes—
117C4×C4⋊C4D412yes—
118C4×C4⋊C4D411yescharacteristic
119C4×C4⋊C4D411yescharacteristic
120C4×C4⋊C4D411yescharacteristic
121C4×C4⋊C4D411yescharacteristic
122C2×#95C2×C411yescharacteristic
123C2×#95C2×C412yes—
124C2×#95C2×C412yes—
125C2×#95C2×C412yes—
126C2×#95C2×C412yes—
127C2×#95C2×C412yes—
128C2×#95C2×C412yes—
129C2×#95C2×C412yes—
130C2×#95C2311yescharacteristic
131C2×#95C2312yes—
132#336—21no—
133#336—22no—
134#336—21no—
135#336—22no—
136#336—21no—
137#336—22no—
138#336—21no—
139#336—22no—
140#324—24no—
141#324—22no—
142#324Q814yescomplemented
143C22×Q8—21no—
144C22×Q8—21no—
145C22×Q8—22no—
146C22×Q8—22no—
147C22×Q8—42no—
148C22×Q8—42no—
149C23×C4C22×C411yescharacteristic
150C23×C4C22×C411yescharacteristic
151C23×C4C22×C412yes—
152C23×C4C22×C411yescharacteristic
153C23×C4C22×C412yes—
154C23×C4C22×C412yes—
155C23×C4C22×C412yes—
156C23×C4C22×C411yescharacteristic
157C23×C4C22×C411yescharacteristic
158C23×C4C22×C411yescharacteristic
159C23×C4C2411yescharacteristic
160C22×C8—21no—
161C22×C8—21no—
162C22×C8—21no—
163C22×C8—21no—
164C22×C8—22no—
165C22×C8—22no—
166C2×C4⋊C4—22no—
167C2×C4⋊C4—22no—
168C2×C4⋊C4—21no—
169C2×C4⋊C4—21no—
170C2×C4⋊C4—21no—
171C2×C4⋊C4—21no—
172C2×C4⋊C4—21no—
173C2×C4⋊C4—21no—
174C2×C4⋊C4—22no—
175C2×C4⋊C4—22no—
176C2×C4⋊C4—21no—
177C2×C4⋊C4—21no—
178C2×C4⋊C4—22no—
179C2×C4⋊C4—24no—
180C2×C4⋊C4—22no—
181C2×C4⋊C4—22no—
182C2×C4⋊C4—22no—
183C2×C4⋊C4—22no—
184C2×C4⋊C4—42no—
185C2×C4⋊C4—41no—
186C2×C4⋊C4—42no—
187C2×C4⋊C4—42no—
188C2×C4⋊C4—82no—
189C2×C4⋊C4—82no—
190C2×C4⋊C4C22⋊C411yescharacteristic
191C2×C4⋊C4C22⋊C411yescharacteristic
192C2×C4⋊C4C22⋊C412yes—
193C2×C4⋊C4C22⋊C411yescharacteristic
194C2×C4⋊C4C22⋊C411yescharacteristic
195C2×C4⋊C4C22⋊C412yes—
196C2×C4⋊C4C2×Q812yes—
197C2×C4⋊C4#4112yes—
198C2×C42—21no—
199C2×C42—21no—
200C2×C42—21no—
201C2×C42—21no—
202C2×C42—22no—
203C2×C42—22no—
204C2×C42—42no—
205C2×C42—42no—
206C2×C42—82no—
207C2×C42C2×D411yescharacteristic
208C2×C42C2×D411yescharacteristic
209C2×C42C2×D411yescharacteristic
210C2×C42C2×D411yescharacteristic
211C2×C42C2×D412yes—
212C2×C42C2×D412yes—
213C2×C42C2×D412yes—
214C2×C42C2×D412yes—
215C4⋊C8—42no—
216C4⋊C8—42no—
217C4⋊C8—82no—
218C4⋊C8—82no—
219C22⋊C8—21no—
220C22⋊C8—21no—
221C22⋊C8—22no—
222C22⋊C8—21no—
223C22⋊C8—21no—
224C22⋊C8—22no—
225C22⋊C8—22no—
226C22⋊C8—21no—
227C22⋊C8—21no—
228C22⋊C8—22no—
229C22⋊C8C2×Q812yes—
230C22⋊C8C2×Q812yes—
231#97—82no—
232C4×C8—42no—
233C4×C8—42no—
234C4×C8—82no—
235#95—21no—
236#95—21no—
237#95—22no—
238#95—22no—
239#95—22no—
240#95—22no—
241#95—22no—
242#95—22no—
243#95—22no—
244#95—22no—
245#95—22no—
246#95—22no—
247#95—22no—
248#95—22no—
249#95—21no—
250#95—21no—
251#95—22no—
252#95—22no—
253#95C22⋊C412yes—
254#95C22⋊C412yes—
255#95C22⋊C412yes—
256#95C22⋊C412yes—
257C24C23×C411yescharacteristic
258C2×Q8—42no—
259C2×Q8—42no—
260C2×Q8—42no—
261C2×Q8—42no—
262C2×Q8—42no—
263C2×Q8—42no—
264C2×Q8—84no—
265C2×Q8—84no—
266C22×C4—22no—
267C22×C4—22no—
268C22×C4—21no—
269C22×C4—21no—
270C22×C4—21no—
271C22×C4—21no—
272C22×C4—21no—
273C22×C4—21no—
274C22×C4—21no—
275C22×C4—21no—
276C22×C4—21no—
277C22×C4—21no—
278C22×C4—22no—
279C22×C4—22no—
280C22×C4—22no—
281C22×C4—21no—
282C22×C4—22no—
283C22×C4—21no—
284C22×C4—22no—
285C22×C4—21no—
286C22×C4—22no—
287C22×C4—21no—
288C22×C4—21no—
289C22×C4—22no—
290C22×C4—22no—
291C22×C4—22no—
292C22×C4—22no—
293C22×C4—41no—
294C22×C4—41no—
295C22×C4—41no—
296C22×C4—42no—
297C22×C4—42no—
298C22×C4—42no—
299C22×C4—42no—
300C22×C4—41no—
301C22×C4—41no—
302C22×C4—41no—
303C22×C4—41no—
304C22×C4—41no—
305C22×C4—42no—
306C22×C4—42no—
307C22×C4—41no—
308C22×C4—41no—
309C22×C4—42no—
310C22×C4—42no—
311C22×C4C2×C22⋊C411yescharacteristic
312C22×C4C2×C22⋊C411yescharacteristic
313C22×C4C2×C22⋊C412yes—
314C22×C4C2×C22⋊C412yes—
315C22×C4C2×C22⋊C412yes—
316C22×C4C2×C22⋊C412yes—
317C22×C4C2×C22⋊C411yescharacteristic
318C22×C4C2×C22⋊C411yescharacteristic
319C22×C4C4×Q812yes—
320C22×C4C4×Q812yes—
321C22×C4#13112yes—
322C22×C4#13112yes—
323C22×C4#13112yes—
324C22×C4C22×D411yescharacteristic
325C22×C4C22×D411yescharacteristic
326C22×C4C22×Q811yescharacteristic
327C22×C4C2×#4111yescharacteristic
328C22×C42−1+411yescharacteristic
329C22×C42−1+411yescharacteristic
330C2×C8—22no—
331C2×C8—22no—
332C2×C8—41no—
333C2×C8—41no—
334C2×C8—41no—
335C2×C8—41no—
336C2×C8—41no—
337C2×C8—41no—
338C2×C8—42no—
339C2×C8—42no—
340C2×C8—42no—
341C2×C8—42no—
342C2×C8—41no—
343C2×C8—41no—
344C2×C8—41no—
345C2×C8—41no—
346C2×C8—42no—
347C2×C8—42no—
348C4⋊C4—22no—
349C4⋊C4—22no—
350C4⋊C4—22no—
351C4⋊C4—22no—
352C4⋊C4—24no—
353C4⋊C4—22no—
354C4⋊C4—22no—
355C4⋊C4—22no—
356C4⋊C4—22no—
357C4⋊C4—42no—
358C4⋊C4—42no—
359C4⋊C4—42no—
360C4⋊C4—44no—
361C4⋊C4—44no—
362C4⋊C4—42no—
363C4⋊C4—42no—
364C4⋊C4—82no—
365C4⋊C4—82no—
366C4⋊C4—82no—
367C4⋊C4—82no—
368C42—22no—
369C42—22no—
370C42—42no—
371C42—42no—
372C42—42no—
373C42—42no—
374C42—41no—
375C42—41no—
376C42—42no—
377C42—42no—
378C42—42no—
379C42—42no—
380C23—21no—
381C23—21no—
382C23—21no—
383C23—21no—
384C23C22×C22⋊C411yescharacteristic
385C23C2×C4×Q811yescharacteristic
386C23#51911yescharacteristic
387C23C2×#13111yescharacteristic
388C23C2×#13111yescharacteristic
389C23C2×#13111yescharacteristic
390C23#56811yescharacteristic
391Q8—84no—
392Q8—84no—
393C2×C4—21no—
394C2×C4—21no—
395C2×C4—21no—
396C2×C4—21no—
397C2×C4—21no—
398C2×C4—21no—
399C2×C4—24no—
400C2×C4—24no—
401C2×C4—22no—
402C2×C4—22no—
403C2×C4—22no—
404C2×C4—22no—
405C2×C4—22no—
406C2×C4—21no—
407C2×C4—21no—
408C2×C4—22no—
409C2×C4—22no—
410C2×C4—21no—
411C2×C4—21no—
412C2×C4—22no—
413C2×C4—22no—
414C2×C4—22no—
415C2×C4—22no—
416C2×C4—22no—
417C2×C4—22no—
418C2×C4—21no—
419C2×C4—21no—
420C2×C4—22no—
421C2×C4—22no—
422C2×C4—22no—
423C2×C4—22no—
424C2×C4—41no—
425C2×C4—41no—
426C2×C4—41no—
427C2×C4—41no—
428C2×C4—42no—
429C2×C4—42no—
430C2×C4—41no—
431C2×C4—41no—
432C2×C4—41no—
433C2×C4—41no—
434C2×C4—41no—
435C2×C4—41no—
436C2×C4—41no—
437C2×C4—41no—
438C2×C4—42no—
439C2×C4—42no—
440C2×C4—41no—
441C2×C4—42no—
442C2×C4—42no—
443C2×C4—42no—
444C2×C4—42no—
445C2×C4—41no—
446C2×C4—42no—
447C2×C4—42no—
448C2×C4—41no—
449C2×C4—81no—
450C2×C4—81no—
451C2×C4—82no—
452C2×C4—82no—
453C2×C4—82no—
454C2×C4—82no—
455C2×C4—82no—
456C2×C4—82no—
457C2×C4—81no—
458C2×C4—81no—
459C2×C4—82no—
460C2×C4—82no—
461C2×C4#40812yes—
462C2×C4#40812yes—
463C2×C4#41012yes—
464C2×C4#41312yes—
465C2×C4#54811yescharacteristic
466C2×C4#54811yescharacteristic
467C2×C4D4×Q811yescharacteristic
468C2×C4D4×Q811yescharacteristic
469C8—42no—
470C8—42no—
471C8—81no—
472C8—81no—
473C8—82no—
474C8—82no—
475C22—21no—
476C22—21no—
477C22—21no—
478C22—21no—
479C22—21no—
480C22—21no—
481C22—21no—
482C22—21no—
483C22—21no—
484C22—21no—
485C22—21no—
486C22—21no—
487C22Q8×C22⋊C411yescharacteristic
488C22C2×#40811yescharacteristic
489C22#288011yescharacteristic
490C22C2×#41011yescharacteristic
491C22#288311yescharacteristic
492C22C2×#41311yescharacteristic
493C22#288911yescharacteristic
494C22#296011yescharacteristic
495C22#296211yescharacteristic
496C22#298511yescharacteristic
497C22#298611yescharacteristic
498C4—22no—
499C4—21no—
500C4—21no—
501C4—21no—
502C4—21no—
503C4—41no—
504C4—41no—
505C4—42no—
506C4—42no—
507C4—42no—
508C4—42no—
509C4—42no—
510C4—42no—
511C4—42no—
512C4—82no—
513C4—82no—
514C4—81no—
515C4#147314yes—
516C2—21no—
517C2—21no—
518C2—21no—
519C2—21no—
520C2C2×#147311yescharacteristic
521C2#1738811yescharacteristic
522C2#1741311yescharacteristic
523C2#1742911yescharacteristic
524C2#2250911yescharacteristic
525C2#2251311yescharacteristic
526C2#2251711yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutator, prime radicalC24C23×C4
Frattini, cosocleC23×C4C24
LCSG ▷ C24 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×C4⋊C4 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
142. #324Q8semidirect

Maximal subgroups

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

subgroupindex
C2×#25112
#226292
#226252
#98812
#98802
#98782
#85622
#85582
#81542
C4×#3242