Finite Group Databasebeta

← #224113

Subgroups — #224113

Order 512.

860 conjugacy classes of subgroups.

The subgroup lattice has 511 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34941C211yesmaximal, characteristic
2#22825C211yesmaximal, characteristic, complemented
3C4×C8×Q8C211yesmaximal, characteristic, complemented
4#9871C212yesmaximal, complemented
5#9870C211yesmaximal, characteristic, complemented
6#9867C212yesmaximal, complemented
7#9864C211yesmaximal, characteristic, complemented
8#8569C212yesmaximal
9#8563C211yesmaximal, characteristic
10#8177C212yesmaximal
11C4×D4⋊C8C211yesmaximal, characteristic
12C2×C8×Q8—21no—
13#2574—22no—
14#2567C411yescharacteristic
15#2563C411yescharacteristic
16#2552C411yescharacteristic
17#2538C411yescharacteristic
18#2508C412yes—
19#2497—21no—
20#2454C412yes—
21#2423—21no—
22#2396—22no—
23C4×C4⋊Q8C2211yescharacteristic
24C4×#125C2212yes—
25C4×C22⋊Q8—21no—
26#2293C2212yes—
27#2285—21no—
28C42×Q8C2211yescharacteristic
29C42×D4C2211yescharacteristic
30#1984—21no—
31#1917—22no—
32#1842C2211yescharacteristic
33#1831C2212yes—
34#1770—21no—
35C8×C4⋊C4—22no—
36C8×C4⋊C4—21no—
37#1767C2212yes—
38C4×C4⋊C8C2211yescharacteristic
39C4×C4⋊C8C2211yescharacteristic
40C4×C4⋊C8C2212yes—
41C42×C8C2211yescharacteristic
42C42×C8C2212yes—
43C2×D4⋊C8C2212yes—
44C2×D4⋊C8C2211yescharacteristic
45C2×D4⋊C8C2211yescharacteristic
46#1279C2212yes—
47#1279C2212yes—
48#1279C2212yes—
49#1279C2212yes—
50#1279C2211yescharacteristic
51#1279C2212yes—
52#1279C2211yescharacteristic
53C2×C4⋊Q8C2×C411yescharacteristic
54C2×#125C2×C412yes—
55C2×C22⋊Q8—21no—
56C2×C4×Q8—21no—
57C2×C4×Q8C2×C411yescharacteristic
58C2×C4×Q8D411yescharacteristic
59C2×C4×Q8D411yescharacteristic
60C2×C4×D4C2×C411yescharacteristic
61C2×C4×D4C2×C411yescharacteristic
62C2×C4×D4C2311yescharacteristic
63C2×#117—22no—
64C22×C42—21no—
65C8×Q8—28no—
66C2×C4⋊C8—21no—
67C2×C4⋊C8—22no—
68C2×C4⋊C8—21no—
69C2×C4⋊C8—21no—
70C2×C4⋊C8C2311yescharacteristic
71C2×C4⋊C8C2312yes—
72C2×C4⋊C8C2311yescharacteristic
73C2×#97—22no—
74C2×C4×C8—21no—
75C2×C4×C8—22no—
76C2×C4×C8—21no—
77C2×C4×C8—22no—
78C2×C4×C8—21no—
79C2×C4×C8C2311yescharacteristic
80C2×C4×C8C2312yes—
81C2×C4×C8C2311yescharacteristic
82#400—22no—
83#399—22no—
84#398—22no—
85#398—22no—
86#396—22no—
87#396—22no—
88#395—22no—
89#393—22no—
90#391—21no—
91#391—21no—
92#390C2×C412yes—
93#390C2×C412yes—
94#389—21no—
95#389—21no—
96#389C2×C412yes—
97#389C2×C412yes—
98#389C2×C412yes—
99#389C2×C412yes—
100#389C2×C412yes—
101#388—22no—
102#388—22no—
103#387—22no—
104#387—22no—
105#387—22no—
106#385—22no—
107#384—21no—
108#384—21no—
109#384C2×C411yescharacteristic
110#382—22no—
111#382C2×C412yes—
112#380—21no—
113#380—21no—
114#380—21no—
115#380—22no—
116C4×C4⋊C4—22no—
117C4×C4⋊C4—22no—
118C4×C4⋊C4—21no—
119C4×C4⋊C4C2×C411yescharacteristic
120C4×C4⋊C4C2×C411yescharacteristic
121C4×C4⋊C4C2×C411yescharacteristic
122C4×C4⋊C4C2×C412yes—
123C4×C4⋊C4D412yes—
124C4×C4⋊C4D412yes—
125C4×C4⋊C4D411yescharacteristic
126C4×C4⋊C4D411yescharacteristic
127C4×C4⋊C4C2311yescharacteristic
128C4×C4⋊C4C2312yes—
129C4×C22⋊C4—21no—
130C4×C22⋊C4—21no—
131C4×C22⋊C4—22no—
132C4×C22⋊C4—21no—
133C43C2312yes—
134C43C2311yescharacteristic
135#336—22no—
136#336—22no—
137#336—21no—
138#336—22no—
139#336—21no—
140#336—22no—
141D4⋊C8—24no—
142D4⋊C8—22no—
143D4⋊C8Q814yescomplemented
144C22×Q8—21no—
145C22×Q8C22⋊C411yescharacteristic
146C22×D4C22×C411yescharacteristic
147C23×C4—21no—
148C23×C4—21no—
149C23×C4—21no—
150C22×C8—21no—
151C22×C8—22no—
152C22×C8—21no—
153C22×C8—21no—
154C22×C8—21no—
155C22×C8—22no—
156C4⋊Q8—22no—
157C4⋊Q8—22no—
158#125—24no—
159#125—42no—
160C22⋊Q8—44no—
161C4×Q8—24no—
162C4×Q8—22no—
163C4×Q8—22no—
164C4×Q8—22no—
165C4×Q8—22no—
166C4×Q8—81no—
167C4×Q8—81no—
168C4×D4—22no—
169C4×D4—22no—
170C4×D4—22no—
171C4×D4—21no—
172C4×D4—21no—
173C4×D4—41no—
174C4×D4C2×Q812yes—
175C4×D4#4112yes—
176#117—82no—
177#117—82no—
178C2×C4⋊C4—21no—
179C2×C4⋊C4—22no—
180C2×C4⋊C4—22no—
181C2×C4⋊C4—22no—
182C2×C4⋊C4—21no—
183C2×C4⋊C4—22no—
184C2×C4⋊C4—22no—
185C2×C4⋊C4—21no—
186C2×C4⋊C4—22no—
187C2×C4⋊C4—22no—
188C2×C4⋊C4C22⋊C411yescharacteristic
189C2×C4⋊C4C22⋊C412yes—
190C2×C4⋊C4C22⋊C411yescharacteristic
191C2×C4⋊C4C22⋊C411yescharacteristic
192C2×C4⋊C4C22⋊C412yes—
193C2×C4⋊C4C22⋊C412yes—
194C2×C4⋊C4C22×C411yescharacteristic
195C2×C4⋊C4C22×C412yes—
196C2×C4⋊C4C22×C412yes—
197C2×C4⋊C4C22×C411yescharacteristic
198C2×C4⋊C4C22×C411yescharacteristic
199C2×C22⋊C4—21no—
200C2×C22⋊C4—21no—
201C2×C22⋊C4—21no—
202C2×C22⋊C4—22no—
203C2×C22⋊C4—22no—
204C2×C22⋊C4—22no—
205C2×C22⋊C4—22no—
206C2×C42—21no—
207C2×C42—21no—
208C2×C42—22no—
209C2×C42—21no—
210C2×C42—42no—
211C2×C42—42no—
212C2×C42C22×C411yescharacteristic
213C2×C42C22×C412yes—
214C2×C42C22×C412yes—
215C2×C42C22×C411yescharacteristic
216C2×C42C2×D411yescharacteristic
217C2×C42C2×D412yes—
218C2×C42C2×D412yes—
219C2×C42C2×D411yescharacteristic
220C2×C42C2×D412yes—
221C2×C42C2×D411yescharacteristic
222C2×C42C2×D412yes—
223C2×C42C2×D411yescharacteristic
224C2×C42C2411yescharacteristic
225C4⋊C8—28no—
226C4⋊C8—24no—
227C4⋊C8—21no—
228C4⋊C8—22no—
229C4⋊C8—21no—
230C4⋊C8—21no—
231C4⋊C8—22no—
232C4⋊C8—42no—
233C4⋊C8—81no—
234C4⋊C8C2×Q812yes—
235#97—82no—
236C4×C8—28no—
237C4×C8—24no—
238C4×C8—22no—
239C4×C8—21no—
240C4×C8—21no—
241C4×C8—21no—
242C4×C8—22no—
243C4×C8—42no—
244C4×C8—44no—
245C4×C8—42no—
246C4×C8C2×Q812yes—
247#95—21no—
248#95—21no—
249#95—22no—
250#95—22no—
251#95—22no—
252#95—22no—
253#95—22no—
254#95—22no—
255#95—22no—
256#95—21no—
257#95—22no—
258#95C22⋊C412yes—
259#95C22⋊C412yes—
260#95C22⋊C412yes—
261C24—21no—
262C2×Q8—22no—
263C2×Q8—22no—
264C2×Q8—42no—
265C2×Q8—81no—
266C2×Q8—81no—
267C2×Q8#10414yes—
268C2×D4—21no—
269C2×D4—21no—
270C2×D4—21no—
271C2×D4—21no—
272C2×D4—21no—
273C2×D4C4×Q812yes—
274C22×C4—21no—
275C22×C4—21no—
276C22×C4—22no—
277C22×C4—22no—
278C22×C4—21no—
279C22×C4—22no—
280C22×C4—22no—
281C22×C4—42no—
282C22×C4—41no—
283C22×C4—41no—
284C22×C4—41no—
285C22×C4—41no—
286C22×C4—42no—
287C22×C4—42no—
288C22×C4—42no—
289C22×C4—42no—
290C22×C4—81no—
291C22×C4C2×C22⋊C412yes—
292C22×C4C2×C22⋊C412yes—
293C22×C4C2×C22⋊C411yescharacteristic
294C22×C4C2×C22⋊C412yes—
295C22×C4C2×C22⋊C411yescharacteristic
296C22×C4C2×C22⋊C411yescharacteristic
297C22×C4C2×C22⋊C412yes—
298C22×C4C2×C22⋊C411yescharacteristic
299C22×C4C23×C411yescharacteristic
300C22×C4C22×D411yescharacteristic
301C22×C4C22×D411yescharacteristic
302C2×C8—22no—
303C2×C8—22no—
304C2×C8—22no—
305C2×C8—24no—
306C2×C8—22no—
307C2×C8—22no—
308C2×C8—41no—
309C2×C8—42no—
310C2×C8—42no—
311C2×C8—41no—
312C2×C8—41no—
313C2×C8—41no—
314C2×C8—41no—
315C2×C8—42no—
316C2×C8—41no—
317C2×C8—42no—
318C2×C8—42no—
319C2×C8—41no—
320C4⋊C4—21no—
321C4⋊C4—21no—
322C4⋊C4—24no—
323C4⋊C4—22no—
324C4⋊C4—24no—
325C4⋊C4—22no—
326C4⋊C4—24no—
327C4⋊C4—22no—
328C4⋊C4—21no—
329C4⋊C4—21no—
330C4⋊C4—22no—
331C4⋊C4—22no—
332C4⋊C4—22no—
333C4⋊C4—22no—
334C4⋊C4—44no—
335C4⋊C4—44no—
336C4⋊C4—42no—
337C4⋊C4—42no—
338C4⋊C4—44no—
339C4⋊C4—82no—
340C4⋊C4—82no—
341C4⋊C4—82no—
342C4⋊C4—81no—
343C4⋊C4—82no—
344C4⋊C4C4×Q812yes—
345C4⋊C4#13112yes—
346C22⋊C4—22no—
347C22⋊C4—22no—
348C22⋊C4—42no—
349C22⋊C4—42no—
350C22⋊C4—44no—
351C22⋊C4—82no—
352C22⋊C4—82no—
353C22⋊C4—82no—
354C42—21no—
355C42—22no—
356C42—22no—
357C42—22no—
358C42—21no—
359C42—22no—
360C42—22no—
361C42—22no—
362C42—22no—
363C42—21no—
364C42—22no—
365C42—21no—
366C42—24no—
367C42—22no—
368C42—41no—
369C42—42no—
370C42—41no—
371C42—41no—
372C42—42no—
373C42—42no—
374C42—42no—
375C42—42no—
376C42#13112yes—
377C42#13112yes—
378C42#13512yes—
379C42#13512yes—
380C42C22×Q811yescharacteristic
381C42C2×#4111yescharacteristic
382C422−1+411yescharacteristic
383C422−1+411yescharacteristic
384C23—22no—
385C23—41no—
386C23—41no—
387C23—41no—
388C23—41no—
389C23—41no—
390C23—41no—
391C23C22×C22⋊C411yescharacteristic
392Q8—28no—
393Q8—84no—
394D4—22no—
395D4—22no—
396D4—41no—
397D4—41no—
398C2×C4—22no—
399C2×C4—21no—
400C2×C4—22no—
401C2×C4—21no—
402C2×C4—21no—
403C2×C4—21no—
404C2×C4—21no—
405C2×C4—21no—
406C2×C4—22no—
407C2×C4—22no—
408C2×C4—21no—
409C2×C4—22no—
410C2×C4—22no—
411C2×C4—22no—
412C2×C4—22no—
413C2×C4—22no—
414C2×C4—22no—
415C2×C4—21no—
416C2×C4—21no—
417C2×C4—22no—
418C2×C4—22no—
419C2×C4—22no—
420C2×C4—41no—
421C2×C4—41no—
422C2×C4—41no—
423C2×C4—41no—
424C2×C4—42no—
425C2×C4—42no—
426C2×C4—42no—
427C2×C4—42no—
428C2×C4—41no—
429C2×C4—42no—
430C2×C4—42no—
431C2×C4—41no—
432C2×C4—42no—
433C2×C4—42no—
434C2×C4—42no—
435C2×C4—41no—
436C2×C4—42no—
437C2×C4—84no—
438C2×C4—84no—
439C2×C4#40812yes—
440C2×C4#40812yes—
441C2×C4#41612yes—
442C2×C4#41612yes—
443C2×C4C2×#10414yes—
444C2×C4C2×#10412yes—
445C2×C4C2×C4×Q811yescharacteristic
446C2×C4#51911yescharacteristic
447C2×C4#54811yescharacteristic
448C2×C4#54811yescharacteristic
449C2×C4D4×Q811yescharacteristic
450C2×C4D4×Q811yescharacteristic
451C2×C4C2×#13111yescharacteristic
452C2×C4C2×#13111yescharacteristic
453C2×C4C2×#13111yescharacteristic
454C2×C4#56811yescharacteristic
455C2×C4C2×#13511yescharacteristic
456C2×C4C2×#13511yescharacteristic
457C2×C4#57611yescharacteristic
458C2×C4#57711yescharacteristic
459C8—24no—
460C8—44no—
461C8—42no—
462C8—42no—
463C8—81no—
464C8—82no—
465C22—42no—
466C22—42no—
467C22—42no—
468C22—81no—
469C22—81no—
470C22—81no—
471C22—81no—
472C22Q8×C22⋊C411yescharacteristic
473C22C2×#40811yescharacteristic
474C22#288011yescharacteristic
475C22C2×#41611yescharacteristic
476C22#289711yescharacteristic
477C22C22×#10411yescharacteristic
478C22#290111yescharacteristic
479C4—21no—
480C4—21no—
481C4—24no—
482C4—22no—
483C4—22no—
484C4—42no—
485C4—42no—
486C4—42no—
487C4—41no—
488C4—42no—
489C4—42no—
490C4—81no—
491C4—82no—
492C4—82no—
493C4#149414yes—
494C4#296011yescharacteristic
495C4#296211yescharacteristic
496C4#298511yescharacteristic
497C4#298611yescharacteristic
498C4#336011yescharacteristic
499C4#336111yescharacteristic
500C4#338011yescharacteristic
501C4#338111yescharacteristic
502C2—82no—
503C2C2×#149411yescharacteristic
504C2#1748311yescharacteristic
505C2#1750811yescharacteristic
506C2#1752011yescharacteristic
507C2#2250911yescharacteristic
508C2#2251911yescharacteristic
509C2Q8×#10411yescharacteristic
subgroupG/H
centerC22×C4C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
socleC23C22×C22⋊C4
prime radicalC22×D4C22×C4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C4 ◁ C42×Q8 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #22825C2semidirect
3. C4×C8×Q8C2semidirect
4. #9871C2semidirect
5. #9870C2semidirect
6. #9867C2semidirect
7. #9864C2semidirect
143. D4⋊C8Q8semidirect

Maximal subgroups

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

subgroupindex
#349412
#228252
C4×C8×Q82
#98712
#98702
#98672
#98642
#85692
#85632
#81772
C4×D4⋊C82