Finite Group Databasebeta

← #224112

Subgroups — #224112

Order 512.

986 conjugacy classes of subgroups.

The subgroup lattice has 529 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34929C211yesmaximal, characteristic
2#22722C211yesmaximal, characteristic, complemented
3C4×C8×Q8C211yesmaximal, characteristic, complemented
4#9870C211yesmaximal, characteristic, complemented
5#9870C212yesmaximal, complemented
6#9869C212yesmaximal, complemented
7#9865C211yesmaximal, characteristic, complemented
8#8567C212yesmaximal
9#8563C211yesmaximal, characteristic
10#8177C212yesmaximal
11C4×D4⋊C8C211yesmaximal, characteristic
12C2×#533—21no—
13C2×#446—21no—
14C2×C8×Q8—21no—
15#2556—22no—
16#2551C411yescharacteristic
17#2546C411yescharacteristic
18#2542C411yescharacteristic
19#2538C411yescharacteristic
20#2506C412yes—
21#2492—21no—
22#2452C412yes—
23#2429—21no—
24#2396—22no—
25C4×C4⋊Q8C2211yescharacteristic
26C4×C4⋊Q8C2212yes—
27C4×C22⋊Q8—21no—
28#2293C2212yes—
29C42×Q8C2211yescharacteristic
30C42×D4C2211yescharacteristic
31#1940—22no—
32#1939—21no—
33#1842C2211yescharacteristic
34#1842C2212yes—
35C8×C4⋊C4—21no—
36C8×C4⋊C4—22no—
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#1279C2211yescharacteristic
52#1279C2212yes—
53#533—44no—
54#533—44no—
55C2×C4⋊Q8—21no—
56C2×C4⋊Q8—21no—
57C2×C4⋊Q8C2×C411yescharacteristic
58C2×C4⋊Q8C2×C412yes—
59C2×#125—22no—
60C2×C22⋊Q8—21no—
61C2×C22⋊Q8—22no—
62C2×C22⋊Q8—21no—
63C2×C22⋊Q8—21no—
64C2×C4×Q8—22no—
65C2×C4×Q8—21no—
66C2×C4×Q8C2×C411yescharacteristic
67C2×C4×Q8D411yescharacteristic
68C2×C4×Q8D411yescharacteristic
69C2×C4×D4C2×C411yescharacteristic
70C2×C4×D4C2×C411yescharacteristic
71C2×C4×D4C2311yescharacteristic
72C2×#117—22no—
73C22×C42—21no—
74#446—44no—
75C8×Q8—28no—
76C2×C4⋊C8—21no—
77C2×C4⋊C8—22no—
78C2×C4⋊C8—21no—
79C2×C4⋊C8—22no—
80C2×C4⋊C8—22no—
81C2×C4⋊C8—21no—
82C2×C4⋊C8C2311yescharacteristic
83C2×C4⋊C8C2312yes—
84C2×C4⋊C8C2311yescharacteristic
85C2×#97—22no—
86C2×C4×C8—21no—
87C2×C4×C8—22no—
88C2×C4×C8—21no—
89C2×C4×C8—21no—
90C2×C4×C8—22no—
91C2×C4×C8C2311yescharacteristic
92C2×C4×C8C2312yes—
93C2×C4×C8C2311yescharacteristic
94#400—22no—
95#399—22no—
96#398—22no—
97#398—22no—
98#396—22no—
99#396—22no—
100#395—22no—
101#393—22no—
102#391—21no—
103#391—21no—
104#391—22no—
105#390C2×C412yes—
106#390C2×C412yes—
107#389—21no—
108#389—22no—
109#389—21no—
110#389C2×C412yes—
111#389C2×C412yes—
112#389C2×C412yes—
113#389C2×C412yes—
114#389C2×C412yes—
115#388—22no—
116#388—22no—
117#387—22no—
118#385—22no—
119#384C2×C412yes—
120#384C2×C411yescharacteristic
121#380—21no—
122C4×C4⋊C4—21no—
123C4×C4⋊C4—22no—
124C4×C4⋊C4C2×C411yescharacteristic
125C4×C4⋊C4C2×C411yescharacteristic
126C4×C4⋊C4C2×C412yes—
127C4×C4⋊C4C2×C411yescharacteristic
128C4×C4⋊C4D412yes—
129C4×C4⋊C4D411yescharacteristic
130C4×C4⋊C4D411yescharacteristic
131C4×C4⋊C4D412yes—
132C4×C4⋊C4C2311yescharacteristic
133C4×C4⋊C4C2312yes—
134C4×C22⋊C4—21no—
135C4×C22⋊C4—22no—
136C4×C22⋊C4—21no—
137C4×C22⋊C4—21no—
138C43C2311yescharacteristic
139C43C2312yes—
140#336—22no—
141#336—21no—
142#336—21no—
143#336—22no—
144D4⋊C8—24no—
145D4⋊C8—22no—
146D4⋊C8Q814yescomplemented
147C22×Q8—21no—
148C22×Q8—22no—
149C22×Q8C22⋊C411yescharacteristic
150C22×D4C22×C411yescharacteristic
151C23×C4—21no—
152C23×C4—21no—
153C23×C4—21no—
154C22×C8—21no—
155C22×C8—22no—
156C22×C8—21no—
157C22×C8—21no—
158C22×C8—21no—
159C22×C8—22no—
160C4⋊Q8—24no—
161C4⋊Q8—22no—
162C4⋊Q8—22no—
163C4⋊Q8—42no—
164C4⋊Q8—44no—
165C4⋊Q8—44no—
166#125—48no—
167C22⋊Q8—44no—
168C22⋊Q8—44no—
169C22⋊Q8—48no—
170C22⋊Q8—44no—
171C4×Q8—24no—
172C4×Q8—22no—
173C4×Q8—22no—
174C4×Q8—44no—
175C4×Q8—42no—
176C4×Q8—44no—
177C4×Q8—42no—
178C4×Q8D814yes—
179C4×Q8SD1614yes—
180C4×D4—22no—
181C4×D4—22no—
182C4×D4—22no—
183C4×D4—21no—
184C4×D4—21no—
185C4×D4—41no—
186C4×D4C2×Q812yes—
187C4×D4#4112yes—
188#117—44no—
189#117—44no—
190C2×C4⋊C4—22no—
191C2×C4⋊C4—22no—
192C2×C4⋊C4—22no—
193C2×C4⋊C4—22no—
194C2×C4⋊C4—22no—
195C2×C4⋊C4—21no—
196C2×C4⋊C4—22no—
197C2×C4⋊C4—22no—
198C2×C4⋊C4—22no—
199C2×C4⋊C4—21no—
200C2×C4⋊C4—21no—
201C2×C4⋊C4C22⋊C411yescharacteristic
202C2×C4⋊C4C22⋊C412yes—
203C2×C4⋊C4C22⋊C412yes—
204C2×C4⋊C4C22⋊C412yes—
205C2×C4⋊C4C22⋊C411yescharacteristic
206C2×C4⋊C4C22⋊C411yescharacteristic
207C2×C4⋊C4C22×C411yescharacteristic
208C2×C4⋊C4C22×C412yes—
209C2×C4⋊C4C22×C412yes—
210C2×C4⋊C4C22×C411yescharacteristic
211C2×C4⋊C4C22×C411yescharacteristic
212C2×C22⋊C4—21no—
213C2×C22⋊C4—22no—
214C2×C22⋊C4—22no—
215C2×C22⋊C4—21no—
216C2×C22⋊C4—21no—
217C2×C22⋊C4—22no—
218C2×C22⋊C4—22no—
219C2×C42—21no—
220C2×C42—21no—
221C2×C42—22no—
222C2×C42—21no—
223C2×C42—42no—
224C2×C42—42no—
225C2×C42C22×C411yescharacteristic
226C2×C42C22×C412yes—
227C2×C42C22×C412yes—
228C2×C42C22×C411yescharacteristic
229C2×C42C2×D411yescharacteristic
230C2×C42C2×D412yes—
231C2×C42C2×D411yescharacteristic
232C2×C42C2×D412yes—
233C2×C42C2×D411yescharacteristic
234C2×C42C2×D412yes—
235C2×C42C2×D412yes—
236C2×C42C2×D411yescharacteristic
237C2×C42C2411yescharacteristic
238C4⋊C8—28no—
239C4⋊C8—24no—
240C4⋊C8—21no—
241C4⋊C8—22no—
242C4⋊C8—21no—
243C4⋊C8—21no—
244C4⋊C8—22no—
245C4⋊C8—42no—
246C4⋊C8—44no—
247C4⋊C8—82no—
248C4⋊C8—81no—
249C4⋊C8C2×Q812yes—
250#97—44no—
251C4×C8—28no—
252C4×C8—24no—
253C4×C8—22no—
254C4×C8—21no—
255C4×C8—21no—
256C4×C8—21no—
257C4×C8—22no—
258C4×C8—42no—
259C4×C8—44no—
260C4×C8—42no—
261C4×C8C2×Q812yes—
262#95—21no—
263#95—21no—
264#95—22no—
265#95—22no—
266#95—22no—
267#95—22no—
268#95—22no—
269#95—21no—
270#95—22no—
271#95C22⋊C412yes—
272#95C22⋊C412yes—
273#95C22⋊C412yes—
274C24—21no—
275C2×Q8—22no—
276C2×Q8—22no—
277C2×Q8—44no—
278C2×Q8—42no—
279C2×Q8—44no—
280C2×Q8—42no—
281C2×Q8—42no—
282C2×Q8—82no—
283C2×Q8#10214yes—
284C2×D4—21no—
285C2×D4—21no—
286C2×D4—21no—
287C2×D4—21no—
288C2×D4—21no—
289C2×D4C4×Q812yes—
290C22×C4—21no—
291C22×C4—21no—
292C22×C4—22no—
293C22×C4—22no—
294C22×C4—21no—
295C22×C4—22no—
296C22×C4—22no—
297C22×C4—42no—
298C22×C4—41no—
299C22×C4—41no—
300C22×C4—41no—
301C22×C4—41no—
302C22×C4—42no—
303C22×C4—42no—
304C22×C4—42no—
305C22×C4—42no—
306C22×C4—81no—
307C22×C4C2×C22⋊C412yes—
308C22×C4C2×C22⋊C411yescharacteristic
309C22×C4C2×C22⋊C411yescharacteristic
310C22×C4C2×C22⋊C411yescharacteristic
311C22×C4C2×C22⋊C412yes—
312C22×C4C2×C22⋊C412yes—
313C22×C4C2×C22⋊C412yes—
314C22×C4C2×C22⋊C411yescharacteristic
315C22×C4C23×C411yescharacteristic
316C22×C4C22×D411yescharacteristic
317C22×C4C22×D411yescharacteristic
318C2×C8—22no—
319C2×C8—22no—
320C2×C8—22no—
321C2×C8—24no—
322C2×C8—22no—
323C2×C8—22no—
324C2×C8—41no—
325C2×C8—42no—
326C2×C8—42no—
327C2×C8—41no—
328C2×C8—41no—
329C2×C8—41no—
330C2×C8—41no—
331C2×C8—42no—
332C2×C8—41no—
333C2×C8—42no—
334C2×C8—41no—
335C2×C8—42no—
336C4⋊C4—21no—
337C4⋊C4—21no—
338C4⋊C4—24no—
339C4⋊C4—22no—
340C4⋊C4—24no—
341C4⋊C4—21no—
342C4⋊C4—21no—
343C4⋊C4—22no—
344C4⋊C4—22no—
345C4⋊C4—22no—
346C4⋊C4—22no—
347C4⋊C4—44no—
348C4⋊C4—42no—
349C4⋊C4—44no—
350C4⋊C4—44no—
351C4⋊C4—44no—
352C4⋊C4—44no—
353C4⋊C4—44no—
354C4⋊C4—44no—
355C4⋊C4—42no—
356C4⋊C4—44no—
357C4⋊C4—81no—
358C4⋊C4#10218yes—
359C4⋊C4#10214yes—
360C4⋊C4#10414yes—
361C4⋊C4C4×Q812yes—
362C4⋊C4#13112yes—
363C22⋊C4—22no—
364C22⋊C4—22no—
365C22⋊C4—42no—
366C22⋊C4—42no—
367C22⋊C4—44no—
368C22⋊C4—44no—
369C22⋊C4—44no—
370C22⋊C4—82no—
371C42—21no—
372C42—22no—
373C42—22no—
374C42—22no—
375C42—21no—
376C42—22no—
377C42—21no—
378C42—22no—
379C42—21no—
380C42—22no—
381C42—24no—
382C42—22no—
383C42—41no—
384C42—42no—
385C42—41no—
386C42—42no—
387C42—41no—
388C42—42no—
389C42—42no—
390C42—42no—
391C42#13112yes—
392C42#13112yes—
393C42C2×D812yes—
394C42C2×D814yes—
395C42C2×SD1612yes—
396C42C2×SD1614yes—
397C42C22×Q811yescharacteristic
398C42C2×#4111yescharacteristic
399C422−1+411yescharacteristic
400C422−1+411yescharacteristic
401C23—22no—
402C23—41no—
403C23—41no—
404C23—41no—
405C23—41no—
406C23—41no—
407C23—41no—
408C23C22×C22⋊C411yescharacteristic
409Q8—28no—
410Q8—88no—
411Q8—84no—
412D4—22no—
413D4—22no—
414D4—41no—
415D4—41no—
416C2×C4—22no—
417C2×C4—22no—
418C2×C4—21no—
419C2×C4—22no—
420C2×C4—21no—
421C2×C4—22no—
422C2×C4—21no—
423C2×C4—22no—
424C2×C4—21no—
425C2×C4—21no—
426C2×C4—21no—
427C2×C4—22no—
428C2×C4—22no—
429C2×C4—22no—
430C2×C4—21no—
431C2×C4—22no—
432C2×C4—22no—
433C2×C4—21no—
434C2×C4—21no—
435C2×C4—22no—
436C2×C4—41no—
437C2×C4—41no—
438C2×C4—41no—
439C2×C4—41no—
440C2×C4—42no—
441C2×C4—42no—
442C2×C4—42no—
443C2×C4—42no—
444C2×C4—41no—
445C2×C4—42no—
446C2×C4—42no—
447C2×C4—41no—
448C2×C4—42no—
449C2×C4—42no—
450C2×C4—42no—
451C2×C4—41no—
452C2×C4—42no—
453C2×C4—84no—
454C2×C4—84no—
455C2×C4#40812yes—
456C2×C4#40812yes—
457C2×C4C2×#10212yes—
458C2×C4C2×#10214yes—
459C2×C4C2×#10214yes—
460C2×C4C2×#10212yes—
461C2×C4C2×#10414yes—
462C2×C4C2×#10412yes—
463C2×C4C2×C4×Q811yescharacteristic
464C2×C4#51911yescharacteristic
465C2×C4#54811yescharacteristic
466C2×C4#54811yescharacteristic
467C2×C4D4×Q811yescharacteristic
468C2×C4D4×Q811yescharacteristic
469C2×C4C2×#13111yescharacteristic
470C2×C4C2×#13111yescharacteristic
471C2×C4C2×#13111yescharacteristic
472C2×C4#56811yescharacteristic
473C2×C4C22×D811yescharacteristic
474C2×C4C22×SD1611yescharacteristic
475C2×C4#57711yescharacteristic
476C2×C4#57811yescharacteristic
477C8—24no—
478C8—44no—
479C8—42no—
480C8—42no—
481C8—81no—
482C8—82no—
483C22—42no—
484C22—42no—
485C22—42no—
486C22—81no—
487C22—81no—
488C22—81no—
489C22—81no—
490C22Q8×C22⋊C411yescharacteristic
491C22C2×#40811yescharacteristic
492C22#288011yescharacteristic
493C22C22×#10211yescharacteristic
494C22#289811yescharacteristic
495C22C22×#10411yescharacteristic
496C22#290111yescharacteristic
497C4—21no—
498C4—21no—
499C4—24no—
500C4—22no—
501C4—22no—
502C4—24no—
503C4—24no—
504C4—42no—
505C4—42no—
506C4—41no—
507C4—42no—
508C4—81no—
509C4—82no—
510C4—82no—
511C4#149214yes—
512C4#296011yescharacteristic
513C4#296211yescharacteristic
514C4#298511yescharacteristic
515C4#298611yescharacteristic
516C4#335811yescharacteristic
517C4#335911yescharacteristic
518C4Q8×D811yescharacteristic
519C4Q8×SD1611yescharacteristic
520C2—82no—
521C2C2×#149211yescharacteristic
522C2#1748411yescharacteristic
523C2#1750711yescharacteristic
524C2#1751911yescharacteristic
525C2#2250911yescharacteristic
526C2Q8×#10211yescharacteristic
527C2Q8×#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. #22722C2semidirect
3. C4×C8×Q8C2semidirect
4. #9870C2semidirect
5. #9870C2semidirect
6. #9869C2semidirect
7. #9865C2semidirect
146. D4⋊C8Q8semidirect

Maximal subgroups

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

subgroupindex
#349292
#227222
C4×C8×Q82
#98702
#98692
#98652
#85672
#85632
#81772
C4×D4⋊C82