Finite Group Databasebeta

← #224320

Subgroups — #224320

Order 512.

950 conjugacy classes of subgroups.

The subgroup lattice has 523 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C4×#554C211yesmaximal, characteristic
2#22825C211yesmaximal, characteristic, complemented
3#21339C211yesmaximal, characteristic, complemented
4#9875C212yesmaximal, complemented
5#9870C211yesmaximal, characteristic, complemented
6#9867C212yesmaximal, complemented
7#9864C211yesmaximal, characteristic, complemented
8#8565C212yesmaximal
9#8563C211yesmaximal, characteristic
10#8174C212yesmaximal
11C4×D4⋊C8C211yesmaximal, characteristic
12C2×#554C411yescharacteristic
13#2370C412yes—
14#2369C411yescharacteristic
15#2368—22no—
16#2360—21no—
17#2355C411yescharacteristic
18#2349C412yes—
19D4×C4⋊C4C411yescharacteristic
20#2345—21no—
21C4×C4⋊Q8C2211yescharacteristic
22C4×#125C2212yes—
23C4×C22⋊Q8—22no—
24C4×C22⋊Q8—21no—
25C42×Q8C2211yescharacteristic
26C42×D4C2211yescharacteristic
27C42×D4C2212yes—
28C2×C4×C4⋊C4—21no—
29#1984—21no—
30#1984—21no—
31#1939—21no—
32#1917—22no—
33#1916—22no—
34#1842C2211yescharacteristic
35#1831C2212yes—
36#1770—21no—
37#1767C2212yes—
38C4×C4⋊C8C2211yescharacteristic
39C4×C4⋊C8C2211yescharacteristic
40C4×C4⋊C8C2212yes—
41#1744C2212yes—
42C42×C8C2211yescharacteristic
43C2×D4⋊C8C2212yes—
44C2×D4⋊C8C2211yescharacteristic
45C2×D4⋊C8C2211yescharacteristic
46#1279C2212yes—
47#1279C2212yes—
48#1279C2212yes—
49#1279C2212yes—
50#1279C2211yescharacteristic
51#1279C2212yes—
52#1279C2211yescharacteristic
53#554—24no—
54#554—24no—
55C2×C4⋊Q8C2×C411yescharacteristic
56C2×#125C2×C412yes—
57C2×C22⋊Q8—21no—
58C2×C22⋊Q8—22no—
59C2×C4×Q8—21no—
60C2×C4×Q8C2×C411yescharacteristic
61C2×C4×Q8D411yescharacteristic
62C2×C4×Q8D411yescharacteristic
63C2×C4×D4C2×C412yes—
64C2×C4×D4C2×C412yes—
65C2×C4×D4C2×C411yescharacteristic
66C2×C4×D4C2×C411yescharacteristic
67C2×C4×D4C2311yescharacteristic
68C22×C4⋊C4—21no—
69C22×C4⋊C4—21no—
70C22×C42—21no—
71C22×C42—22no—
72C2×C4⋊C8—21no—
73C2×C4⋊C8—22no—
74C2×C4⋊C8—21no—
75C2×C4⋊C8—21no—
76C2×C4⋊C8C2311yescharacteristic
77C2×C4⋊C8C2312yes—
78C2×C4⋊C8C2311yescharacteristic
79C2×#97—22no—
80C2×C4×C8—21no—
81C2×C4×C8—21no—
82C2×C4×C8—21no—
83C2×C4×C8C2311yescharacteristic
84C2×C4×C8C2312yes—
85C2×C4×C8C2311yescharacteristic
86#391—21no—
87#391—21no—
88#391—22no—
89#389—21no—
90#389—22no—
91#389—21no—
92#389C2×C412yes—
93#389C2×C412yes—
94#389C2×C412yes—
95#387—22no—
96#387—22no—
97#387—22no—
98#387—22no—
99#387—22no—
100#385—22no—
101#385—22no—
102#385—22no—
103#384C2×C411yescharacteristic
104#382C2×C412yes—
105#380—21no—
106#380—22no—
107C4×C4⋊C4—21no—
108C4×C4⋊C4—21no—
109C4×C4⋊C4—22no—
110C4×C4⋊C4—22no—
111C4×C4⋊C4—21no—
112C4×C4⋊C4—22no—
113C4×C4⋊C4C2×C411yescharacteristic
114C4×C4⋊C4C2×C412yes—
115C4×C4⋊C4C2×C412yes—
116C4×C4⋊C4C2×C411yescharacteristic
117C4×C4⋊C4C2×C411yescharacteristic
118C4×C4⋊C4C2×C412yes—
119C4×C4⋊C4D411yescharacteristic
120C4×C4⋊C4D412yes—
121C4×C4⋊C4D412yes—
122C4×C4⋊C4D411yescharacteristic
123C4×C4⋊C4C2311yescharacteristic
124C4×C4⋊C4C2312yes—
125C4×C22⋊C4—21no—
126C4×C22⋊C4—22no—
127C4×C22⋊C4—22no—
128C4×C22⋊C4—21no—
129C4×C22⋊C4—21no—
130C4×C22⋊C4—22no—
131C2×#95—22no—
132C43C2312yes—
133C43C2311yescharacteristic
134#336—22no—
135#336—22no—
136#336—21no—
137#336—22no—
138#336—22no—
139#336—22no—
140#336—21no—
141#336—22no—
142D4⋊C8—24no—
143D4⋊C8—22no—
144D4⋊C8Q814yescomplemented
145C22×Q8—21no—
146C22×Q8C22⋊C411yescharacteristic
147C22×D4C22×C411yescharacteristic
148C23×C4—21no—
149C23×C4—21no—
150C23×C4—22no—
151C23×C4—22no—
152C23×C4—21no—
153C22×C8—21no—
154C22×C8—21no—
155C22×C8—21no—
156C22×C8—22no—
157C22×C8—21no—
158C22×C8—22no—
159C4⋊Q8—22no—
160C4⋊Q8M1614yes—
161#125—24no—
162#125—24no—
163C22⋊Q8—28no—
164C22⋊Q8—216no—
165C4×Q8—22no—
166C4×Q8—22no—
167C4×Q8—22no—
168C4×Q8—22no—
169C4×Q8—42no—
170C4×Q8—81no—
171C4×Q8—81no—
172C4×D4—24no—
173C4×D4—24no—
174C4×D4—24no—
175C4×D4—22no—
176C4×D4—22no—
177C4×D4—21no—
178C4×D4—21no—
179C4×D4—42no—
180C4×D4—41no—
181C4×D4M1614yes—
182C4×D4C2×Q812yes—
183C4×D4#4112yes—
184C2×C4⋊C4—28no—
185C2×C4⋊C4—22no—
186C2×C4⋊C4—21no—
187C2×C4⋊C4—22no—
188C2×C4⋊C4—21no—
189C2×C4⋊C4—22no—
190C2×C4⋊C4—22no—
191C2×C4⋊C4—21no—
192C2×C4⋊C4—22no—
193C2×C4⋊C4—44no—
194C2×C4⋊C4C22⋊C411yescharacteristic
195C2×C4⋊C4C22⋊C411yescharacteristic
196C2×C4⋊C4C22⋊C412yes—
197C2×C4⋊C4C22⋊C411yescharacteristic
198C2×C4⋊C4C22⋊C412yes—
199C2×C4⋊C4C22⋊C412yes—
200C2×C4⋊C4C22×C411yescharacteristic
201C2×C4⋊C4C22×C412yes—
202C2×C4⋊C4C22×C411yescharacteristic
203C2×C4⋊C4C22×C411yescharacteristic
204C2×C4⋊C4C22×C412yes—
205C2×C22⋊C4—22no—
206C2×C22⋊C4—21no—
207C2×C22⋊C4—21no—
208C2×C22⋊C4—21no—
209C2×C22⋊C4—22no—
210C2×C42—21no—
211C2×C42—21no—
212C2×C42—22no—
213C2×C42—21no—
214C2×C42—22no—
215C2×C42—22no—
216C2×C42—42no—
217C2×C42—42no—
218C2×C42—82no—
219C2×C42—82no—
220C2×C42C22×C412yes—
221C2×C42C22×C412yes—
222C2×C42C22×C411yescharacteristic
223C2×C42C22×C411yescharacteristic
224C2×C42C2×D411yescharacteristic
225C2×C42C2×D411yescharacteristic
226C2×C42C2×D412yes—
227C2×C42C2×D412yes—
228C2×C42C2×D412yes—
229C2×C42C2×D412yes—
230C2×C42C2×D411yescharacteristic
231C2×C42C2×D411yescharacteristic
232C2×C42C2411yescharacteristic
233C4⋊C8—21no—
234C4⋊C8—22no—
235C4⋊C8—21no—
236C4⋊C8—21no—
237C4⋊C8—22no—
238C4⋊C8—42no—
239C4⋊C8—44no—
240C4⋊C8—42no—
241C4⋊C8—81no—
242C4⋊C8C2×Q812yes—
243#97—82no—
244C4×C8—21no—
245C4×C8—22no—
246C4×C8—21no—
247C4×C8—21no—
248C4×C8—22no—
249C4×C8—42no—
250C4×C8—42no—
251C4×C8—42no—
252C4×C8C2×Q812yes—
253#95—22no—
254#95—22no—
255#95—22no—
256#95—21no—
257#95—22no—
258#95—22no—
259#95—21no—
260#95—22no—
261#95—22no—
262#95—21no—
263#95—22no—
264#95C22⋊C412yes—
265#95C22⋊C412yes—
266#95C22⋊C412yes—
267C24—21no—
268C2×Q8—22no—
269C2×Q8—24no—
270C2×Q8—22no—
271C2×Q8—22no—
272C2×Q8—81no—
273C2×Q8—81no—
274C2×D4—21no—
275C2×D4—21no—
276C2×D4—21no—
277C2×D4C4×Q812yes—
278C2×D4C2×M1612yes—
279C2×D4#13112yes—
280C22×C4—21no—
281C22×C4—21no—
282C22×C4—22no—
283C22×C4—28no—
284C22×C4—22no—
285C22×C4—22no—
286C22×C4—24no—
287C22×C4—21no—
288C22×C4—22no—
289C22×C4—42no—
290C22×C4—44no—
291C22×C4—44no—
292C22×C4—41no—
293C22×C4—41no—
294C22×C4—41no—
295C22×C4—41no—
296C22×C4—42no—
297C22×C4—42no—
298C22×C4—42no—
299C22×C4—82no—
300C22×C4—82no—
301C22×C4—81no—
302C22×C4—82no—
303C22×C4C2×C22⋊C412yes—
304C22×C4C2×C22⋊C412yes—
305C22×C4C2×C22⋊C412yes—
306C22×C4C2×C22⋊C412yes—
307C22×C4C2×C22⋊C411yescharacteristic
308C22×C4C2×C22⋊C411yescharacteristic
309C22×C4C2×C22⋊C411yescharacteristic
310C22×C4C2×C22⋊C411yescharacteristic
311C22×C4C23×C411yescharacteristic
312C22×C4C22×D411yescharacteristic
313C22×C4C22×D411yescharacteristic
314C2×C8—22no—
315C2×C8—22no—
316C2×C8—22no—
317C2×C8—41no—
318C2×C8—41no—
319C2×C8—41no—
320C2×C8—41no—
321C2×C8—41no—
322C2×C8—41no—
323C2×C8—42no—
324C2×C8—42no—
325C2×C8—42no—
326C2×C8—41no—
327C2×C8—42no—
328C2×C8—41no—
329C2×C8—42no—
330C2×C8—42no—
331C2×C8—41no—
332C4⋊C4—28no—
333C4⋊C4—28no—
334C4⋊C4—21no—
335C4⋊C4—21no—
336C4⋊C4—22no—
337C4⋊C4—22no—
338C4⋊C4—24no—
339C4⋊C4—24no—
340C4⋊C4—22no—
341C4⋊C4—24no—
342C4⋊C4—22no—
343C4⋊C4—21no—
344C4⋊C4—22no—
345C4⋊C4—21no—
346C4⋊C4—22no—
347C4⋊C4—42no—
348C4⋊C4—44no—
349C4⋊C4—42no—
350C4⋊C4—82no—
351C4⋊C4—81no—
352C4⋊C4—82no—
353C4⋊C4C4×Q812yes—
354C4⋊C4C2×M1614yes—
355C4⋊C4C2×M1612yes—
356C22⋊C4—28no—
357C22⋊C4—22no—
358C22⋊C4—22no—
359C22⋊C4—24no—
360C22⋊C4—44no—
361C22⋊C4—42no—
362C42—22no—
363C42—22no—
364C42—21no—
365C42—22no—
366C42—21no—
367C42—22no—
368C42—22no—
369C42—22no—
370C42—22no—
371C42—21no—
372C42—22no—
373C42—21no—
374C42—22no—
375C42—41no—
376C42—42no—
377C42—42no—
378C42—41no—
379C42—41no—
380C42—42no—
381C42—42no—
382C42—42no—
383C42—42no—
384C42—82no—
385C42—82no—
386C42C2×M1612yes—
387C42C2×M1612yes—
388C42#13512yes—
389C42#13512yes—
390C42C22×Q811yescharacteristic
391C42C2×#4111yescharacteristic
392C422−1+411yescharacteristic
393C422−1+411yescharacteristic
394C23—22no—
395C23—22no—
396C23—22no—
397C23—41no—
398C23—41no—
399C23—41no—
400C23—41no—
401C23C22×C22⋊C411yescharacteristic
402Q8—44no—
403Q8—84no—
404D4—22no—
405D4—22no—
406D4—22no—
407D4#44614yes—
408C2×C4—21no—
409C2×C4—21no—
410C2×C4—21no—
411C2×C4—21no—
412C2×C4—21no—
413C2×C4—21no—
414C2×C4—22no—
415C2×C4—22no—
416C2×C4—22no—
417C2×C4—22no—
418C2×C4—24no—
419C2×C4—22no—
420C2×C4—21no—
421C2×C4—22no—
422C2×C4—22no—
423C2×C4—22no—
424C2×C4—22no—
425C2×C4—22no—
426C2×C4—21no—
427C2×C4—21no—
428C2×C4—22no—
429C2×C4—22no—
430C2×C4—22no—
431C2×C4—22no—
432C2×C4—41no—
433C2×C4—41no—
434C2×C4—42no—
435C2×C4—41no—
436C2×C4—41no—
437C2×C4—42no—
438C2×C4—42no—
439C2×C4—42no—
440C2×C4—42no—
441C2×C4—42no—
442C2×C4—48no—
443C2×C4—41no—
444C2×C4—42no—
445C2×C4—41no—
446C2×C4—42no—
447C2×C4—42no—
448C2×C4—41no—
449C2×C4—84no—
450C2×C4—88no—
451C2×C4—88no—
452C2×C4#40712yes—
453C2×C4#40712yes—
454C2×C4#41612yes—
455C2×C4#41612yes—
456C2×C4#42112yes—
457C2×C4C2×C4×Q811yescharacteristic
458C2×C4#51911yescharacteristic
459C2×C4#54811yescharacteristic
460C2×C4#54811yescharacteristic
461C2×C4D4×Q811yescharacteristic
462C2×C4D4×Q811yescharacteristic
463C2×C4C22×M1611yescharacteristic
464C2×C4C22×M1611yescharacteristic
465C2×C4C2×#13111yescharacteristic
466C2×C4#56811yescharacteristic
467C2×C4C2×#13511yescharacteristic
468C2×C4C2×#13511yescharacteristic
469C2×C4#57611yescharacteristic
470C2×C4#57711yescharacteristic
471C8—42no—
472C8—42no—
473C8—42no—
474C8—82no—
475C8—81no—
476C8—82no—
477C22—24no—
478C22—42no—
479C22—42no—
480C22—42no—
481C22—42no—
482C22—42no—
483C22—42no—
484C22Q8×C22⋊C411yescharacteristic
485C22C2×#40711yescharacteristic
486C22#288011yescharacteristic
487C22C2×#41611yescharacteristic
488C22#289711yescharacteristic
489C22C2×#42111yescharacteristic
490C22#290111yescharacteristic
491C4—21no—
492C4—21no—
493C4—22no—
494C4—22no—
495C4—42no—
496C4—42no—
497C4—42no—
498C4—44no—
499C4—42no—
500C4—41no—
501C4—42no—
502C4—42no—
503C4—81no—
504C4—82no—
505C4#148814yes—
506C4C2×#44611yescharacteristic
507C4Q8×M1611yescharacteristic
508C4#298311yescharacteristic
509C4#298411yescharacteristic
510C4#336011yescharacteristic
511C4#336111yescharacteristic
512C4#338011yescharacteristic
513C4#338111yescharacteristic
514C2—44no—
515C2C2×#148811yescharacteristic
516C2#1745311yescharacteristic
517C2#1750811yescharacteristic
518C2#1752211yescharacteristic
519C2#2251011yescharacteristic
520C2#2251911yescharacteristic
521C2#2252911yescharacteristic
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. #21339C2semidirect
4. #9875C2semidirect
5. #9870C2semidirect
6. #9867C2semidirect
7. #9864C2semidirect
144. D4⋊C8Q8semidirect

Maximal subgroups

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

subgroupindex
C4×#5542
#228252
#213392
#98752
#98702
#98672
#98642
#85652
#85632
#81742
C4×D4⋊C82