Finite Group Databasebeta

← #224334

Subgroups — #224334

Order 512.

1010 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
1C4×#554C211yesmaximal, characteristic
2#22722C211yesmaximal, characteristic, complemented
3C4×#446C211yesmaximal, characteristic, complemented
4#9876C212yesmaximal, complemented
5#9870C211yesmaximal, characteristic, complemented
6#9867C212yesmaximal, complemented
7#9864C211yesmaximal, characteristic, complemented
8#8563C211yesmaximal, characteristic
9#8563C212yesmaximal
10#8172C212yesmaximal
11C4×D4⋊C8C211yesmaximal, characteristic
12C2×#554C411yescharacteristic
13C2×#446—21no—
14C2×#446—21no—
15#2370C412yes—
16#2369C411yescharacteristic
17#2368—22no—
18#2360—21no—
19#2355C411yescharacteristic
20#2349C412yes—
21D4×C4⋊C4C411yescharacteristic
22#2345—21no—
23C4×C4⋊Q8C2211yescharacteristic
24C4×#125C2212yes—
25C4×C22⋊Q8—22no—
26C4×C22⋊Q8—21no—
27C42×Q8C2211yescharacteristic
28C42×D4C2211yescharacteristic
29C42×D4C2212yes—
30C2×C4×C4⋊C4—21no—
31#1940—22no—
32#1939—21no—
33#1842C2211yescharacteristic
34#1842C2212yes—
35#1771—22no—
36#1770—21no—
37#1767C2212yes—
38C4×C4⋊C8C2211yescharacteristic
39C4×C4⋊C8C2211yescharacteristic
40C4×C4⋊C8C2212yes—
41C4×#97C2212yes—
42C42×C8C2211yescharacteristic
43C2×D4⋊C8C2211yescharacteristic
44C2×D4⋊C8C2212yes—
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—
72#446—28no—
73#446—44no—
74C2×C4⋊C8—21no—
75C2×C4⋊C8—22no—
76C2×C4⋊C8—21no—
77C2×C4⋊C8—22no—
78C2×C4⋊C8—22no—
79C2×C4⋊C8—21no—
80C2×C4⋊C8C2311yescharacteristic
81C2×C4⋊C8C2312yes—
82C2×C4⋊C8C2311yescharacteristic
83C2×#97—22no—
84C2×#97—22no—
85C2×#97—22no—
86C2×C4×C8—21no—
87C2×C4×C8—21no—
88C2×C4×C8—21no—
89C2×C4×C8C2311yescharacteristic
90C2×C4×C8C2312yes—
91C2×C4×C8C2311yescharacteristic
92#391—21no—
93#391—21no—
94#391—22no—
95#389—21no—
96#389—22no—
97#389—21no—
98#389C2×C412yes—
99#389C2×C412yes—
100#389C2×C412yes—
101#387—22no—
102#387—22no—
103#387—22no—
104#387—22no—
105#387—22no—
106#385—22no—
107#385—22no—
108#385—22no—
109#384C2×C411yescharacteristic
110#382C2×C412yes—
111#380—21no—
112#380—22no—
113C4×C4⋊C4—21no—
114C4×C4⋊C4—21no—
115C4×C4⋊C4—22no—
116C4×C4⋊C4—22no—
117C4×C4⋊C4—21no—
118C4×C4⋊C4—22no—
119C4×C4⋊C4C2×C411yescharacteristic
120C4×C4⋊C4C2×C412yes—
121C4×C4⋊C4C2×C412yes—
122C4×C4⋊C4C2×C411yescharacteristic
123C4×C4⋊C4C2×C411yescharacteristic
124C4×C4⋊C4C2×C412yes—
125C4×C4⋊C4D411yescharacteristic
126C4×C4⋊C4D412yes—
127C4×C4⋊C4D411yescharacteristic
128C4×C4⋊C4D412yes—
129C4×C4⋊C4C2311yescharacteristic
130C4×C4⋊C4C2312yes—
131C4×C22⋊C4—21no—
132C4×C22⋊C4—22no—
133C4×C22⋊C4—22no—
134C4×C22⋊C4—21no—
135C4×C22⋊C4—21no—
136C4×C22⋊C4—22no—
137C2×#95—22no—
138C43C2312yes—
139C43C2311yescharacteristic
140#336—22no—
141#336—21no—
142#336—22no—
143#336—21no—
144D4⋊C8—22no—
145D4⋊C8—24no—
146D4⋊C8Q814yescomplemented
147C22×Q8—21no—
148C22×Q8C22⋊C411yescharacteristic
149C22×D4C22×C411yescharacteristic
150C23×C4—21no—
151C23×C4—21no—
152C23×C4—22no—
153C23×C4—22no—
154C23×C4—21no—
155C22×C8—21no—
156C22×C8—21no—
157C22×C8—22no—
158C22×C8—21no—
159C22×C8—21no—
160C22×C8—22no—
161C4⋊Q8—22no—
162C4⋊Q8M1614yes—
163#125—24no—
164#125—24no—
165C22⋊Q8—28no—
166C22⋊Q8—216no—
167C4×Q8—24no—
168C4×Q8—22no—
169C4×Q8—22no—
170C4×Q8—22no—
171C4×Q8—22no—
172C4×Q8—81no—
173C4×Q8—81no—
174C4×D4—24no—
175C4×D4—24no—
176C4×D4—24no—
177C4×D4—22no—
178C4×D4—22no—
179C4×D4—21no—
180C4×D4—21no—
181C4×D4—42no—
182C4×D4—41no—
183C4×D4M1614yes—
184C4×D4C2×Q812yes—
185C4×D4#4112yes—
186C2×C4⋊C4—28no—
187C2×C4⋊C4—22no—
188C2×C4⋊C4—21no—
189C2×C4⋊C4—22no—
190C2×C4⋊C4—21no—
191C2×C4⋊C4—22no—
192C2×C4⋊C4—22no—
193C2×C4⋊C4—21no—
194C2×C4⋊C4—22no—
195C2×C4⋊C4—44no—
196C2×C4⋊C4C22⋊C411yescharacteristic
197C2×C4⋊C4C22⋊C412yes—
198C2×C4⋊C4C22⋊C411yescharacteristic
199C2×C4⋊C4C22⋊C412yes—
200C2×C4⋊C4C22⋊C412yes—
201C2×C4⋊C4C22⋊C411yescharacteristic
202C2×C4⋊C4C22×C411yescharacteristic
203C2×C4⋊C4C22×C412yes—
204C2×C4⋊C4C22×C411yescharacteristic
205C2×C4⋊C4C22×C411yescharacteristic
206C2×C4⋊C4C22×C412yes—
207C2×C22⋊C4—22no—
208C2×C22⋊C4—21no—
209C2×C22⋊C4—21no—
210C2×C22⋊C4—21no—
211C2×C22⋊C4—22no—
212C2×C42—21no—
213C2×C42—21no—
214C2×C42—22no—
215C2×C42—21no—
216C2×C42—22no—
217C2×C42—22no—
218C2×C42—42no—
219C2×C42—42no—
220C2×C42—82no—
221C2×C42—82no—
222C2×C42C22×C412yes—
223C2×C42C22×C412yes—
224C2×C42C22×C411yescharacteristic
225C2×C42C22×C411yescharacteristic
226C2×C42C2×D412yes—
227C2×C42C2×D411yescharacteristic
228C2×C42C2×D411yescharacteristic
229C2×C42C2×D412yes—
230C2×C42C2×D412yes—
231C2×C42C2×D412yes—
232C2×C42C2×D411yescharacteristic
233C2×C42C2×D411yescharacteristic
234C2×C42C2411yescharacteristic
235C4⋊C8—24no—
236C4⋊C8—28no—
237C4⋊C8—21no—
238C4⋊C8—22no—
239C4⋊C8—21no—
240C4⋊C8—21no—
241C4⋊C8—22no—
242C4⋊C8—44no—
243C4⋊C8—42no—
244C4⋊C8—82no—
245C4⋊C8—81no—
246C4⋊C8C2×Q812yes—
247#97—28no—
248#97—44no—
249#97—44no—
250C4×C8—24no—
251C4×C8—21no—
252C4×C8—22no—
253C4×C8—21no—
254C4×C8—21no—
255C4×C8—22no—
256C4×C8—42no—
257C4×C8—42no—
258C4×C8C2×Q812yes—
259#95—22no—
260#95—22no—
261#95—22no—
262#95—21no—
263#95—22no—
264#95—22no—
265#95—21no—
266#95—22no—
267#95—22no—
268#95—21no—
269#95—22no—
270#95C22⋊C412yes—
271#95C22⋊C412yes—
272#95C22⋊C412yes—
273C24—21no—
274C2×Q8—22no—
275C2×Q8—24no—
276C2×Q8—22no—
277C2×Q8—22no—
278C2×Q8—81no—
279C2×Q8—81no—
280C2×D4—21no—
281C2×D4—21no—
282C2×D4—21no—
283C2×D4C4×Q812yes—
284C2×D4C2×M1612yes—
285C2×D4#13112yes—
286C22×C4—21no—
287C22×C4—21no—
288C22×C4—22no—
289C22×C4—28no—
290C22×C4—22no—
291C22×C4—22no—
292C22×C4—24no—
293C22×C4—21no—
294C22×C4—22no—
295C22×C4—44no—
296C22×C4—42no—
297C22×C4—41no—
298C22×C4—41no—
299C22×C4—41no—
300C22×C4—41no—
301C22×C4—42no—
302C22×C4—42no—
303C22×C4—42no—
304C22×C4—44no—
305C22×C4—82no—
306C22×C4—82no—
307C22×C4—81no—
308C22×C4—82no—
309C22×C4C2×C22⋊C412yes—
310C22×C4C2×C22⋊C412yes—
311C22×C4C2×C22⋊C412yes—
312C22×C4C2×C22⋊C411yescharacteristic
313C22×C4C2×C22⋊C411yescharacteristic
314C22×C4C2×C22⋊C411yescharacteristic
315C22×C4C2×C22⋊C412yes—
316C22×C4C2×C22⋊C411yescharacteristic
317C22×C4C23×C411yescharacteristic
318C22×C4C22×D411yescharacteristic
319C22×C4C22×D411yescharacteristic
320C2×C8—22no—
321C2×C8—22no—
322C2×C8—22no—
323C2×C8—22no—
324C2×C8—24no—
325C2×C8—22no—
326C2×C8—41no—
327C2×C8—42no—
328C2×C8—41no—
329C2×C8—42no—
330C2×C8—41no—
331C2×C8—41no—
332C2×C8—41no—
333C2×C8—42no—
334C2×C8—41no—
335C2×C8—42no—
336C2×C8—41no—
337C2×C8—42no—
338C4⋊C4—28no—
339C4⋊C4—28no—
340C4⋊C4—21no—
341C4⋊C4—21no—
342C4⋊C4—22no—
343C4⋊C4—22no—
344C4⋊C4—24no—
345C4⋊C4—24no—
346C4⋊C4—22no—
347C4⋊C4—24no—
348C4⋊C4—24no—
349C4⋊C4—21no—
350C4⋊C4—22no—
351C4⋊C4—21no—
352C4⋊C4—22no—
353C4⋊C4—44no—
354C4⋊C4—42no—
355C4⋊C4—82no—
356C4⋊C4—81no—
357C4⋊C4—82no—
358C4⋊C4#10414yes—
359C4⋊C4C4×Q812yes—
360C4⋊C4C2×M1614yes—
361C4⋊C4C2×M1612yes—
362C22⋊C4—28no—
363C22⋊C4—22no—
364C22⋊C4—22no—
365C22⋊C4—24no—
366C22⋊C4—44no—
367C22⋊C4—42no—
368C42—22no—
369C42—22no—
370C42—21no—
371C42—22no—
372C42—21no—
373C42—22no—
374C42—22no—
375C42—22no—
376C42—22no—
377C42—21no—
378C42—22no—
379C42—21no—
380C42—22no—
381C42—24no—
382C42—41no—
383C42—42no—
384C42—41no—
385C42—41no—
386C42—42no—
387C42—42no—
388C42—42no—
389C42—42no—
390C42—82no—
391C42—82no—
392C42C2×M1612yes—
393C42C2×M1612yes—
394C42AGL(1,ℤ/8)12yes—
395C42#13712yes—
396C42C22×Q811yescharacteristic
397C42C2×#4111yescharacteristic
398C422−1+411yescharacteristic
399C422−1+411yescharacteristic
400C23—22no—
401C23—22no—
402C23—22no—
403C23—41no—
404C23—41no—
405C23—41no—
406C23—41no—
407C23C22×C22⋊C411yescharacteristic
408Q8—44no—
409Q8—84no—
410D4—22no—
411D4—22no—
412D4—22no—
413D4#44614yescomplemented
414C2×C4—21no—
415C2×C4—22no—
416C2×C4—22no—
417C2×C4—21no—
418C2×C4—21no—
419C2×C4—21no—
420C2×C4—21no—
421C2×C4—21no—
422C2×C4—22no—
423C2×C4—22no—
424C2×C4—22no—
425C2×C4—22no—
426C2×C4—24no—
427C2×C4—22no—
428C2×C4—21no—
429C2×C4—22no—
430C2×C4—22no—
431C2×C4—22no—
432C2×C4—21no—
433C2×C4—21no—
434C2×C4—22no—
435C2×C4—22no—
436C2×C4—22no—
437C2×C4—41no—
438C2×C4—41no—
439C2×C4—42no—
440C2×C4—41no—
441C2×C4—41no—
442C2×C4—42no—
443C2×C4—42no—
444C2×C4—42no—
445C2×C4—42no—
446C2×C4—42no—
447C2×C4—48no—
448C2×C4—41no—
449C2×C4—42no—
450C2×C4—41no—
451C2×C4—42no—
452C2×C4—42no—
453C2×C4—41no—
454C2×C4—84no—
455C2×C4—88no—
456C2×C4—88no—
457C2×C4#40712yes—
458C2×C4#40712yes—
459C2×C4#41712yes—
460C2×C4#41712yes—
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×C4C22×M1611yescharacteristic
470C2×C4C22×M1611yescharacteristic
471C2×C4C2×#13111yescharacteristic
472C2×C4#56811yescharacteristic
473C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
474C2×C4C2×#13711yescharacteristic
475C2×C4#57611yescharacteristic
476C2×C4#57711yescharacteristic
477C8—24no—
478C8—42no—
479C8—44no—
480C8—42no—
481C8—81no—
482C8—82no—
483C22—24no—
484C22—42no—
485C22—42no—
486C22—42no—
487C22—42no—
488C22—42no—
489C22—42no—
490C22Q8×C22⋊C411yescharacteristic
491C22C2×#40711yescharacteristic
492C22#288011yescharacteristic
493C22C2×#41711yescharacteristic
494C22#289711yescharacteristic
495C22C22×#10411yescharacteristic
496C22#290111yescharacteristic
497C4—21no—
498C4—21no—
499C4—22no—
500C4—22no—
501C4—24no—
502C4—42no—
503C4—42no—
504C4—42no—
505C4—44no—
506C4—42no—
507C4—41no—
508C4—42no—
509C4—81no—
510C4—82no—
511C4#148914yes—
512C4C2×#44611yescharacteristic
513C4Q8×M1611yescharacteristic
514C4#298311yescharacteristic
515C4#298411yescharacteristic
516C4#336611yescharacteristic
517C4#337211yescharacteristic
518C4#338411yescharacteristic
519C4#338511yescharacteristic
520C2—44no—
521C2C2×#148911yescharacteristic
522C2#1745311yescharacteristic
523C2#1751211yescharacteristic
524C2#1751711yescharacteristic
525C2#2251011yescharacteristic
526C2#2252211yescharacteristic
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×#446C2semidirect
4. #9876C2semidirect
5. #9870C2semidirect
6. #9867C2semidirect
7. #9864C2semidirect
146. D4⋊C8Q8semidirect
413. D4#446semidirect

Maximal subgroups

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

subgroupindex
C4×#5542
#227222
C4×#4462
#98762
#98702
#98672
#98642
#85632
#81722
C4×D4⋊C82