Finite Group Databasebeta

← #224106

Subgroups — #224106

Order 512.

872 conjugacy classes of subgroups.

The subgroup lattice has 514 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34941C211yesmaximal, characteristic
2Q8×C4⋊C8C211yesmaximal, characteristic, complemented
3C4×#446C211yesmaximal, characteristic, complemented
4#9876C212yesmaximal, complemented
5#9870C211yesmaximal, characteristic, complemented
6#9866C212yesmaximal, complemented
7#9864C211yesmaximal, characteristic, complemented
8#8567C212yesmaximal
9#8563C211yesmaximal, characteristic
10#8181C212yesmaximal
11C4×D4⋊C8C211yesmaximal, characteristic
12C2×#446—21no—
13C2×C8×Q8—21no—
14#2574—22no—
15#2567C411yescharacteristic
16#2563C411yescharacteristic
17#2552C411yescharacteristic
18#2538C411yescharacteristic
19#2508C412yes—
20#2497—21no—
21#2454C412yes—
22#2423—21no—
23#2396—22no—
24C4×C4⋊Q8C2211yescharacteristic
25C4×#125C2212yes—
26C4×C22⋊Q8—21no—
27#2293C2212yes—
28#2285—21no—
29C42×Q8C2211yescharacteristic
30C42×D4C2211yescharacteristic
31#1939—22no—
32#1939—21no—
33#1842C2211yescharacteristic
34#1842C2212yes—
35#1771—22no—
36#1770—21no—
37C4×C4⋊C8C2211yescharacteristic
38C4×C4⋊C8C2212yes—
39C4×C4⋊C8C2211yescharacteristic
40C4×C4⋊C8C2212yes—
41C4×#97C2212yes—
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
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—
65#446—28no—
66C8×Q8—44no—
67C2×C4⋊C8—21no—
68C2×C4⋊C8—22no—
69C2×C4⋊C8—22no—
70C2×C4⋊C8—21no—
71C2×C4⋊C8—22no—
72C2×C4⋊C8—21no—
73C2×C4⋊C8C2312yes—
74C2×C4⋊C8C2311yescharacteristic
75C2×C4⋊C8C2311yescharacteristic
76C2×#97—22no—
77C2×#97—22no—
78C2×C4×C8—21no—
79C2×C4×C8—22no—
80C2×C4×C8—21no—
81C2×C4×C8—21no—
82C2×C4×C8C2311yescharacteristic
83C2×C4×C8C2312yes—
84C2×C4×C8C2311yescharacteristic
85#400—22no—
86#399—22no—
87#398—22no—
88#398—22no—
89#396—22no—
90#396—22no—
91#395—22no—
92#393—22no—
93#391—21no—
94#391—21no—
95#390C2×C412yes—
96#390C2×C412yes—
97#389—21no—
98#389—21no—
99#389C2×C412yes—
100#389C2×C412yes—
101#389C2×C412yes—
102#389C2×C412yes—
103#389C2×C412yes—
104#388—22no—
105#388—22no—
106#387—22no—
107#387—22no—
108#387—22no—
109#385—22no—
110#384—21no—
111#384—21no—
112#384C2×C411yescharacteristic
113#382—22no—
114#382C2×C412yes—
115#380—21no—
116#380—21no—
117#380—21no—
118#380—22no—
119C4×C4⋊C4—22no—
120C4×C4⋊C4—22no—
121C4×C4⋊C4—21no—
122C4×C4⋊C4C2×C411yescharacteristic
123C4×C4⋊C4C2×C411yescharacteristic
124C4×C4⋊C4C2×C411yescharacteristic
125C4×C4⋊C4C2×C412yes—
126C4×C4⋊C4D412yes—
127C4×C4⋊C4D412yes—
128C4×C4⋊C4D411yescharacteristic
129C4×C4⋊C4D411yescharacteristic
130C4×C4⋊C4C2311yescharacteristic
131C4×C4⋊C4C2312yes—
132C4×C22⋊C4—21no—
133C4×C22⋊C4—21no—
134C4×C22⋊C4—22no—
135C4×C22⋊C4—21no—
136C43C2312yes—
137C43C2311yescharacteristic
138#336—22no—
139#336—21no—
140#336—22no—
141#336—21no—
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—21no—
151C22×C8—21no—
152C22×C8—21no—
153C22×C8—22no—
154C22×C8—21no—
155C22×C8—22no—
156C22×C8—21no—
157C4⋊Q8—22no—
158C4⋊Q8—22no—
159#125—24no—
160#125—42no—
161C22⋊Q8—44no—
162C4×Q8—24no—
163C4×Q8—22no—
164C4×Q8—22no—
165C4×Q8—22no—
166C4×Q8—22no—
167C4×Q8—81no—
168C4×Q8—81no—
169C4×D4—22no—
170C4×D4—22no—
171C4×D4—22no—
172C4×D4—21no—
173C4×D4—21no—
174C4×D4—41no—
175C4×D4C2×Q812yes—
176C4×D4#4112yes—
177#117—82no—
178#117—82no—
179C2×C4⋊C4—21no—
180C2×C4⋊C4—22no—
181C2×C4⋊C4—22no—
182C2×C4⋊C4—22no—
183C2×C4⋊C4—21no—
184C2×C4⋊C4—22no—
185C2×C4⋊C4—22no—
186C2×C4⋊C4—21no—
187C2×C4⋊C4—22no—
188C2×C4⋊C4—22no—
189C2×C4⋊C4C22⋊C411yescharacteristic
190C2×C4⋊C4C22⋊C412yes—
191C2×C4⋊C4C22⋊C411yescharacteristic
192C2×C4⋊C4C22⋊C411yescharacteristic
193C2×C4⋊C4C22⋊C412yes—
194C2×C4⋊C4C22⋊C412yes—
195C2×C4⋊C4C22×C411yescharacteristic
196C2×C4⋊C4C22×C412yes—
197C2×C4⋊C4C22×C412yes—
198C2×C4⋊C4C22×C411yescharacteristic
199C2×C4⋊C4C22×C411yescharacteristic
200C2×C22⋊C4—21no—
201C2×C22⋊C4—21no—
202C2×C22⋊C4—21no—
203C2×C22⋊C4—22no—
204C2×C22⋊C4—22no—
205C2×C22⋊C4—22no—
206C2×C22⋊C4—22no—
207C2×C42—21no—
208C2×C42—21no—
209C2×C42—22no—
210C2×C42—21no—
211C2×C42—42no—
212C2×C42—42no—
213C2×C42C22×C411yescharacteristic
214C2×C42C22×C412yes—
215C2×C42C22×C412yes—
216C2×C42C22×C411yescharacteristic
217C2×C42C2×D411yescharacteristic
218C2×C42C2×D412yes—
219C2×C42C2×D412yes—
220C2×C42C2×D411yescharacteristic
221C2×C42C2×D412yes—
222C2×C42C2×D411yescharacteristic
223C2×C42C2×D412yes—
224C2×C42C2×D411yescharacteristic
225C2×C42C2411yescharacteristic
226C4⋊C8—24no—
227C4⋊C8—28no—
228C4⋊C8—22no—
229C4⋊C8—21no—
230C4⋊C8—21no—
231C4⋊C8—22no—
232C4⋊C8—21no—
233C4⋊C8—44no—
234C4⋊C8—42no—
235C4⋊C8—82no—
236C4⋊C8—81no—
237C4⋊C8C2×Q812yes—
238#97—28no—
239#97—44no—
240C4×C8—24no—
241C4×C8—21no—
242C4×C8—22no—
243C4×C8—21no—
244C4×C8—21no—
245C4×C8—22no—
246C4×C8—44no—
247C4×C8—42no—
248C4×C8—42no—
249C4×C8C2×Q812yes—
250#95—21no—
251#95—21no—
252#95—22no—
253#95—22no—
254#95—22no—
255#95—22no—
256#95—22no—
257#95—22no—
258#95—22no—
259#95—21no—
260#95—22no—
261#95C22⋊C412yes—
262#95C22⋊C412yes—
263#95C22⋊C412yes—
264C24—21no—
265C2×Q8—22no—
266C2×Q8—22no—
267C2×Q8—42no—
268C2×Q8—81no—
269C2×Q8—81no—
270C2×Q8#10414yes—
271C2×D4—21no—
272C2×D4—21no—
273C2×D4—21no—
274C2×D4—21no—
275C2×D4—21no—
276C2×D4C4×Q812yes—
277C22×C4—21no—
278C22×C4—21no—
279C22×C4—22no—
280C22×C4—22no—
281C22×C4—21no—
282C22×C4—22no—
283C22×C4—22no—
284C22×C4—42no—
285C22×C4—41no—
286C22×C4—41no—
287C22×C4—41no—
288C22×C4—41no—
289C22×C4—42no—
290C22×C4—42no—
291C22×C4—42no—
292C22×C4—42no—
293C22×C4—81no—
294C22×C4C2×C22⋊C412yes—
295C22×C4C2×C22⋊C412yes—
296C22×C4C2×C22⋊C411yescharacteristic
297C22×C4C2×C22⋊C412yes—
298C22×C4C2×C22⋊C411yescharacteristic
299C22×C4C2×C22⋊C411yescharacteristic
300C22×C4C2×C22⋊C412yes—
301C22×C4C2×C22⋊C411yescharacteristic
302C22×C4C23×C411yescharacteristic
303C22×C4C22×D411yescharacteristic
304C22×C4C22×D411yescharacteristic
305C2×C8—22no—
306C2×C8—22no—
307C2×C8—22no—
308C2×C8—22no—
309C2×C8—24no—
310C2×C8—22no—
311C2×C8—41no—
312C2×C8—42no—
313C2×C8—41no—
314C2×C8—42no—
315C2×C8—41no—
316C2×C8—41no—
317C2×C8—42no—
318C2×C8—41no—
319C2×C8—42no—
320C2×C8—41no—
321C2×C8—41no—
322C2×C8—42no—
323C4⋊C4—21no—
324C4⋊C4—21no—
325C4⋊C4—24no—
326C4⋊C4—22no—
327C4⋊C4—24no—
328C4⋊C4—22no—
329C4⋊C4—24no—
330C4⋊C4—22no—
331C4⋊C4—21no—
332C4⋊C4—21no—
333C4⋊C4—22no—
334C4⋊C4—22no—
335C4⋊C4—22no—
336C4⋊C4—22no—
337C4⋊C4—44no—
338C4⋊C4—44no—
339C4⋊C4—42no—
340C4⋊C4—42no—
341C4⋊C4—44no—
342C4⋊C4—82no—
343C4⋊C4—82no—
344C4⋊C4—82no—
345C4⋊C4—81no—
346C4⋊C4—82no—
347C4⋊C4C4×Q812yes—
348C4⋊C4#13112yes—
349C22⋊C4—22no—
350C22⋊C4—22no—
351C22⋊C4—42no—
352C22⋊C4—42no—
353C22⋊C4—44no—
354C22⋊C4—82no—
355C22⋊C4—82no—
356C22⋊C4—82no—
357C42—21no—
358C42—22no—
359C42—22no—
360C42—22no—
361C42—21no—
362C42—22no—
363C42—22no—
364C42—22no—
365C42—22no—
366C42—21no—
367C42—22no—
368C42—21no—
369C42—22no—
370C42—24no—
371C42—41no—
372C42—42no—
373C42—41no—
374C42—41no—
375C42—42no—
376C42—42no—
377C42—42no—
378C42—42no—
379C42#13112yes—
380C42#13112yes—
381C42AGL(1,ℤ/8)12yes—
382C42#13712yes—
383C42C22×Q811yescharacteristic
384C42C2×#4111yescharacteristic
385C422−1+411yescharacteristic
386C422−1+411yescharacteristic
387C23—22no—
388C23—41no—
389C23—41no—
390C23—41no—
391C23—41no—
392C23—41no—
393C23—41no—
394C23C22×C22⋊C411yescharacteristic
395Q8—28no—
396Q8—84no—
397D4—22no—
398D4—22no—
399D4—41no—
400D4—41no—
401C2×C4—22no—
402C2×C4—21no—
403C2×C4—22no—
404C2×C4—21no—
405C2×C4—21no—
406C2×C4—21no—
407C2×C4—21no—
408C2×C4—21no—
409C2×C4—22no—
410C2×C4—22no—
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—41no—
424C2×C4—41no—
425C2×C4—41no—
426C2×C4—41no—
427C2×C4—42no—
428C2×C4—42no—
429C2×C4—42no—
430C2×C4—42no—
431C2×C4—41no—
432C2×C4—42no—
433C2×C4—42no—
434C2×C4—41no—
435C2×C4—42no—
436C2×C4—42no—
437C2×C4—42no—
438C2×C4—41no—
439C2×C4—42no—
440C2×C4—84no—
441C2×C4—84no—
442C2×C4#40812yes—
443C2×C4#40812yes—
444C2×C4#41712yes—
445C2×C4#41712yes—
446C2×C4C2×#10414yes—
447C2×C4C2×#10412yes—
448C2×C4C2×C4×Q811yescharacteristic
449C2×C4#51911yescharacteristic
450C2×C4#54811yescharacteristic
451C2×C4#54811yescharacteristic
452C2×C4D4×Q811yescharacteristic
453C2×C4D4×Q811yescharacteristic
454C2×C4C2×#13111yescharacteristic
455C2×C4C2×#13111yescharacteristic
456C2×C4C2×#13111yescharacteristic
457C2×C4#56811yescharacteristic
458C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
459C2×C4C2×#13711yescharacteristic
460C2×C4#57611yescharacteristic
461C2×C4#57711yescharacteristic
462C8—24no—
463C8—42no—
464C8—44no—
465C8—42no—
466C8—82no—
467C8—81no—
468C22—42no—
469C22—42no—
470C22—42no—
471C22—81no—
472C22—81no—
473C22—81no—
474C22—81no—
475C22Q8×C22⋊C411yescharacteristic
476C22C2×#40811yescharacteristic
477C22#288011yescharacteristic
478C22C2×#41711yescharacteristic
479C22#289711yescharacteristic
480C22C22×#10411yescharacteristic
481C22#290111yescharacteristic
482C4—21no—
483C4—21no—
484C4—24no—
485C4—22no—
486C4—22no—
487C4—42no—
488C4—42no—
489C4—42no—
490C4—41no—
491C4—42no—
492C4—42no—
493C4—81no—
494C4—82no—
495C4—82no—
496C4#148314yes—
497C4#296011yescharacteristic
498C4#296211yescharacteristic
499C4#298511yescharacteristic
500C4#298611yescharacteristic
501C4#336611yescharacteristic
502C4#337211yescharacteristic
503C4#338411yescharacteristic
504C4#338511yescharacteristic
505C2—82no—
506C2C2×#148311yescharacteristic
507C2#1746811yescharacteristic
508C2#1751211yescharacteristic
509C2#1752011yescharacteristic
510C2#2250911yescharacteristic
511C2#2252211yescharacteristic
512C2Q8×#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. Q8×C4⋊C8C2semidirect
3. C4×#446C2semidirect
4. #9876C2semidirect
5. #9870C2semidirect
6. #9866C2semidirect
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
#349412
Q8×C4⋊C82
C4×#4462
#98762
#98702
#98662
#98642
#85672
#85632
#81812
C4×D4⋊C82