Finite Group Databasebeta

← #224569

Subgroups — #224569

Order 512.

852 conjugacy classes of subgroups.

The subgroup lattice has 527 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34954C211yesmaximal, characteristic
2#22722C211yesmaximal, characteristic, complemented
3#21339C211yesmaximal, characteristic, complemented
4#9676C212yesmaximal, complemented
5#9674C211yesmaximal, characteristic
6#9640C212yesmaximal, complemented
7#9638C211yesmaximal, characteristic
8#8567C212yesmaximal, complemented
9#8564C211yesmaximal, characteristic, complemented
10#8179C212yesmaximal, complemented
11C4×Q8⋊C8C211yesmaximal, characteristic, complemented
12C2×#446—21no—
13#2593—22no—
14#2582C411yescharacteristic
15#2571C411yescharacteristic
16#2548C411yescharacteristic
17#2548C411yescharacteristic
18#2513—21no—
19#2513C412yes—
20#2448—21no—
21#2448C412yes—
22#2395—22no—
23C4×#127C2211yescharacteristic
24C4×#124—21no—
25C4×#124C2212yes—
26#2293—21no—
27#2293C2212yes—
28C42×Q8C2211yescharacteristic
29C42×Q8C2211yescharacteristic
30#1984—21no—
31#1940—22no—
32#1939—21no—
33#1939—21no—
34#1916—22no—
35#1842C2211yescharacteristic
36#1842C2212yes—
37#1767C2212yes—
38C4×C4⋊C8C2211yescharacteristic
39C4×C4⋊C8C2211yescharacteristic
40C4×C4⋊C8C2212yes—
41#1744C2212yes—
42C42×C8C2211yescharacteristic
43C2×Q8⋊C8C2211yescharacteristic
44C2×Q8⋊C8C2211yescharacteristic
45C2×Q8⋊C8C2212yes—
46C2×D4⋊C8C2212yes—
47C2×D4⋊C8C2212yes—
48C2×D4⋊C8C2212yes—
49C2×D4⋊C8C2212yes—
50#1279C2212yes—
51#1279C2211yescharacteristic
52#1279C2211yescharacteristic
53C2×#127C2×C411yescharacteristic
54C2×#124—21no—
55C2×#124C2×C412yes—
56C2×C4×Q8C2×C411yescharacteristic
57C2×C4×Q8C2×C411yescharacteristic
58C2×C4×Q8C2×C411yescharacteristic
59C2×C4×Q8D411yescharacteristic
60C2×C4×Q8D411yescharacteristic
61C2×C4×Q8C2311yescharacteristic
62C2×C4×D4—21no—
63C2×C4×D4—21no—
64C2×C4×D4C2312yes—
65C2×#117—22no—
66#446—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⋊C8C2311yescharacteristic
74C2×C4⋊C8C2312yes—
75C2×C4⋊C8C2311yescharacteristic
76C2×#97—22no—
77C2×C4×C8—21no—
78C2×C4×C8—21no—
79C2×C4×C8—21no—
80C2×C4×C8C2311yescharacteristic
81C2×C4×C8C2312yes—
82C2×C4×C8C2311yescharacteristic
83#400—22no—
84#399—22no—
85#398—22no—
86#397—22no—
87#397—22no—
88#397—22no—
89#394—22no—
90#394—22no—
91#391—21no—
92#391C2×C412yes—
93#390—21no—
94#390—21no—
95#390—21no—
96#390—21no—
97#390—21no—
98#390C2×C412yes—
99#390C2×C412yes—
100#390C2×C412yes—
101#390C2×C412yes—
102#389—21no—
103#389—21no—
104#389C2×C412yes—
105#389C2×C412yes—
106#388—22no—
107#388—22no—
108#388—22no—
109#388—22no—
110#388—22no—
111#388—22no—
112#388—22no—
113#388—22no—
114#384C2×C411yescharacteristic
115#382—21no—
116#382C2×C412yes—
117C4×C4⋊C4—21no—
118C4×C4⋊C4—21no—
119C4×C4⋊C4—21no—
120C4×C4⋊C4C2×C411yescharacteristic
121C4×C4⋊C4C2×C411yescharacteristic
122C4×C4⋊C4C2×C411yescharacteristic
123C4×C4⋊C4C2×C412yes—
124C4×C4⋊C4D412yes—
125C4×C4⋊C4D412yes—
126C4×C4⋊C4D411yescharacteristic
127C4×C4⋊C4D411yescharacteristic
128C4×C4⋊C4C2311yescharacteristic
129C4×C22⋊C4—22no—
130C4×C22⋊C4—22no—
131C4×C22⋊C4—22no—
132C43—21no—
133C43C2311yescharacteristic
134C43C2312yes—
135#336—21no—
136#336—22no—
137#336—22no—
138#336—22no—
139#336—22no—
140#336—21no—
141Q8⋊C8—22no—
142Q8⋊C8—22no—
143Q8⋊C8—42no—
144D4⋊C8—24no—
145D4⋊C8—24no—
146D4⋊C8—42no—
147D4⋊C8—42no—
148C22×Q8C22⋊C411yescharacteristic
149C22×Q8C22×C411yescharacteristic
150C22×D4—21no—
151C22×D4—21no—
152C22×D4C22×C412yes—
153C23×C4—22no—
154C22×C8—21no—
155C22×C8—21no—
156C22×C8—21no—
157C22×C8—22no—
158C22×C8—21no—
159C22×C8—22no—
160#127—22no—
161#127—22no—
162#124—24no—
163#124—42no—
164#124—44no—
165C4×Q8—22no—
166C4×Q8—22no—
167C4×Q8—22no—
168C4×Q8—22no—
169C4×Q8—22no—
170C4×Q8—22no—
171C4×Q8—22no—
172C4×Q8—21no—
173C4×Q8—21no—
174C4×Q8—21no—
175C4×Q8—21no—
176C4×Q8—41no—
177C4×Q8—42no—
178C4×D4—22no—
179C4×D4—22no—
180C4×D4—22no—
181C4×D4—22no—
182C4×D4—42no—
183C4×D4—42no—
184C4×D4—81no—
185C4×D4—81no—
186#117—82no—
187#117—82no—
188C2×C4⋊C4—21no—
189C2×C4⋊C4—21no—
190C2×C4⋊C4—21no—
191C2×C4⋊C4—21no—
192C2×C4⋊C4—22no—
193C2×C4⋊C4—22no—
194C2×C4⋊C4—21no—
195C2×C4⋊C4C22⋊C411yescharacteristic
196C2×C4⋊C4C22⋊C412yes—
197C2×C4⋊C4C22⋊C411yescharacteristic
198C2×C4⋊C4C22⋊C412yes—
199C2×C4⋊C4C22⋊C411yescharacteristic
200C2×C4⋊C4C22⋊C412yes—
201C2×C4⋊C4C22×C412yes—
202C2×C4⋊C4C22×C411yescharacteristic
203C2×C4⋊C4C22×C411yescharacteristic
204C2×C4⋊C4C22×C411yescharacteristic
205C2×C22⋊C4—22no—
206C2×C22⋊C4—22no—
207C2×C22⋊C4—22no—
208C2×C22⋊C4—22no—
209C2×C22⋊C4—22no—
210C2×C22⋊C4—22no—
211C2×C22⋊C4—22no—
212C2×C22⋊C4—22no—
213C2×C22⋊C4—22no—
214C2×C22⋊C4—22no—
215C2×C22⋊C4—22no—
216C2×C42—21no—
217C2×C42—21no—
218C2×C42—21no—
219C2×C42—21no—
220C2×C42—21no—
221C2×C42—21no—
222C2×C42C22×C411yescharacteristic
223C2×C42C22×C412yes—
224C2×C42C22×C412yes—
225C2×C42C22×C411yescharacteristic
226C2×C42C2×D411yescharacteristic
227C2×C42C2×D412yes—
228C2×C42C2×D411yescharacteristic
229C2×C42C2×D412yes—
230C2×C42C2×D412yes—
231C2×C42C2×D411yescharacteristic
232C2×C42C2×D412yes—
233C2×C42C2×D411yescharacteristic
234C2×C42C2411yescharacteristic
235C4⋊C8—22no—
236C4⋊C8—21no—
237C4⋊C8—21no—
238C4⋊C8—22no—
239C4⋊C8—21no—
240C4⋊C8—42no—
241C4⋊C8—44no—
242C4⋊C8—44no—
243C4⋊C8—42no—
244C4⋊C8—82no—
245C4⋊C8—81no—
246C4⋊C8#4112yes—
247#97—44no—
248C4×C8—22no—
249C4×C8—21no—
250C4×C8—21no—
251C4×C8—22no—
252C4×C8—21no—
253C4×C8—42no—
254C4×C8—42no—
255C4×C8—42no—
256C4×C8#4112yes—
257#95—22no—
258#95—22no—
259#95—22no—
260#95—22no—
261#95—22no—
262#95—22no—
263#95—21no—
264#95—21no—
265#95—21no—
266#95C22⋊C412yes—
267#95C22⋊C412yes—
268#95C22⋊C412yes—
269C24—22no—
270C2×Q8—21no—
271C2×Q8—22no—
272C2×Q8—22no—
273C2×Q8—22no—
274C2×Q8—21no—
275C2×Q8—21no—
276C2×Q8—21no—
277C2×Q8—21no—
278C2×Q8#13112yes—
279C2×D4—22no—
280C2×D4—22no—
281C2×D4—22no—
282C2×D4—22no—
283C2×D4—22no—
284C2×D4—22no—
285C2×D4—42no—
286C2×D4—42no—
287C2×D4—42no—
288C2×D4—42no—
289C2×D4—81no—
290C2×D4—81no—
291C22×C4—21no—
292C22×C4—21no—
293C22×C4—22no—
294C22×C4—21no—
295C22×C4—21no—
296C22×C4—42no—
297C22×C4—42no—
298C22×C4—42no—
299C22×C4—42no—
300C22×C4—42no—
301C22×C4—42no—
302C22×C4C2×C22⋊C412yes—
303C22×C4C2×C22⋊C412yes—
304C22×C4C2×C22⋊C412yes—
305C22×C4C2×C22⋊C412yes—
306C22×C4C2×C22⋊C411yescharacteristic
307C22×C4C2×C22⋊C411yescharacteristic
308C22×C4C2×C22⋊C411yescharacteristic
309C22×C4C2×C22⋊C411yescharacteristic
310C22×C4C23×C411yescharacteristic
311C22×C4C22×D411yescharacteristic
312C22×C4C22×D411yescharacteristic
313C2×C8—22no—
314C2×C8—22no—
315C2×C8—22no—
316C2×C8—41no—
317C2×C8—41no—
318C2×C8—41no—
319C2×C8—41no—
320C2×C8—41no—
321C2×C8—41no—
322C2×C8—42no—
323C2×C8—42no—
324C2×C8—42no—
325C2×C8—42no—
326C2×C8—41no—
327C2×C8—41no—
328C2×C8—42no—
329C2×C8—41no—
330C2×C8—42no—
331C4⋊C4—22no—
332C4⋊C4—24no—
333C4⋊C4—22no—
334C4⋊C4—24no—
335C4⋊C4—22no—
336C4⋊C4—22no—
337C4⋊C4—21no—
338C4⋊C4—22no—
339C4⋊C4—21no—
340C4⋊C4—21no—
341C4⋊C4—21no—
342C4⋊C4—21no—
343C4⋊C4—21no—
344C4⋊C4—42no—
345C4⋊C4—42no—
346C4⋊C4—41no—
347C4⋊C4—41no—
348C4⋊C4—42no—
349C4⋊C4—42no—
350C4⋊C4—82no—
351C4⋊C4—82no—
352C4⋊C4—81no—
353C22⋊C4—44no—
354C22⋊C4—44no—
355C22⋊C4—44no—
356C22⋊C4—44no—
357C22⋊C4—42no—
358C22⋊C4—42no—
359C22⋊C4—44no—
360C22⋊C4—82no—
361C22⋊C4—82no—
362C22⋊C4—82no—
363C22⋊C4—82no—
364C22⋊C4—82no—
365C42—21no—
366C42—22no—
367C42—22no—
368C42—22no—
369C42—21no—
370C42—22no—
371C42—22no—
372C42—22no—
373C42—22no—
374C42—21no—
375C42—22no—
376C42—21no—
377C42—22no—
378C42—41no—
379C42—42no—
380C42—42no—
381C42—42no—
382C42—41no—
383C42—41no—
384C42—41no—
385C42—41no—
386C42—42no—
387C42—42no—
388C42—81no—
389C42#13112yes—
390C42#13112yes—
391C42AGL(1,ℤ/8)12yes—
392C42#13712yes—
393C42C2×#4111yescharacteristic
394C42C2×#4111yescharacteristic
395C42H(5,𝔽2)11yescharacteristic
396C422−1+411yescharacteristic
397C23—42no—
398C23—42no—
399C23—42no—
400C23—42no—
401C23—42no—
402C23—42no—
403C23—42no—
404C23C22×C22⋊C411yescharacteristic
405Q8—22no—
406Q8—22no—
407Q8—41no—
408Q8—41no—
409Q8—44no—
410D4—42no—
411D4—42no—
412D4—42no—
413D4—42no—
414D4—84no—
415D4—84no—
416C2×C4—21no—
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—21no—
427C2×C4—22no—
428C2×C4—22no—
429C2×C4—22no—
430C2×C4—21no—
431C2×C4—22no—
432C2×C4—22no—
433C2×C4—21no—
434C2×C4—22no—
435C2×C4—22no—
436C2×C4—22no—
437C2×C4—22no—
438C2×C4—41no—
439C2×C4—41no—
440C2×C4—41no—
441C2×C4—42no—
442C2×C4—41no—
443C2×C4—41no—
444C2×C4—42no—
445C2×C4—41no—
446C2×C4—41no—
447C2×C4—42no—
448C2×C4—41no—
449C2×C4—41no—
450C2×C4—41no—
451C2×C4—41no—
452C2×C4—82no—
453C2×C4—82no—
454C2×C4—82no—
455C2×C4—82no—
456C2×C4#40812yes—
457C2×C4#40812yes—
458C2×C4#41712yes—
459C2×C4#41712yes—
460C2×C4#42112yes—
461C2×C4C4×#4111yescharacteristic
462C2×C4#52011yescharacteristic
463C2×C4#54811yescharacteristic
464C2×C4#54811yescharacteristic
465C2×C4#55011yescharacteristic
466C2×C4#55011yescharacteristic
467C2×C4C2×#13111yescharacteristic
468C2×C4C2×#13111yescharacteristic
469C2×C4C2×#13111yescharacteristic
470C2×C4#56811yescharacteristic
471C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
472C2×C4C2×#13711yescharacteristic
473C2×C4#57611yescharacteristic
474C2×C4#57711yescharacteristic
475C8—42no—
476C8—42no—
477C8—42no—
478C8—82no—
479C8—81no—
480C8—82no—
481C22—82no—
482C22—82no—
483C22—82no—
484C22—82no—
485C22—82no—
486C22—82no—
487C22—82no—
488C22#234111yescharacteristic
489C22C2×#40811yescharacteristic
490C22#288011yescharacteristic
491C22C2×#41711yescharacteristic
492C22#289711yescharacteristic
493C22C2×#42111yescharacteristic
494C22#290111yescharacteristic
495C4—21no—
496C4—21no—
497C4—22no—
498C4—22no—
499C4—42no—
500C4—42no—
501C4—42no—
502C4—42no—
503C4—42no—
504C4—41no—
505C4—82no—
506C4—81no—
507C4—81no—
508C4—81no—
509C4#149514yes—
510C4#296611yescharacteristic
511C4#296911yescharacteristic
512C4#299311yescharacteristic
513C4#299411yescharacteristic
514C4#337311yescharacteristic
515C4#337511yescharacteristic
516C4#339911yescharacteristic
517C4#340011yescharacteristic
518C2—162no—
519C2C2×#149511yescharacteristic
520C2#1748311yescharacteristic
521C2#1751511yescharacteristic
522C2#1752611yescharacteristic
523C2#2253111yescharacteristic
524C2#2254611yescharacteristic
525C2#2255111yescharacteristic
subgroupG/H
centerC22×C4C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
socleC23C22×C22⋊C4
prime radicalC2×#127C2×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. #21339C2semidirect
4. #9676C2semidirect
6. #9640C2semidirect
8. #8567C2semidirect
9. #8564C2semidirect
10. #8179C2semidirect
11. C4×Q8⋊C8C2semidirect

Maximal subgroups

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

subgroupindex
#349542
#227222
#213392
#96762
#96742
#96402
#96382
#85672
#85642
#81792
C4×Q8⋊C82