Finite Group Databasebeta

← #214655

Subgroups — #214655

Order 512.

777 conjugacy classes of subgroups.

The subgroup lattice has 512 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#37423C211yesmaximal, characteristic
2#21832C211yesmaximal, characteristic, complemented
3#21828C211yesmaximal, characteristic, complemented
4#9465C212yesmaximal
5#9460C212yesmaximal, complemented
6#9446C212yesmaximal
7#9444C212yesmaximal, complemented
8#8741C211yesmaximal, characteristic, complemented
9#8740C211yesmaximal, characteristic, complemented
10#8519C211yesmaximal, characteristic
11#8519C211yesmaximal, characteristic
12C2×#544C2211yescharacteristic
13C2×#542C2211yescharacteristic
14#2921C2212yes—
15#2862C411yescharacteristic, complemented
16#2861C411yescharacteristic, complemented
17#2716—22no—
18#2700—22no—
19#2690—22no—
20#2687—22no—
21#2660C411yescharacteristic, complemented
22#2640C411yescharacteristic, complemented
23#2568C411yescharacteristic, complemented
24#2566C411yescharacteristic, complemented
25#2565C411yescharacteristic, complemented
26#2563C411yescharacteristic, complemented
27#2504C2211yescharacteristic
28#2503C2211yescharacteristic
29#2500C2211yescharacteristic
30#2499C2211yescharacteristic
31#2313C2212yes—
32#2298C2211yescharacteristic
33#1998—22no—
34#1997—22no—
35#1996—22no—
36#1996—22no—
37#1822C2211yescharacteristic
38#1822C2211yescharacteristic
39#1821C2212yes—
40#1820C2211yescharacteristic
41#1820C2211yescharacteristic
42#1820C2212yes—
43#1511C2212yes—
44#1511C2212yes—
45#1510C2212yes—
46#1510C2212yes—
47#1284C2212yes—
48#1284C2212yes—
49#1284C2212yes—
50#1284C2212yes—
51#544—42no—
52#544—42no—
53#542—42no—
54#542—42no—
55C2×C4⋊Q8C2×C411yescharacteristic
56C2×C4⋊Q8C2×C411yescharacteristic
57C2×C4⋊Q8C2×C411yescharacteristic
58C2×C4⋊Q8D412yes—
59C2×#126—21no—
60C2×#125C2×C411yescharacteristic
61C2×#125C2×C411yescharacteristic
62C2×#125D412yes—
63C2×#124—21no—
64C2×#124C2×C411yescharacteristic
65C2×#123—21no—
66C2×C22⋊Q8—21no—
67C2×C22⋊Q8D412yes—
68C2×C22⋊Q8D412yes—
69C2×C22⋊Q8C2311yescharacteristic
70C2×#121C2311yescharacteristic
71C2×C4×Q8C2×C411yescharacteristic
72C2×C4×D4C2×C411yescharacteristic
73C2×#117C2311yescharacteristic
74C2×#117C2311yescharacteristic
75C22×C4⋊C4C2311yescharacteristic
76#433—42no—
77#433—42no—
78#423—22no—
79#407C2312yes—
80#407C2312yes—
81C4×M16—22no—
82#400—22no—
83#400—22no—
84#398—22no—
85#398—22no—
86#397—22no—
87#397—22no—
88#397—22no—
89#396—22no—
90#395—22no—
91#394—22no—
92#393—22no—
93#393—22no—
94#391—22no—
95#391—22no—
96#391—22no—
97#391C2×C411yescharacteristic
98#391C2×C411yescharacteristic
99#391C2×C411yescharacteristic
100#391C2×C411yescharacteristic
101#390C2×C411yescharacteristic
102#390C2×C411yescharacteristic
103#390C2×C411yescharacteristic
104#390C2×C411yescharacteristic
105#388—22no—
106#388—22no—
107#388—21no—
108#388—21no—
109#388—22no—
110#387—22no—
111#387—21no—
112#387—21no—
113#387—22no—
114#387—42no—
115#387—42no—
116#386—22no—
117#385—22no—
118#384C2×C411yescharacteristic
119#384C2×C411yescharacteristic
120#382C2×C411yescharacteristic
121#382C2×C411yescharacteristic
122#382C2×C411yescharacteristic
123#382C2×C411yescharacteristic
124#382C2×C411yescharacteristic
125#382C2×C411yescharacteristic
126#381—22no—
127#381C2312yes—
128#381C2312yes—
129#380—22no—
130#380—22no—
131C4×C4⋊C4C2×C411yescharacteristic
132C4×C4⋊C4C2×C411yescharacteristic
133C4×C22⋊C4—22no—
134C4×C22⋊C4C2311yescharacteristic
135C4×C22⋊C4C2311yescharacteristic
136#376C2×C411yescharacteristic
137C43C2×C411yescharacteristic
138#343—24no—
139#343—21no—
140#343—21no—
141#343—21no—
142#343—21no—
143#342—21no—
144#342—21no—
145#342—21no—
146#342—21no—
147#342—24no—
148#328—42no—
149#328—42no—
150#327—42no—
151#327—42no—
152C22×Q8—21no—
153C22×Q8C22⋊C412yes—
154C22×Q8C22⋊C412yes—
155C22×Q8C22×C411yescharacteristic
156C22×D4—21no—
157C22×D4C22×C411yescharacteristic
158C23×C4—21no—
159C23×C4C2×D412yes—
160C23×C4C2411yescharacteristic
161C2×M16—22no—
162C2×M16—22no—
163C4⋊Q8—41no—
164C4⋊Q8—41no—
165C4⋊Q8—41no—
166C4⋊Q8—41no—
167C4⋊Q8—41no—
168C4⋊Q8—41no—
169C4⋊Q8—42no—
170C4⋊Q8—42no—
171#126—44no—
172#125—41no—
173#125—41no—
174#125—41no—
175#125—41no—
176#125—42no—
177#125—42no—
178#124—41no—
179#124—41no—
180#124—44no—
181#123—44no—
182C22⋊Q8—22no—
183C22⋊Q8—22no—
184C22⋊Q8—42no—
185C22⋊Q8—42no—
186C22⋊Q8—42no—
187C22⋊Q8—42no—
188C22⋊Q8—44no—
189#121—22no—
190#121—22no—
191C4×Q8—41no—
192C4×Q8—41no—
193C4×D4—41no—
194C4×D4—41no—
195#117—22no—
196#117—22no—
197#117—22no—
198#117—22no—
199C2×C4⋊C4—22no—
200C2×C4⋊C4—22no—
201C2×C4⋊C4—21no—
202C2×C4⋊C4—21no—
203C2×C4⋊C4—21no—
204C2×C4⋊C4—21no—
205C2×C4⋊C4—21no—
206C2×C4⋊C4—22no—
207C2×C4⋊C4—22no—
208C2×C4⋊C4—42no—
209C2×C4⋊C4C22⋊C412yes—
210C2×C4⋊C4C22⋊C412yes—
211C2×C4⋊C4C22⋊C412yes—
212C2×C4⋊C4C22⋊C412yes—
213C2×C4⋊C4C22⋊C412yes—
214C2×C4⋊C4C22⋊C412yes—
215C2×C4⋊C4C22×C411yescharacteristic
216C2×C4⋊C4C22×C411yescharacteristic
217C2×C4⋊C4C22×C411yescharacteristic
218C2×C4⋊C4C22×C411yescharacteristic
219C2×C4⋊C4C22×C411yescharacteristic
220C2×C4⋊C4C22×C411yescharacteristic
221C2×C4⋊C4C2×D412yes—
222C2×C4⋊C4C2×D411yescharacteristic
223C2×C4⋊C4C2×D411yescharacteristic
224C2×C22⋊C4—21no—
225C2×C22⋊C4—21no—
226C2×C22⋊C4—21no—
227C2×C22⋊C4—21no—
228C2×C22⋊C4—21no—
229C2×C22⋊C4—22no—
230C2×C22⋊C4—22no—
231C2×C22⋊C4—21no—
232C2×C22⋊C4—21no—
233C2×C22⋊C4—22no—
234C2×C22⋊C4—22no—
235C2×C22⋊C4—22no—
236C2×C22⋊C4—22no—
237C2×C22⋊C4C2×D411yescharacteristic
238C2×C22⋊C4C2×D411yescharacteristic
239C2×C22⋊C4C2×D412yes—
240C2×C42—21no—
241C2×C42—21no—
242C2×C42—21no—
243C2×C42—42no—
244C2×C42C22×C411yescharacteristic
245C2×C42C22×C411yescharacteristic
246C2×C42C22×C411yescharacteristic
247C2×C42C22×C411yescharacteristic
248C2×C42C22×C411yescharacteristic
249C2×C42C22×C411yescharacteristic
250C2×C42C2×D412yes—
251C4⋊C8—44no—
252C22⋊C8—22no—
253C22⋊C8—22no—
254C22⋊C8—22no—
255C22⋊C8—22no—
256#97—42no—
257C4×C8—42no—
258#95—21no—
259#95—21no—
260#95—21no—
261#95—21no—
262#95—22no—
263#95—22no—
264#95—22no—
265#95—22no—
266#95—22no—
267#95—22no—
268#95—22no—
269#95—22no—
270#95—22no—
271#95—22no—
272#95—21no—
273#95—21no—
274#95—22no—
275#95—22no—
276#95—22no—
277#95—22no—
278#95—42no—
279#95—42no—
280C24—21no—
281C24C22×D411yescharacteristic
282C2×Q8—21no—
283C2×Q8—21no—
284C2×Q8—42no—
285C2×Q8—42no—
286C2×Q8—42no—
287C2×Q8—41no—
288C2×Q8—41no—
289C2×Q8—82no—
290C2×Q8—82no—
291C2×Q8—82no—
292C2×D4—21no—
293C2×D4—21no—
294C2×D4—41no—
295C2×D4—41no—
296C2×D4—42no—
297C2×D4—82no—
298C22×C4—22no—
299C22×C4—22no—
300C22×C4—21no—
301C22×C4—21no—
302C22×C4—21no—
303C22×C4—21no—
304C22×C4—21no—
305C22×C4—21no—
306C22×C4—21no—
307C22×C4—41no—
308C22×C4—41no—
309C22×C4—42no—
310C22×C4—42no—
311C22×C4—42no—
312C22×C4—82no—
313C22×C4—82no—
314C22×C4C2×C22⋊C412yes—
315C22×C4C2×C22⋊C412yes—
316C22×C4C2×C22⋊C412yes—
317C22×C4C2×C22⋊C411yescharacteristic
318C22×C4C2×C22⋊C411yescharacteristic
319C22×C4C2×C22⋊C411yescharacteristic
320C22×C4C2×C22⋊C411yescharacteristic
321C22×C4C2×C22⋊C412yes—
322C22×C4C23×C411yescharacteristic
323C22×C4C22×D411yescharacteristic
324C22×C4H(5,𝔽2)12yes—
325C22×C4H(5,𝔽2)12yes—
326M16—42no—
327M16—42no—
328M16—42no—
329M16—42no—
330C2×C8—42no—
331C2×C8—42no—
332C4⋊C4—21no—
333C4⋊C4—21no—
334C4⋊C4—21no—
335C4⋊C4—21no—
336C4⋊C4—21no—
337C4⋊C4—21no—
338C4⋊C4—21no—
339C4⋊C4—21no—
340C4⋊C4—22no—
341C4⋊C4—22no—
342C4⋊C4—22no—
343C4⋊C4—22no—
344C4⋊C4—21no—
345C4⋊C4—21no—
346C4⋊C4—21no—
347C4⋊C4—21no—
348C4⋊C4—42no—
349C4⋊C4—42no—
350C4⋊C4—42no—
351C4⋊C4—42no—
352C4⋊C4—42no—
353C4⋊C4—42no—
354C4⋊C4—42no—
355C4⋊C4—42no—
356C4⋊C4—42no—
357C4⋊C4—82no—
358C4⋊C4—82no—
359C4⋊C4—81no—
360C4⋊C4—81no—
361C4⋊C4—162no—
362C22⋊C4—22no—
363C22⋊C4—22no—
364C22⋊C4—22no—
365C22⋊C4—22no—
366C22⋊C4—42no—
367C22⋊C4—42no—
368C22⋊C4—42no—
369C22⋊C4—42no—
370C22⋊C4—42no—
371C22⋊C4—42no—
372C22⋊C4—42no—
373C22⋊C4—42no—
374C22⋊C4—42no—
375C22⋊C4—82no—
376C22⋊C4—82no—
377C22⋊C4—82no—
378C22⋊C4—82no—
379C22⋊C4—81no—
380C22⋊C4—81no—
381C42—21no—
382C42—21no—
383C42—21no—
384C42—21no—
385C42—22no—
386C42—22no—
387C42—41no—
388C42—41no—
389C42—41no—
390C42—41no—
391C42—42no—
392C42—81no—
393C42—81no—
394C42—162no—
395C23—21no—
396C23—21no—
397C23—41no—
398C23—41no—
399C23—42no—
400C23—42no—
401C23—41no—
402C23—41no—
403C23—41no—
404C23C22×C22⋊C411yescharacteristic
405C23#53412yes—
406Q8—81no—
407Q8—81no—
408Q8—162no—
409Q8—162no—
410Q8—162no—
411D4—81no—
412D4—81no—
413D4—162no—
414C2×C4—21no—
415C2×C4—21no—
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—42no—
427C2×C4—42no—
428C2×C4—41no—
429C2×C4—41no—
430C2×C4—42no—
431C2×C4—42no—
432C2×C4—42no—
433C2×C4—42no—
434C2×C4—41no—
435C2×C4—41no—
436C2×C4—41no—
437C2×C4—41no—
438C2×C4—41no—
439C2×C4—41no—
440C2×C4—41no—
441C2×C4—41no—
442C2×C4—41no—
443C2×C4—41no—
444C2×C4—42no—
445C2×C4—42no—
446C2×C4—41no—
447C2×C4—41no—
448C2×C4—41no—
449C2×C4—41no—
450C2×C4—41no—
451C2×C4—41no—
452C2×C4—41no—
453C2×C4—41no—
454C2×C4—41no—
455C2×C4—82no—
456C2×C4—82no—
457C2×C4—82no—
458C2×C4—82no—
459C2×C4—82no—
460C2×C4—82no—
461C2×C4—82no—
462C2×C4—82no—
463C2×C4—162no—
464C2×C4#51811yescharacteristic
465C2×C4#51811yescharacteristic
466C2×C4#53412yes—
467C8—82no—
468C8—82no—
469C22—21no—
470C22—21no—
471C22—22no—
472C22—22no—
473C22—22no—
474C22—81no—
475C22—81no—
476C22—82no—
477C22—82no—
478C22—81no—
479C22—81no—
480C22—81no—
481C22#230811yescharacteristic
482C22#288411yescharacteristic
483C22#288411yescharacteristic
484C4—22no—
485C4—21no—
486C4—21no—
487C4—42no—
488C4—42no—
489C4—41no—
490C4—41no—
491C4—42no—
492C4—42no—
493C4—41no—
494C4—41no—
495C4—81no—
496C4—81no—
497C4—81no—
498C4—81no—
499C4—81no—
500C4—81no—
501C4—81no—
502C4—162no—
503C4—162no—
504C2—21no—
505C2—21no—
506C2—42no—
507C2—161no—
508C2#2175111yescharacteristic
509C2#2178411yescharacteristic
510C2#2178511yescharacteristic
subgroupG/H
center, socleC22#2308
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC2×#121C23
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C23×C4 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
2. #21832C2semidirect
3. #21828C2semidirect
5. #9460C2semidirect
7. #9444C2semidirect
8. #8741C2semidirect
9. #8740C2semidirect
15. #2862C4semidirect
16. #2861C4semidirect
21. #2660C4semidirect
22. #2640C4semidirect
23. #2568C4semidirect
24. #2566C4semidirect
25. #2565C4semidirect
26. #2563C4semidirect

Maximal subgroups

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

subgroupindex
#374232
#218322
#218282
#94652
#94602
#94462
#94442
#87412
#87402
#85192