Finite Group Databasebeta

← #214654

Subgroups — #214654

Order 512.

815 conjugacy classes of subgroups.

The subgroup lattice has 519 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#37303C211yesmaximal, characteristic
2#21832C211yesmaximal, characteristic, complemented
3#21828C211yesmaximal, characteristic, complemented
4#9464C212yesmaximal
5#9461C212yesmaximal, 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#2868C411yescharacteristic, complemented
16#2861C411yescharacteristic, complemented
17#2721—22no—
18#2680—22no—
19#2677—22no—
20#2674—22no—
21#2660C411yescharacteristic, complemented
22#2640C411yescharacteristic, complemented
23#2567C411yescharacteristic, complemented
24#2566C411yescharacteristic, complemented
25#2565C411yescharacteristic, complemented
26#2542C411yescharacteristic, complemented
27#2502C2211yescharacteristic
28#2501C2211yescharacteristic
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⋊Q8—22no—
56C2×C4⋊Q8C2×C411yescharacteristic
57C2×C4⋊Q8C2×C411yescharacteristic
58C2×C4⋊Q8C2×C411yescharacteristic
59C2×C4⋊Q8D412yes—
60C2×#126—21no—
61C2×#125C2×C411yescharacteristic
62C2×#125C2×C411yescharacteristic
63C2×#125D412yes—
64C2×#124—21no—
65C2×#124C2×C411yescharacteristic
66C2×#123—22no—
67C2×#123—21no—
68C2×C22⋊Q8—22no—
69C2×C22⋊Q8—21no—
70C2×C22⋊Q8D412yes—
71C2×C22⋊Q8D412yes—
72C2×C22⋊Q8C2311yescharacteristic
73C2×#121C2311yescharacteristic
74C2×C4×Q8C2×C411yescharacteristic
75C2×C4×D4C2×C411yescharacteristic
76C2×#117C2311yescharacteristic
77C2×#117C2311yescharacteristic
78C22×C4⋊C4C2311yescharacteristic
79#433—42no—
80#433—42no—
81#423—22no—
82#407C2312yes—
83#407C2312yes—
84C4×M16—22no—
85#399—22no—
86#399—22no—
87#398—22no—
88#398—22no—
89#398—22no—
90#397—22no—
91#397—22no—
92#396—22no—
93#395—22no—
94#395—22no—
95#394—22no—
96#393—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#389—22no—
106#388—22no—
107#388—22no—
108#388—21no—
109#388—21no—
110#388—22no—
111#387—22no—
112#387—21no—
113#387—21no—
114#387—22no—
115#387—42no—
116#387—42no—
117#385—22no—
118#385—22no—
119#384—22no—
120#384C2×C411yescharacteristic
121#384C2×C411yescharacteristic
122#384C2×C411yescharacteristic
123#384C2×C411yescharacteristic
124#384C2×C411yescharacteristic
125#384C2×C411yescharacteristic
126#382C2×C411yescharacteristic
127#382C2×C411yescharacteristic
128#381C2312yes—
129#381C2312yes—
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—
171C4⋊Q8—84no—
172#126—44no—
173#125—41no—
174#125—41no—
175#125—41no—
176#125—41no—
177#125—42no—
178#125—42no—
179#124—41no—
180#124—41no—
181#124—44no—
182#123—44no—
183#123—84no—
184C22⋊Q8—22no—
185C22⋊Q8—22no—
186C22⋊Q8—42no—
187C22⋊Q8—42no—
188C22⋊Q8—42no—
189C22⋊Q8—42no—
190C22⋊Q8—44no—
191C22⋊Q8—84no—
192#121—22no—
193#121—22no—
194C4×Q8—41no—
195C4×Q8—41no—
196C4×D4—41no—
197C4×D4—41no—
198#117—22no—
199#117—22no—
200#117—22no—
201#117—22no—
202C2×C4⋊C4—22no—
203C2×C4⋊C4—22no—
204C2×C4⋊C4—21no—
205C2×C4⋊C4—21no—
206C2×C4⋊C4—22no—
207C2×C4⋊C4—21no—
208C2×C4⋊C4—22no—
209C2×C4⋊C4—21no—
210C2×C4⋊C4—21no—
211C2×C4⋊C4—22no—
212C2×C4⋊C4—22no—
213C2×C4⋊C4—22no—
214C2×C4⋊C4—22no—
215C2×C4⋊C4—42no—
216C2×C4⋊C4C22⋊C412yes—
217C2×C4⋊C4C22⋊C412yes—
218C2×C4⋊C4C22⋊C412yes—
219C2×C4⋊C4C22⋊C412yes—
220C2×C4⋊C4C22⋊C412yes—
221C2×C4⋊C4C22⋊C412yes—
222C2×C4⋊C4C22×C411yescharacteristic
223C2×C4⋊C4C22×C411yescharacteristic
224C2×C4⋊C4C22×C411yescharacteristic
225C2×C4⋊C4C22×C411yescharacteristic
226C2×C4⋊C4C22×C411yescharacteristic
227C2×C4⋊C4C22×C411yescharacteristic
228C2×C4⋊C4C2×D412yes—
229C2×C4⋊C4C2×D411yescharacteristic
230C2×C4⋊C4C2×D411yescharacteristic
231C2×C22⋊C4—21no—
232C2×C22⋊C4—21no—
233C2×C22⋊C4—21no—
234C2×C22⋊C4—21no—
235C2×C22⋊C4—22no—
236C2×C22⋊C4—21no—
237C2×C22⋊C4—21no—
238C2×C22⋊C4—22no—
239C2×C22⋊C4—21no—
240C2×C22⋊C4—22no—
241C2×C22⋊C4—22no—
242C2×C22⋊C4—22no—
243C2×C22⋊C4—22no—
244C2×C22⋊C4C2×D411yescharacteristic
245C2×C22⋊C4C2×D411yescharacteristic
246C2×C22⋊C4C2×D412yes—
247C2×C42—21no—
248C2×C42—21no—
249C2×C42—21no—
250C2×C42—42no—
251C2×C42C22×C411yescharacteristic
252C2×C42C22×C411yescharacteristic
253C2×C42C22×C411yescharacteristic
254C2×C42C22×C411yescharacteristic
255C2×C42C22×C411yescharacteristic
256C2×C42C22×C411yescharacteristic
257C2×C42C2×D412yes—
258C4⋊C8—44no—
259C22⋊C8—22no—
260C22⋊C8—22no—
261C22⋊C8—22no—
262C22⋊C8—22no—
263#97—42no—
264C4×C8—42no—
265#95—21no—
266#95—21no—
267#95—21no—
268#95—21no—
269#95—22no—
270#95—22no—
271#95—22no—
272#95—22no—
273#95—22no—
274#95—22no—
275#95—21no—
276#95—21no—
277#95—22no—
278#95—22no—
279#95—22no—
280#95—22no—
281#95—42no—
282#95—42no—
283C24—21no—
284C24C22×D411yescharacteristic
285C2×Q8—21no—
286C2×Q8—21no—
287C2×Q8—42no—
288C2×Q8—42no—
289C2×Q8—42no—
290C2×Q8—41no—
291C2×Q8—41no—
292C2×Q8—82no—
293C2×Q8—82no—
294C2×Q8—82no—
295C2×D4—21no—
296C2×D4—21no—
297C2×D4—41no—
298C2×D4—41no—
299C2×D4—42no—
300C2×D4—82no—
301C22×C4—21no—
302C22×C4—21no—
303C22×C4—22no—
304C22×C4—22no—
305C22×C4—21no—
306C22×C4—21no—
307C22×C4—21no—
308C22×C4—21no—
309C22×C4—21no—
310C22×C4—41no—
311C22×C4—41no—
312C22×C4—42no—
313C22×C4—42no—
314C22×C4—42no—
315C22×C4—82no—
316C22×C4—82no—
317C22×C4C2×C22⋊C412yes—
318C22×C4C2×C22⋊C411yescharacteristic
319C22×C4C2×C22⋊C411yescharacteristic
320C22×C4C2×C22⋊C411yescharacteristic
321C22×C4C2×C22⋊C411yescharacteristic
322C22×C4C2×C22⋊C412yes—
323C22×C4C2×C22⋊C412yes—
324C22×C4C2×C22⋊C412yes—
325C22×C4C23×C411yescharacteristic
326C22×C4C22×D411yescharacteristic
327C22×C4H(5,𝔽2)12yes—
328C22×C4H(5,𝔽2)12yes—
329M16—42no—
330M16—42no—
331M16—42no—
332M16—42no—
333C2×C8—42no—
334C2×C8—42no—
335C4⋊C4—21no—
336C4⋊C4—21no—
337C4⋊C4—21no—
338C4⋊C4—21no—
339C4⋊C4—21no—
340C4⋊C4—21no—
341C4⋊C4—21no—
342C4⋊C4—21no—
343C4⋊C4—22no—
344C4⋊C4—22no—
345C4⋊C4—22no—
346C4⋊C4—22no—
347C4⋊C4—21no—
348C4⋊C4—21no—
349C4⋊C4—21no—
350C4⋊C4—21no—
351C4⋊C4—42no—
352C4⋊C4—42no—
353C4⋊C4—42no—
354C4⋊C4—42no—
355C4⋊C4—42no—
356C4⋊C4—42no—
357C4⋊C4—42no—
358C4⋊C4—42no—
359C4⋊C4—42no—
360C4⋊C4—44no—
361C4⋊C4—42no—
362C4⋊C4—44no—
363C4⋊C4—42no—
364C4⋊C4—82no—
365C4⋊C4—82no—
366C4⋊C4—82no—
367C4⋊C4—82no—
368C4⋊C4—162no—
369C22⋊C4—22no—
370C22⋊C4—22no—
371C22⋊C4—22no—
372C22⋊C4—22no—
373C22⋊C4—44no—
374C22⋊C4—42no—
375C22⋊C4—42no—
376C22⋊C4—42no—
377C22⋊C4—44no—
378C22⋊C4—42no—
379C22⋊C4—42no—
380C22⋊C4—42no—
381C22⋊C4—42no—
382C22⋊C4—42no—
383C22⋊C4—42no—
384C22⋊C4—42no—
385C22⋊C4—42no—
386C22⋊C4—82no—
387C22⋊C4—82no—
388C42—21no—
389C42—21no—
390C42—21no—
391C42—21no—
392C42—22no—
393C42—22no—
394C42—41no—
395C42—41no—
396C42—41no—
397C42—41no—
398C42—42no—
399C42—42no—
400C42—42no—
401C42—162no—
402C23—21no—
403C23—21no—
404C23—41no—
405C23—41no—
406C23—42no—
407C23—42no—
408C23—41no—
409C23—41no—
410C23—41no—
411C23C22×C22⋊C411yescharacteristic
412C23#53412yes—
413Q8—81no—
414Q8—81no—
415Q8—162no—
416Q8—162no—
417Q8—162no—
418D4—81no—
419D4—81no—
420D4—162no—
421C2×C4—21no—
422C2×C4—21no—
423C2×C4—21no—
424C2×C4—21no—
425C2×C4—22no—
426C2×C4—22no—
427C2×C4—21no—
428C2×C4—21no—
429C2×C4—21no—
430C2×C4—21no—
431C2×C4—22no—
432C2×C4—22no—
433C2×C4—22no—
434C2×C4—22no—
435C2×C4—42no—
436C2×C4—42no—
437C2×C4—41no—
438C2×C4—41no—
439C2×C4—41no—
440C2×C4—41no—
441C2×C4—41no—
442C2×C4—41no—
443C2×C4—42no—
444C2×C4—42no—
445C2×C4—42no—
446C2×C4—42no—
447C2×C4—41no—
448C2×C4—41no—
449C2×C4—41no—
450C2×C4—41no—
451C2×C4—41no—
452C2×C4—42no—
453C2×C4—42no—
454C2×C4—41no—
455C2×C4—41no—
456C2×C4—41no—
457C2×C4—41no—
458C2×C4—41no—
459C2×C4—41no—
460C2×C4—41no—
461C2×C4—41no—
462C2×C4—82no—
463C2×C4—82no—
464C2×C4—82no—
465C2×C4—82no—
466C2×C4—82no—
467C2×C4—82no—
468C2×C4—82no—
469C2×C4—82no—
470C2×C4—162no—
471C2×C4#51811yescharacteristic
472C2×C4#51811yescharacteristic
473C2×C4#53412yes—
474C8—82no—
475C8—82no—
476C22—21no—
477C22—21no—
478C22—22no—
479C22—22no—
480C22—22no—
481C22—81no—
482C22—81no—
483C22—82no—
484C22—82no—
485C22—81no—
486C22—81no—
487C22—81no—
488C22#230811yescharacteristic
489C22#288411yescharacteristic
490C22#288411yescharacteristic
491C4—22no—
492C4—21no—
493C4—21no—
494C4—42no—
495C4—42no—
496C4—42no—
497C4—42no—
498C4—41no—
499C4—42no—
500C4—41no—
501C4—41no—
502C4—41no—
503C4—42no—
504C4—81no—
505C4—81no—
506C4—81no—
507C4—81no—
508C4—81no—
509C4—162no—
510C4—162no—
511C2—21no—
512C2—21no—
513C2—42no—
514C2—161no—
515C2#2175111yescharacteristic
516C2#2178211yescharacteristic
517C2#2178311yescharacteristic
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. #9461C2semidirect
7. #9444C2semidirect
8. #8741C2semidirect
9. #8740C2semidirect
15. #2868C4semidirect
16. #2861C4semidirect
21. #2660C4semidirect
22. #2640C4semidirect
23. #2567C4semidirect
24. #2566C4semidirect
25. #2565C4semidirect
26. #2542C4semidirect

Maximal subgroups

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

subgroupindex
#373032
#218322
#218282
#94642
#94612
#94462
#94442
#87412
#87402
#85192