Finite Group Databasebeta

← #230096

Subgroups — #230096

Order 512.

546 conjugacy classes of subgroups.

The subgroup lattice has 528 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23235C211yesmaximal, characteristic
2Q8×#108C211yesmaximal, characteristic, complemented
3#19123C211yesmaximal, characteristic
4#18004C211yesmaximal, characteristic
5#17998C211yesmaximal, characteristic, complemented
6#9858C211yesmaximal, characteristic
7#9630C211yesmaximal, characteristic, complemented
8#9591C211yesmaximal, characteristic, complemented
9#9590C211yesmaximal, characteristic
10#8992C211yesmaximal, characteristic
11#8990C211yesmaximal, characteristic, complemented
12#8933C211yesmaximal, characteristic, complemented
13#8930C211yesmaximal, characteristic
14#8297C211yesmaximal, characteristic, complemented
15#8205C211yesmaximal, characteristic, complemented
16#3131C2211yescharacteristic
17#3104C2211yescharacteristic
18#3103C2211yescharacteristic
19Q8×M16C2211yescharacteristic
20Q8×M16C2211yescharacteristic
21#2962C2211yescharacteristic
22C2×C8×Q8C2211yescharacteristic
23#2935C2211yescharacteristic
24#1986C2211yescharacteristic
25#1980C2211yescharacteristic
26#1950—21no—
27#1950—21no—
28#1950C2211yescharacteristic
29#1950C2211yescharacteristic
30#1938—21no—
31#1929—21no—
32#1866—21no—
33#1848—21no—
34#1848C2211yescharacteristic
35C4×#108—21no—
36C4×#108C2211yescharacteristic
37#1770C2211yescharacteristic
38#1765—21no—
39C4×#103C2211yescharacteristic
40#1684C411yescharacteristic, complemented
41#1684C411yescharacteristic, complemented
42#1682C411yescharacteristic, complemented
43#1682C411yescharacteristic, complemented
44#1680C411yescharacteristic, complemented
45#1680C411yescharacteristic, complemented
46#1678C411yescharacteristic, complemented
47#1678C411yescharacteristic, complemented
48#1592—21no—
49#1591—21no—
50#1588—21no—
51#1585—21no—
52#1584—21no—
53#1584—21no—
54SD16⋊C8—21no—
55C8×Q16—21no—
56#1571C2211yescharacteristic
57#1571C2211yescharacteristic
58#1564C2211yescharacteristic
59#1542C2211yescharacteristic
60#1540C2211yescharacteristic
61#1493C2211yescharacteristic
62#1493C2211yescharacteristic
63#1492C2211yescharacteristic
64#1492C2211yescharacteristic
65C2×Q8⋊C8C2211yescharacteristic
66#1451C2211yescharacteristic
67C8×M16C2211yescharacteristic
68C2×#322C2211yescharacteristic
69#1303C2211yescharacteristic
70#1303C2211yescharacteristic
71#1279C2211yescharacteristic
72#1278C2211yescharacteristic
73#1278C2211yescharacteristic
74#1277C2211yescharacteristic
75C2×#131—21no—
76#533C2311yescharacteristic
77#532C2311yescharacteristic
78C2×C4×Q8C2311yescharacteristic
79#488—21no—
80#488—21no—
81#487C2×C411yescharacteristic
82#487C2×C411yescharacteristic
83#486C2×C411yescharacteristic
84#486C2×C411yescharacteristic
85#484—21no—
86#481—21no—
87#477C2×C411yescharacteristic
88#477C2×C411yescharacteristic
89#475C2×C411yescharacteristic
90#475C2×C411yescharacteristic
91#462C2×C411yescharacteristic
92#462C2×C411yescharacteristic
93#460C2×C411yescharacteristic
94#460C2×C411yescharacteristic
95#451—21no—
96#448—21no—
97#446—21no—
98#446—21no—
99#446—21no—
100#446—41no—
101#446—41no—
102#446—41no—
103C8×Q8—21no—
104C8×Q8—21no—
105C8×Q8—21no—
106C8×Q8—21no—
107C8×Q8—21no—
108C8×Q8C2×C411yescharacteristic
109C8×Q8C2×C411yescharacteristic
110C8×Q8C2×C411yescharacteristic
111C8×Q8C2×C411yescharacteristic
112C4×Q16—21no—
113C4×Q16—21no—
114C4×SD16—21no—
115C4×SD16—21no—
116#436—21no—
117#435—41no—
118C8×D4—21no—
119#431C2311yescharacteristic
120#431C2311yescharacteristic
121C2×#108—21no—
122C2×#108—21no—
123C2×#108—21no—
124C2×#108—21no—
125C2×#108D411yescharacteristic
126C2×#108D411yescharacteristic
127#424—21no—
128#423—21no—
129#423—21no—
130#423C2311yescharacteristic
131#423C2311yescharacteristic
132C2×C4⋊C8—21no—
133C2×C4⋊C8C2311yescharacteristic
134C2×C4⋊C8C2311yescharacteristic
135C2×#104—21no—
136C2×#104—21no—
137#416—21no—
138#416—21no—
139C2×#103—21no—
140#408—21no—
141#405—21no—
142C4×M16—21no—
143C4×M16—21no—
144C4×M16C2311yescharacteristic
145C4×M16C2311yescharacteristic
146C2×#97—21no—
147C2×C4×C8—21no—
148C2×C4×C8C2311yescharacteristic
149C2×C4×C8C2311yescharacteristic
150C2×C4×C8C2311yescharacteristic
151C4×C4⋊C4C2311yescharacteristic
152#341—21no—
153#341—21no—
154#341D411yescharacteristic
155#341D411yescharacteristic
156#340—21no—
157#339—21no—
158#339—21no—
159#336—21no—
160#335—21no—
161#335—21no—
162#335—21no—
163#335—21no—
164#335C2×C411yescharacteristic
165#335C2×C411yescharacteristic
166#334C2×C411yescharacteristic
167#334C2×C411yescharacteristic
168#333C2×C411yescharacteristic
169#333C2×C411yescharacteristic
170#332C2×C411yescharacteristic
171#332C2×C411yescharacteristic
172#332C2×C411yescharacteristic
173#332C2×C411yescharacteristic
174#331C2×C411yescharacteristic
175#331C2×C411yescharacteristic
176Q8⋊C8—21no—
177Q8⋊C8—21no—
178Q8⋊C8—41no—
179Q8⋊C8D411yescharacteristic
180Q8⋊C8D411yescharacteristic
181Q8⋊C8D411yescharacteristic
182Q8⋊C8D411yescharacteristic
183D4⋊C8—21no—
184D4⋊C8—21no—
185D4⋊C8—41no—
186#322—21no—
187#322—21no—
188#322—21no—
189#322—21no—
190#322—21no—
191C82—21no—
192C2×#41—21no—
193C22×Q8#4111yescharacteristic
194C2×Q16—21no—
195C2×Q16—21no—
196C2×SD16—21no—
197C2×SD16—21no—
198#131—42no—
199#131—81no—
200C2×M16—21no—
201C2×M16—21no—
202C2×M16—41no—
203C2×M16C2×D411yescharacteristic
204C2×M16C2×D411yescharacteristic
205C2×M16C2×D411yescharacteristic
206C2×M16C2×D411yescharacteristic
207C22×C8—21no—
208C22×C8—21no—
209C22×C8—21no—
210C22×C8—21no—
211C22×C8—41no—
212C22×C8C2×D411yescharacteristic
213C22×C8C2×D411yescharacteristic
214C4⋊Q8C22×C411yescharacteristic
215C4⋊Q8C22×C411yescharacteristic
216C4⋊Q8C22×C411yescharacteristic
217#127C22×C411yescharacteristic
218#125—21no—
219#124—21no—
220C22⋊Q8—21no—
221C22⋊Q8—21no—
222#121—21no—
223#121—21no—
224C4×Q8—21no—
225C4×Q8—21no—
226C4×Q8—21no—
227C4×Q8—21no—
228C4×Q8C22×C411yescharacteristic
229C4×Q8C22×C411yescharacteristic
230C4×Q8C2×D411yescharacteristic
231C4×Q8C2×D411yescharacteristic
232C4×D4—21no—
233C4×D4—21no—
234#117—21no—
235C2×C4⋊C4—21no—
236C2×C4⋊C4—21no—
237C2×C4⋊C4#4111yescharacteristic
238C2×C42—21no—
239C2×C42—21no—
240C2×C42C2411yescharacteristic
241#108—22no—
242#108—41no—
243#108—42no—
244#108—81no—
245#108—81no—
246#108—81no—
247#107—21no—
248#107—21no—
249#106—21no—
250#106—21no—
251C4⋊C8—21no—
252C4⋊C8—21no—
253C4⋊C8—21no—
254C4⋊C8—21no—
255C4⋊C8—21no—
256C4⋊C8—21no—
257C4⋊C8—21no—
258C4⋊C8—21no—
259C4⋊C8—21no—
260C4⋊C8—21no—
261C4⋊C8—41no—
262C4⋊C8—41no—
263C4⋊C8—41no—
264C4⋊C8—41no—
265C4⋊C8—41no—
266C4⋊C8—41no—
267C4⋊C8C22×C411yescharacteristic
268C4⋊C8C22×C411yescharacteristic
269C4⋊C8C22×C411yescharacteristic
270C4⋊C8C22×C411yescharacteristic
271C4⋊C8C22×C411yescharacteristic
272C4⋊C8C22×C411yescharacteristic
273C4⋊C8C2×D411yescharacteristic
274C4⋊C8C2×D411yescharacteristic
275#104—42no—
276#104—81no—
277#103—21no—
278#103—21no—
279#103—21no—
280#103—21no—
281#103—21no—
282#103—21no—
283#103—21no—
284#103—21no—
285#103—21no—
286#103—21no—
287#103—42no—
288#102—21no—
289#102—21no—
290#102—21no—
291#102—21no—
292#102—21no—
293#102—21no—
294C22⋊C8—21no—
295C22⋊C8—21no—
296C22⋊C8—41no—
297C22⋊C8—41no—
298#97—21no—
299#97—21no—
300#97—41no—
301#97—41no—
302#97—41no—
303#97—81no—
304C4×C8—21no—
305C4×C8—21no—
306C4×C8—21no—
307C4×C8—21no—
308C4×C8—21no—
309C4×C8—21no—
310C4×C8—21no—
311C4×C8—21no—
312C4×C8—21no—
313C4×C8—21no—
314C4×C8—21no—
315C4×C8—41no—
316C4×C8—41no—
317C4×C8—41no—
318C4×C8C22×C411yescharacteristic
319C4×C8C22×C411yescharacteristic
320C4×C8C2×D411yescharacteristic
321C4×C8C2×D411yescharacteristic
322C4×C8#4111yescharacteristic
323C4×C8#4111yescharacteristic
324#95—21no—
325#41—41no—
326#41—41no—
327#41—41no—
328#41—41no—
329C2×Q8—21no—
330C2×Q8—22no—
331C2×Q8—21no—
332C2×Q8—21no—
333C2×Q8—21no—
334C2×Q8—41no—
335C2×Q8C4×D411yescharacteristic
336C2×Q8C4×D411yescharacteristic
337C2×D4—21no—
338C2×D4—21no—
339C2×D4—21no—
340C2×D4—41no—
341C22×C4—21no—
342C22×C4—21no—
343C22×C4—41no—
344C22×C4—41no—
345C22×C4C22×D411yescharacteristic
346C22×C4C2×#4111yescharacteristic
347C22×C42−1+411yescharacteristic
348Q16—42no—
349Q16—81no—
350SD16—42no—
351SD16—81no—
352M16—21no—
353M16—21no—
354M16—21no—
355M16—21no—
356M16—21no—
357M16—21no—
358M16—21no—
359M16—41no—
360M16—41no—
361M16—41no—
362M16—41no—
363M16—82no—
364M16#13712yes—
365C2×C8—21no—
366C2×C8—21no—
367C2×C8—21no—
368C2×C8—21no—
369C2×C8—21no—
370C2×C8—21no—
371C2×C8—21no—
372C2×C8—21no—
373C2×C8—21no—
374C2×C8—21no—
375C2×C8—21no—
376C2×C8—21no—
377C2×C8—21no—
378C2×C8—21no—
379C2×C8—21no—
380C2×C8—21no—
381C2×C8—21no—
382C2×C8—21no—
383C2×C8—42no—
384C2×C8—41no—
385C2×C8—41no—
386C2×C8—41no—
387C2×C8—41no—
388C2×C8—41no—
389C2×C8—41no—
390C2×C8—41no—
391C2×C8—41no—
392C2×C8—41no—
393C2×C8—82no—
394C2×C8C4×D411yescharacteristic
395C2×C8C4×D411yescharacteristic
396C2×C8C4×D411yescharacteristic
397C2×C8C4×D411yescharacteristic
398C4⋊C4—21no—
399C4⋊C4—21no—
400C4⋊C4—21no—
401C4⋊C4—21no—
402C4⋊C4—21no—
403C4⋊C4—21no—
404C4⋊C4—21no—
405C4⋊C4—21no—
406C4⋊C4—21no—
407C4⋊C4—21no—
408C4⋊C4—21no—
409C4⋊C4—41no—
410C4⋊C4—41no—
411C4⋊C4—41no—
412C4⋊C4—41no—
413C4⋊C4C4×D411yescharacteristic
414C4⋊C4C4×D411yescharacteristic
415C22⋊C4—41no—
416C22⋊C4—41no—
417C22⋊C4—41no—
418C42—21no—
419C42—21no—
420C42—21no—
421C42—21no—
422C42—41no—
423C42—41no—
424C42—41no—
425C42C23×C411yescharacteristic
426C42C22×D411yescharacteristic
427C42C2×#4111yescharacteristic
428C42H(5,𝔽2)11yescharacteristic
429C23—41no—
430C23#54811yescharacteristic
431Q8—21no—
432Q8—21no—
433Q8—21no—
434Q8—42no—
435Q8—41no—
436Q8—41no—
437Q8—41no—
438Q8—41no—
439Q8—81no—
440Q8#44412yes—
441D4—41no—
442D4—41no—
443D4—41no—
444D4—41no—
445D4—81no—
446D4—81no—
447D4—81no—
448C2×C4—21no—
449C2×C4—21no—
450C2×C4—21no—
451C2×C4—21no—
452C2×C4—21no—
453C2×C4—21no—
454C2×C4—41no—
455C2×C4—42no—
456C2×C4—41no—
457C2×C4—41no—
458C2×C4—41no—
459C2×C4—81no—
460C2×C4—81no—
461C2×C4—81no—
462C2×C4—81no—
463C2×C4—81no—
464C2×C4—81no—
465C2×C4C2×C4×D411yescharacteristic
466C2×C4C2×C4×D411yescharacteristic
467C2×C4#52011yescharacteristic
468C2×C4D4211yescharacteristic
469C2×C4#54711yescharacteristic
470C2×C4#55311yescharacteristic
471C2×C4C2×#13711yescharacteristic
472C2×C4#57511yescharacteristic
473C2×C4#57611yescharacteristic
474C2×C4#57811yescharacteristic
475C8—22no—
476C8—21no—
477C8—21no—
478C8—21no—
479C8—21no—
480C8—22no—
481C8—41no—
482C8—41no—
483C8—41no—
484C8—41no—
485C8—41no—
486C8—41no—
487C8—41no—
488C8—41no—
489C8—41no—
490C8—41no—
491C8—41no—
492C8—41no—
493C8—81no—
494C8—81no—
495C8—81no—
496C22—21no—
497C22—21no—
498C22—81no—
499C22—81no—
500C22—81no—
501C22#236111yescharacteristic
502C22#337511yescharacteristic
503C22#337711yescharacteristic
504C4—21no—
505C4—21no—
506C4—21no—
507C4—22no—
508C4—41no—
509C4—41no—
510C4—41no—
511C4—81no—
512C4—81no—
513C4—81no—
514C4—161no—
515C4C4×#13711yescharacteristic
516C4#295111yescharacteristic
517C4C2×#44411yescharacteristic
518C4#295711yescharacteristic
519C4#326811yescharacteristic
520C4#327011yescharacteristic
521C2—21no—
522C2—21no—
523C2—161no—
524C2#2300911yescharacteristic
525C2#2301911yescharacteristic
526C2#2304211yescharacteristic
subgroupG/H
centerC2×C4D42
commutatorC42C23×C4
Frattini, cosocleC2×C42C24
socleC22#2361
prime radical#532C23
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. Q8×#108C2semidirect
5. #17998C2semidirect
7. #9630C2semidirect
8. #9591C2semidirect
11. #8990C2semidirect
12. #8933C2semidirect
14. #8297C2semidirect
15. #8205C2semidirect
40. #1684C4semidirect
41. #1684C4semidirect
42. #1682C4semidirect
43. #1682C4semidirect
44. #1680C4semidirect
45. #1680C4semidirect
46. #1678C4semidirect
47. #1678C4semidirect

Maximal subgroups

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

subgroupindex
#232352
Q8×#1082
#191232
#180042
#179982
#98582
#96302
#95912
#95902
#89922
#89902
#89332
#89302
#82972
#82052