Finite Group Databasebeta

← #230432

Subgroups — #230432

Order 512.

532 conjugacy classes of subgroups.

The subgroup lattice has 521 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23080C211yesmaximal, characteristic
2#22983C211yesmaximal, characteristic
3#19158C211yesmaximal, characteristic
4#18072C211yesmaximal, characteristic, complemented
5#17830C211yesmaximal, characteristic
6#9848C211yesmaximal, characteristic, complemented
7#9626C211yesmaximal, characteristic, complemented
8#9584C211yesmaximal, characteristic, complemented
9#9579C211yesmaximal, characteristic, complemented
10#8994C211yesmaximal, characteristic
11#8993C211yesmaximal, characteristic
12#8933C211yesmaximal, characteristic, complemented
13#8931C211yesmaximal, characteristic, complemented
14#8277C211yesmaximal, characteristic, complemented
15#8195C211yesmaximal, characteristic
16#3170C2211yescharacteristic
17#3076C2211yescharacteristic
18#3076C2211yescharacteristic
19#2965C2211yescharacteristic
20#2962C2211yescharacteristic
21#2962C2211yescharacteristic
22#2925C2211yescharacteristic
23#2906C2211yescharacteristic
24#1994C2211yescharacteristic
25#1994C2211yescharacteristic
26#1950C2211yescharacteristic
27#1950C2211yescharacteristic
28#1946—21no—
29#1937—21no—
30#1929—21no—
31#1922—21no—
32#1922—21no—
33#1866—21no—
34#1772—21no—
35#1767C2211yescharacteristic
36C4×C4⋊C8C2211yescharacteristic
37#1765—21no—
38C4×#104C2211yescharacteristic
39#1744C2211yescharacteristic
40#1704C411yescharacteristic, complemented
41#1704C411yescharacteristic, complemented
42#1703C411yescharacteristic, complemented
43#1703C411yescharacteristic, complemented
44#1696C411yescharacteristic, complemented
45#1695C411yescharacteristic, complemented
46#1690C411yescharacteristic, complemented
47#1689C411yescharacteristic, complemented
48#1593—21no—
49#1587—21no—
50#1587—21no—
51#1579—21no—
52#1574—21no—
53#1573—21no—
54#1571C2211yescharacteristic
55#1570C2211yescharacteristic
56#1569—21no—
57#1567C2211yescharacteristic
58#1565—21no—
59C2×#334C2211yescharacteristic
60C2×#334C2211yescharacteristic
61#1542C2211yescharacteristic
62#1542C2211yescharacteristic
63#1493C2211yescharacteristic
64#1493C2211yescharacteristic
65#1481C2211yescharacteristic
66#1481C2211yescharacteristic
67#1476C2211yescharacteristic
68#1451C2211yescharacteristic
69#1301C2211yescharacteristic
70#1301C2211yescharacteristic
71#1278C2211yescharacteristic
72#1277C2211yescharacteristic
73#1277C2211yescharacteristic
74#1276C2211yescharacteristic
75C2×#131—21no—
76#533C2311yescharacteristic
77#533C2311yescharacteristic
78C4×#41C2311yescharacteristic
79#500C2×C411yescharacteristic
80#494C2×C411yescharacteristic
81#487C2×C411yescharacteristic
82#487C2×C411yescharacteristic
83#479—21no—
84#479—21no—
85#478—21no—
86#478—21no—
87#477C2×C411yescharacteristic
88#477C2×C411yescharacteristic
89#476C2×C411yescharacteristic
90#476C2×C411yescharacteristic
91#475C2×C411yescharacteristic
92#475C2×C411yescharacteristic
93#474C2×C411yescharacteristic
94#474C2×C411yescharacteristic
95#467—21no—
96#467—21no—
97#446—21no—
98#446—21no—
99#446—41no—
100#446—41no—
101#446C2×C411yescharacteristic
102#446C2×C411yescharacteristic
103C8×Q8—21no—
104C8×Q8—21no—
105C4×Q16—21no—
106C4×Q16—21no—
107#436C2×C411yescharacteristic
108#436C2×C411yescharacteristic
109#435—21no—
110#435—21no—
111#433—41no—
112#432—21no—
113#431—21no—
114#431—21no—
115#431C2311yescharacteristic
116#431C2311yescharacteristic
117C2×#108—21no—
118C2×#108—21no—
119#427—21no—
120#427—21no—
121#424—21no—
122#424—21no—
123#424—21no—
124#424—41no—
125#423—21no—
126#423C2311yescharacteristic
127#423C2311yescharacteristic
128C2×C4⋊C8—21no—
129C2×C4⋊C8C2311yescharacteristic
130C2×C4⋊C8C2311yescharacteristic
131C2×C4⋊C8C2311yescharacteristic
132C2×C4⋊C8C2311yescharacteristic
133C2×#104D411yescharacteristic
134C2×#104D411yescharacteristic
135#416—21no—
136#416—21no—
137#408—21no—
138#405—21no—
139#405—21no—
140#405—21no—
141C4×M16—21no—
142C4×M16C2311yescharacteristic
143C2×#97—21no—
144C2×C4×C8—21no—
145C2×C4×C8C2311yescharacteristic
146C2×C4×C8C2311yescharacteristic
147C43C2311yescharacteristic
148#341—21no—
149#341—21no—
150#339—21no—
151#339—21no—
152#339—21no—
153#339—21no—
154#339D411yescharacteristic
155#339D411yescharacteristic
156#336—21no—
157#336—21no—
158#336—21no—
159#336—21no—
160#335—21no—
161#335—21no—
162#335—21no—
163#335—21no—
164#334—21no—
165#334—21no—
166#334C2×C411yescharacteristic
167#334C2×C411yescharacteristic
168#334C2×C411yescharacteristic
169#334C2×C411yescharacteristic
170#334D411yescharacteristic
171#334D411yescharacteristic
172#334D411yescharacteristic
173#334D411yescharacteristic
174#333C2×C411yescharacteristic
175#333C2×C411yescharacteristic
176#333C2×C411yescharacteristic
177#333C2×C411yescharacteristic
178#332C2×C411yescharacteristic
179#332C2×C411yescharacteristic
180#332C2×C411yescharacteristic
181#332C2×C411yescharacteristic
182Q8⋊C8—21no—
183Q8⋊C8—21no—
184Q8⋊C8—21no—
185Q8⋊C8—21no—
186Q8⋊C8—21no—
187Q8⋊C8—41no—
188Q8⋊C8—41no—
189D4⋊C8—21no—
190#322—21no—
191#322—21no—
192C2×#41C2×D411yescharacteristic
193C2×Q16—21no—
194C2×Q16—21no—
195#131—42no—
196#131—81no—
197C2×M16—21no—
198C2×M16—21no—
199C2×M16—21no—
200C2×M16—21no—
201C2×M16—41no—
202C2×M16C2×D411yescharacteristic
203C2×M16C2×D411yescharacteristic
204C22×C8—21no—
205C22×C8—21no—
206C22×C8—21no—
207C22×C8—21no—
208C22×C8—21no—
209C22×C8—41no—
210C22×C8#4111yescharacteristic
211C22×C8#4111yescharacteristic
212C4⋊Q8C22×C411yescharacteristic
213C4⋊Q8C22×C411yescharacteristic
214C4⋊Q8C22×C411yescharacteristic
215C4⋊Q8C22×C411yescharacteristic
216#125—21no—
217#125—21no—
218C22⋊Q8—21no—
219C22⋊Q8—21no—
220C22⋊Q8—21no—
221C22⋊Q8—21no—
222C4×Q8—21no—
223C4×Q8—21no—
224C4×Q8—21no—
225C4×Q8—21no—
226C4×Q8C22×C411yescharacteristic
227C4×Q8#4111yescharacteristic
228C4×D4—21no—
229C4×D4—21no—
230C4×D4C22×C411yescharacteristic
231C4×D4#4111yescharacteristic
232#117—21no—
233#117—21no—
234#117—21no—
235#117C2×D411yescharacteristic
236C2×C42—21no—
237C2×C42—21no—
238C2×C42—21no—
239C2×C42C2×D411yescharacteristic
240C2×C42C2×D411yescharacteristic
241C2×C42C2411yescharacteristic
242#108—42no—
243#108—81no—
244#107—21no—
245#107—21no—
246#107—21no—
247#107—21no—
248#107—21no—
249#107—21no—
250#106—21no—
251#106—21no—
252#106—21no—
253#106—21no—
254C4⋊C8—21no—
255C4⋊C8—21no—
256C4⋊C8—21no—
257C4⋊C8—21no—
258C4⋊C8—21no—
259C4⋊C8—21no—
260C4⋊C8—21no—
261C4⋊C8—21no—
262C4⋊C8—21no—
263C4⋊C8—21no—
264C4⋊C8—21no—
265C4⋊C8—41no—
266C4⋊C8—41no—
267C4⋊C8—41no—
268C4⋊C8—41no—
269C4⋊C8—41no—
270C4⋊C8—41no—
271C4⋊C8C22×C411yescharacteristic
272C4⋊C8C22×C411yescharacteristic
273C4⋊C8C22×C411yescharacteristic
274C4⋊C8C22×C411yescharacteristic
275C4⋊C8C22×C411yescharacteristic
276C4⋊C8C22×C411yescharacteristic
277C4⋊C8C2×D411yescharacteristic
278C4⋊C8C2×D411yescharacteristic
279C4⋊C8C2×D411yescharacteristic
280C4⋊C8C2×D411yescharacteristic
281#104—22no—
282#104—41no—
283#103—21no—
284#103—21no—
285#103—21no—
286#103—21no—
287#103—21no—
288#103—21no—
289#103—21no—
290#103—21no—
291#102—21no—
292#102—21no—
293#102—21no—
294#102—21no—
295C22⋊C8—21no—
296C22⋊C8—21no—
297C22⋊C8—21no—
298C22⋊C8—21no—
299C22⋊C8—41no—
300C22⋊C8—41no—
301#97—21no—
302#97—21no—
303#97—21no—
304#97—41no—
305#97—41no—
306#97—41no—
307#97—81no—
308C4×C8—21no—
309C4×C8—21no—
310C4×C8—21no—
311C4×C8—21no—
312C4×C8—21no—
313C4×C8—41no—
314C4×C8—41no—
315C4×C8—41no—
316C4×C8—42no—
317C4×C8C22×C411yescharacteristic
318C4×C8C22×C411yescharacteristic
319C4×C8C2×D411yescharacteristic
320C4×C8C2×D411yescharacteristic
321#41—21no—
322#41—21no—
323#41—21no—
324#41—21no—
325C2×Q8—21no—
326C2×Q8—21no—
327C2×Q8—21no—
328C2×Q8—21no—
329C2×Q8C4×D411yescharacteristic
330C2×D4—21no—
331C2×D4C4×D411yescharacteristic
332C22×C4—21no—
333C22×C4—21no—
334C22×C4—21no—
335C22×C4—21no—
336C22×C4—21no—
337C22×C4C22×D411yescharacteristic
338C22×C4C2×#4111yescharacteristic
339C22×C42−1+411yescharacteristic
340Q16—42no—
341Q16—81no—
342M16—21no—
343M16—21no—
344M16—21no—
345M16—21no—
346M16—42no—
347M16—41no—
348M16—41no—
349M16—41no—
350M16—41no—
351M16—81no—
352M16—82no—
353C2×C8—21no—
354C2×C8—21no—
355C2×C8—21no—
356C2×C8—21no—
357C2×C8—21no—
358C2×C8—21no—
359C2×C8—21no—
360C2×C8—21no—
361C2×C8—21no—
362C2×C8—21no—
363C2×C8—21no—
364C2×C8—21no—
365C2×C8—41no—
366C2×C8—41no—
367C2×C8—41no—
368C2×C8—41no—
369C2×C8—41no—
370C2×C8—41no—
371C2×C8—41no—
372C2×C8—41no—
373C2×C8—41no—
374C2×C8—41no—
375C2×C8—41no—
376C2×C8—41no—
377C2×C8—41no—
378C2×C8—41no—
379C2×C8—41no—
380C2×C8—41no—
381C2×C8—82no—
382C2×C8C4×D411yescharacteristic
383C2×C8C4×D411yescharacteristic
384C2×C8C4×D411yescharacteristic
385C2×C8C4×D411yescharacteristic
386C4⋊C4—21no—
387C4⋊C4—21no—
388C4⋊C4—21no—
389C4⋊C4—21no—
390C4⋊C4—21no—
391C4⋊C4—21no—
392C4⋊C4—21no—
393C4⋊C4—21no—
394C4⋊C4—21no—
395C4⋊C4—21no—
396C4⋊C4—41no—
397C4⋊C4—41no—
398C4⋊C4—41no—
399C4⋊C4—41no—
400C4⋊C4C4×D411yescharacteristic
401C4⋊C4C4×D411yescharacteristic
402C22⋊C4—21no—
403C22⋊C4—21no—
404C22⋊C4—21no—
405C22⋊C4—41no—
406C22⋊C4—41no—
407C42—21no—
408C42—21no—
409C42—21no—
410C42—21no—
411C42—21no—
412C42—21no—
413C42—41no—
414C42—41no—
415C42—41no—
416C42—41no—
417C42—81no—
418C42#13712yes—
419C42C23×C411yescharacteristic
420C42C22×D411yescharacteristic
421C42C2×#4111yescharacteristic
422C42H(5,𝔽2)11yescharacteristic
423C23—21no—
424C23#54711yescharacteristic
425Q8—21no—
426Q8—21no—
427Q8—41no—
428Q8—41no—
429Q8—41no—
430Q8—41no—
431Q8—81no—
432Q8—81no—
433D4—21no—
434D4—21no—
435D4—41no—
436D4—41no—
437C2×C4—21no—
438C2×C4—21no—
439C2×C4—22no—
440C2×C4—21no—
441C2×C4—21no—
442C2×C4—21no—
443C2×C4—41no—
444C2×C4—41no—
445C2×C4—41no—
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—41no—
456C2×C4—41no—
457C2×C4—81no—
458C2×C4C2×C4×D411yescharacteristic
459C2×C4C2×C4×D411yescharacteristic
460C2×C4#52011yescharacteristic
461C2×C4D4211yescharacteristic
462C2×C4#54811yescharacteristic
463C2×C4#55311yescharacteristic
464C2×C4C2×#13711yescharacteristic
465C2×C4#57511yescharacteristic
466C2×C4#57711yescharacteristic
467C2×C4#57811yescharacteristic
468C8—21no—
469C8—21no—
470C8—21no—
471C8—21no—
472C8—41no—
473C8—41no—
474C8—41no—
475C8—41no—
476C8—41no—
477C8—41no—
478C8—41no—
479C8—41no—
480C8—41no—
481C8—41no—
482C8—41no—
483C8—41no—
484C8—81no—
485C8—81no—
486C8—81no—
487C8—81no—
488C22—21no—
489C22—21no—
490C22—41no—
491C22—41no—
492C22—41no—
493C22#236111yescharacteristic
494C22#334411yescharacteristic
495C22#334911yescharacteristic
496C4—21no—
497C4—21no—
498C4—21no—
499C4—42no—
500C4—41no—
501C4—41no—
502C4—41no—
503C4—81no—
504C4—81no—
505C4—81no—
506C4—81no—
507C4—81no—
508C4C4×#13711yescharacteristic
509C4#294711yescharacteristic
510C4#295511yescharacteristic
511C4#295711yescharacteristic
512C4#327511yescharacteristic
513C4#327611yescharacteristic
514C2—21no—
515C2—21no—
516C2—81no—
517C2#2301711yescharacteristic
518C2#2303811yescharacteristic
519C2#2304311yescharacteristic
subgroupG/H
centerC2×C4D42
commutatorC42C23×C4
Frattini, cosocleC2×C42C24
socleC22#2361
prime radicalC2×#41C2×D4
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
4. #18072C2semidirect
6. #9848C2semidirect
7. #9626C2semidirect
8. #9584C2semidirect
9. #9579C2semidirect
12. #8933C2semidirect
13. #8931C2semidirect
14. #8277C2semidirect
40. #1704C4semidirect
41. #1704C4semidirect
42. #1703C4semidirect
43. #1703C4semidirect
44. #1696C4semidirect
45. #1695C4semidirect
46. #1690C4semidirect
47. #1689C4semidirect

Maximal subgroups

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

subgroupindex
#230802
#229832
#191582
#180722
#178302
#98482
#96262
#95842
#95792
#89942
#89932
#89332
#89312
#82772
#81952