Finite Group Databasebeta

← #230414

Subgroups — #230414

Order 512.

549 conjugacy classes of subgroups.

The subgroup lattice has 525 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22983C211yesmaximal, characteristic
2#22982C211yesmaximal, characteristic
3#19126C211yesmaximal, characteristic
4#18072C211yesmaximal, characteristic, complemented
5#17826C211yesmaximal, characteristic
6#9853C211yesmaximal, characteristic, complemented
7#9625C211yesmaximal, characteristic, complemented
8#9583C211yesmaximal, characteristic, complemented
9#9579C211yesmaximal, characteristic, complemented
10#8993C211yesmaximal, characteristic
11#8993C211yesmaximal, characteristic
12#8931C211yesmaximal, characteristic, complemented
13#8931C211yesmaximal, characteristic, complemented
14#8279C211yesmaximal, characteristic, complemented
15#8193C211yesmaximal, characteristic
16#3170C2211yescharacteristic
17#3076C2211yescharacteristic
18#3076C2211yescharacteristic
19C8×#41C2211yescharacteristic
20#2962C2211yescharacteristic
21#2962C2211yescharacteristic
22#2906C2211yescharacteristic
23#2906C2211yescharacteristic
24#1994C2211yescharacteristic
25#1994C2211yescharacteristic
26#1950C2211yescharacteristic
27#1950C2211yescharacteristic
28#1946—21no—
29#1945—21no—
30#1922—21no—
31#1922—21no—
32#1866—21no—
33#1866—21no—
34#1767C2211yescharacteristic
35C4×C4⋊C8C2211yescharacteristic
36#1765—21no—
37#1764—21no—
38C4×#104C2211yescharacteristic
39#1744C2211yescharacteristic
40#1684C411yescharacteristic, complemented
41#1684C411yescharacteristic, complemented
42#1683C411yescharacteristic, complemented
43#1683C411yescharacteristic, complemented
44#1676C411yescharacteristic, complemented
45#1675C411yescharacteristic, complemented
46#1670C411yescharacteristic, complemented
47#1669C411yescharacteristic, complemented
48#1593—21no—
49#1587—21no—
50#1587—21no—
51#1579—21no—
52#1570—21no—
53#1570—21no—
54#1570C2211yescharacteristic
55#1570C2211yescharacteristic
56#1566—21no—
57#1566C2211yescharacteristic
58C2×#335C2211yescharacteristic
59C2×#334—21no—
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—
76C2×#131—21no—
77#533C2311yescharacteristic
78#533C2311yescharacteristic
79C4×#41C2311yescharacteristic
80#500C2×C411yescharacteristic
81#494C2×C411yescharacteristic
82#487C2×C411yescharacteristic
83#487C2×C411yescharacteristic
84#479—21no—
85#479—21no—
86#478—21no—
87#478—21no—
88#477C2×C411yescharacteristic
89#477C2×C411yescharacteristic
90#476C2×C411yescharacteristic
91#476C2×C411yescharacteristic
92#475C2×C411yescharacteristic
93#475C2×C411yescharacteristic
94#474C2×C411yescharacteristic
95#474C2×C411yescharacteristic
96#467—21no—
97#467—21no—
98#446—21no—
99#446—21no—
100#446—41no—
101#446—41no—
102C8×Q8—21no—
103C8×Q8—21no—
104C8×Q8C2×C411yescharacteristic
105C8×Q8C2×C411yescharacteristic
106C4×Q16—21no—
107C4×Q16—21no—
108C8×D4—21no—
109C8×D4—21no—
110C8×D4C2×C411yescharacteristic
111C8×D4C2×C411yescharacteristic
112#431—21no—
113#431C2311yescharacteristic
114#431C2311yescharacteristic
115C2×#108—21no—
116C2×#108—21no—
117#427—21no—
118#427—21no—
119#424—21no—
120#424—21no—
121#424—21no—
122#424—21no—
123#424—41no—
124#424—41no—
125#423—21no—
126#423—21no—
127#423C2311yescharacteristic
128#423C2311yescharacteristic
129C2×C4⋊C8—21no—
130C2×C4⋊C8—21no—
131C2×C4⋊C8C2311yescharacteristic
132C2×C4⋊C8C2311yescharacteristic
133C2×C4⋊C8C2311yescharacteristic
134C2×C4⋊C8C2311yescharacteristic
135C2×#104D411yescharacteristic
136C2×#104D411yescharacteristic
137#416—21no—
138#416—21no—
139#405—21no—
140#405—21no—
141C4×M16C2311yescharacteristic
142C2×#97—21no—
143C2×C4×C8—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#335C2×C411yescharacteristic
161#335C2×C411yescharacteristic
162#335C2×C411yescharacteristic
163#335C2×C411yescharacteristic
164#334—21no—
165#334—21no—
166#334—21no—
167#334—21no—
168#334—21no—
169#334—21no—
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—42no—
197#131—81no—
198#131—81no—
199C2×M16—21no—
200C2×M16—21no—
201C2×M16—41no—
202C2×M16—41no—
203C2×M16C2×D411yescharacteristic
204C2×M16C2×D411yescharacteristic
205C22×C8—21no—
206C22×C8—21no—
207C22×C8—21no—
208C22×C8—21no—
209C22×C8—21no—
210C22×C8—21no—
211C22×C8—21no—
212C22×C8—41no—
213C22×C8—41no—
214C22×C8#4111yescharacteristic
215C22×C8#4111yescharacteristic
216C4⋊Q8C22×C411yescharacteristic
217C4⋊Q8C22×C411yescharacteristic
218C4⋊Q8C22×C411yescharacteristic
219C4⋊Q8C22×C411yescharacteristic
220#125—21no—
221#125—21no—
222C22⋊Q8—21no—
223C22⋊Q8—21no—
224C22⋊Q8—21no—
225C22⋊Q8—21no—
226C4×Q8—21no—
227C4×Q8—21no—
228C4×Q8—21no—
229C4×Q8—21no—
230C4×Q8C22×C411yescharacteristic
231C4×Q8#4111yescharacteristic
232C4×D4—21no—
233C4×D4—21no—
234C4×D4C22×C411yescharacteristic
235C4×D4#4111yescharacteristic
236#117—21no—
237#117—21no—
238#117—21no—
239#117C2×D411yescharacteristic
240C2×C42—21no—
241C2×C42—21no—
242C2×C42—21no—
243C2×C42C2×D411yescharacteristic
244C2×C42C2×D411yescharacteristic
245C2×C42C2411yescharacteristic
246#108—81no—
247#108—81no—
248#107—21no—
249#107—21no—
250#107—21no—
251#107—21no—
252#107—21no—
253#107—21no—
254#106—21no—
255#106—21no—
256#106—21no—
257#106—21no—
258C4⋊C8—21no—
259C4⋊C8—21no—
260C4⋊C8—21no—
261C4⋊C8—21no—
262C4⋊C8—21no—
263C4⋊C8—21no—
264C4⋊C8—21no—
265C4⋊C8—21no—
266C4⋊C8—21no—
267C4⋊C8—21no—
268C4⋊C8—21no—
269C4⋊C8—41no—
270C4⋊C8—41no—
271C4⋊C8—41no—
272C4⋊C8—41no—
273C4⋊C8—41no—
274C4⋊C8—41no—
275C4⋊C8—41no—
276C4⋊C8—41no—
277C4⋊C8C22×C411yescharacteristic
278C4⋊C8C22×C411yescharacteristic
279C4⋊C8C22×C411yescharacteristic
280C4⋊C8C22×C411yescharacteristic
281C4⋊C8C22×C411yescharacteristic
282C4⋊C8C22×C411yescharacteristic
283C4⋊C8C2×D411yescharacteristic
284C4⋊C8C2×D411yescharacteristic
285C4⋊C8C2×D411yescharacteristic
286C4⋊C8C2×D411yescharacteristic
287#104—22no—
288#104—22no—
289#103—21no—
290#103—21no—
291#103—21no—
292#103—21no—
293#103—21no—
294#103—21no—
295#103—21no—
296#103—21no—
297#102—21no—
298#102—21no—
299#102—21no—
300#102—21no—
301C22⋊C8—21no—
302C22⋊C8—21no—
303C22⋊C8—21no—
304C22⋊C8—21no—
305#97—21no—
306#97—41no—
307#97—41no—
308#97—81no—
309C4×C8—21no—
310C4×C8—21no—
311C4×C8—21no—
312C4×C8—21no—
313C4×C8—21no—
314C4×C8—21no—
315C4×C8—21no—
316C4×C8—41no—
317C4×C8—41no—
318C4×C8—81no—
319C4×C8C22×C411yescharacteristic
320C4×C8C22×C411yescharacteristic
321C4×C8C2×D411yescharacteristic
322C4×C8C2×D411yescharacteristic
323#41—21no—
324#41—21no—
325#41#13712yes—
326#41#13712yes—
327C2×Q8—21no—
328C2×Q8—21no—
329C2×Q8—21no—
330C2×Q8—21no—
331C2×Q8C4×D411yescharacteristic
332C2×D4—21no—
333C2×D4C4×D411yescharacteristic
334C22×C4—21no—
335C22×C4—21no—
336C22×C4—21no—
337C22×C4—21no—
338C22×C4—21no—
339C22×C4C22×D411yescharacteristic
340C22×C4C2×#4111yescharacteristic
341C22×C42−1+411yescharacteristic
342Q16—42no—
343Q16—81no—
344M16—21no—
345M16—21no—
346M16—21no—
347M16—21no—
348M16—41no—
349M16—41no—
350M16—41no—
351M16—41no—
352M16—82no—
353M16—82no—
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—21no—
366C2×C8—41no—
367C2×C8—41no—
368C2×C8—42no—
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—41no—
382C2×C8—41no—
383C2×C8—81no—
384C2×C8—82no—
385C2×C8—82no—
386C2×C8C4×D411yescharacteristic
387C2×C8C4×D411yescharacteristic
388C2×C8C4×D411yescharacteristic
389C2×C8C4×D411yescharacteristic
390C4⋊C4—21no—
391C4⋊C4—21no—
392C4⋊C4—21no—
393C4⋊C4—21no—
394C4⋊C4—21no—
395C4⋊C4—21no—
396C4⋊C4—21no—
397C4⋊C4—21no—
398C4⋊C4—21no—
399C4⋊C4—21no—
400C4⋊C4—41no—
401C4⋊C4—41no—
402C4⋊C4—41no—
403C4⋊C4—41no—
404C4⋊C4C4×D411yescharacteristic
405C4⋊C4C4×D411yescharacteristic
406C22⋊C4—21no—
407C22⋊C4—21no—
408C22⋊C4—21no—
409C22⋊C4—41no—
410C22⋊C4—41no—
411C42—21no—
412C42—21no—
413C42—21no—
414C42—21no—
415C42—21no—
416C42—21no—
417C42—21no—
418C42—41no—
419C42—41no—
420C42—42no—
421C42—41no—
422C42—41no—
423C42C23×C411yescharacteristic
424C42C22×D411yescharacteristic
425C42C2×#4111yescharacteristic
426C42H(5,𝔽2)11yescharacteristic
427C23—21no—
428C23#54711yescharacteristic
429Q8—21no—
430Q8—41no—
431Q8—41no—
432Q8—41no—
433Q8—41no—
434Q8—81no—
435Q8—81no—
436Q8#44112yes—
437D4—22no—
438D4—22no—
439D4—21no—
440D4#44112yes—
441C2×C4—21no—
442C2×C4—21no—
443C2×C4—21no—
444C2×C4—22no—
445C2×C4—22no—
446C2×C4—22no—
447C2×C4—21no—
448C2×C4—21no—
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—41no—
458C2×C4—41no—
459C2×C4—41no—
460C2×C4—41no—
461C2×C4—81no—
462C2×C4C2×C4×D411yescharacteristic
463C2×C4C2×C4×D411yescharacteristic
464C2×C4#52011yescharacteristic
465C2×C4D4211yescharacteristic
466C2×C4#54811yescharacteristic
467C2×C4#55311yescharacteristic
468C2×C4C2×#13711yescharacteristic
469C2×C4C2×#13711yescharacteristic
470C2×C4#57711yescharacteristic
471C2×C4#57711yescharacteristic
472C8—21no—
473C8—21no—
474C8—21no—
475C8—21no—
476C8—41no—
477C8—41no—
478C8—41no—
479C8—41no—
480C8—41no—
481C8—41no—
482C8—41no—
483C8—41no—
484C8—41no—
485C8—41no—
486C8—41no—
487C8—41no—
488C8—81no—
489C8—81no—
490C8—81no—
491C8—81no—
492C22—21no—
493C22—21no—
494C22—22no—
495C22—41no—
496C22—41no—
497C22#236111yescharacteristic
498C22#334411yescharacteristic
499C22#334411yescharacteristic
500C4—21no—
501C4—21no—
502C4—21no—
503C4—22no—
504C4—42no—
505C4—41no—
506C4—41no—
507C4—81no—
508C4—81no—
509C4—81no—
510C4—81no—
511C4—81no—
512C4C2×#44111yescharacteristic
513C4#295211yescharacteristic
514C4#295511yescharacteristic
515C4#295711yescharacteristic
516C4#326611yescharacteristic
517C4#326611yescharacteristic
518C2—21no—
519C2—21no—
520C2—42no—
521C2#2301711yescharacteristic
522C2#2302811yescharacteristic
523C2#2304211yescharacteristic
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. #9853C2semidirect
7. #9625C2semidirect
8. #9583C2semidirect
9. #9579C2semidirect
12. #8931C2semidirect
13. #8931C2semidirect
14. #8279C2semidirect
40. #1684C4semidirect
41. #1684C4semidirect
42. #1683C4semidirect
43. #1683C4semidirect
44. #1676C4semidirect
45. #1675C4semidirect
46. #1670C4semidirect
47. #1669C4semidirect

Maximal subgroups

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

subgroupindex
#229832
#229822
#191262
#180722
#178262
#98532
#96252
#95832
#95792
#89932
#89312
#82792
#81932