Finite Group Databasebeta

← #230416

Subgroups — #230416

Order 512.

526 conjugacy classes of subgroups.

The subgroup lattice has 517 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23081C211yesmaximal, characteristic
2#23080C211yesmaximal, characteristic
3#19126C211yesmaximal, characteristic
4#18072C211yesmaximal, characteristic, complemented
5#17827C211yesmaximal, characteristic
6#9853C211yesmaximal, characteristic, complemented
7#9625C211yesmaximal, characteristic, complemented
8#9584C211yesmaximal, characteristic, complemented
9#9580C211yesmaximal, characteristic, complemented
10#8994C211yesmaximal, characteristic
11#8994C211yesmaximal, characteristic
12#8933C211yesmaximal, characteristic, complemented
13#8933C211yesmaximal, characteristic, complemented
14#8279C211yesmaximal, characteristic, complemented
15#8193C211yesmaximal, characteristic
16#3170C2211yescharacteristic
17#3076C2211yescharacteristic
18#3076C2211yescharacteristic
19C8×#41C2211yescharacteristic
20#2962C2211yescharacteristic
21#2962C2211yescharacteristic
22#2925C2211yescharacteristic
23#2925C2211yescharacteristic
24#1994C2211yescharacteristic
25#1994C2211yescharacteristic
26#1950C2211yescharacteristic
27#1950C2211yescharacteristic
28#1938—21no—
29#1937—21no—
30#1929—21no—
31#1929—21no—
32#1922—21no—
33#1922—21no—
34#1773—21no—
35#1772—21no—
36#1767C2211yescharacteristic
37C4×C4⋊C8C2211yescharacteristic
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#1572—21no—
53#1572—21no—
54#1571C2211yescharacteristic
55#1571C2211yescharacteristic
56#1568—21no—
57#1566C2211yescharacteristic
58#1564—21no—
59C2×#335C2211yescharacteristic
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
75#533C2311yescharacteristic
76#533C2311yescharacteristic
77C4×#41C2311yescharacteristic
78#500C2×C411yescharacteristic
79#494C2×C411yescharacteristic
80#487C2×C411yescharacteristic
81#487C2×C411yescharacteristic
82#479—21no—
83#479—21no—
84#478—21no—
85#478—21no—
86#477C2×C411yescharacteristic
87#477C2×C411yescharacteristic
88#476C2×C411yescharacteristic
89#476C2×C411yescharacteristic
90#475C2×C411yescharacteristic
91#475C2×C411yescharacteristic
92#474C2×C411yescharacteristic
93#474C2×C411yescharacteristic
94#467—21no—
95#467—21no—
96#446—21no—
97#446—21no—
98#446—41no—
99#446—41no—
100C8×Q8—21no—
101C8×Q8—21no—
102C8×Q8C2×C411yescharacteristic
103C8×Q8C2×C411yescharacteristic
104C4×Q16—21no—
105C4×Q16—21no—
106C8×D4—21no—
107C8×D4—21no—
108C8×D4C2×C411yescharacteristic
109C8×D4C2×C411yescharacteristic
110#433—41no—
111#433—41no—
112#432—21no—
113#432—21no—
114#431—21no—
115#431—21no—
116#431—21no—
117#431C2311yescharacteristic
118#431C2311yescharacteristic
119C2×#108—21no—
120C2×#108—21no—
121#427—21no—
122#427—21no—
123#424—21no—
124#424—21no—
125#423C2311yescharacteristic
126#423C2311yescharacteristic
127C2×C4⋊C8C2311yescharacteristic
128C2×C4⋊C8C2311yescharacteristic
129C2×C4⋊C8C2311yescharacteristic
130C2×C4⋊C8C2311yescharacteristic
131C2×#104D411yescharacteristic
132C2×#104D411yescharacteristic
133#416—21no—
134#416—21no—
135#408—21no—
136#408—21no—
137#405—21no—
138#405—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#335—21no—
161#335—21no—
162#335—21no—
163#335—21no—
164#335C2×C411yescharacteristic
165#335C2×C411yescharacteristic
166#335C2×C411yescharacteristic
167#335C2×C411yescharacteristic
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—
195C2×M16—21no—
196C2×M16—21no—
197C2×M16C2×D411yescharacteristic
198C2×M16C2×D411yescharacteristic
199C22×C8—21no—
200C22×C8—21no—
201C22×C8—21no—
202C22×C8—21no—
203C22×C8—21no—
204C22×C8—21no—
205C22×C8—21no—
206C22×C8#4111yescharacteristic
207C22×C8#4111yescharacteristic
208C4⋊Q8C22×C411yescharacteristic
209C4⋊Q8C22×C411yescharacteristic
210C4⋊Q8C22×C411yescharacteristic
211C4⋊Q8C22×C411yescharacteristic
212#125—21no—
213#125—21no—
214C22⋊Q8—21no—
215C22⋊Q8—21no—
216C22⋊Q8—21no—
217C22⋊Q8—21no—
218C4×Q8—21no—
219C4×Q8—21no—
220C4×Q8—21no—
221C4×Q8—21no—
222C4×Q8C22×C411yescharacteristic
223C4×Q8#4111yescharacteristic
224C4×D4—21no—
225C4×D4—21no—
226C4×D4C22×C411yescharacteristic
227C4×D4#4111yescharacteristic
228#117—21no—
229#117—21no—
230#117—21no—
231#117C2×D411yescharacteristic
232C2×C42—21no—
233C2×C42—21no—
234C2×C42—21no—
235C2×C42C2×D411yescharacteristic
236C2×C42C2×D411yescharacteristic
237C2×C42C2411yescharacteristic
238#108—42no—
239#108—42no—
240#107—21no—
241#107—21no—
242#107—21no—
243#107—21no—
244#107—21no—
245#107—21no—
246#106—21no—
247#106—21no—
248#106—21no—
249#106—21no—
250C4⋊C8—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⋊C8C22×C411yescharacteristic
266C4⋊C8C22×C411yescharacteristic
267C4⋊C8C22×C411yescharacteristic
268C4⋊C8C22×C411yescharacteristic
269C4⋊C8C22×C411yescharacteristic
270C4⋊C8C22×C411yescharacteristic
271C4⋊C8C2×D411yescharacteristic
272C4⋊C8C2×D411yescharacteristic
273C4⋊C8C2×D411yescharacteristic
274C4⋊C8C2×D411yescharacteristic
275#104—41no—
276#104—41no—
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#102—21no—
286#102—21no—
287#102—21no—
288#102—21no—
289C22⋊C8—21no—
290C22⋊C8—21no—
291C22⋊C8—21no—
292C22⋊C8—21no—
293C22⋊C8—41no—
294C22⋊C8—41no—
295C22⋊C8—41no—
296C22⋊C8—41no—
297#97—21no—
298#97—41no—
299#97—41no—
300#97—42no—
301#97—41no—
302#97—41no—
303C4×C8—21no—
304C4×C8—21no—
305C4×C8—21no—
306C4×C8—21no—
307C4×C8—21no—
308C4×C8—21no—
309C4×C8—21no—
310C4×C8—41no—
311C4×C8—41no—
312C4×C8—42no—
313C4×C8—41no—
314C4×C8—41no—
315C4×C8C22×C411yescharacteristic
316C4×C8C22×C411yescharacteristic
317C4×C8C2×D411yescharacteristic
318C4×C8C2×D411yescharacteristic
319#41—21no—
320#41—21no—
321#41—21no—
322#41—21no—
323C2×Q8—21no—
324C2×Q8—21no—
325C2×Q8—21no—
326C2×Q8—21no—
327C2×Q8C4×D411yescharacteristic
328C2×D4—21no—
329C2×D4C4×D411yescharacteristic
330C22×C4—21no—
331C22×C4—21no—
332C22×C4—21no—
333C22×C4—21no—
334C22×C4—21no—
335C22×C4C22×D411yescharacteristic
336C22×C4C2×#4111yescharacteristic
337C22×C42−1+411yescharacteristic
338Q16—42no—
339Q16—81no—
340M16—21no—
341M16—21no—
342M16—21no—
343M16—21no—
344M16—41no—
345M16—41no—
346M16—41no—
347M16—41no—
348C2×C8—21no—
349C2×C8—21no—
350C2×C8—21no—
351C2×C8—21no—
352C2×C8—21no—
353C2×C8—21no—
354C2×C8—21no—
355C2×C8—21no—
356C2×C8—21no—
357C2×C8—21no—
358C2×C8—21no—
359C2×C8—21no—
360C2×C8—41no—
361C2×C8—41no—
362C2×C8—42no—
363C2×C8—41no—
364C2×C8—41no—
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—81no—
378C2×C8C4×D411yescharacteristic
379C2×C8C4×D411yescharacteristic
380C2×C8C4×D411yescharacteristic
381C2×C8C4×D411yescharacteristic
382C4⋊C4—21no—
383C4⋊C4—21no—
384C4⋊C4—21no—
385C4⋊C4—21no—
386C4⋊C4—21no—
387C4⋊C4—21no—
388C4⋊C4—21no—
389C4⋊C4—21no—
390C4⋊C4—21no—
391C4⋊C4—21no—
392C4⋊C4—41no—
393C4⋊C4—41no—
394C4⋊C4—41no—
395C4⋊C4—41no—
396C4⋊C4C4×D411yescharacteristic
397C4⋊C4C4×D411yescharacteristic
398C22⋊C4—21no—
399C22⋊C4—21no—
400C22⋊C4—21no—
401C22⋊C4—41no—
402C22⋊C4—41no—
403C42—21no—
404C42—21no—
405C42—21no—
406C42—21no—
407C42—21no—
408C42—21no—
409C42—21no—
410C42—41no—
411C42—41no—
412C42—42no—
413C42—41no—
414C42—41no—
415C42C23×C411yescharacteristic
416C42C22×D411yescharacteristic
417C42C2×#4111yescharacteristic
418C42H(5,𝔽2)11yescharacteristic
419C23—21no—
420C23#54711yescharacteristic
421Q8—21no—
422Q8—21no—
423Q8—41no—
424Q8—41no—
425Q8—41no—
426Q8—41no—
427Q8—81no—
428Q8—81no—
429D4—21no—
430D4—21no—
431D4—41no—
432D4—41no—
433C2×C4—21no—
434C2×C4—21no—
435C2×C4—22no—
436C2×C4—21no—
437C2×C4—21no—
438C2×C4—21no—
439C2×C4—41no—
440C2×C4—41no—
441C2×C4—41no—
442C2×C4—41no—
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—81no—
454C2×C4C2×C4×D411yescharacteristic
455C2×C4C2×C4×D411yescharacteristic
456C2×C4#52011yescharacteristic
457C2×C4D4211yescharacteristic
458C2×C4#54811yescharacteristic
459C2×C4#55311yescharacteristic
460C2×C4#57511yescharacteristic
461C2×C4#57511yescharacteristic
462C2×C4#57811yescharacteristic
463C2×C4#57811yescharacteristic
464C8—21no—
465C8—21no—
466C8—21no—
467C8—21no—
468C8—41no—
469C8—41no—
470C8—41no—
471C8—41no—
472C8—41no—
473C8—41no—
474C8—41no—
475C8—41no—
476C8—41no—
477C8—41no—
478C8—41no—
479C8—41no—
480C8—81no—
481C8—81no—
482C8—81no—
483C8—81no—
484C22—21no—
485C22—21no—
486C22—41no—
487C22—41no—
488C22—41no—
489C22#236111yescharacteristic
490C22#334911yescharacteristic
491C22#334911yescharacteristic
492C4—21no—
493C4—21no—
494C4—21no—
495C4—42no—
496C4—41no—
497C4—41no—
498C4—41no—
499C4—81no—
500C4—81no—
501C4—81no—
502C4—81no—
503C4—81no—
504C4#294311yescharacteristic
505C4#295011yescharacteristic
506C4#295511yescharacteristic
507C4#295711yescharacteristic
508C4#326911yescharacteristic
509C4#326911yescharacteristic
510C2—21no—
511C2—21no—
512C2—81no—
513C2#2301711yescharacteristic
514C2#2303011yescharacteristic
515C2#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. #9584C2semidirect
9. #9580C2semidirect
12. #8933C2semidirect
13. #8933C2semidirect
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
#230812
#230802
#191262
#180722
#178272
#98532
#96252
#95842
#95802
#89942
#89332
#82792
#81932