Finite Group Databasebeta

← #230093

Subgroups — #230093

Order 512.

526 conjugacy classes of subgroups.

The subgroup lattice has 505 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23170C211yesmaximal, characteristic
2#22746C211yesmaximal, characteristic
3#19118C211yesmaximal, characteristic
4#18003C211yesmaximal, characteristic
5#17998C211yesmaximal, characteristic
6#9859C211yesmaximal, characteristic
7#9629C211yesmaximal, characteristic
8#9591C211yesmaximal, characteristic
9#9589C211yesmaximal, characteristic
10#8990C211yesmaximal, characteristic
11#8990C211yesmaximal, characteristic
12#8933C211yesmaximal, characteristic
13#8931C211yesmaximal, characteristic
14#8296C211yesmaximal, characteristic
15#8201C211yesmaximal, characteristic
16#3160C2211yescharacteristic
17#3104C2211yescharacteristic
18#3104C2211yescharacteristic
19Q8×M16C2211yescharacteristic
20Q8×M16C2211yescharacteristic
21#2962C2211yescharacteristic
22#2962C2211yescharacteristic
23C2×C8×Q8C2211yescharacteristic
24#1986C2211yescharacteristic
25#1986C2211yescharacteristic
26#1950—21no—
27#1950—21no—
28#1950C2211yescharacteristic
29#1950C2211yescharacteristic
30#1945—21no—
31#1922—21no—
32#1922—21no—
33#1837—21no—
34#1837C2211yescharacteristic
35C4×#108—21no—
36C4×#108C2211yescharacteristic
37#1772—21no—
38C8×C4⋊C4C2211yescharacteristic
39C4×#103C2211yescharacteristic
40#1676C411yescharacteristic, complemented
41#1676C411yescharacteristic, complemented
42#1674C411yescharacteristic, complemented
43#1674C411yescharacteristic, complemented
44#1672C411yescharacteristic, complemented
45#1672C411yescharacteristic, complemented
46#1672C411yescharacteristic, complemented
47#1672C411yescharacteristic, complemented
48#1593—21no—
49#1591—21no—
50#1587—21no—
51#1587—21no—
52#1584—21no—
53#1584—21no—
54#1579—21no—
55C8×Q16—21no—
56#1571C2211yescharacteristic
57#1570C2211yescharacteristic
58#1564C2211yescharacteristic
59#1542C2211yescharacteristic
60#1542C2211yescharacteristic
61#1493C2211yescharacteristic
62#1493C2211yescharacteristic
63#1493C2211yescharacteristic
64#1493C2211yescharacteristic
65C2×Q8⋊C8C2211yescharacteristic
66#1451C2211yescharacteristic
67C8×M16C2211yescharacteristic
68C2×#322C2211yescharacteristic
69#1303C2211yescharacteristic
70#1303C2211yescharacteristic
71#1279C2211yescharacteristic
72#1278C2211yescharacteristic
73#1278C2211yescharacteristic
74#1277C2211yescharacteristic
75#533C2311yescharacteristic
76#533C2311yescharacteristic
77C2×C4×Q8C2311yescharacteristic
78#500C2×C411yescharacteristic
79#500C2×C411yescharacteristic
80#499—21no—
81#499—21no—
82#498C2×C411yescharacteristic
83#498C2×C411yescharacteristic
84#484—21no—
85#484—21no—
86#477C2×C411yescharacteristic
87#477C2×C411yescharacteristic
88#477C2×C411yescharacteristic
89#477C2×C411yescharacteristic
90#462C2×C411yescharacteristic
91#462C2×C411yescharacteristic
92#462C2×C411yescharacteristic
93#462C2×C411yescharacteristic
94#451—21no—
95#451—21no—
96#446—21no—
97#446—21no—
98#446—21no—
99#446—21no—
100#446—41no—
101#446—41no—
102#446—41no—
103#446—41no—
104C8×Q8—21no—
105C8×Q8—21no—
106C8×Q8—21no—
107C8×Q8—21no—
108C8×Q8—21no—
109C8×Q8—21no—
110C8×Q8C2×C411yescharacteristic
111C8×Q8C2×C411yescharacteristic
112C8×Q8C2×C411yescharacteristic
113C8×Q8C2×C411yescharacteristic
114C4×Q16—21no—
115C4×Q16—21no—
116C4×Q16—21no—
117C4×Q16—21no—
118#431C2311yescharacteristic
119#431C2311yescharacteristic
120C2×#108—21no—
121C2×#108—21no—
122C2×#108D411yescharacteristic
123C2×#108D411yescharacteristic
124#427—21no—
125#427—21no—
126#424—21no—
127#424—21no—
128#423—21no—
129#423—21no—
130#423C2311yescharacteristic
131#423C2311yescharacteristic
132C2×C4⋊C8—21no—
133C2×C4⋊C8C2311yescharacteristic
134C2×C4⋊C8C2311yescharacteristic
135C2×#103—21no—
136#405—21no—
137#405—21no—
138C4×M16—21no—
139C4×M16—21no—
140C4×M16C2311yescharacteristic
141C4×M16C2311yescharacteristic
142C2×C4×C8—21no—
143C2×C4×C8—21no—
144C2×C4×C8C2311yescharacteristic
145C2×C4×C8C2311yescharacteristic
146C2×C4×C8C2311yescharacteristic
147C4×C4⋊C4C2311yescharacteristic
148#341—21no—
149#341—21no—
150#341—21no—
151#341—21no—
152#341D411yescharacteristic
153#341D411yescharacteristic
154#340—21no—
155#339—21no—
156#339—21no—
157#339—21no—
158#339—21no—
159#336—21no—
160#335—21no—
161#335—21no—
162#335C2×C411yescharacteristic
163#335C2×C411yescharacteristic
164#334—21no—
165#334—21no—
166#334C2×C411yescharacteristic
167#334C2×C411yescharacteristic
168#333C2×C411yescharacteristic
169#333C2×C411yescharacteristic
170#333C2×C411yescharacteristic
171#333C2×C411yescharacteristic
172#332C2×C411yescharacteristic
173#332C2×C411yescharacteristic
174#332C2×C411yescharacteristic
175#332C2×C411yescharacteristic
176Q8⋊C8—21no—
177Q8⋊C8—21no—
178Q8⋊C8—21no—
179Q8⋊C8—21no—
180Q8⋊C8—41no—
181Q8⋊C8—41no—
182Q8⋊C8D411yescharacteristic
183Q8⋊C8D411yescharacteristic
184Q8⋊C8D411yescharacteristic
185Q8⋊C8D411yescharacteristic
186#322—21no—
187#322—21no—
188#322—21no—
189#322—21no—
190#322—21no—
191C82—21no—
192C22×Q8#4111yescharacteristic
193C2×Q16—21no—
194C2×Q16—21no—
195C2×Q16—21no—
196C2×Q16—21no—
197C2×M16—21no—
198C2×M16—21no—
199C2×M16C2×D411yescharacteristic
200C2×M16C2×D411yescharacteristic
201C2×M16C2×D411yescharacteristic
202C2×M16C2×D411yescharacteristic
203C22×C8—21no—
204C22×C8—21no—
205C22×C8—21no—
206C22×C8—21no—
207C22×C8C2×D411yescharacteristic
208C22×C8C2×D411yescharacteristic
209C4⋊Q8C22×C411yescharacteristic
210C4⋊Q8C22×C411yescharacteristic
211C4⋊Q8C22×C411yescharacteristic
212C4⋊Q8C22×C411yescharacteristic
213#125—21no—
214#125—21no—
215C22⋊Q8—21no—
216C22⋊Q8—21no—
217C22⋊Q8—21no—
218C22⋊Q8—21no—
219C4×Q8—21no—
220C4×Q8—21no—
221C4×Q8—21no—
222C4×Q8—21no—
223C4×Q8—21no—
224C4×Q8—21no—
225C4×Q8C22×C411yescharacteristic
226C4×Q8C22×C411yescharacteristic
227C4×Q8C2×D411yescharacteristic
228C4×Q8C2×D411yescharacteristic
229#117—21no—
230#117—21no—
231C2×C4⋊C4—21no—
232C2×C4⋊C4—21no—
233C2×C4⋊C4#4111yescharacteristic
234C2×C42—21no—
235C2×C42—21no—
236C2×C42C2411yescharacteristic
237#108—22no—
238#108—41no—
239#108—42no—
240#108—42no—
241#107—21no—
242#107—21no—
243#107—21no—
244#107—21no—
245#107—21no—
246#107—21no—
247#107—21no—
248#107—21no—
249#106—21no—
250#106—21no—
251#106—21no—
252#106—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—21no—
262C4⋊C8—21no—
263C4⋊C8—41no—
264C4⋊C8—41no—
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
279#103—24no—
280#103—21no—
281#103—21no—
282#103—21no—
283#103—21no—
284#103—21no—
285#103—21no—
286#103—21no—
287#103—21no—
288C22⋊C8—21no—
289C22⋊C8—21no—
290#97—21no—
291#97—21no—
292#97—41no—
293#97—41no—
294#97—41no—
295#97—41no—
296C4×C8—21no—
297C4×C8—21no—
298C4×C8—21no—
299C4×C8—21no—
300C4×C8—21no—
301C4×C8—21no—
302C4×C8—21no—
303C4×C8—21no—
304C4×C8—21no—
305C4×C8—21no—
306C4×C8—21no—
307C4×C8—41no—
308C4×C8—41no—
309C4×C8—41no—
310C4×C8—41no—
311C4×C8—42no—
312C4×C8C22×C411yescharacteristic
313C4×C8C22×C411yescharacteristic
314C4×C8C2×D411yescharacteristic
315C4×C8C2×D411yescharacteristic
316C4×C8#4111yescharacteristic
317C4×C8#4111yescharacteristic
318#95—21no—
319C2×Q8—21no—
320C2×Q8—22no—
321C2×Q8—21no—
322C2×Q8—21no—
323C2×Q8—21no—
324C2×Q8—21no—
325C2×Q8—41no—
326C2×Q8C4×D411yescharacteristic
327C2×Q8C4×D411yescharacteristic
328C22×C4—21no—
329C22×C4—21no—
330C22×C4—41no—
331C22×C4C22×D411yescharacteristic
332C22×C4C2×#4111yescharacteristic
333C22×C42−1+411yescharacteristic
334Q16—42no—
335Q16—42no—
336Q16—81no—
337Q16—81no—
338M16—21no—
339M16—21no—
340M16—21no—
341M16—21no—
342M16—21no—
343M16—21no—
344M16—21no—
345M16—41no—
346M16—41no—
347M16—41no—
348M16—41no—
349M16#13712yes—
350C2×C8—21no—
351C2×C8—21no—
352C2×C8—21no—
353C2×C8—21no—
354C2×C8—21no—
355C2×C8—22no—
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—21no—
367C2×C8—21no—
368C2×C8—21no—
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—22no—
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⋊C4—41no—
401C4⋊C4C4×D411yescharacteristic
402C4⋊C4C4×D411yescharacteristic
403C22⋊C4—41no—
404C22⋊C4—41no—
405C42—21no—
406C42—21no—
407C42—21no—
408C42—21no—
409C42—41no—
410C42—41no—
411C42—41no—
412C42—41no—
413C42C23×C411yescharacteristic
414C42C22×D411yescharacteristic
415C42C2×#4111yescharacteristic
416C42H(5,𝔽2)11yescharacteristic
417C23#54811yescharacteristic
418Q8—21no—
419Q8—21no—
420Q8—21no—
421Q8—42no—
422Q8—41no—
423Q8—41no—
424Q8—41no—
425Q8—41no—
426Q8—81no—
427Q8—81no—
428Q8#44312yes—
429C2×C4—21no—
430C2×C4—21no—
431C2×C4—21no—
432C2×C4—21no—
433C2×C4—21no—
434C2×C4—21no—
435C2×C4—41no—
436C2×C4—42no—
437C2×C4—42no—
438C2×C4—41no—
439C2×C4—41no—
440C2×C4—41no—
441C2×C4—41no—
442C2×C4—81no—
443C2×C4—81no—
444C2×C4—81no—
445C2×C4C2×C4×D411yescharacteristic
446C2×C4C2×C4×D411yescharacteristic
447C2×C4#52011yescharacteristic
448C2×C4D4211yescharacteristic
449C2×C4#54711yescharacteristic
450C2×C4#55311yescharacteristic
451C2×C4C2×#13711yescharacteristic
452C2×C4#57511yescharacteristic
453C2×C4#57811yescharacteristic
454C2×C4#57811yescharacteristic
455C8—21no—
456C8—21no—
457C8—21no—
458C8—21no—
459C8—22no—
460C8—42no—
461C8—41no—
462C8—41no—
463C8—41no—
464C8—41no—
465C8—41no—
466C8—41no—
467C8—41no—
468C8—41no—
469C8—41no—
470C8—41no—
471C8—41no—
472C8—41no—
473C8—41no—
474C8—81no—
475C8—81no—
476C22—21no—
477C22—21no—
478C22#236111yescharacteristic
479C22#337411yescharacteristic
480C22#337711yescharacteristic
481C4—21no—
482C4—21no—
483C4—21no—
484C4—22no—
485C4—41no—
486C4—41no—
487C4—41no—
488C4—82no—
489C4—81no—
490C4—81no—
491C4—81no—
492C4—81no—
493C4C4×#13711yescharacteristic
494C4#295011yescharacteristic
495C4C2×#44311yescharacteristic
496C4#295711yescharacteristic
497C4#326911yescharacteristic
498C4#327011yescharacteristic
499C2—21no—
500C2—21no—
501C2#2301411yescharacteristic
502C2#2301811yescharacteristic
503C2#2304311yescharacteristic
subgroupG/H
centerC2×C4D42
commutatorC42C23×C4
Frattini, cosocleC2×C42C24
socleC22#2361
prime radicalC23#548
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
40. #1676C4semidirect
41. #1676C4semidirect
42. #1674C4semidirect
43. #1674C4semidirect
44. #1672C4semidirect
45. #1672C4semidirect
46. #1672C4semidirect
47. #1672C4semidirect

Maximal subgroups

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

subgroupindex
#231702
#227462
#191182
#180032
#179982
#98592
#96292
#95912
#95892
#89902
#89332
#89312
#82962
#82012