Finite Group Databasebeta

← #191231

Subgroups — #191231

Order 512.

532 conjugacy classes of subgroups.

The subgroup lattice has 473 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21993C211yesmaximal, characteristic
2#21983C211yesmaximal, characteristic
3#17326C211yesmaximal, characteristic
4#8795C211yesmaximal, characteristic, complemented
5#8795C211yesmaximal, characteristic, complemented
6#8794C211yesmaximal, characteristic, complemented
7#8794C211yesmaximal, characteristic, complemented
8#8793C211yesmaximal, characteristic, complemented
9#8792C211yesmaximal, characteristic, complemented
10#8791C211yesmaximal, characteristic, complemented
11#8790C211yesmaximal, characteristic, complemented
12#7962C211yesmaximal, characteristic
13#7959C211yesmaximal, characteristic
14#7956C211yesmaximal, characteristic
15C2×#1286C211yesmaximal, characteristic
16#2920C2211yescharacteristic
17#2920C2211yescharacteristic
18C2×#431C2211yescharacteristic
19C2×#431C2211yescharacteristic
20C2×#423C2211yescharacteristic
21#2283C2211yescharacteristic
22C2×#382C2211yescharacteristic
23#1950—21no—
24#1950—21no—
25#1949—21no—
26#1949—21no—
27#1855—21no—
28#1855—21no—
29#1853—21no—
30#1852—21no—
31#1848C2211yescharacteristic
32#1848C2211yescharacteristic
33#1846C2211yescharacteristic
34#1846C2211yescharacteristic
35#1837C2211yescharacteristic
36#1837C2211yescharacteristic
37#1835C2211yescharacteristic
38#1835C2211yescharacteristic
39#1454C412yescomplemented
40#1454C412yescomplemented
41#1452C2211yescharacteristic
42#1452C2211yescharacteristic
43#1451C412yescomplemented
44#1450C2211yescharacteristic
45#1450C2211yescharacteristic
46C8×M16C412yescomplemented
47#1286C2211yescharacteristic
48#1286C2211yescharacteristic
49#1286C2211yescharacteristic
50#1286C2211yescharacteristic
51#1286C2211yescharacteristic
52#1286C2211yescharacteristic
53#1286C2211yescharacteristic
54#1286C2211yescharacteristic
55#1286C2211yescharacteristic
56#1286C2211yescharacteristic
57#1286C2211yescharacteristic
58#1286C2211yescharacteristic
59#1286C2211yescharacteristic
60#1286C2211yescharacteristic
61#1286C2211yescharacteristic
62#1286C2211yescharacteristic
63C22×M16—21no—
64C22×C4⋊C4—21no—
65C22×C42C2311yescharacteristic
66#432—21no—
67#432—21no—
68#432—21no—
69#432—21no—
70#431—21no—
71#431—21no—
72#431—21no—
73#431—21no—
74#431C2×C412yes—
75#431C2×C412yes—
76#431C2×C411yescharacteristic
77#431C2×C411yescharacteristic
78#431C2×C411yescharacteristic
79#431C2×C411yescharacteristic
80#431Q811yescharacteristic
81#431Q811yescharacteristic
82#431Q811yescharacteristic
83#431Q811yescharacteristic
84#431C2311yescharacteristic
85#431C2311yescharacteristic
86#431C2311yescharacteristic
87#431C2311yescharacteristic
88C2×#108—21no—
89C2×#108—21no—
90C2×#108—21no—
91C2×#108—21no—
92C2×#107—21no—
93C2×#107—21no—
94C2×#106—21no—
95C2×#106—21no—
96#423—21no—
97#423—21no—
98#423—21no—
99#423—21no—
100#423C2311yescharacteristic
101#423C2311yescharacteristic
102#423C2311yescharacteristic
103#423C2311yescharacteristic
104C2×C4⋊C8—21no—
105C2×C4⋊C8—21no—
106C2×C4⋊C8—21no—
107C2×C4⋊C8D411yescharacteristic
108C2×C4⋊C8D411yescharacteristic
109#407—21no—
110#407—21no—
111C2×C22⋊C8—21no—
112C2×C22⋊C8D411yescharacteristic
113C2×C22⋊C8D411yescharacteristic
114C4×M16—21no—
115C4×M16C2×C412yes—
116C2×C4×C8C2×C412yes—
117C2×C4×C8C2311yescharacteristic
118C2×C4×C8C2311yescharacteristic
119#383—21no—
120#382—21no—
121#382—21no—
122#382—21no—
123#382C2311yescharacteristic
124#382C2311yescharacteristic
125#382C2311yescharacteristic
126#382C2311yescharacteristic
127#381—21no—
128#380—21no—
129C4×C4⋊C4—21no—
130C4×C22⋊C4—21no—
131#376—21no—
132C2×#95—21no—
133C2×#95—21no—
134#341—21no—
135#341—21no—
136#341—21no—
137#341—21no—
138#340—21no—
139#340—21no—
140#340—21no—
141#340—21no—
142#322C2×C412yescomplemented
143#322C2×C412yescomplemented
144#322C2×C412yescomplemented
145#322C2×C412yescomplemented
146#322C2×C412yescomplemented
147#322C2×C412yescomplemented
148C82C2×C412yescomplemented
149C82C2×C412yescomplemented
150C23×C4—41no—
151C23×C4C2×D411yescharacteristic
152C23×C4#4111yescharacteristic
153C23×C4#4111yescharacteristic
154C2×M16—22no—
155C2×M16—21no—
156C2×M16—21no—
157C2×M16—21no—
158C2×M16—21no—
159C2×M16—41no—
160C2×M16—81no—
161C2×M16—81no—
162C22×C8—22no—
163C22×C8—41no—
164C22×C8C2×D411yescharacteristic
165C22×C8C2×D411yescharacteristic
166C22×C8C2×D411yescharacteristic
167C22×C8C2×D411yescharacteristic
168C2×C4⋊C4—21no—
169C2×C4⋊C4—21no—
170C2×C4⋊C4—21no—
171C2×C4⋊C4—21no—
172C2×C4⋊C4—41no—
173C2×C4⋊C4—81no—
174C2×C4⋊C4—81no—
175C2×C22⋊C4—41no—
176C2×C42—41no—
177C2×C42C22×C411yescharacteristic
178C2×C42C22×C411yescharacteristic
179C2×C42C2×D411yescharacteristic
180C2×C42C2×Q811yescharacteristic
181C2×C42C2×Q811yescharacteristic
182C2×C42#4111yescharacteristic
183C2×C42#4111yescharacteristic
184C2×C42#4111yescharacteristic
185C2×C42#4111yescharacteristic
186C2×C42#4111yescharacteristic
187C2×C42#4111yescharacteristic
188C2×C42C2411yescharacteristic
189#108—81no—
190#108—81no—
191#108—81no—
192#108—81no—
193#107—81no—
194#107—81no—
195#106—81no—
196#106—81no—
197C4⋊C8—22no—
198C4⋊C8—22no—
199C4⋊C8—21no—
200C4⋊C8—21no—
201C4⋊C8—21no—
202C4⋊C8—21no—
203C4⋊C8—21no—
204C4⋊C8—21no—
205C4⋊C8—21no—
206C4⋊C8—21no—
207C4⋊C8—21no—
208C4⋊C8—21no—
209C4⋊C8—21no—
210C4⋊C8—21no—
211C4⋊C8—41no—
212C4⋊C8—41no—
213C4⋊C8C4⋊C411yescharacteristic
214C4⋊C8C4⋊C411yescharacteristic
215C4⋊C8C4⋊C411yescharacteristic
216C4⋊C8C4⋊C411yescharacteristic
217C4⋊C8C4⋊C411yescharacteristic
218C4⋊C8C4⋊C411yescharacteristic
219C4⋊C8C4⋊C411yescharacteristic
220C4⋊C8C4⋊C411yescharacteristic
221C22⋊C8—22no—
222C22⋊C8—22no—
223C22⋊C8—21no—
224C22⋊C8—21no—
225C22⋊C8—21no—
226C22⋊C8—21no—
227C22⋊C8—41no—
228C22⋊C8—41no—
229C22⋊C8C4⋊C411yescharacteristic
230C22⋊C8C4⋊C411yescharacteristic
231C22⋊C8C4⋊C411yescharacteristic
232C22⋊C8C4⋊C411yescharacteristic
233C22⋊C8C4⋊C411yescharacteristic
234C22⋊C8C4⋊C411yescharacteristic
235C22⋊C8C4⋊C411yescharacteristic
236C22⋊C8C4⋊C411yescharacteristic
237#97—22no—
238#97—41no—
239C4×C8—22no—
240C4×C8—41no—
241C4×C8C22×C412yes—
242C4×C8C22×C412yes—
243C4×C8C22×C412yes—
244C4×C8C22×C412yes—
245C4×C8C22×C411yescharacteristic
246C4×C8C22×C411yescharacteristic
247C4×C8C22×C411yescharacteristic
248C4×C8C22×C411yescharacteristic
249C4×C8C2×Q811yescharacteristic
250C4×C8C2×Q811yescharacteristic
251C4×C8C2×Q811yescharacteristic
252C4×C8C2×Q811yescharacteristic
253#95—21no—
254#95—21no—
255#95—21no—
256#95—21no—
257#95—21no—
258#95—21no—
259#95—21no—
260#95—21no—
261#95—41no—
262#95—41no—
263#95—41no—
264#95—41no—
265C24#12411yescharacteristic
266C22×C4—21no—
267C22×C4—21no—
268C22×C4—21no—
269C22×C4—21no—
270C22×C4—42no—
271C22×C4—41no—
272C22×C4—41no—
273C22×C4—41no—
274C22×C4—41no—
275C22×C4—81no—
276C22×C4—82no—
277C22×C4—82no—
278C22×C4—82no—
279C22×C4C2×C4⋊C411yescharacteristic
280C22×C4C2×C4⋊C411yescharacteristic
281C22×C4#11711yescharacteristic
282C22×C4#11711yescharacteristic
283C22×C4#11711yescharacteristic
284C22×C4#11711yescharacteristic
285C22×C4#12411yescharacteristic
286C22×C4#12411yescharacteristic
287C22×C4#12411yescharacteristic
288C22×C4#12411yescharacteristic
289C22×C4#12411yescharacteristic
290C22×C4#12511yescharacteristic
291C22×C4#12511yescharacteristic
292C22×C4#12511yescharacteristic
293C22×C4#12511yescharacteristic
294C22×C4C22×D411yescharacteristic
295C22×C4C2×#4111yescharacteristic
296C22×C4C2×#4111yescharacteristic
297M16—44no—
298M16—41no—
299M16—41no—
300M16—41no—
301M16—41no—
302M16—41no—
303M16—41no—
304M16—41no—
305M16—41no—
306M16—82no—
307C2×C8—21no—
308C2×C8—21no—
309C2×C8—21no—
310C2×C8—21no—
311C2×C8—21no—
312C2×C8—21no—
313C2×C8—21no—
314C2×C8—21no—
315C2×C8—22no—
316C2×C8—22no—
317C2×C8—22no—
318C2×C8—22no—
319C2×C8—41no—
320C2×C8—41no—
321C2×C8—41no—
322C2×C8—41no—
323C2×C8—82no—
324C2×C8—82no—
325C2×C8C2×C4⋊C411yescharacteristic
326C2×C8C2×C4⋊C411yescharacteristic
327C2×C8C2×C4⋊C411yescharacteristic
328C2×C8C2×C4⋊C411yescharacteristic
329C2×C8C2×C4⋊C411yescharacteristic
330C2×C8C2×C4⋊C411yescharacteristic
331C2×C8C2×C4⋊C411yescharacteristic
332C2×C8C2×C4⋊C411yescharacteristic
333C4⋊C4—41no—
334C4⋊C4—41no—
335C4⋊C4—41no—
336C4⋊C4—41no—
337C4⋊C4—41no—
338C4⋊C4—41no—
339C4⋊C4—41no—
340C4⋊C4—41no—
341C4⋊C4—82no—
342C22⋊C4—82no—
343C42—82no—
344C42C2×C4⋊C411yescharacteristic
345C42C2×C4⋊C411yescharacteristic
346C42#11711yescharacteristic
347C42#11711yescharacteristic
348C42#11711yescharacteristic
349C42#11711yescharacteristic
350C42#12411yescharacteristic
351C42#12411yescharacteristic
352C42#12511yescharacteristic
353C42#12511yescharacteristic
354C42#12511yescharacteristic
355C42#12511yescharacteristic
356C42C23×C411yescharacteristic
357C42C22×Q811yescharacteristic
358C42C2×#4111yescharacteristic
359C42C2×#4111yescharacteristic
360C23—22no—
361C23—21no—
362C23—21no—
363C23—41no—
364C23#38211yescharacteristic
365C23#38211yescharacteristic
366C23C2×#12411yescharacteristic
367C2×C4—22no—
368C2×C4—22no—
369C2×C4—21no—
370C2×C4—21no—
371C2×C4—21no—
372C2×C4—21no—
373C2×C4—21no—
374C2×C4—21no—
375C2×C4—21no—
376C2×C4—21no—
377C2×C4—21no—
378C2×C4—21no—
379C2×C4—41no—
380C2×C4—41no—
381C2×C4—41no—
382C2×C4—41no—
383C2×C4—41no—
384C2×C4—41no—
385C2×C4—41no—
386C2×C4—41no—
387C2×C4—41no—
388C2×C4—41no—
389C2×C4—81no—
390C2×C4—81no—
391C2×C4—81no—
392C2×C4—81no—
393C2×C4—81no—
394C2×C4—81no—
395C2×C4—81no—
396C2×C4—81no—
397C2×C4—81no—
398C2×C4—81no—
399C2×C4—81no—
400C2×C4—81no—
401C2×C4—164no—
402C2×C4#38211yescharacteristic
403C2×C4#38211yescharacteristic
404C2×C4#38211yescharacteristic
405C2×C4#38211yescharacteristic
406C2×C4#38211yescharacteristic
407C2×C4#38211yescharacteristic
408C2×C4C22×C4⋊C411yescharacteristic
409C2×C4C2×#11711yescharacteristic
410C2×C4C2×#11711yescharacteristic
411C2×C4C2×#12411yescharacteristic
412C2×C4C2×#12511yescharacteristic
413C2×C4C2×#12511yescharacteristic
414C2×C4#57611yescharacteristic
415C2×C4#57711yescharacteristic
416C2×C4#57711yescharacteristic
417C2×C4#57811yescharacteristic
418C8—21no—
419C8—21no—
420C8—21no—
421C8—21no—
422C8—21no—
423C8—21no—
424C8—21no—
425C8—21no—
426C8—42no—
427C8—42no—
428C8—42no—
429C8—42no—
430C8—81no—
431C8—81no—
432C8—81no—
433C8—81no—
434C22—21no—
435C22—21no—
436C22—21no—
437C22—21no—
438C22—21no—
439C22—21no—
440C22—21no—
441C22—42no—
442C22—42no—
443C22#183312yes—
444C22C2×#38211yescharacteristic
445C22#313611yescharacteristic
446C22#313711yescharacteristic
447C4—21no—
448C4—21no—
449C4—21no—
450C4—21no—
451C4—21no—
452C4—21no—
453C4—21no—
454C4—21no—
455C4—21no—
456C4—21no—
457C4—161no—
458C4—161no—
459C4—161no—
460C4—161no—
461C4#291211yescharacteristic
462C4#291311yescharacteristic
463C4#313611yescharacteristic
464C4#313711yescharacteristic
465C2—21no—
466C2—21no—
467C2—22no—
468C2—41no—
469C2C2×#183311yescharacteristic
470C2#2075711yescharacteristic
471C2#2076311yescharacteristic
subgroupG/H
center, socleC22C2×#382
commutatorC42C23×C4
Frattini, cosocleC2×C42C24
prime radicalC24#124
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C22×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #8795C2semidirect
5. #8795C2semidirect
6. #8794C2semidirect
7. #8794C2semidirect
8. #8793C2semidirect
9. #8792C2semidirect
10. #8791C2semidirect
11. #8790C2semidirect
39. #1454C4semidirect
40. #1454C4semidirect
43. #1451C4semidirect
46. C8×M16C4semidirect
142. #322C2×C4semidirect
143. #322C2×C4semidirect
144. #322C2×C4semidirect
145. #322C2×C4semidirect
146. #322C2×C4semidirect
147. #322C2×C4semidirect
148. C82C2×C4semidirect
149. C82C2×C4semidirect

Maximal subgroups

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

subgroupindex
#219932
#219832
#173262
#87952
#87942
#87932
#87922
#87912
#87902
#79622
#79592
#79562
C2×#12862