Finite Group Databasebeta

← #219573

Subgroups — #219573

Order 512.

539 conjugacy classes of subgroups.

The subgroup lattice has 462 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21733C211yesmaximal, characteristic, complemented
2#21644C211yesmaximal, characteristic
3#19705C211yesmaximal, characteristic
4#9774C211yesmaximal, characteristic
5#9759C211yesmaximal, characteristic, complemented
6#9298C211yesmaximal, characteristic, complemented
7#9297C211yesmaximal, characteristic, complemented
8#9295C211yesmaximal, characteristic
9#9295C211yesmaximal, characteristic
10#9294C211yesmaximal, characteristic
11#9294C211yesmaximal, characteristic
12#9290C211yesmaximal, characteristic, complemented
13#9288C211yesmaximal, characteristic, complemented
14#8794C211yesmaximal, characteristic, complemented
15#8139C211yesmaximal, characteristic, complemented
16#3230C2211yescharacteristic
17#3209C2211yescharacteristic
18#3208C2211yescharacteristic
19#3208C2211yescharacteristic
20#3160C2211yescharacteristic
21C2×#486C2211yescharacteristic
22#2961C2211yescharacteristic
23#2961C2211yescharacteristic
24#2331C2211yescharacteristic
25#2328C2211yescharacteristic
26#1995C2211yescharacteristic
27#1995C2211yescharacteristic
28#1993C2211yescharacteristic
29#1961C2211yescharacteristic
30#1946—21no—
31#1945—21no—
32#1936—21no—
33#1934—21no—
34#1848C2211yescharacteristic
35#1835C2211yescharacteristic
36#1773—21no—
37#1772—21no—
38#1762—21no—
39C4×#102—21no—
40#1726C2211yescharacteristic
41#1722C412yescomplemented
42#1720C412yescomplemented
43#1719C412yescomplemented
44#1714C412yescomplemented
45#1549C2211yescharacteristic
46#1548C2211yescharacteristic
47#1545C2211yescharacteristic
48#1545C2211yescharacteristic
49#1518C2211yescharacteristic
50#1518C2211yescharacteristic
51#1518C2211yescharacteristic
52#1518C2211yescharacteristic
53#1452C2211yescharacteristic
54#1452C2211yescharacteristic
55#1286C2211yescharacteristic
56#1286C2211yescharacteristic
57#1286C2211yescharacteristic
58#1286C2211yescharacteristic
59#1275C2211yescharacteristic
60#1275C2211yescharacteristic
61#1274C2211yescharacteristic
62#1274C2211yescharacteristic
63#533C2311yescharacteristic
64#533C2311yescharacteristic
65C2×C4⋊Q8C2311yescharacteristic
66C2×#127C2311yescharacteristic
67C2×C4×D4—21no—
68#500C2×C411yescharacteristic, complemented
69#500C2×C411yescharacteristic, complemented
70#500C2×C412yescomplemented
71#499—21no—
72#499—21no—
73#498C2×C411yescharacteristic, complemented
74#498C2×C411yescharacteristic, complemented
75#494C2×C412yescomplemented
76#492C2×C412yes—
77#492C2×C412yes—
78#487C2×C412yescomplemented
79#487C2×C412yescomplemented
80#486—21no—
81#486—21no—
82#486C2×C412yes—
83#486C2×C412yes—
84#486C2×C412yes—
85#486C2×C412yes—
86#484—41no—
87#477—41no—
88#462—41no—
89#460—41no—
90#460—41no—
91#451—41no—
92#450—41no—
93#450—41no—
94#446—41no—
95#446—41no—
96#446—41no—
97#446—41no—
98#431C2311yescharacteristic
99#431C2311yescharacteristic
100#431C2311yescharacteristic
101#431C2311yescharacteristic
102C2×#108—21no—
103C2×#108—21no—
104#427—21no—
105#427—21no—
106#424—21no—
107#424—21no—
108#423C2311yescharacteristic
109#423C2311yescharacteristic
110C2×#102—21no—
111C2×#102—21no—
112C2×#102—21no—
113C2×#102—21no—
114#405—21no—
115#405—21no—
116C2×#97D411yescharacteristic
117C2×#97D411yescharacteristic
118C2×#97C2311yescharacteristic
119C2×C4×C8D411yescharacteristic
120C2×C4×C8D411yescharacteristic
121C2×C4×C8C2311yescharacteristic
122#391—41no—
123#390—21no—
124#389—21no—
125#387—41no—
126#386—41no—
127#386—41no—
128#382C2311yescharacteristic
129#382C2311yescharacteristic
130C4×C4⋊C4—21no—
131#376C2311yescharacteristic
132#340—21no—
133#340—21no—
134#340—21no—
135#340—21no—
136#339—21no—
137#339—21no—
138#339—21no—
139#339—21no—
140#333—41no—
141#332—41no—
142#331—41no—
143#331—41no—
144#328—41no—
145#328—41no—
146#328—41no—
147#328—41no—
148#322—21no—
149#322—21no—
150#322C2×C412yescomplemented
151#322C2×C412yescomplemented
152C22×Q8—21no—
153C22×D4—21no—
154C22×D4—21no—
155C22×D4—21no—
156C23×C4—41no—
157C2×Q16—42no—
158C2×SD16—42no—
159C2×SD16—42no—
160C2×M16—21no—
161C2×M16—21no—
162C22×C8C2×D411yescharacteristic
163C22×C8C2×D411yescharacteristic
164C22×C8C2×D411yescharacteristic
165C22×C8C2×D411yescharacteristic
166C4⋊Q8—21no—
167C4⋊Q8—21no—
168C4⋊Q8—21no—
169C4⋊Q8C22×C412yes—
170C4⋊Q8C22×C411yescharacteristic
171C4⋊Q8C22×C411yescharacteristic
172C4⋊Q8C22×C411yescharacteristic
173C4⋊Q8C22×C411yescharacteristic
174#127—21no—
175#127—21no—
176#127—21no—
177#127C22×C412yes—
178#125—21no—
179#125—21no—
180C22⋊Q8—21no—
181C22⋊Q8—21no—
182C22⋊Q8—21no—
183C22⋊Q8—21no—
184C4×Q8—21no—
185C4×Q8—21no—
186C4×Q8—21no—
187C4×Q8—21no—
188C4×D4—42no—
189C4×D4—42no—
190#117—21no—
191#117—21no—
192C2×C4⋊C4—21no—
193C2×C4⋊C4—21no—
194C2×C4⋊C4—21no—
195C2×C4⋊C4—21no—
196C2×C4⋊C4—41no—
197C2×C22⋊C4—41no—
198C2×C22⋊C4—41no—
199C2×C22⋊C4—41no—
200C2×C42—41no—
201C2×C42C2×D411yescharacteristic
202C2×C42C2×D411yescharacteristic
203C2×C42C2411yescharacteristic
204#108—42no—
205#108—42no—
206#107—21no—
207#107—21no—
208#107—21no—
209#107—21no—
210#107—42no—
211#106—21no—
212#106—21no—
213#106—21no—
214#106—21no—
215C4⋊C8—21no—
216C4⋊C8—21no—
217C4⋊C8—21no—
218C4⋊C8—21no—
219C4⋊C8—21no—
220C4⋊C8—21no—
221C4⋊C8—21no—
222C4⋊C8—21no—
223C4⋊C8—42no—
224C4⋊C8—42no—
225#103—42no—
226#103—42no—
227#102—22no—
228#102—22no—
229#102—22no—
230#102—22no—
231#102—42no—
232#102—42no—
233#102—42no—
234#102—42no—
235C22⋊C8—21no—
236C22⋊C8—21no—
237C22⋊C8—21no—
238C22⋊C8—21no—
239#97—22no—
240#97—21no—
241#97—21no—
242#97—41no—
243#97—41no—
244#97—41no—
245C4×C8—22no—
246C4×C8—21no—
247C4×C8—21no—
248C4×C8—41no—
249C4×C8—41no—
250C4×C8—41no—
251C4×C8C22×C412yes—
252C4×C8C22×C412yes—
253C4×C8C22×C411yescharacteristic
254C4×C8C22×C411yescharacteristic
255C4×C8#4111yescharacteristic
256C4×C8#4111yescharacteristic
257#95—21no—
258#95—21no—
259#95—21no—
260#95—21no—
261#95—21no—
262#95—21no—
263#95—41no—
264#95—41no—
265C24—41no—
266C2×Q8—21no—
267C2×Q8—22no—
268C2×Q8—21no—
269C2×Q8—21no—
270C2×Q8—21no—
271C2×Q8—41no—
272C2×Q8—161no—
273C2×D4—22no—
274C2×D4—22no—
275C2×D4—22no—
276C2×D4—41no—
277C2×D4—41no—
278C2×D4—41no—
279C2×D4—41no—
280C2×D4—41no—
281C2×D4—41no—
282C2×D4—41no—
283C2×D4—161no—
284C2×D4—161no—
285C22×C4—21no—
286C22×C4—21no—
287C22×C4—41no—
288C22×C4—41no—
289C22×C4—41no—
290C22×C4—82no—
291C22×C4—161no—
292C22×C4—161no—
293C22×C4C22×D411yescharacteristic
294C22×C4H(5,𝔽2)11yescharacteristic
295C22×C4H(5,𝔽2)11yescharacteristic
296Q16—162no—
297SD16—162no—
298SD16—162no—
299M16—41no—
300M16—41no—
301M16—41no—
302M16—41no—
303C2×C8—21no—
304C2×C8—21no—
305C2×C8—21no—
306C2×C8—21no—
307C2×C8—21no—
308C2×C8—21no—
309C2×C8—22no—
310C2×C8—22no—
311C2×C8—21no—
312C2×C8—21no—
313C2×C8—21no—
314C2×C8—41no—
315C2×C8—41no—
316C2×C8—41no—
317C2×C8—41no—
318C2×C8C4×D411yescharacteristic
319C2×C8C4×D411yescharacteristic
320C2×C8C4×D411yescharacteristic
321C2×C8C4×D411yescharacteristic
322C2×C8#13712yes—
323C4⋊C4—22no—
324C4⋊C4—21no—
325C4⋊C4—21no—
326C4⋊C4—21no—
327C4⋊C4—21no—
328C4⋊C4—21no—
329C4⋊C4—21no—
330C4⋊C4—21no—
331C4⋊C4—22no—
332C4⋊C4—21no—
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—41no—
342C4⋊C4—41no—
343C4⋊C4—161no—
344C22⋊C4—41no—
345C22⋊C4—41no—
346C22⋊C4—82no—
347C22⋊C4—161no—
348C22⋊C4—161no—
349C42—21no—
350C42—21no—
351C42—21no—
352C42—41no—
353C42—41no—
354C42—161no—
355C42#13712yes—
356C42C23×C411yescharacteristic
357C42C2×#4111yescharacteristic
358C42H(5,𝔽2)11yescharacteristic
359C42H(5,𝔽2)11yescharacteristic
360C23—42no—
361C23—81no—
362C23—81no—
363C23—81no—
364C23—81no—
365C23—81no—
366C23—81no—
367C23#53511yescharacteristic
368Q8—81no—
369Q8—81no—
370Q8—81no—
371Q8—82no—
372Q8—81no—
373Q8—161no—
374D4—42no—
375D4—42no—
376D4—82no—
377D4—82no—
378D4—81no—
379D4—81no—
380D4—161no—
381D4—161no—
382C2×C4—22no—
383C2×C4—41no—
384C2×C4—41no—
385C2×C4—41no—
386C2×C4—41no—
387C2×C4—41no—
388C2×C4—41no—
389C2×C4—41no—
390C2×C4—41no—
391C2×C4—41no—
392C2×C4—41no—
393C2×C4—42no—
394C2×C4—41no—
395C2×C4—81no—
396C2×C4—81no—
397C2×C4—81no—
398C2×C4—81no—
399C2×C4—81no—
400C2×C4—81no—
401C2×C4—81no—
402C2×C4—81no—
403C2×C4—164no—
404C2×C4C2×C4×D411yescharacteristic
405C2×C4#51811yescharacteristic
406C2×C4#51811yescharacteristic
407C2×C4#53511yescharacteristic
408C2×C4#54011yescharacteristic
409C2×C4#54011yescharacteristic
410C2×C4C2×#13711yescharacteristic
411C2×C4C2×#13711yescharacteristic
412C2×C4#57511yescharacteristic
413C2×C4#57511yescharacteristic
414C8—21no—
415C8—21no—
416C8—21no—
417C8—21no—
418C8—22no—
419C8—42no—
420C8—42no—
421C8—41no—
422C8—81no—
423C8—81no—
424C8—81no—
425C8—81no—
426C22—21no—
427C22—21no—
428C22—82no—
429C22—82no—
430C22—82no—
431C22—161no—
432C22—161no—
433C22—161no—
434C22—161no—
435C22#232811yescharacteristic
436C22#321511yescharacteristic
437C22#321611yescharacteristic
438C4—21no—
439C4—21no—
440C4—21no—
441C4—21no—
442C4—42no—
443C4—81no—
444C4—81no—
445C4—81no—
446C4—82no—
447C4—81no—
448C4—81no—
449C4—161no—
450C4—161no—
451C4C4×#13711yescharacteristic
452C4C4×#13711yescharacteristic
453C4#321811yescharacteristic
454C4#321911yescharacteristic
455C2—21no—
456C2—21no—
457C2—162no—
458C2#2227311yescharacteristic
459C2#2228411yescharacteristic
460C2#2228611yescharacteristic
subgroupG/H
center, socleC22#2328
commutatorC42C23×C4
Frattini, cosocleC2×C42C24
prime radicalC2×#127C23
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C2×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #21733C2semidirect
5. #9759C2semidirect
6. #9298C2semidirect
7. #9297C2semidirect
12. #9290C2semidirect
13. #9288C2semidirect
14. #8794C2semidirect
15. #8139C2semidirect
41. #1722C4semidirect
42. #1720C4semidirect
43. #1719C4semidirect
44. #1714C4semidirect
68. #500C2×C4semidirect
69. #500C2×C4semidirect
70. #500C2×C4semidirect
73. #498C2×C4semidirect
74. #498C2×C4semidirect
75. #494C2×C4semidirect
78. #487C2×C4semidirect
79. #487C2×C4semidirect
150. #322C2×C4semidirect
151. #322C2×C4semidirect

Maximal subgroups

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

subgroupindex
#217332
#216442
#197052
#97742
#97592
#92982
#92972
#92952
#92942
#92902
#92882
#87942
#81392