Finite Group Databasebeta

← #219645

Subgroups — #219645

Order 512.

501 conjugacy classes of subgroups.

The subgroup lattice has 447 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22254C211yesmaximal, characteristic, complemented
2#22236C211yesmaximal, characteristic
3#19704C211yesmaximal, characteristic
4#9771C211yesmaximal, characteristic, complemented
5#9757C211yesmaximal, characteristic
6#9302C211yesmaximal, characteristic
7#9301C211yesmaximal, characteristic
8#9298C211yesmaximal, characteristic, complemented
9#9297C211yesmaximal, characteristic, complemented
10#9290C211yesmaximal, characteristic, complemented
11#9289C211yesmaximal, characteristic
12#9288C211yesmaximal, characteristic, complemented
13#9287C211yesmaximal, characteristic
14#8794C211yesmaximal, characteristic, complemented
15#8141C211yesmaximal, characteristic, complemented
16#3230C2211yescharacteristic
17#3227C2211yescharacteristic
18#3209C2211yescharacteristic
19#3206C2211yescharacteristic
20C2×#500C2211yescharacteristic
21#3131C2211yescharacteristic
22#2961C2211yescharacteristic
23#2932C2211yescharacteristic
24#2331C2211yescharacteristic
25#2331C2211yescharacteristic
26#1995C2211yescharacteristic
27#1993C2211yescharacteristic
28#1993C2211yescharacteristic
29#1963C2211yescharacteristic
30#1929—21no—
31#1929—21no—
32#1921—21no—
33#1921—21no—
34#1918—21no—
35#1918—21no—
36#1866—21no—
37#1866—21no—
38#1848C2211yescharacteristic
39#1835C2211yescharacteristic
40#1746C2211yescharacteristic
41#1723C412yescomplemented
42#1722C412yescomplemented
43#1719C412yescomplemented
44#1718C412yescomplemented
45#1549C2211yescharacteristic
46#1549C2211yescharacteristic
47#1548C2211yescharacteristic
48#1548C2211yescharacteristic
49#1518C2211yescharacteristic
50#1518C2211yescharacteristic
51#1517C2211yescharacteristic
52#1517C2211yescharacteristic
53#1452C2211yescharacteristic
54#1450C2211yescharacteristic
55#1286C2211yescharacteristic
56#1286C2211yescharacteristic
57#1286C2211yescharacteristic
58#1286C2211yescharacteristic
59#1275C2211yescharacteristic
60#1275C2211yescharacteristic
61#1274C2211yescharacteristic
62#1274C2211yescharacteristic
63C2×#131—21no—
64#533C2311yescharacteristic
65#532C2311yescharacteristic
66C2×C4⋊Q8C2311yescharacteristic
67C2×C4⋊Q8C2311yescharacteristic
68#500—21no—
69#500—21no—
70#500C2×C412yes—
71#500C2×C412yes—
72#500C2×C412yes—
73#498C2×C412yes—
74#494C2×C412yes—
75#492C2×C412yes—
76#488—21no—
77#488—21no—
78#487C2×C411yescharacteristic
79#487C2×C411yescharacteristic
80#487C2×C412yes—
81#487C2×C412yes—
82#487C2×C412yes—
83#486C2×C411yescharacteristic
84#486C2×C411yescharacteristic
85#486C2×C412yes—
86#484—41no—
87#481—41no—
88#477—41no—
89#476—41no—
90#462—41no—
91#461—41no—
92#451—41no—
93#448—41no—
94#446—41no—
95#446—41no—
96#435—41no—
97#435—41no—
98#431C2311yescharacteristic
99#431C2311yescharacteristic
100#431C2311yescharacteristic
101#431C2311yescharacteristic
102C2×#108—21no—
103C2×#108—21no—
104C2×#107—21no—
105C2×#107—21no—
106#424—21no—
107#423C2311yescharacteristic
108#423C2311yescharacteristic
109C2×#104—21no—
110C2×#104—21no—
111#416—21no—
112#416—21no—
113#408—21no—
114#405—21no—
115C2×#97C2311yescharacteristic
116C2×C4×C8C2311yescharacteristic
117#391—41no—
118#391—41no—
119#389—21no—
120#389—21no—
121#387—41no—
122#387—41no—
123#382C2311yescharacteristic
124#382C2311yescharacteristic
125C4×C4⋊C4—21no—
126C4×C4⋊C4—21no—
127#376C2311yescharacteristic
128#340—21no—
129#340—21no—
130#340—21no—
131#340—21no—
132#340—21no—
133#340—21no—
134#339—21no—
135#339—21no—
136#336D411yescharacteristic
137#336D411yescharacteristic
138#336D411yescharacteristic
139#336D411yescharacteristic
140#333—41no—
141#333—41no—
142#332—41no—
143#332—41no—
144#328—41no—
145#328—41no—
146#327—41no—
147#327—41no—
148#322—21no—
149#322C2×C412yes—
150#322C2×C412yes—
151C82—21no—
152C2×#41—21no—
153C22×Q8—21no—
154C22×Q8—21no—
155C2×Q16—42no—
156C2×SD16—42no—
157#131—81no—
158#131—81no—
159C2×M16—21no—
160C2×M16—21no—
161C2×M16—41no—
162C22×C8—41no—
163C22×C8C2×D411yescharacteristic
164C22×C8C2×D411yescharacteristic
165C22×C8C2×D411yescharacteristic
166C22×C8C2×D411yescharacteristic
167C4⋊Q8—21no—
168C4⋊Q8—21no—
169C4⋊Q8—21no—
170C4⋊Q8—21no—
171C4⋊Q8—21no—
172C4⋊Q8—21no—
173C4⋊Q8C22×C412yes—
174C4⋊Q8C22×C412yes—
175C4⋊Q8C22×C411yescharacteristic
176C4⋊Q8C22×C411yescharacteristic
177C4⋊Q8C22×C411yescharacteristic
178#127C22×C411yescharacteristic
179#125—21no—
180#124—21no—
181C22⋊Q8—21no—
182C22⋊Q8—21no—
183#121—21no—
184#121—21no—
185C4×Q8—21no—
186C4×Q8—21no—
187C4×D4—21no—
188C4×D4—21no—
189#117—21no—
190C2×C4⋊C4—21no—
191C2×C4⋊C4—21no—
192C2×C4⋊C4—21no—
193C2×C4⋊C4—21no—
194C2×C4⋊C4—21no—
195C2×C4⋊C4—21no—
196C2×C4⋊C4—41no—
197C2×C4⋊C4—41no—
198C2×C42—41no—
199C2×C42—41no—
200C2×C42C2×D411yescharacteristic
201C2×C42C2×D411yescharacteristic
202C2×C42C2411yescharacteristic
203#108—81no—
204#108—81no—
205#107—22no—
206#107—22no—
207#107—22no—
208#107—22no—
209#107—42no—
210#106—42no—
211C4⋊C8—21no—
212C4⋊C8—21no—
213C4⋊C8—21no—
214C4⋊C8—21no—
215C4⋊C8—21no—
216C4⋊C8—21no—
217C4⋊C8—21no—
218C4⋊C8—21no—
219C4⋊C8—42no—
220#104—81no—
221#104—81no—
222#103—21no—
223#103—21no—
224#103—21no—
225#103—21no—
226#103—42no—
227#103—42no—
228#103—42no—
229#102—21no—
230#102—21no—
231#102—21no—
232#102—21no—
233#102—42no—
234C22⋊C8—21no—
235C22⋊C8—21no—
236C22⋊C8—21no—
237C22⋊C8—21no—
238C22⋊C8—42no—
239#97—21no—
240#97—21no—
241#97—41no—
242C4×C8—21no—
243C4×C8—21no—
244C4×C8—41no—
245C4×C8C22×C412yes—
246C4×C8C22×C412yes—
247C4×C8C22×C411yescharacteristic
248C4×C8C22×C411yescharacteristic
249C4×C8#4111yescharacteristic
250C4×C8#4111yescharacteristic
251#95—21no—
252#95—21no—
253#95—21no—
254#95—21no—
255#95—21no—
256#95—21no—
257#95—41no—
258#95—41no—
259#95—41no—
260#95—41no—
261#41—41no—
262#41—41no—
263#41—41no—
264#41—41no—
265C2×Q8—21no—
266C2×Q8—21no—
267C2×Q8—22no—
268C2×Q8—22no—
269C2×Q8—21no—
270C2×Q8—41no—
271C2×Q8—41no—
272C2×Q8—161no—
273C2×Q8—161no—
274C2×D4—21no—
275C2×D4—21no—
276C2×D4—21no—
277C2×D4—41no—
278C22×C4—21no—
279C22×C4—21no—
280C22×C4—41no—
281C22×C4—41no—
282C22×C4—41no—
283C22×C4—41no—
284C22×C4—41no—
285C22×C4C22×D411yescharacteristic
286C22×C4H(5,𝔽2)11yescharacteristic
287C22×C4H(5,𝔽2)11yescharacteristic
288Q16—162no—
289SD16—162no—
290M16—41no—
291M16—41no—
292M16—41no—
293M16—41no—
294M16—82no—
295C2×C8—21no—
296C2×C8—21no—
297C2×C8—21no—
298C2×C8—21no—
299C2×C8—22no—
300C2×C8—22no—
301C2×C8—21no—
302C2×C8—21no—
303C2×C8—21no—
304C2×C8—21no—
305C2×C8—21no—
306C2×C8—21no—
307C2×C8—41no—
308C2×C8—41no—
309C2×C8—41no—
310C2×C8—41no—
311C2×C8—82no—
312C2×C8C4×D411yescharacteristic
313C2×C8C4×D411yescharacteristic
314C2×C8C4×D411yescharacteristic
315C2×C8C4×D411yescharacteristic
316C4⋊C4—21no—
317C4⋊C4—21no—
318C4⋊C4—22no—
319C4⋊C4—22no—
320C4⋊C4—22no—
321C4⋊C4—21no—
322C4⋊C4—21no—
323C4⋊C4—22no—
324C4⋊C4—21no—
325C4⋊C4—21no—
326C4⋊C4—41no—
327C4⋊C4—41no—
328C4⋊C4—41no—
329C4⋊C4—41no—
330C4⋊C4—41no—
331C4⋊C4—41no—
332C4⋊C4—41no—
333C4⋊C4—41no—
334C4⋊C4—41no—
335C4⋊C4—41no—
336C4⋊C4—161no—
337C4⋊C4—161no—
338C22⋊C4—41no—
339C22⋊C4—41no—
340C22⋊C4—41no—
341C42—21no—
342C42—21no—
343C42—21no—
344C42—21no—
345C42—41no—
346C42—161no—
347C42—161no—
348C42C23×C411yescharacteristic
349C42C2×#4111yescharacteristic
350C42H(5,𝔽2)11yescharacteristic
351C42H(5,𝔽2)11yescharacteristic
352C23—41no—
353C23#53511yescharacteristic
354Q8—41no—
355Q8—41no—
356Q8—82no—
357Q8—81no—
358Q8—82no—
359Q8—81no—
360Q8—161no—
361Q8—161no—
362D4—41no—
363D4—41no—
364D4—81no—
365D4—81no—
366D4—81no—
367D4—81no—
368C2×C4—41no—
369C2×C4—41no—
370C2×C4—41no—
371C2×C4—41no—
372C2×C4—41no—
373C2×C4—41no—
374C2×C4—41no—
375C2×C4—41no—
376C2×C4—41no—
377C2×C4—42no—
378C2×C4—42no—
379C2×C4—41no—
380C2×C4—41no—
381C2×C4—81no—
382C2×C4—81no—
383C2×C4—81no—
384C2×C4—81no—
385C2×C4—81no—
386C2×C4—81no—
387C2×C4—81no—
388C2×C4—81no—
389C2×C4—81no—
390C2×C4—81no—
391C2×C4—81no—
392C2×C4—81no—
393C2×C4C2×C4×D411yescharacteristic
394C2×C4#51811yescharacteristic
395C2×C4#51811yescharacteristic
396C2×C4#53511yescharacteristic
397C2×C4#54011yescharacteristic
398C2×C4#54011yescharacteristic
399C2×C4#57711yescharacteristic
400C2×C4#57711yescharacteristic
401C2×C4#57811yescharacteristic
402C2×C4#57811yescharacteristic
403C8—21no—
404C8—21no—
405C8—21no—
406C8—21no—
407C8—42no—
408C8—42no—
409C8—41no—
410C8—41no—
411C8—81no—
412C8—81no—
413C8—81no—
414C8—81no—
415C22—21no—
416C22—21no—
417C22—81no—
418C22—81no—
419C22—81no—
420C22#232811yescharacteristic
421C22#322511yescharacteristic
422C22#322511yescharacteristic
423C4—21no—
424C4—21no—
425C4—21no—
426C4—21no—
427C4—81no—
428C4—81no—
429C4—81no—
430C4—81no—
431C4—82no—
432C4—82no—
433C4—81no—
434C4—161no—
435C4—161no—
436C4#294711yescharacteristic
437C4#294711yescharacteristic
438C4#322211yescharacteristic
439C4#322311yescharacteristic
440C2—21no—
441C2—21no—
442C2—161no—
443C2#2227811yescharacteristic
444C2#2228511yescharacteristic
445C2#2228611yescharacteristic
subgroupG/H
center, socleC22#2328
commutatorC42C23×C4
Frattini, cosocleC2×C42C24
prime radical#532C23
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. #22254C2semidirect
4. #9771C2semidirect
8. #9298C2semidirect
9. #9297C2semidirect
10. #9290C2semidirect
12. #9288C2semidirect
14. #8794C2semidirect
15. #8141C2semidirect
41. #1723C4semidirect
42. #1722C4semidirect
43. #1719C4semidirect
44. #1718C4semidirect

Maximal subgroups

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

subgroupindex
#222542
#222362
#197042
#97712
#97572
#93022
#93012
#92982
#92972
#92902
#92892
#92882
#92872
#87942
#81412