Finite Group Databasebeta

← #219571

Subgroups — #219571

Order 512.

518 conjugacy classes of subgroups.

The subgroup lattice has 451 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21725C211yesmaximal, characteristic, complemented
2#21659C211yesmaximal, characteristic
3#19704C211yesmaximal, characteristic
4#9778C211yesmaximal, characteristic, complemented
5#9758C211yesmaximal, 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#8795C211yesmaximal, characteristic, complemented
15#8140C211yesmaximal, 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#1943—21no—
31#1942—21no—
32#1938—21no—
33#1937—21no—
34#1846C2211yescharacteristic
35#1837C2211yescharacteristic
36#1776—21no—
37C4×#107—21no—
38#1765—21no—
39#1764—21no—
40#1726C2211yescharacteristic
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#1452C2211yescharacteristic
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×#107—21no—
103C2×#107—21no—
104C2×#107—21no—
105C2×#106—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×#97D411yescharacteristic
116C2×#97D411yescharacteristic
117C2×#97C2311yescharacteristic
118C2×C4×C8D411yescharacteristic
119C2×C4×C8D411yescharacteristic
120C2×C4×C8C2311yescharacteristic
121#391—41no—
122#391—41no—
123#389—21no—
124#389—21no—
125#387—41no—
126#387—41no—
127#382C2311yescharacteristic
128#382C2311yescharacteristic
129C4×C4⋊C4—21no—
130C4×C4⋊C4—21no—
131#376C2311yescharacteristic
132#341—21no—
133#341—21no—
134#340—21no—
135#340—21no—
136#340—21no—
137#340—21no—
138#339—21no—
139#339—21no—
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#322—21no—
150#322C2×C412yes—
151#322C2×C412yes—
152C2×#41—21no—
153C22×Q8—21no—
154C22×Q8—21no—
155C2×Q16—42no—
156C2×SD16—42no—
157#131—42no—
158#131—42no—
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#107—22no—
204#107—22no—
205#107—22no—
206#107—22no—
207#107—42no—
208#107—42no—
209#106—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—42no—
221#104—42no—
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—22no—
240#97—21no—
241#97—21no—
242#97—41no—
243#97—41no—
244C4×C8—22no—
245C4×C8—21no—
246C4×C8—21no—
247C4×C8—41no—
248C4×C8—41no—
249C4×C8C22×C412yes—
250C4×C8C22×C412yes—
251C4×C8C22×C411yescharacteristic
252C4×C8C22×C411yescharacteristic
253C4×C8#4111yescharacteristic
254C4×C8#4111yescharacteristic
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—
265#41—41no—
266#41—41no—
267#41—41no—
268#41—41no—
269C2×Q8—21no—
270C2×Q8—21no—
271C2×Q8—22no—
272C2×Q8—22no—
273C2×Q8—21no—
274C2×Q8—41no—
275C2×Q8—41no—
276C2×Q8—161no—
277C2×Q8—161no—
278C2×D4—21no—
279C2×D4—21no—
280C2×D4—21no—
281C2×D4—41no—
282C22×C4—21no—
283C22×C4—21no—
284C22×C4—41no—
285C22×C4—41no—
286C22×C4—41no—
287C22×C4—41no—
288C22×C4—41no—
289C22×C4C22×D411yescharacteristic
290C22×C4H(5,𝔽2)11yescharacteristic
291C22×C4H(5,𝔽2)11yescharacteristic
292Q16—162no—
293SD16—162no—
294M16—41no—
295M16—41no—
296M16—41no—
297M16—41no—
298M16—82no—
299C2×C8—21no—
300C2×C8—21no—
301C2×C8—21no—
302C2×C8—21no—
303C2×C8—21no—
304C2×C8—21no—
305C2×C8—22no—
306C2×C8—22no—
307C2×C8—21no—
308C2×C8—21no—
309C2×C8—21no—
310C2×C8—41no—
311C2×C8—41no—
312C2×C8—41no—
313C2×C8—41no—
314C2×C8—82no—
315C2×C8C4×D411yescharacteristic
316C2×C8C4×D411yescharacteristic
317C2×C8C4×D411yescharacteristic
318C2×C8C4×D411yescharacteristic
319C2×C8#13712yes—
320C4⋊C4—21no—
321C4⋊C4—21no—
322C4⋊C4—22no—
323C4⋊C4—22no—
324C4⋊C4—22no—
325C4⋊C4—21no—
326C4⋊C4—21no—
327C4⋊C4—22no—
328C4⋊C4—21no—
329C4⋊C4—21no—
330C4⋊C4—41no—
331C4⋊C4—41no—
332C4⋊C4—41no—
333C4⋊C4—41no—
334C4⋊C4—41no—
335C4⋊C4—41no—
336C4⋊C4—41no—
337C4⋊C4—41no—
338C4⋊C4—41no—
339C4⋊C4—41no—
340C4⋊C4—161no—
341C4⋊C4—161no—
342C22⋊C4—41no—
343C22⋊C4—41no—
344C22⋊C4—41no—
345C42—21no—
346C42—21no—
347C42—21no—
348C42—41no—
349C42—161no—
350C42—161no—
351C42#13712yes—
352C42C23×C411yescharacteristic
353C42C2×#4111yescharacteristic
354C42H(5,𝔽2)11yescharacteristic
355C42H(5,𝔽2)11yescharacteristic
356C23—41no—
357C23#53511yescharacteristic
358Q8—41no—
359Q8—41no—
360Q8—82no—
361Q8—81no—
362Q8—82no—
363Q8—81no—
364Q8—161no—
365Q8—161no—
366D4—41no—
367D4—41no—
368D4—81no—
369D4—81no—
370D4—81no—
371D4—81no—
372C2×C4—22no—
373C2×C4—41no—
374C2×C4—41no—
375C2×C4—41no—
376C2×C4—41no—
377C2×C4—41no—
378C2×C4—41no—
379C2×C4—41no—
380C2×C4—41no—
381C2×C4—41no—
382C2×C4—42no—
383C2×C4—42no—
384C2×C4—41no—
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×C4—81no—
394C2×C4—81no—
395C2×C4—81no—
396C2×C4—81no—
397C2×C4C2×C4×D411yescharacteristic
398C2×C4#51811yescharacteristic
399C2×C4#51811yescharacteristic
400C2×C4#53511yescharacteristic
401C2×C4#54011yescharacteristic
402C2×C4#54011yescharacteristic
403C2×C4C2×#13711yescharacteristic
404C2×C4C2×#13711yescharacteristic
405C2×C4#57511yescharacteristic
406C2×C4#57511yescharacteristic
407C8—21no—
408C8—21no—
409C8—21no—
410C8—21no—
411C8—22no—
412C8—42no—
413C8—42no—
414C8—41no—
415C8—81no—
416C8—81no—
417C8—81no—
418C8—81no—
419C22—21no—
420C22—21no—
421C22—81no—
422C22—81no—
423C22—81no—
424C22#232811yescharacteristic
425C22#321811yescharacteristic
426C22#321911yescharacteristic
427C4—21no—
428C4—21no—
429C4—21no—
430C4—21no—
431C4—42no—
432C4—81no—
433C4—81no—
434C4—81no—
435C4—82no—
436C4—82no—
437C4—81no—
438C4—161no—
439C4—161no—
440C4C4×#13711yescharacteristic
441C4C4×#13711yescharacteristic
442C4#321511yescharacteristic
443C4#321611yescharacteristic
444C2—21no—
445C2—21no—
446C2—161no—
447C2#2227311yescharacteristic
448C2#2228511yescharacteristic
449C2#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. #21725C2semidirect
4. #9778C2semidirect
8. #9298C2semidirect
9. #9297C2semidirect
10. #9290C2semidirect
12. #9288C2semidirect
14. #8795C2semidirect
15. #8140C2semidirect
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
#217252
#216592
#197042
#97782
#97582
#93022
#93012
#92982
#92972
#92902
#92892
#92882
#92872
#87952
#81402