Finite Group Databasebeta

← #219601

Subgroups — #219601

Order 512.

547 conjugacy classes of subgroups.

The subgroup lattice has 481 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22236C211yesmaximal, characteristic, complemented
2#22230C211yesmaximal, characteristic, complemented
3#19656C211yesmaximal, characteristic
4#9774C211yesmaximal, characteristic, complemented
5#9758C211yesmaximal, characteristic, complemented
6#9302C211yesmaximal, characteristic, complemented
7#9299C211yesmaximal, characteristic
8#9297C211yesmaximal, characteristic, complemented
9#9295C211yesmaximal, characteristic, complemented
10#9294C211yesmaximal, characteristic, complemented
11#9293C211yesmaximal, characteristic
12#9290C211yesmaximal, characteristic, complemented
13#9287C211yesmaximal, characteristic, complemented
14#8793C211yesmaximal, characteristic, complemented
15#8141C211yesmaximal, characteristic, complemented
16#3227C2211yescharacteristic
17#3209C2211yescharacteristic
18#3208C2211yescharacteristic
19#3205C2211yescharacteristic
20#3131C2211yescharacteristic
21C2×#486C2211yescharacteristic
22#2961C2211yescharacteristic
23#2932C2211yescharacteristic
24#2331C2211yescharacteristic
25#2328C2211yescharacteristic
26#1995C2211yescharacteristic
27#1993C2211yescharacteristic
28#1963C2211yescharacteristic
29#1961C2211yescharacteristic
30#1929—21no—
31#1929—21no—
32#1926—21no—
33#1925—21no—
34#1866—21no—
35#1866—21no—
36#1864—21no—
37#1862—21no—
38#1848C2211yescharacteristic
39#1846C2211yescharacteristic
40#1746C2211yescharacteristic
41#1720C412yescomplemented
42#1717C412yescomplemented
43#1709C412yescomplemented
44#1708C412yescomplemented
45#1549C2211yescharacteristic
46#1548C2211yescharacteristic
47#1545C2211yescharacteristic
48#1545C2211yescharacteristic
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×#127C2311yescharacteristic
68C2×C4×D4—21no—
69#498C2×C412yescomplemented
70#494C2×C412yescomplemented
71#493C2×C412yes—
72#492C2×C412yes—
73#488—21no—
74#488—21no—
75#487C2×C411yescharacteristic, complemented
76#487C2×C411yescharacteristic, complemented
77#487C2×C412yescomplemented
78#486—21no—
79#486—21no—
80#486C2×C412yes—
81#486C2×C412yes—
82#486C2×C412yes—
83#486C2×C412yes—
84#486C2×C411yescharacteristic, complemented
85#486C2×C411yescharacteristic, complemented
86#486C2×C412yescomplemented
87#481—41no—
88#476—41no—
89#462—41no—
90#460—41no—
91#459—41no—
92#451—41no—
93#450—41no—
94#447—41no—
95#446—41no—
96#446—41no—
97#435—41no—
98#435—41no—
99#431C2311yescharacteristic
100#431C2311yescharacteristic
101#431C2311yescharacteristic
102#431C2311yescharacteristic
103C2×#108—21no—
104C2×#108—21no—
105C2×#107—21no—
106C2×#106—21no—
107#424—21no—
108#423C2311yescharacteristic
109#423C2311yescharacteristic
110C2×#104—21no—
111C2×#104—21no—
112#416—21no—
113#416—21no—
114C2×#102—21no—
115C2×#102—21no—
116C2×#102—21no—
117C2×#102—21no—
118#408—21no—
119#405—21no—
120C2×#97C2311yescharacteristic
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#339—21no—
135#339—21no—
136#336D411yescharacteristic
137#336D411yescharacteristic
138#336D411yescharacteristic
139#336D411yescharacteristic
140#333—41no—
141#332—41no—
142#331—41no—
143#331—41no—
144#328—41no—
145#328—41no—
146#327—41no—
147#327—41no—
148#322—21no—
149#322—21no—
150#322C2×C412yescomplemented
151C82C2×C412yescomplemented
152C2×#41—21no—
153C22×Q8—21no—
154C22×D4—21no—
155C22×D4—21no—
156C22×D4—21no—
157C23×C4—41no—
158C2×Q16—42no—
159C2×SD16—42no—
160C2×D8—42no—
161#131—81no—
162#131—81no—
163C2×M16—21no—
164C2×M16—21no—
165C2×M16—41no—
166C22×C8—41no—
167C22×C8C2×D411yescharacteristic
168C22×C8C2×D411yescharacteristic
169C22×C8C2×D411yescharacteristic
170C22×C8C2×D411yescharacteristic
171C4⋊Q8—21no—
172C4⋊Q8—21no—
173C4⋊Q8—21no—
174C4⋊Q8C22×C412yes—
175C4⋊Q8C22×C411yescharacteristic
176C4⋊Q8C22×C411yescharacteristic
177C4⋊Q8C22×C411yescharacteristic
178#127—21no—
179#127—21no—
180#127—21no—
181#127C22×C412yes—
182#127C22×C411yescharacteristic
183#125—21no—
184#124—21no—
185C22⋊Q8—21no—
186C22⋊Q8—21no—
187#121—21no—
188#121—21no—
189C4×Q8—21no—
190C4×Q8—21no—
191C4×D4—21no—
192C4×D4—21no—
193C4×D4—81no—
194C4×D4—81no—
195#117—21no—
196C2×C4⋊C4—21no—
197C2×C4⋊C4—21no—
198C2×C4⋊C4—21no—
199C2×C4⋊C4—21no—
200C2×C4⋊C4—41no—
201C2×C22⋊C4—41no—
202C2×C22⋊C4—41no—
203C2×C22⋊C4—41no—
204C2×C42—41no—
205C2×C42C2×D411yescharacteristic
206C2×C42C2×D411yescharacteristic
207C2×C42C2411yescharacteristic
208#108—81no—
209#108—81no—
210#107—81no—
211#106—42no—
212#106—81no—
213C4⋊C8—21no—
214C4⋊C8—21no—
215C4⋊C8—21no—
216C4⋊C8—21no—
217C4⋊C8—21no—
218C4⋊C8—21no—
219C4⋊C8—21no—
220C4⋊C8—21no—
221C4⋊C8—42no—
222#104—81no—
223#104—81no—
224#103—21no—
225#103—21no—
226#103—21no—
227#103—21no—
228#103—42no—
229#102—22no—
230#102—22no—
231#102—22no—
232#102—22no—
233#102—21no—
234#102—21no—
235#102—21no—
236#102—21no—
237#102—42no—
238#102—42no—
239#102—42no—
240#102—81no—
241#102—81no—
242C22⋊C8—21no—
243C22⋊C8—21no—
244C22⋊C8—21no—
245C22⋊C8—21no—
246C22⋊C8—42no—
247#97—21no—
248#97—21no—
249#97—41no—
250C4×C8—21no—
251C4×C8—21no—
252C4×C8—41no—
253C4×C8C22×C412yes—
254C4×C8C22×C412yes—
255C4×C8C22×C411yescharacteristic
256C4×C8C22×C411yescharacteristic
257C4×C8#4111yescharacteristic
258C4×C8#4111yescharacteristic
259#95—21no—
260#95—21no—
261#95—21no—
262#95—21no—
263#95—21no—
264#95—21no—
265#95—41no—
266#95—41no—
267C24—41no—
268#41—41no—
269#41—41no—
270#41—41no—
271#41—41no—
272C2×Q8—21no—
273C2×Q8—22no—
274C2×Q8—21no—
275C2×Q8—21no—
276C2×Q8—41no—
277C2×Q8—161no—
278C2×D4—21no—
279C2×D4—22no—
280C2×D4—21no—
281C2×D4—22no—
282C2×D4—22no—
283C2×D4—21no—
284C2×D4—41no—
285C2×D4—41no—
286C2×D4—41no—
287C2×D4—41no—
288C2×D4—41no—
289C2×D4—41no—
290C2×D4—41no—
291C2×D4—41no—
292C2×D4—161no—
293C2×D4—161no—
294C22×C4—21no—
295C22×C4—21no—
296C22×C4—41no—
297C22×C4—41no—
298C22×C4—41no—
299C22×C4—41no—
300C22×C4—82no—
301C22×C4—161no—
302C22×C4—161no—
303C22×C4C22×D411yescharacteristic
304C22×C4H(5,𝔽2)11yescharacteristic
305C22×C4H(5,𝔽2)11yescharacteristic
306Q16—162no—
307SD16—162no—
308D8—162no—
309M16—41no—
310M16—41no—
311M16—41no—
312M16—41no—
313M16—82no—
314C2×C8—21no—
315C2×C8—21no—
316C2×C8—21no—
317C2×C8—21no—
318C2×C8—22no—
319C2×C8—22no—
320C2×C8—21no—
321C2×C8—21no—
322C2×C8—21no—
323C2×C8—21no—
324C2×C8—21no—
325C2×C8—21no—
326C2×C8—41no—
327C2×C8—41no—
328C2×C8—41no—
329C2×C8—41no—
330C2×C8—82no—
331C2×C8C4×D411yescharacteristic
332C2×C8C4×D411yescharacteristic
333C2×C8C4×D411yescharacteristic
334C2×C8C4×D411yescharacteristic
335C4⋊C4—22no—
336C4⋊C4—21no—
337C4⋊C4—21no—
338C4⋊C4—21no—
339C4⋊C4—21no—
340C4⋊C4—22no—
341C4⋊C4—21no—
342C4⋊C4—21no—
343C4⋊C4—41no—
344C4⋊C4—41no—
345C4⋊C4—41no—
346C4⋊C4—41no—
347C4⋊C4—41no—
348C4⋊C4—41no—
349C4⋊C4—41no—
350C4⋊C4—41no—
351C4⋊C4—161no—
352C22⋊C4—41no—
353C22⋊C4—41no—
354C22⋊C4—41no—
355C22⋊C4—82no—
356C22⋊C4—161no—
357C22⋊C4—161no—
358C42—21no—
359C42—21no—
360C42—21no—
361C42—21no—
362C42—41no—
363C42—161no—
364C42C23×C411yescharacteristic
365C42C2×#4111yescharacteristic
366C42H(5,𝔽2)11yescharacteristic
367C42H(5,𝔽2)11yescharacteristic
368C23—41no—
369C23—42no—
370C23—81no—
371C23—81no—
372C23—81no—
373C23—81no—
374C23—81no—
375C23—81no—
376C23#53511yescharacteristic
377Q8—41no—
378Q8—41no—
379Q8—81no—
380Q8—81no—
381Q8—82no—
382Q8—161no—
383D4—42no—
384D4—41no—
385D4—42no—
386D4—41no—
387D4—81no—
388D4—82no—
389D4—82no—
390D4—81no—
391D4—81no—
392D4—81no—
393D4—81no—
394D4—81no—
395D4—161no—
396D4—161no—
397C2×C4—41no—
398C2×C4—41no—
399C2×C4—41no—
400C2×C4—41no—
401C2×C4—41no—
402C2×C4—41no—
403C2×C4—41no—
404C2×C4—41no—
405C2×C4—41no—
406C2×C4—42no—
407C2×C4—41no—
408C2×C4—41no—
409C2×C4—81no—
410C2×C4—81no—
411C2×C4—81no—
412C2×C4—81no—
413C2×C4—81no—
414C2×C4—81no—
415C2×C4—81no—
416C2×C4—81no—
417C2×C4—81no—
418C2×C4—81no—
419C2×C4—164no—
420C2×C4C2×C4×D411yescharacteristic
421C2×C4#51811yescharacteristic
422C2×C4#51811yescharacteristic
423C2×C4#53511yescharacteristic
424C2×C4#54011yescharacteristic
425C2×C4#54011yescharacteristic
426C2×C4#57611yescharacteristic
427C2×C4#57711yescharacteristic
428C2×C4#57711yescharacteristic
429C2×C4#57811yescharacteristic
430C8—21no—
431C8—21no—
432C8—21no—
433C8—21no—
434C8—42no—
435C8—42no—
436C8—41no—
437C8—41no—
438C8—81no—
439C8—81no—
440C8—81no—
441C8—81no—
442C22—21no—
443C22—21no—
444C22—81no—
445C22—82no—
446C22—82no—
447C22—81no—
448C22—82no—
449C22—81no—
450C22—161no—
451C22—161no—
452C22—161no—
453C22—161no—
454C22#232811yescharacteristic
455C22#322411yescharacteristic
456C22#322511yescharacteristic
457C4—21no—
458C4—21no—
459C4—21no—
460C4—21no—
461C4—81no—
462C4—81no—
463C4—81no—
464C4—81no—
465C4—82no—
466C4—81no—
467C4—161no—
468C4—161no—
469C4#295111yescharacteristic
470C4#295211yescharacteristic
471C4#322411yescharacteristic
472C4#322511yescharacteristic
473C2—21no—
474C2—21no—
475C2—161no—
476C2—162no—
477C2#2227411yescharacteristic
478C2#2228411yescharacteristic
479C2#2228611yescharacteristic
subgroupG/H
center, socleC22#2328
commutatorC42C23×C4
Frattini, cosocleC2×C42C24
prime radical#3205C22
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. #22236C2semidirect
2. #22230C2semidirect
4. #9774C2semidirect
5. #9758C2semidirect
6. #9302C2semidirect
8. #9297C2semidirect
9. #9295C2semidirect
10. #9294C2semidirect
12. #9290C2semidirect
13. #9287C2semidirect
14. #8793C2semidirect
15. #8141C2semidirect
41. #1720C4semidirect
42. #1717C4semidirect
43. #1709C4semidirect
44. #1708C4semidirect
69. #498C2×C4semidirect
70. #494C2×C4semidirect
75. #487C2×C4semidirect
76. #487C2×C4semidirect
77. #487C2×C4semidirect
84. #486C2×C4semidirect
85. #486C2×C4semidirect
86. #486C2×C4semidirect
150. #322C2×C4semidirect
151. C82C2×C4semidirect

Maximal subgroups

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

subgroupindex
#222362
#222302
#196562
#97742
#97582
#93022
#92992
#92972
#92952
#92942
#92932
#92902
#92872
#87932
#81412