Finite Group Databasebeta

← #512206

Subgroups — #512206

Order 512.

641 conjugacy classes of subgroups.

The subgroup lattice has 488 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2243C211yesmaximal, characteristic, complemented
2C2×#2108C211yesmaximal, characteristic, complemented
3#33215C211yesmaximal, characteristic, complemented
4#33182C211yesmaximal, characteristic
5#13282C211yesmaximal, characteristic, complemented
6#13204C211yesmaximal, characteristic, complemented
7#13150C211yesmaximal, characteristic, complemented
8#13149C211yesmaximal, characteristic, complemented
9#13148C211yesmaximal, characteristic, complemented
10#13147C211yesmaximal, characteristic, complemented
11#13062C211yesmaximal, characteristic, complemented
12#13061C211yesmaximal, characteristic
13#13060C211yesmaximal, characteristic, complemented
14#13059C211yesmaximal, characteristic
15#12890C211yesmaximal, characteristic, complemented
16C22×M32C2211yescharacteristic
17C23×C16C2211yescharacteristic
18C2×#500C2211yescharacteristic
19C2×#495C2211yescharacteristic
20C2×#468—21no—
21C2×#467C2211yescharacteristic
22C2×#466C2211yescharacteristic
23C2×#405C2211yescharacteristic
24#2628—21no—
25#2620—21no—
26#2243—24no—
27C2×#358C2211yescharacteristic
28C2×#358C2211yescharacteristic
29C2×#358C2211yescharacteristic
30C2×#358C2211yescharacteristic
31C2×#357C2211yescharacteristic
32C2×#357C2211yescharacteristic
33C2×#357C2211yescharacteristic
34C2×#357C2211yescharacteristic
35#2108—24no—
36C2×#346C2211yescharacteristic
37C2×#346C2211yescharacteristic
38C2×C4×C16C2211yescharacteristic
39C2×C4×C16C2211yescharacteristic
40#1993—21no—
41#1973—21no—
42#1970C2211yescharacteristic
43#1969C2211yescharacteristic
44#1959—21no—
45#1944—21no—
46#1942C2211yescharacteristic
47#1942C2211yescharacteristic
48#1846C2211yescharacteristic
49#1845C2211yescharacteristic
50#1809—21no—
51#1808C2211yescharacteristic
52#1808C2211yescharacteristic
53#1380C2211yescharacteristic
54#1380C2211yescharacteristic
55#1380C2211yescharacteristic
56#1380C2211yescharacteristic
57#1380C2211yescharacteristic
58#1380C2211yescharacteristic
59#1380C2211yescharacteristic
60#1380C2211yescharacteristic
61C22×Q16C2311yescharacteristic
62C22×SD16—21no—
63C22×D8C2311yescharacteristic
64C22×M16D411yescharacteristic
65C23×C8C2311yescharacteristic
66C2×C4⋊Q8—21no—
67C2×#124—21no—
68C2×C22⋊Q8—21no—
69C2×#121—21no—
70C2×#117D411yescharacteristic
71C2×M32—22no—
72C2×M32—22no—
73C2×M32—22no—
74C22×C16—22no—
75C22×C16—22no—
76C22×C16—22no—
77C22×C16C2311yescharacteristic
78C22×C16C2311yescharacteristic
79C22×C16C2311yescharacteristic
80C22×C16C2311yescharacteristic
81#500—22no—
82#500—22no—
83#495—22no—
84#495—22no—
85#468—44no—
86#467—22no—
87#467—22no—
88#466—22no—
89#466—22no—
90C2×#107—21no—
91C2×#107—21no—
92C2×#107C2311yescharacteristic
93C2×#107C2311yescharacteristic
94C2×#107C2311yescharacteristic
95C2×#107C2311yescharacteristic
96C2×#107C2311yescharacteristic
97C2×#107C2311yescharacteristic
98C2×#106—21no—
99C2×#103—21no—
100C2×#103—21no—
101C2×#102—21no—
102C2×#102—21no—
103#405—22no—
104#405—22no—
105C2×#97D411yescharacteristic
106C2×#97D411yescharacteristic
107C2×C4×C8C2311yescharacteristic
108C2×C4×C8C2311yescharacteristic
109#400—41no—
110#399—41no—
111#396—41no—
112#396—41no—
113#395—41no—
114#394—41no—
115#394—41no—
116#393—41no—
117#391—21no—
118#390—21no—
119#389—21no—
120#389—21no—
121#384—21no—
122#382—21no—
123#380—21no—
124#380—21no—
125#358—22no—
126#358—22no—
127#358—22no—
128#358—22no—
129#357—22no—
130#357—22no—
131#357—22no—
132#357—22no—
133#346—22no—
134#346D414yescomplemented
135C4×C16—22no—
136C4×C16—22no—
137#340—21no—
138#340—21no—
139#340—21no—
140#340—21no—
141C22×Q8—21no—
142C22×Q8—41no—
143C22×D4—21no—
144C22×D4—41no—
145C23×C4C2×D411yescharacteristic
146C2×Q16—21no—
147C2×Q16—21no—
148C2×Q16—21no—
149C2×Q16—21no—
150C2×Q16—21no—
151C2×Q16—21no—
152C2×SD16—42no—
153C2×SD16—42no—
154C2×SD16—42no—
155C2×D8—21no—
156C2×D8—21no—
157C2×D8—21no—
158C2×D8—21no—
159C2×D8—21no—
160C2×D8—21no—
161C2×M16—22no—
162C2×M16—22no—
163C2×M16SD1614yes—
164C22×C8—21no—
165C22×C8—21no—
166C22×C8—21no—
167C22×C8—21no—
168C22×C8C2×D411yescharacteristic
169C22×C8C2×D411yescharacteristic
170C22×C8C2×D411yescharacteristic
171C22×C8#4112yes—
172C22×C8#4112yes—
173C22×C8C2411yescharacteristic
174C4⋊Q8—42no—
175C4⋊Q8—42no—
176#124—42no—
177#124—42no—
178C22⋊Q8—42no—
179C22⋊Q8—42no—
180#121—42no—
181#121—42no—
182#117—22no—
183#117—22no—
184C2×C4⋊C4—21no—
185C2×C4⋊C4—21no—
186C2×C4⋊C4—21no—
187C2×C4⋊C4—21no—
188C2×C4⋊C4—21no—
189C2×C4⋊C4—21no—
190C2×C4⋊C4—41no—
191C2×C4⋊C4—41no—
192C2×C4⋊C4—41no—
193C2×C4⋊C4—41no—
194C2×C4⋊C4—41no—
195C2×C4⋊C4—41no—
196C2×C4⋊C4D811yescharacteristic
197C2×C4⋊C4D811yescharacteristic
198C2×C22⋊C4—41no—
199C2×C22⋊C4—41no—
200C2×C22⋊C4—41no—
201C2×C22⋊C4—41no—
202C2×C22⋊C4—41no—
203C2×C22⋊C4—41no—
204C2×C22⋊C4—41no—
205C2×C22⋊C4—41no—
206C2×C22⋊C4D811yescharacteristic
207C2×C22⋊C4D811yescharacteristic
208C2×C42C2×D411yescharacteristic
209C2×C42C2×D411yescharacteristic
210M32—28no—
211C2×C16—28no—
212C2×C16—21no—
213C2×C16—21no—
214C2×C16—21no—
215C2×C16—21no—
216C2×C16—21no—
217C2×C16—21no—
218C2×C16—21no—
219C2×C16—21no—
220C2×C16C2×D412yes—
221C2×C16C2×D412yes—
222C2×C16#4112yes—
223C2×C16#4112yes—
224#107—21no—
225#107—21no—
226#107—21no—
227#107—21no—
228#107—21no—
229#107—21no—
230#107—21no—
231#107—21no—
232#107—21no—
233#107—21no—
234#107—21no—
235#107—21no—
236#107—42no—
237#107—42no—
238#106—42no—
239#103—42no—
240#103—42no—
241#102—42no—
242#102—42no—
243#97—22no—
244#97—22no—
245C4×C8—21no—
246C4×C8—21no—
247C4×C8C2×D412yes—
248C4×C8#4112yes—
249#95—41no—
250#95—41no—
251#95—41no—
252#95—41no—
253#95—41no—
254#95—41no—
255#95—41no—
256#95—41no—
257C24—41no—
258C24C2×D811yescharacteristic
259C2×Q8—41no—
260C2×Q8—41no—
261C2×Q8—41no—
262C2×Q8—41no—
263C2×Q8—41no—
264C2×Q8—41no—
265C2×Q8—82no—
266C2×Q8—161no—
267C2×Q8—161no—
268C2×D4—41no—
269C2×D4—41no—
270C2×D4—41no—
271C2×D4—41no—
272C2×D4—41no—
273C2×D4—41no—
274C2×D4—82no—
275C2×D4—161no—
276C2×D4—161no—
277C22×C4—21no—
278C22×C4—21no—
279C22×C4—21no—
280C22×C4—21no—
281C22×C4—41no—
282C22×C4—41no—
283C22×C4—41no—
284C22×C4—41no—
285C22×C4—41no—
286C22×C4—41no—
287C22×C4—41no—
288C22×C4#12412yes—
289C22×C4C2×D811yescharacteristic
290C22×C4C2×D811yescharacteristic
291C22×C4C2×D811yescharacteristic
292C22×C4C2×D811yescharacteristic
293C22×C4C2×D811yescharacteristic
294C22×C4C2×SD1612yes—
295C22×C4C22×D411yescharacteristic
296Q16—41no—
297Q16—41no—
298Q16—41no—
299Q16—41no—
300SD16—84no—
301D8—41no—
302D8—41no—
303D8—41no—
304D8—41no—
305M16—28no—
306C2×C8—21no—
307C2×C8—21no—
308C2×C8—21no—
309C2×C8—21no—
310C2×C8—22no—
311C2×C8—22no—
312C2×C8—22no—
313C2×C8—22no—
314C2×C8#12112yes—
315C2×C8#12412yes—
316C2×C8C2×SD1612yes—
317C2×C8C2×SD1612yes—
318C2×C8#13512yes—
319C2×C8C22×D411yescharacteristic
320C2×C8C2×#4111yescharacteristic
321C2×C8C2×#4111yescharacteristic
322C2×C8C2×#4111yescharacteristic
323C2×C8C2×#4111yescharacteristic
324C2×C8C2×#4111yescharacteristic
325C4⋊C4—22no—
326C4⋊C4—22no—
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—41no—
337C4⋊C4—41no—
338C4⋊C4—41no—
339C4⋊C4—82no—
340C4⋊C4—82no—
341C4⋊C4—161no—
342C4⋊C4—161no—
343C4⋊C4—161no—
344C4⋊C4—161no—
345C22⋊C4—22no—
346C22⋊C4—22no—
347C22⋊C4—82no—
348C22⋊C4—82no—
349C22⋊C4—82no—
350C22⋊C4—82no—
351C22⋊C4—161no—
352C22⋊C4—161no—
353C22⋊C4—161no—
354C22⋊C4—161no—
355C42—21no—
356C42—21no—
357C42#12112yes—
358C42#13512yes—
359C16—22no—
360C16—22no—
361C16—22no—
362C16—22no—
363C23—21no—
364C23—21no—
365C23—21no—
366C23—21no—
367C23—21no—
368C23—21no—
369C23—81no—
370C23—81no—
371C23—81no—
372C23—81no—
373C23—81no—
374C23—81no—
375C23—81no—
376C23#48612yes—
377C23C22×D811yescharacteristic
378Q8—81no—
379Q8—81no—
380Q8—81no—
381Q8—81no—
382Q8—164no—
383D4—81no—
384D4—81no—
385D4—81no—
386D4—81no—
387D4—164no—
388C2×C4—21no—
389C2×C4—21no—
390C2×C4—21no—
391C2×C4—21no—
392C2×C4—21no—
393C2×C4—21no—
394C2×C4—22no—
395C2×C4—22no—
396C2×C4—22no—
397C2×C4—22no—
398C2×C4—21no—
399C2×C4—21no—
400C2×C4—21no—
401C2×C4—21no—
402C2×C4—21no—
403C2×C4—21no—
404C2×C4—81no—
405C2×C4—81no—
406C2×C4—81no—
407C2×C4—81no—
408C2×C4—81no—
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—81no—
420C2×C4—81no—
421C2×C4—81no—
422C2×C4—81no—
423C2×C4—81no—
424C2×C4—81no—
425C2×C4#46612yes—
426C2×C4#46612yes—
427C2×C4#48612yes—
428C2×C4C2×#12111yescharacteristic
429C2×C4C2×#12411yescharacteristic
430C2×C4#53211yescharacteristic
431C2×C4C22×SD1611yescharacteristic
432C2×C4C2×#13511yescharacteristic
433C2×C4C2×#13511yescharacteristic
434C8—22no—
435C8—22no—
436C8#46514yes—
437C8#52511yescharacteristic
438C8#52511yescharacteristic
439C8#52911yescharacteristic
440C8#52911yescharacteristic
441C22—22no—
442C22—22no—
443C22—22no—
444C22—41no—
445C22—41no—
446C22—41no—
447C22—41no—
448C22—161no—
449C22—161no—
450C22—161no—
451C22—161no—
452C22—161no—
453C22—161no—
454C22—161no—
455C22C2×#46611yescharacteristic
456C22C2×#48611yescharacteristic
457C22#316911yescharacteristic
458C22#341511yescharacteristic
459C22#341611yescharacteristic
460C22#341611yescharacteristic
461C22#341711yescharacteristic
462C4—22no—
463C4—22no—
464C4—22no—
465C4—41no—
466C4—41no—
467C4—161no—
468C4—161no—
469C4—161no—
470C4—161no—
471C4—161no—
472C4—161no—
473C4—161no—
474C4#285011yescharacteristic
475C4C2×#46511yescharacteristic
476C4#313111yescharacteristic
477C4#316711yescharacteristic
478C2—42no—
479C2—321no—
480C2#3304211yescharacteristic
481C2#3361211yescharacteristic
482C2#3361311yescharacteristic
483C2#3370211yescharacteristic
484C2#3370311yescharacteristic
485C2#3373911yescharacteristic
486C2#3374211yescharacteristic
subgroupG/H
center, socleC23C22×D8
commutator, Frattini, cosocleC22×C8C24
prime radicalC2×#466C22
LCSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×#117 ◁ C2×#405 ◁ G
LECSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#2243C2semidirect
2. C2×#2108C2semidirect
3. #33215C2semidirect
5. #13282C2semidirect
6. #13204C2semidirect
7. #13150C2semidirect
8. #13149C2semidirect
9. #13148C2semidirect
10. #13147C2semidirect
11. #13062C2semidirect
13. #13060C2semidirect
15. #12890C2semidirect
134. #346D4semidirect

Maximal subgroups

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

subgroupindex
C2×#22432
C2×#21082
#332152
#331822
#132822
#132042
#131502
#131492
#131482
#131472
#130622
#130612
#130602
#130592
#128902