Finite Group Databasebeta

← #213097

Subgroups — #213097

Order 512.

786 conjugacy classes of subgroups.

The subgroup lattice has 469 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22221C211yesmaximal, characteristic, complemented
2#22220C211yesmaximal, characteristic
3#22219C211yesmaximal, characteristic
4#22218C211yesmaximal, characteristic, complemented
5C4×#487C211yesmaximal, characteristic, complemented
6C4×#486C211yesmaximal, characteristic
7C2×#1711C211yesmaximal, characteristic
8#9833C211yesmaximal, characteristic, complemented
9#9830C211yesmaximal, characteristic
10#9827C211yesmaximal, characteristic
11#9826C211yesmaximal, characteristic, complemented
12#9822C211yesmaximal, characteristic
13#9821C211yesmaximal, characteristic, complemented
14#8814C211yesmaximal, characteristic, complemented
15#7841C211yesmaximal, characteristic, complemented
16C2×#487C2211yescharacteristic
17C2×#487C2211yescharacteristic
18C2×#487C2211yescharacteristic
19C2×#486C2211yescharacteristic
20C2×#486C2211yescharacteristic
21C2×#486C2211yescharacteristic
22C4×C4⋊Q8C2211yescharacteristic
23C4×C4⋊Q8C2211yescharacteristic
24C4×C4⋊Q8C2211yescharacteristic
25C4×#127C2211yescharacteristic
26#1990C2211yescharacteristic
27#1990C2211yescharacteristic
28#1990C2211yescharacteristic
29#1990C2211yescharacteristic
30#1990C2211yescharacteristic
31#1990C2211yescharacteristic
32#1957C2211yescharacteristic
33#1957C2211yescharacteristic
34#1957C2211yescharacteristic
35#1956C2211yescharacteristic
36#1956C2211yescharacteristic
37#1956C2211yescharacteristic
38#1934—22no—
39#1933—22no—
40#1928—22no—
41#1927—22no—
42#1926—22no—
43#1925—22no—
44#1865—22no—
45#1864—22no—
46#1863—22no—
47#1862—22no—
48#1845C2211yescharacteristic
49#1845C2211yescharacteristic
50#1844C2211yescharacteristic
51#1844C2211yescharacteristic
52#1834C2211yescharacteristic
53#1834C2211yescharacteristic
54C4×#103—22no—
55C4×#102—22no—
56#1744C2211yescharacteristic
57#1744C2211yescharacteristic
58C42×C8C2211yescharacteristic
59#1711C418yescomplemented
60C2×#322C2211yescharacteristic
61C2×#322C2211yescharacteristic
62C2×C82C2211yescharacteristic
63C2×C82C2211yescharacteristic
64C2×C4⋊Q8C2311yescharacteristic
65C2×C4⋊Q8C2311yescharacteristic
66C2×C4⋊Q8C2311yescharacteristic
67C2×#127C2311yescharacteristic
68C2×C4×Q8—21no—
69C2×C4×Q8—21no—
70C2×C4×Q8—21no—
71C2×C4×D4—21no—
72C2×C4×D4—21no—
73C2×C4×D4—21no—
74#487—22no—
75#487—22no—
76#487—22no—
77#487C2×C414yescomplemented
78#487C2×C414yescomplemented
79#487C2×C414yescomplemented
80#486—22no—
81#486—22no—
82#486—22no—
83#486C2×C414yes—
84#486C2×C414yes—
85#486C2×C414yes—
86C2×#107—22no—
87C2×#107—22no—
88C2×#106—22no—
89C2×#106—22no—
90C2×#103—22no—
91C2×#103—22no—
92C2×#103—22no—
93C2×#103—22no—
94C2×#103—22no—
95C2×#103—22no—
96C2×#102—22no—
97C2×#102—22no—
98C2×#102—22no—
99C2×#102—22no—
100C2×#102—22no—
101C2×#102—22no—
102C2×C4×C8D412yes—
103C2×C4×C8D412yes—
104C2×C4×C8C2311yescharacteristic
105C2×C4×C8C2311yescharacteristic
106C2×C4×C8C2311yescharacteristic
107C2×C4×C8C2311yescharacteristic
108C2×C4×C8C2311yescharacteristic
109C2×C4×C8C2311yescharacteristic
110#391—21no—
111#391—21no—
112#391—21no—
113#390—21no—
114#390—21no—
115#390—21no—
116#389—21no—
117#389—21no—
118#389—21no—
119#389—21no—
120#389—21no—
121#389—21no—
122#384C2311yescharacteristic
123#384C2311yescharacteristic
124#384C2311yescharacteristic
125#384C2311yescharacteristic
126C4×C4⋊C4—21no—
127C4×C4⋊C4—21no—
128C4×C4⋊C4—21no—
129C4×C4⋊C4—21no—
130C4×C4⋊C4—21no—
131C4×C4⋊C4—21no—
132C43C2311yescharacteristic
133#340—22no—
134#340—22no—
135#340—22no—
136#340—22no—
137#340—22no—
138#340—22no—
139#340—22no—
140#340—22no—
141#336D412yes—
142#336D412yes—
143#336D412yes—
144#336D412yes—
145#322—22no—
146#322—22no—
147C82—22no—
148C82C2×C414yescomplemented
149C22×Q8—21no—
150C22×Q8—21no—
151C22×Q8—21no—
152C22×D4—21no—
153C22×D4—21no—
154C22×D4—21no—
155C23×C4—41no—
156C22×C8C2×D412yes—
157C22×C8C2×D412yes—
158C22×C8C2×D412yes—
159C22×C8C2×D412yes—
160C22×C8C2×D412yes—
161C22×C8C2×D412yes—
162C4⋊Q8—21no—
163C4⋊Q8—21no—
164C4⋊Q8—21no—
165C4⋊Q8—21no—
166C4⋊Q8—21no—
167C4⋊Q8—21no—
168C4⋊Q8—21no—
169C4⋊Q8—21no—
170C4⋊Q8—21no—
171C4⋊Q8C22×C412yes—
172C4⋊Q8C22×C412yes—
173C4⋊Q8C22×C412yes—
174#127—21no—
175#127—21no—
176#127—21no—
177#127C22×C412yes—
178C4×Q8—44no—
179C4×Q8—82no—
180C4×Q8—82no—
181C4×D4—44no—
182C4×D4—82no—
183C4×D4—82no—
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—21no—
191C2×C4⋊C4—21no—
192C2×C4⋊C4—21no—
193C2×C4⋊C4—21no—
194C2×C4⋊C4—21no—
195C2×C4⋊C4—21no—
196C2×C4⋊C4—21no—
197C2×C4⋊C4—21no—
198C2×C4⋊C4—21no—
199C2×C4⋊C4—21no—
200C2×C4⋊C4—21no—
201C2×C4⋊C4—21no—
202C2×C4⋊C4—41no—
203C2×C4⋊C4—41no—
204C2×C4⋊C4—41no—
205C2×C22⋊C4—41no—
206C2×C22⋊C4—41no—
207C2×C22⋊C4—41no—
208C2×C42—41no—
209C2×C42—41no—
210C2×C42—41no—
211C2×C42C2×D411yescharacteristic
212C2×C42C2×D411yescharacteristic
213C2×C42C2×D411yescharacteristic
214C2×C42C2×D411yescharacteristic
215C2×C42C2×D411yescharacteristic
216C2×C42C2×D411yescharacteristic
217C2×C42C2411yescharacteristic
218#107—82no—
219#107—82no—
220#106—82no—
221#106—82no—
222#103—28no—
223#103—28no—
224#103—28no—
225#103—44no—
226#103—82no—
227#103—82no—
228#102—28no—
229#102—28no—
230#102—28no—
231#102—44no—
232#102—82no—
233#102—82no—
234C4×C8—24no—
235C4×C8—24no—
236C4×C8—21no—
237C4×C8—21no—
238C4×C8—21no—
239C4×C8—21no—
240C4×C8—21no—
241C4×C8—21no—
242C4×C8C22×C412yes—
243C4×C8C22×C412yes—
244C4×C8C22×C412yes—
245C4×C8#4112yes—
246C4×C8#4112yes—
247C4×C8#4112yes—
248#95—41no—
249#95—41no—
250#95—41no—
251#95—41no—
252#95—41no—
253#95—41no—
254C24—41no—
255C2×Q8—22no—
256C2×Q8—22no—
257C2×Q8—22no—
258C2×Q8—41no—
259C2×Q8—41no—
260C2×Q8—41no—
261C2×Q8—42no—
262C2×Q8—42no—
263C2×Q8—42no—
264C2×Q8—42no—
265C2×Q8—42no—
266C2×Q8—42no—
267C2×D4—22no—
268C2×D4—22no—
269C2×D4—22no—
270C2×D4—41no—
271C2×D4—41no—
272C2×D4—41no—
273C2×D4—42no—
274C2×D4—42no—
275C2×D4—42no—
276C2×D4—42no—
277C2×D4—42no—
278C2×D4—42no—
279C22×C4—41no—
280C22×C4—41no—
281C22×C4—41no—
282C22×C4—41no—
283C22×C4—41no—
284C22×C4—41no—
285C22×C4—41no—
286C22×C4—82no—
287C22×C4—82no—
288C22×C4—82no—
289C22×C4#12711yescharacteristic
290C22×C4#12711yescharacteristic
291C22×C4#12711yescharacteristic
292C22×C4#12711yescharacteristic
293C22×C4C22×D411yescharacteristic
294C22×C4C22×D411yescharacteristic
295C22×C4C22×D411yescharacteristic
296C2×C8—22no—
297C2×C8—22no—
298C2×C8—22no—
299C2×C8—22no—
300C2×C8—22no—
301C2×C8—22no—
302C2×C8—22no—
303C2×C8—22no—
304C2×C8—22no—
305C2×C8—22no—
306C2×C8—22no—
307C2×C8—22no—
308C2×C8—22no—
309C2×C8—22no—
310C2×C8C4×D414yes—
311C2×C8C4×D414yes—
312C2×C8C4×D414yes—
313C2×C8#13514yes—
314C4⋊C4—22no—
315C4⋊C4—22no—
316C4⋊C4—22no—
317C4⋊C4—22no—
318C4⋊C4—22no—
319C4⋊C4—22no—
320C4⋊C4—41no—
321C4⋊C4—41no—
322C4⋊C4—41no—
323C4⋊C4—41no—
324C4⋊C4—41no—
325C4⋊C4—41no—
326C4⋊C4—42no—
327C4⋊C4—42no—
328C4⋊C4—42no—
329C4⋊C4—42no—
330C4⋊C4—42no—
331C4⋊C4—42no—
332C4⋊C4—42no—
333C4⋊C4—42no—
334C4⋊C4—42no—
335C4⋊C4—42no—
336C4⋊C4—42no—
337C4⋊C4—42no—
338C4⋊C4—82no—
339C4⋊C4—82no—
340C4⋊C4—82no—
341C22⋊C4—82no—
342C22⋊C4—82no—
343C22⋊C4—82no—
344C42—22no—
345C42—22no—
346C42—22no—
347C42—22no—
348C42—22no—
349C42—82no—
350C42—82no—
351C42—82no—
352C42#13514yes—
353C42C23×C411yescharacteristic
354C42C2×#4111yescharacteristic
355C42C2×#4111yescharacteristic
356C42C2×#4111yescharacteristic
357C23—42no—
358C23—81no—
359C23—81no—
360C23—81no—
361C23—81no—
362C23—81no—
363C23—81no—
364C23C2×#12711yescharacteristic
365Q8—44no—
366Q8—44no—
367Q8—44no—
368Q8—82no—
369Q8—82no—
370Q8—82no—
371D4—44no—
372D4—44no—
373D4—44no—
374D4—82no—
375D4—82no—
376D4—82no—
377C2×C4—21no—
378C2×C4—21no—
379C2×C4—21no—
380C2×C4—21no—
381C2×C4—21no—
382C2×C4—21no—
383C2×C4—21no—
384C2×C4—21no—
385C2×C4—21no—
386C2×C4—21no—
387C2×C4—42no—
388C2×C4—42no—
389C2×C4—42no—
390C2×C4—81no—
391C2×C4—81no—
392C2×C4—81no—
393C2×C4—81no—
394C2×C4—81no—
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—81no—
404C2×C4—81no—
405C2×C4—81no—
406C2×C4—81no—
407C2×C4—81no—
408C2×C4—164no—
409C2×C4#49512yes—
410C2×C4#49512yes—
411C2×C4C2×C4×D411yescharacteristic
412C2×C4C2×C4×D411yescharacteristic
413C2×C4C2×C4×D411yescharacteristic
414C2×C4#53211yescharacteristic
415C2×C4#53211yescharacteristic
416C2×C4#53211yescharacteristic
417C2×C4C2×#13512yes—
418C2×C4C2×#13512yes—
419C2×C4#57612yes—
420C2×C4#57712yes—
421C2×C4#57712yes—
422C2×C4#57812yes—
423C8—24no—
424C8—24no—
425C8—24no—
426C8—24no—
427C8—42no—
428C8—42no—
429C22—82no—
430C22—82no—
431C22—82no—
432C22—161no—
433C22—161no—
434C22—161no—
435C22—161no—
436C22C4×#12711yescharacteristic
437C22C2×#49511yescharacteristic
438C22#314711yescharacteristic
439C22#315411yescharacteristic
440C22#315411yescharacteristic
441C22#315511yescharacteristic
442C22#315611yescharacteristic
443C4—22no—
444C4—22no—
445C4—41no—
446C4—41no—
447C4—82no—
448C4—82no—
449C4—82no—
450C4—161no—
451C4—161no—
452C4—161no—
453C4—161no—
454C4C4×#13512yes—
455C4#295112yes—
456C4#295212yes—
457C4#316812yes—
458C4#317312yes—
459C4#317512yes—
460C2—162no—
461C2#1965811yescharacteristic
462C2#1965911yescharacteristic
463C2#1966911yescharacteristic
464C2#1967011yescharacteristic
465C2C4×#49511yescharacteristic
466C2#2169411yescharacteristic
467C2#2169611yescharacteristic
subgroupG/H
center, socleC23C2×#127
commutatorC42C23×C4
Frattini, cosocleC2×C42C24
prime radicalC2×#127C23
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C43 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #22221C2semidirect
4. #22218C2semidirect
5. C4×#487C2semidirect
8. #9833C2semidirect
11. #9826C2semidirect
13. #9821C2semidirect
14. #8814C2semidirect
15. #7841C2semidirect
59. #1711C4semidirect
77. #487C2×C4semidirect
78. #487C2×C4semidirect
79. #487C2×C4semidirect
148. C82C2×C4semidirect

Maximal subgroups

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

subgroupindex
#222212
#222202
#222192
#222182
C4×#4872
C4×#4862
C2×#17112
#98332
#98302
#98272
#98262
#98222
#98212
#88142
#78412