Finite Group Databasebeta

← #497011

Subgroups — #497011

Order 512.

660 conjugacy classes of subgroups.

The subgroup lattice has 489 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2220C211yesmaximal, characteristic
2C4×#467C211yesmaximal, characteristic, complemented
3C4×#466C211yesmaximal, characteristic
4#13132C211yesmaximal, characteristic, complemented
5#13131C211yesmaximal, characteristic, complemented
6#13112C211yesmaximal, characteristic, complemented
7#13109C211yesmaximal, characteristic, complemented
8#13062C211yesmaximal, characteristic, complemented
9#13061C211yesmaximal, characteristic
10#13014C211yesmaximal, characteristic, complemented
11#13013C211yesmaximal, characteristic, complemented
12#12936C211yesmaximal, characteristic, complemented
13#12935C211yesmaximal, characteristic, complemented
14#12836C211yesmaximal, characteristic, complemented
15#12544C211yesmaximal, characteristic, complemented
16C22×SD32C2211yescharacteristic
17C22×M32C2211yescharacteristic
18C22×M32C2211yescharacteristic
19C2×#467C2211yescharacteristic
20C2×#466C2211yescharacteristic
21C2×C4×Q16C2211yescharacteristic
22C2×C4×D8C2211yescharacteristic
23C22×C4×C8C2211yescharacteristic
24C4×C22⋊Q8—21no—
25C4×#121—21no—
26#2220C418yescomplemented
27C2×#367C2211yescharacteristic
28C2×#366C2211yescharacteristic
29C2×#366C2211yescharacteristic
30C2×#358C2211yescharacteristic
31C2×#358C2211yescharacteristic
32C2×#358C2211yescharacteristic
33C2×#357C2211yescharacteristic
34C2×#357C2211yescharacteristic
35C2×#357C2211yescharacteristic
36#1970C2211yescharacteristic
37#1969C2211yescharacteristic
38#1942C2211yescharacteristic
39#1942C2211yescharacteristic
40#1934—21no—
41#1933—21no—
42#1893—21no—
43#1892—21no—
44#1808C2211yescharacteristic
45#1808C2211yescharacteristic
46C4×#107C2211yescharacteristic
47C4×#107C2211yescharacteristic
48C4×#103—21no—
49C4×#102—21no—
50#1380C2211yescharacteristic
51#1380C2211yescharacteristic
52#1380C2211yescharacteristic
53#1380C2211yescharacteristic
54#1380C2211yescharacteristic
55#1380C2211yescharacteristic
56#1374C2211yescharacteristic
57#1374C2211yescharacteristic
58#1374C2211yescharacteristic
59#1374C2211yescharacteristic
60C22×Q16C2311yescharacteristic
61C22×D8C2311yescharacteristic
62C23×C8D411yescharacteristic
63C23×C8C2311yescharacteristic
64C2×C22⋊Q8—21no—
65C2×#121—21no—
66C2×C4×Q8—21no—
67C2×C4×D4—21no—
68C2×C4×D4—41no—
69C22×C42D411yescharacteristic
70C2×SD32—41no—
71C2×SD32—41no—
72C2×SD32C2×C414yescomplemented
73C2×M32—22no—
74C2×M32—41no—
75C2×M32—41no—
76C2×M32—41no—
77C2×M32—41no—
78C2×M32C2×C414yescomplemented
79C22×C16C2311yescharacteristic
80C22×C16C2311yescharacteristic
81C22×C16C2311yescharacteristic
82C22×C16C2311yescharacteristic
83#467—22no—
84#467C2×C414yescomplemented
85#466—22no—
86#466C2×C414yes—
87C4×Q16—41no—
88C4×Q16—41no—
89C4×D8—41no—
90C4×D8—41no—
91C2×#107C2311yescharacteristic
92C2×#107C2311yescharacteristic
93C2×#107C2311yescharacteristic
94C2×#107C2311yescharacteristic
95C2×#107C2311yescharacteristic
96C2×#107C2311yescharacteristic
97C2×#103—21no—
98C2×#103—21no—
99C2×#103—21no—
100C2×#102—21no—
101C2×#102—21no—
102C2×#102—21no—
103C2×C4×C8—41no—
104C2×C4×C8—41no—
105C2×C4×C8D411yescharacteristic
106C2×C4×C8D411yescharacteristic
107C2×C4×C8C2311yescharacteristic
108C2×C4×C8C2311yescharacteristic
109#391—21no—
110#390—21no—
111#389—21no—
112#389—21no—
113#388—41no—
114#387—41no—
115#386—41no—
116#385—41no—
117#380—21no—
118#380—21no—
119C4×C4⋊C4—21no—
120C4×C4⋊C4—21no—
121C4×C4⋊C4—41no—
122C4×C22⋊C4—41no—
123C4×C22⋊C4—41no—
124#367—41no—
125#366—41no—
126#366C2×C414yescomplemented
127#358—41no—
128#358—41no—
129#358C2×C414yescomplemented
130#357—41no—
131#357—41no—
132#357C2×C414yescomplemented
133#340—21no—
134#340—21no—
135#340—21no—
136#340—21no—
137#340—21no—
138#340—21no—
139C22×Q8—21no—
140C22×D4—21no—
141C22×D4—41no—
142C23×C4—41no—
143C23×C4D811yescharacteristic
144C23×C4D811yescharacteristic
145C23×C4C2×D411yescharacteristic
146C2×Q16—21no—
147C2×Q16—21no—
148C2×Q16—21no—
149C2×Q16—21no—
150C2×Q16—21no—
151C2×Q16C22×C412yes—
152C2×D8—21no—
153C2×D8—21no—
154C2×D8—21no—
155C2×D8—21no—
156C2×D8—21no—
157C2×D8C22×C412yes—
158C22×C8—21no—
159C22×C8—21no—
160C22×C8—21no—
161C22×C8—21no—
162C22×C8—22no—
163C22×C8—41no—
164C22×C8—41no—
165C22×C8C22×C412yes—
166C22×C8C2×D411yescharacteristic
167C22×C8C2×D411yescharacteristic
168C22×C8C2×D411yescharacteristic
169C22×C8#4112yes—
170C22×C8C2411yescharacteristic
171C22⋊Q8—24no—
172C22⋊Q8—42no—
173#121—24no—
174#121—42no—
175C4×Q8—42no—
176C4×Q8—42no—
177C4×D4—42no—
178C4×D4—42no—
179C4×D4—84no—
180C2×C4⋊C4—21no—
181C2×C4⋊C4—21no—
182C2×C4⋊C4—21no—
183C2×C4⋊C4—21no—
184C2×C4⋊C4—21no—
185C2×C4⋊C4—21no—
186C2×C4⋊C4—41no—
187C2×C4⋊C4—41no—
188C2×C4⋊C4—41no—
189C2×C4⋊C4—41no—
190C2×C22⋊C4—41no—
191C2×C22⋊C4—41no—
192C2×C22⋊C4—41no—
193C2×C22⋊C4—41no—
194C2×C22⋊C4—41no—
195C2×C22⋊C4—41no—
196C2×C22⋊C4—41no—
197C2×C42—41no—
198C2×C42—41no—
199C2×C42—41no—
200C2×C42—41no—
201C2×C42—41no—
202C2×C42D811yescharacteristic
203C2×C42D811yescharacteristic
204C2×C42C2×D411yescharacteristic
205C2×C42C2×D411yescharacteristic
206SD32—44no—
207M32—44no—
208M32—44no—
209C2×C16—21no—
210C2×C16—21no—
211C2×C16—21no—
212C2×C16—21no—
213C2×C16—21no—
214C2×C16—21no—
215C2×C16—21no—
216C2×C16—21no—
217C2×C16—21no—
218C2×C16—21no—
219C2×C16C22×C412yes—
220C2×C16C22×C412yes—
221#107—21no—
222#107—21no—
223#107—21no—
224#107—21no—
225#107—21no—
226#107—21no—
227#107—21no—
228#107—21no—
229#107—21no—
230#107—21no—
231#107C22×C412yes—
232#107C22×C412yes—
233#103—24no—
234#103—42no—
235#103—42no—
236#102—24no—
237#102—42no—
238#102—42no—
239C4×C8—21no—
240C4×C8—21no—
241C4×C8—21no—
242C4×C8—21no—
243C4×C8—41no—
244C4×C8—41no—
245#95—41no—
246#95—41no—
247#95—41no—
248#95—41no—
249#95—41no—
250#95—41no—
251#95—41no—
252#95—41no—
253C24—41no—
254C24C2×D811yescharacteristic
255C2×Q8—22no—
256C2×Q8—41no—
257C2×Q8—41no—
258C2×Q8—41no—
259C2×Q8—41no—
260C2×Q8—41no—
261C2×D4—22no—
262C2×D4—44no—
263C2×D4—41no—
264C2×D4—41no—
265C2×D4—41no—
266C2×D4—41no—
267C2×D4—41no—
268C2×D4—82no—
269C2×D4—82no—
270C22×C4—21no—
271C22×C4—21no—
272C22×C4—21no—
273C22×C4—21no—
274C22×C4—22no—
275C22×C4—22no—
276C22×C4—41no—
277C22×C4—41no—
278C22×C4—41no—
279C22×C4—41no—
280C22×C4—41no—
281C22×C4—41no—
282C22×C4—41no—
283C22×C4—41no—
284C22×C4—41no—
285C22×C4—41no—
286C22×C4—41no—
287C22×C4—82no—
288C22×C4—82no—
289C22×C4—82no—
290C22×C4C4×D412yes—
291C22×C4C2×D811yescharacteristic
292C22×C4C2×D811yescharacteristic
293C22×C4C2×D811yescharacteristic
294C22×C4C2×D811yescharacteristic
295C22×C4C2×D811yescharacteristic
296C22×C4#13512yes—
297C22×C4C22×D411yescharacteristic
298Q16—22no—
299Q16—22no—
300Q16—41no—
301Q16—41no—
302D8—22no—
303D8—22no—
304D8—41no—
305D8—41no—
306C2×C8—21no—
307C2×C8—21no—
308C2×C8—22no—
309C2×C8—22no—
310C2×C8—22no—
311C2×C8—22no—
312C2×C8—21no—
313C2×C8—21no—
314C2×C8—21no—
315C2×C8—21no—
316C2×C8—21no—
317C2×C8—21no—
318C2×C8—44no—
319C2×C8C4×D412yes—
320C2×C8C23×C411yescharacteristic
321C2×C8C2×#4111yescharacteristic
322C2×C8H(5,𝔽2)11yescharacteristic
323C2×C8H(5,𝔽2)11yescharacteristic
324C2×C8H(5,𝔽2)11yescharacteristic
325C2×C82−1+411yescharacteristic
326C4⋊C4—22no—
327C4⋊C4—22no—
328C4⋊C4—41no—
329C4⋊C4—41no—
330C4⋊C4—44no—
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—82no—
342C22⋊C4—44no—
343C22⋊C4—44no—
344C22⋊C4—82no—
345C22⋊C4—82no—
346C22⋊C4—82no—
347C22⋊C4—82no—
348C22⋊C4—82no—
349C42—21no—
350C42—21no—
351C42—21no—
352C42—21no—
353C42—41no—
354C42—41no—
355C42—82no—
356C42—82no—
357C42—82no—
358C16—22no—
359C16—22no—
360C16—41no—
361C16—41no—
362C23—21no—
363C23—21no—
364C23—21no—
365C23—21no—
366C23—21no—
367C23—21no—
368C23—42no—
369C23—81no—
370C23—81no—
371C23—81no—
372C23—81no—
373C23—81no—
374C23—81no—
375C23C4×D812yes—
376C23C22×D811yescharacteristic
377Q8—42no—
378Q8—42no—
379Q8—81no—
380Q8—81no—
381D4—42no—
382D4—42no—
383D4—88no—
384D4—81no—
385D4—81no—
386C2×C4—21no—
387C2×C4—21no—
388C2×C4—21no—
389C2×C4—21no—
390C2×C4—21no—
391C2×C4—21no—
392C2×C4—22no—
393C2×C4—22no—
394C2×C4—22no—
395C2×C4—22no—
396C2×C4—21no—
397C2×C4—21no—
398C2×C4—21no—
399C2×C4—21no—
400C2×C4—21no—
401C2×C4—21no—
402C2×C4—44no—
403C2×C4—44no—
404C2×C4—42no—
405C2×C4—42no—
406C2×C4—42no—
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—164no—
426C2×C4C4×D812yes—
427C2×C4#50912yes—
428C2×C4#51012yes—
429C2×C4C2×C4×D411yescharacteristic
430C2×C4#53411yescharacteristic
431C2×C4#53711yescharacteristic
432C2×C4C2×#13511yescharacteristic
433C2×C4#57511yescharacteristic
434C2×C4#57511yescharacteristic
435C8—22no—
436C8—41no—
437C8—41no—
438C8#51811yescharacteristic
439C8#52011yescharacteristic
440C8#53811yescharacteristic
441C8#53911yescharacteristic
442C22—22no—
443C22—22no—
444C22—22no—
445C22—41no—
446C22—41no—
447C22—41no—
448C22—41no—
449C22—82no—
450C22—82no—
451C22—82no—
452C22—161no—
453C22—161no—
454C22—161no—
455C22—161no—
456C22C2×C4×D811yescharacteristic
457C22#318611yescharacteristic
458C22#324511yescharacteristic
459C22C2×#50911yescharacteristic
460C22C2×#51011yescharacteristic
461C22#341411yescharacteristic
462C22#341411yescharacteristic
463C4—22no—
464C4—22no—
465C4—22no—
466C4—41no—
467C4—41no—
468C4—82no—
469C4—82no—
470C4—82no—
471C4—161no—
472C4—161no—
473C4—161no—
474C4—161no—
475C4#232511yescharacteristic
476C4#294311yescharacteristic
477C4#319111yescharacteristic
478C4#325011yescharacteristic
479C2—42no—
480C2—162no—
481C2#2218111yescharacteristic
482C2C4×#50911yescharacteristic
483C2C4×#51011yescharacteristic
484C2#3376011yescharacteristic
485C2#3376211yescharacteristic
486C2#3382011yescharacteristic
487C2#3382211yescharacteristic
subgroupG/H
center, socleC23C22×D8
commutatorC2×C8C23×C4
Frattini, cosocleC22×C8C24
prime radicalC2×#466C22
LCSG ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ C22×C4×C8 ◁ G
LECSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C4×#467C2semidirect
4. #13132C2semidirect
5. #13131C2semidirect
6. #13112C2semidirect
7. #13109C2semidirect
8. #13062C2semidirect
10. #13014C2semidirect
11. #13013C2semidirect
12. #12936C2semidirect
13. #12935C2semidirect
14. #12836C2semidirect
15. #12544C2semidirect
26. #2220C4semidirect
72. C2×SD32C2×C4semidirect
78. C2×M32C2×C4semidirect
84. #467C2×C4semidirect
126. #366C2×C4semidirect
129. #358C2×C4semidirect
132. #357C2×C4semidirect

Maximal subgroups

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

subgroupindex
C2×#22202
C4×#4672
C4×#4662
#131322
#131312
#131122
#131092
#130622
#130612
#130142
#130132
#129362
#129352
#128362
#125442