Finite Group Databasebeta

← #502350

Subgroups — #502350

Order 512.

555 conjugacy classes of subgroups.

The subgroup lattice has 464 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2211C211yesmaximal, characteristic
2C2×#2183C211yesmaximal, characteristic, complemented
3#28740C211yesmaximal, characteristic
4#28265C211yesmaximal, characteristic, complemented
5#13477C211yesmaximal, characteristic, complemented
6#13476C211yesmaximal, characteristic
7#13450C211yesmaximal, characteristic, complemented
8#13449C211yesmaximal, characteristic
9#13240C211yesmaximal, characteristic, complemented
10#13237C211yesmaximal, characteristic
11#13205C211yesmaximal, characteristic
12#13177C211yesmaximal, characteristic, complemented
13#13149C211yesmaximal, characteristic
14#13148C211yesmaximal, characteristic, complemented
15#12638C211yesmaximal, characteristic, complemented
16C22×Q32C2211yescharacteristic
17C22×SD32C2211yescharacteristic
18C2×#495C2211yescharacteristic
19C2×C8×Q8C2211yescharacteristic
20C2×C4×Q16C2211yescharacteristic
21#2592—21no—
22#2582—21no—
23#2211—24no—
24#2183—24no—
25C2×#367C2211yescharacteristic
26C2×#367C2211yescharacteristic
27C2×C4⋊C16C2211yescharacteristic
28C2×C4⋊C16C2211yescharacteristic
29C2×C4⋊C16C2211yescharacteristic
30C2×#358C2211yescharacteristic
31C2×#358C2211yescharacteristic
32C2×#358C2211yescharacteristic
33C2×#358C2211yescharacteristic
34C2×#358C2211yescharacteristic
35C2×#357C2211yescharacteristic
36C2×#357C2211yescharacteristic
37C2×#357C2211yescharacteristic
38C2×#346C2211yescharacteristic
39C2×#346C2211yescharacteristic
40C2×C4×C16C2211yescharacteristic
41#2068—21no—
42#2068—21no—
43#2055—21no—
44#2055—21no—
45#1983—21no—
46#1970C2211yescharacteristic
47#1970C2211yescharacteristic
48#1969C2211yescharacteristic
49#1969C2211yescharacteristic
50#1969C2211yescharacteristic
51#1969C2211yescharacteristic
52#1942C2211yescharacteristic
53#1942C2211yescharacteristic
54#1845C2211yescharacteristic
55#1803—21no—
56C4×#107C2211yescharacteristic
57#1380C2211yescharacteristic
58#1380C2211yescharacteristic
59#1380C2211yescharacteristic
60#1380C2211yescharacteristic
61C22×Q16C2311yescharacteristic
62C22×Q16C2311yescharacteristic
63C22×SD16—21no—
64C22×D8C2311yescharacteristic
65C2×#124—21no—
66C2×C4×Q8—21no—
67C2×C4×Q8D411yescharacteristic
68C2×Q32—22no—
69C2×Q32—22no—
70C2×Q32—41no—
71C2×SD32—22no—
72C2×SD32—22no—
73C2×SD32—41no—
74C22×C16C2311yescharacteristic
75C22×C16C2311yescharacteristic
76C22×C16C2311yescharacteristic
77C22×C16C2311yescharacteristic
78#495—22no—
79#495—22no—
80C8×Q8—22no—
81C8×Q8—22no—
82C4×Q16—22no—
83C4×Q16—22no—
84C2×#107—21no—
85C2×#107C2311yescharacteristic
86C2×#107C2311yescharacteristic
87C2×#107C2311yescharacteristic
88C2×#107C2311yescharacteristic
89C2×#107C2311yescharacteristic
90C2×C4⋊C8D411yescharacteristic
91C2×C4⋊C8D411yescharacteristic
92C2×C4⋊C8D411yescharacteristic
93C2×#103—21no—
94C2×#103—21no—
95C2×#103—21no—
96C2×#103—21no—
97C2×#103—21no—
98C2×#102—21no—
99C2×#102—21no—
100C2×C4×C8C2311yescharacteristic
101C2×C4×C8C2311yescharacteristic
102C2×C4×C8C2311yescharacteristic
103#398—41no—
104#398—41no—
105#397—41no—
106#393—41no—
107#391—21no—
108#391—21no—
109#391—21no—
110#391—21no—
111#391—41no—
112#390—21no—
113#390—21no—
114#389—21no—
115#389—21no—
116#389—41no—
117#388—41no—
118#388—41no—
119#384—21no—
120C4×C4⋊C4—21no—
121#367—41no—
122#367—41no—
123C4⋊C16—22no—
124C4⋊C16—22no—
125C4⋊C16D414yes—
126#358—22no—
127#358—22no—
128#358—22no—
129#358—22no—
130#358—41no—
131#357—22no—
132#357—22no—
133#357—41no—
134#346—22no—
135#346—22no—
136C4×C16—22no—
137#340—21no—
138#340—21no—
139#340—21no—
140C22×Q8—21no—
141C22×Q8—21no—
142C22×Q8D811yescharacteristic
143C22×D4—21no—
144C2×Q16—21no—
145C2×Q16—21no—
146C2×Q16—21no—
147C2×Q16—21no—
148C2×Q16—21no—
149C2×Q16—21no—
150C2×Q16—21no—
151C2×Q16—21no—
152C2×Q16—21no—
153C2×Q16—21no—
154C2×Q16—21no—
155C2×Q16#4112yes—
156C2×SD16—42no—
157C2×SD16—81no—
158C2×SD16—81no—
159C2×D8—21no—
160C2×D8—21no—
161C2×D8—21no—
162C2×D8—21no—
163C2×D8—21no—
164C2×D8—21no—
165C22×C8C2×D411yescharacteristic
166C22×C8C2×D411yescharacteristic
167C22×C8C2×D411yescharacteristic
168C22×C8C2411yescharacteristic
169#124—42no—
170#124—42no—
171C4×Q8—22no—
172C4×Q8—22no—
173C4×Q8—42no—
174C4×Q8—42no—
175C2×C4⋊C4—21no—
176C2×C4⋊C4—21no—
177C2×C4⋊C4—21no—
178C2×C4⋊C4—21no—
179C2×C4⋊C4—21no—
180C2×C4⋊C4—41no—
181C2×C4⋊C4—41no—
182C2×C4⋊C4—41no—
183C2×C4⋊C4—41no—
184C2×C4⋊C4—41no—
185C2×C4⋊C4D811yescharacteristic
186C2×C4⋊C4D811yescharacteristic
187C2×C4⋊C4D811yescharacteristic
188C2×C22⋊C4—41no—
189C2×C22⋊C4—41no—
190C2×C22⋊C4—41no—
191C2×C22⋊C4—41no—
192C2×C22⋊C4—41no—
193C2×C22⋊C4—41no—
194C2×C42—41no—
195C2×C42—41no—
196C2×C42C2×D411yescharacteristic
197C2×C42C2×D411yescharacteristic
198C2×C42C2×D411yescharacteristic
199Q32—44no—
200SD32—44no—
201C2×C16—21no—
202C2×C16—21no—
203C2×C16—21no—
204C2×C16—21no—
205C2×C16—21no—
206C2×C16—21no—
207C2×C16—21no—
208C2×C16—21no—
209C2×C16—21no—
210C2×C16C2×D412yes—
211C2×C16C2×D412yes—
212C2×C16#4112yes—
213#107—21no—
214#107—21no—
215#107—21no—
216#107—21no—
217#107—21no—
218#107—21no—
219#107—21no—
220#107—21no—
221#107—21no—
222#107—21no—
223#107—81no—
224C4⋊C8—22no—
225C4⋊C8—22no—
226C4⋊C8—22no—
227#103—42no—
228#103—81no—
229#103—81no—
230#103—81no—
231#103—81no—
232#102—81no—
233#102—81no—
234C4×C8—21no—
235C4×C8—21no—
236C4×C8—21no—
237C4×C8—21no—
238C4×C8C2×D412yes—
239C4×C8#4112yes—
240#95—41no—
241#95—41no—
242#95—41no—
243#95—41no—
244#95—41no—
245#95—41no—
246#95—41no—
247#95—41no—
248#95—41no—
249#95—41no—
250#95—41no—
251C24—41no—
252C2×Q8—22no—
253C2×Q8—22no—
254C2×Q8—22no—
255C2×Q8—41no—
256C2×Q8—41no—
257C2×Q8—41no—
258C2×Q8—41no—
259C2×Q8—41no—
260C2×Q8—41no—
261C2×Q8—41no—
262C2×Q8—41no—
263C2×Q8—41no—
264C2×Q8—41no—
265C2×Q8—41no—
266C2×Q8—41no—
267C2×D4—41no—
268C2×D4—41no—
269C2×D4—41no—
270C2×D4—41no—
271C2×D4—41no—
272C2×D4—41no—
273C22×C4—41no—
274C22×C4—41no—
275C22×C4—41no—
276C22×C4—41no—
277C22×C4—41no—
278C22×C4—41no—
279C22×C4—41no—
280C22×C4C2×D811yescharacteristic
281C22×C4C2×D811yescharacteristic
282C22×C4C2×D811yescharacteristic
283C22×C4C2×D811yescharacteristic
284C22×C4C2×D811yescharacteristic
285C22×C4C2×D811yescharacteristic
286C22×C4C22×D411yescharacteristic
287Q16—22no—
288Q16—22no—
289Q16—41no—
290Q16—41no—
291Q16—41no—
292Q16—41no—
293Q16—41no—
294Q16—41no—
295SD16—84no—
296D8—41no—
297D8—41no—
298D8—41no—
299D8—41no—
300C2×C8—21no—
301C2×C8—21no—
302C2×C8—21no—
303C2×C8—21no—
304C2×C8—21no—
305C2×C8—21no—
306C2×C8—21no—
307C2×C8—21no—
308C2×C8#12112yes—
309C2×C8C22×D411yescharacteristic
310C2×C8C2×#4111yescharacteristic
311C2×C8C2×#4111yescharacteristic
312C2×C8C2×#4111yescharacteristic
313C2×C8H(5,𝔽2)11yescharacteristic
314C2×C82−1+411yescharacteristic
315C4⋊C4—22no—
316C4⋊C4—22no—
317C4⋊C4—22no—
318C4⋊C4—41no—
319C4⋊C4—41no—
320C4⋊C4—41no—
321C4⋊C4—41no—
322C4⋊C4—41no—
323C4⋊C4—41no—
324C4⋊C4—41no—
325C4⋊C4—41no—
326C4⋊C4—41no—
327C4⋊C4—41no—
328C4⋊C4—82no—
329C4⋊C4—82no—
330C4⋊C4—161no—
331C4⋊C4—161no—
332C4⋊C4—161no—
333C22⋊C4—82no—
334C22⋊C4—82no—
335C22⋊C4—82no—
336C22⋊C4—82no—
337C22⋊C4—161no—
338C22⋊C4—161no—
339C42—21no—
340C42—21no—
341C42—21no—
342C42—21no—
343C42—82no—
344C42—161no—
345C42#12112yes—
346C42#13512yes—
347C16—22no—
348C16—22no—
349C16—22no—
350C16—41no—
351C23—81no—
352C23—81no—
353C23—81no—
354C23—81no—
355C23—81no—
356C23—81no—
357C23—81no—
358C23C22×D811yescharacteristic
359Q8—44no—
360Q8—42no—
361Q8—42no—
362Q8—81no—
363Q8—81no—
364Q8—81no—
365Q8—81no—
366Q8—81no—
367Q8—81no—
368D4—81no—
369D4—81no—
370D4—81no—
371D4—81no—
372C2×C4—21no—
373C2×C4—21no—
374C2×C4—21no—
375C2×C4—21no—
376C2×C4—21no—
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—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×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#46612yes—
409C2×C4#46612yes—
410C2×C4#51012yes—
411C2×C4C2×#12111yescharacteristic
412C2×C4#54811yescharacteristic
413C2×C4#55011yescharacteristic
414C2×C4C2×#13511yescharacteristic
415C2×C4#57611yescharacteristic
416C2×C4#57711yescharacteristic
417C8—22no—
418C8—41no—
419C8—41no—
420C8#52911yescharacteristic
421C8#54211yescharacteristic
422C8#54611yescharacteristic
423C8#54811yescharacteristic
424C22—161no—
425C22—161no—
426C22—161no—
427C22—161no—
428C22—161no—
429C22—161no—
430C22—161no—
431C22C2×#46611yescharacteristic
432C22#335811yescharacteristic
433C22#339111yescharacteristic
434C22C2×#51011yescharacteristic
435C22#341411yescharacteristic
436C22#341611yescharacteristic
437C22#341611yescharacteristic
438C4—22no—
439C4—22no—
440C4—22no—
441C4—41no—
442C4—41no—
443C4—41no—
444C4—82no—
445C4—161no—
446C4—161no—
447C4—161no—
448C4—161no—
449C4—161no—
450C4—161no—
451C4#242611yescharacteristic
452C4#306211yescharacteristic
453C4#336111yescharacteristic
454C4#339411yescharacteristic
455C2—321no—
456C2#2529211yescharacteristic
457C2#3360611yescharacteristic
458C2#3361111yescharacteristic
459C2#3393611yescharacteristic
460C2#3393811yescharacteristic
461C2#3396711yescharacteristic
462C2#3396811yescharacteristic
subgroupG/H
center, socleC23C22×D8
commutator, Frattini, cosocleC22×C8C24
prime radicalC22×D8C23
LCSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×Q8 ◁ C2×C8×Q8 ◁ G
LECSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#2183C2semidirect
4. #28265C2semidirect
5. #13477C2semidirect
7. #13450C2semidirect
9. #13240C2semidirect
12. #13177C2semidirect
14. #13148C2semidirect
15. #12638C2semidirect

Maximal subgroups

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

subgroupindex
C2×#22112
C2×#21832
#287402
#282652
#134772
#134762
#134502
#134492
#132402
#132372
#132052
#131772
#131492
#131482
#126382