Finite Group Databasebeta

← #498364

Subgroups — #498364

Order 512.

694 conjugacy classes of subgroups.

The subgroup lattice has 490 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2220C211yesmaximal, characteristic
2C2×#2175C211yesmaximal, characteristic, complemented
3C2×#2159C211yesmaximal, characteristic, complemented
4#23026C211yesmaximal, characteristic, complemented
5#23025C211yesmaximal, characteristic
6#13237C211yesmaximal, characteristic, complemented
7#13168C211yesmaximal, characteristic, complemented
8#13132C211yesmaximal, characteristic, complemented
9#13130C211yesmaximal, characteristic, complemented
10#12993C211yesmaximal, characteristic, complemented
11#12992C211yesmaximal, characteristic
12#12937C211yesmaximal, characteristic, complemented
13#12936C211yesmaximal, characteristic, complemented
14C4×#366C211yesmaximal, characteristic, complemented
15#12612C211yesmaximal, characteristic, complemented
16C22×SD32C2211yescharacteristic
17C22×M32C2211yescharacteristic
18C2×#467C2211yescharacteristic
19C2×#466C2211yescharacteristic
20C2×C4×Q16C2211yescharacteristic
21C2×C4×D8C2211yescharacteristic
22C2×#427C2211yescharacteristic
23C22×#107C2211yescharacteristic
24#2362—21no—
25#2361—21no—
26#2220C418yescomplemented
27#2175—24no—
28#2159—24no—
29C2×#367C2211yescharacteristic
30C2×#367C2211yescharacteristic
31C2×#366C2211yescharacteristic
32C2×#366C2211yescharacteristic
33C2×#366C2211yescharacteristic
34C2×#358C2211yescharacteristic
35C2×#358C2211yescharacteristic
36C2×#358C2211yescharacteristic
37C2×#357C2211yescharacteristic
38C2×#357C2211yescharacteristic
39C2×#357C2211yescharacteristic
40C2×C22⋊C16C2211yescharacteristic
41C2×#346C2211yescharacteristic
42C2×C4×C16C2211yescharacteristic
43#1970C2211yescharacteristic
44#1969C2211yescharacteristic
45#1942C2211yescharacteristic
46#1942C2211yescharacteristic
47#1927—21no—
48#1925—21no—
49#1874—21no—
50#1873—21no—
51#1865—21no—
52#1862—21no—
53C4×#107C2211yescharacteristic
54C4×#107C2211yescharacteristic
55C8×C22⋊C4C2211yescharacteristic
56#1380C2211yescharacteristic
57#1380C2211yescharacteristic
58#1380C2211yescharacteristic
59#1380C2211yescharacteristic
60#1374C2211yescharacteristic
61#1374C2211yescharacteristic
62C22×Q16C2311yescharacteristic
63C22×D8C2311yescharacteristic
64C23×C8C2311yescharacteristic
65C2×C22⋊Q8—21no—
66C2×#121—21no—
67C2×C4×Q8—21no—
68C2×C4×D4—21no—
69C2×C4×D4—41no—
70C2×#117—21no—
71C22×C4⋊C4—21no—
72C2×SD32—22no—
73C2×SD32—22no—
74C2×SD32C2×C414yescomplemented
75C2×M32—22no—
76C2×M32—22no—
77C2×M32C2×C414yescomplemented
78C22×C16C2311yescharacteristic
79C22×C16C2311yescharacteristic
80C22×C16C2311yescharacteristic
81C22×C16C2311yescharacteristic
82#467—22no—
83#467C2×C414yescomplemented
84#466—22no—
85#466C2×C414yes—
86C4×Q16—22no—
87C4×Q16—41no—
88C4×D8—22no—
89C4×D8—41no—
90#427—22no—
91#427—41no—
92C2×#107—22no—
93C2×#107—21no—
94C2×#107—41no—
95C2×#107C2311yescharacteristic
96C2×#107C2311yescharacteristic
97C2×#107C2311yescharacteristic
98C2×#107C2311yescharacteristic
99C2×#107C2311yescharacteristic
100C2×#107C2311yescharacteristic
101C2×#106—21no—
102C2×#103—21no—
103C2×#103—21no—
104C2×#103—21no—
105C2×#102—21no—
106C2×#102—21no—
107C2×#102—21no—
108C2×C22⋊C8D411yescharacteristic
109C2×C4×C8C2311yescharacteristic
110C2×C4×C8C2311yescharacteristic
111#391—21no—
112#390—21no—
113#389—21no—
114#389—21no—
115#388—41no—
116#388—41no—
117#388—41no—
118#387—41no—
119#386—41no—
120#385—41no—
121C4×C4⋊C4—21no—
122C4×C4⋊C4—21no—
123C4×C4⋊C4—41no—
124C4×C22⋊C4D411yescharacteristic
125#367—22no—
126#367—22no—
127#366—22no—
128#366C2×C414yescomplemented
129#366D414yescomplemented
130#358—22no—
131#358—41no—
132#358C2×C414yescomplemented
133#357—22no—
134#357—41no—
135#357C2×C414yescomplemented
136C22⋊C16—22no—
137#346—22no—
138C4×C16—22no—
139#340—21no—
140#340—21no—
141#340—21no—
142#340—21no—
143#336D411yescharacteristic
144#336D411yescharacteristic
145C22×Q8—21no—
146C22×D4—21no—
147C22×D4—41no—
148C23×C4—41no—
149C23×C4—41no—
150C23×C4C2×D411yescharacteristic
151C2×Q16—21no—
152C2×Q16—21no—
153C2×Q16—21no—
154C2×Q16—21no—
155C2×Q16—21no—
156C2×Q16C22×C412yes—
157C2×D8—21no—
158C2×D8—21no—
159C2×D8—21no—
160C2×D8—21no—
161C2×D8—21no—
162C2×D8C22×C412yes—
163C22×C8—21no—
164C22×C8—21no—
165C22×C8—21no—
166C22×C8—21no—
167C22×C8C22×C412yes—
168C22×C8C2×D411yescharacteristic
169C22×C8C2×D411yescharacteristic
170C22×C8C2×D411yescharacteristic
171C22×C8#4112yes—
172C22×C8C2411yescharacteristic
173C22⋊Q8—24no—
174C22⋊Q8—42no—
175#121—24no—
176#121—42no—
177C4×Q8—42no—
178C4×Q8—81no—
179C4×D4—42no—
180C4×D4—81no—
181C4×D4—84no—
182#117—42no—
183#117—81no—
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—42no—
191C2×C4⋊C4—41no—
192C2×C4⋊C4—41no—
193C2×C4⋊C4—41no—
194C2×C4⋊C4—41no—
195C2×C4⋊C4—41no—
196C2×C4⋊C4—81no—
197C2×C22⋊C4—41no—
198C2×C22⋊C4—41no—
199C2×C22⋊C4—41no—
200C2×C22⋊C4—41no—
201C2×C22⋊C4—41no—
202C2×C22⋊C4—41no—
203C2×C22⋊C4D811yescharacteristic
204C2×C22⋊C4D811yescharacteristic
205C2×C42—41no—
206C2×C42—41no—
207C2×C42—41no—
208C2×C42—41no—
209C2×C42C2×D411yescharacteristic
210C2×C42C2×D411yescharacteristic
211SD32—28no—
212M32—28no—
213C2×C16—21no—
214C2×C16—21no—
215C2×C16—21no—
216C2×C16—21no—
217C2×C16—21no—
218C2×C16—21no—
219C2×C16—21no—
220C2×C16—21no—
221C2×C16C22×C412yes—
222C2×C16C22×C412yes—
223C2×C16C2×D412yes—
224C2×C16#4112yes—
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—42no—
234#107C22×C412yes—
235#107C22×C412yes—
236#107C2×D412yes—
237#107C2×D412yes—
238#106—42no—
239#103—24no—
240#103—42no—
241#103—81no—
242#102—24no—
243#102—42no—
244#102—81no—
245C22⋊C8—22no—
246C4×C8—21no—
247C4×C8—21no—
248C4×C8—21no—
249C4×C8—21no—
250#95—41no—
251#95—41no—
252#95—41no—
253#95—41no—
254#95—41no—
255#95—41no—
256#95D811yescharacteristic
257#95D811yescharacteristic
258C24—41no—
259C24C2×D811yescharacteristic
260C2×Q8—22no—
261C2×Q8—41no—
262C2×Q8—41no—
263C2×Q8—41no—
264C2×Q8—41no—
265C2×Q8—41no—
266C2×D4—22no—
267C2×D4—44no—
268C2×D4—41no—
269C2×D4—41no—
270C2×D4—41no—
271C2×D4—41no—
272C2×D4—41no—
273C2×D4—82no—
274C2×D4—82no—
275C22×C4—21no—
276C22×C4—21no—
277C22×C4—21no—
278C22×C4—21no—
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—82no—
290C22×C4—161no—
291C22×C4—161no—
292C22×C4C4×D412yes—
293C22×C4C2×D811yescharacteristic
294C22×C4C2×D811yescharacteristic
295C22×C4C2×D811yescharacteristic
296C22×C4C2×D811yescharacteristic
297C22×C4C2×D811yescharacteristic
298C22×C4#13512yes—
299C22×C4C22×D411yescharacteristic
300Q16—22no—
301Q16—22no—
302Q16—41no—
303Q16—41no—
304D8—22no—
305D8—22no—
306D8—41no—
307D8—41no—
308C2×C8—21no—
309C2×C8—21no—
310C2×C8—22no—
311C2×C8—22no—
312C2×C8—22no—
313C2×C8—22no—
314C2×C8—21no—
315C2×C8—21no—
316C2×C8—21no—
317C2×C8—21no—
318C2×C8—21no—
319C2×C8—21no—
320C2×C8C4×D412yes—
321C2×C8C23×C411yescharacteristic
322C2×C8C22×D411yescharacteristic
323C2×C8C2×#4111yescharacteristic
324C2×C8C2×#4111yescharacteristic
325C2×C8H(5,𝔽2)11yescharacteristic
326C2×C82−1+411yescharacteristic
327C4⋊C4—22no—
328C4⋊C4—22no—
329C4⋊C4—22no—
330C4⋊C4—22no—
331C4⋊C4—41no—
332C4⋊C4—41no—
333C4⋊C4—41no—
334C4⋊C4—41no—
335C4⋊C4—44no—
336C4⋊C4—41no—
337C4⋊C4—41no—
338C4⋊C4—41no—
339C4⋊C4—41no—
340C4⋊C4—82no—
341C4⋊C4—82no—
342C4⋊C4—82no—
343C4⋊C4—82no—
344C22⋊C4—22no—
345C22⋊C4—41no—
346C22⋊C4—44no—
347C22⋊C4—44no—
348C22⋊C4—82no—
349C22⋊C4—82no—
350C22⋊C4—82no—
351C22⋊C4—161no—
352C42—21no—
353C42—21no—
354C42—21no—
355C42—21no—
356C42—82no—
357C42—82no—
358C42—82no—
359C42—82no—
360C16—22no—
361C16—41no—
362C16—41no—
363C16C4×D414yescomplemented
364C23—21no—
365C23—21no—
366C23—21no—
367C23—21no—
368C23—21no—
369C23—21no—
370C23—42no—
371C23—81no—
372C23—81no—
373C23—81no—
374C23—81no—
375C23—81no—
376C23—81no—
377C23C4×D812yes—
378C23C22×D811yescharacteristic
379Q8—42no—
380Q8—42no—
381Q8—81no—
382Q8—81no—
383D4—42no—
384D4—42no—
385D4—88no—
386D4—81no—
387D4—81no—
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—21no—
403C2×C4—21no—
404C2×C4—21no—
405C2×C4—42no—
406C2×C4—42no—
407C2×C4—42no—
408C2×C4—42no—
409C2×C4—42no—
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—81no—
426C2×C4—164no—
427C2×C4—164no—
428C2×C4C4×D812yes—
429C2×C4#50912yes—
430C2×C4C2×C4×D411yescharacteristic
431C2×C4D4211yescharacteristic
432C2×C4#54711yescharacteristic
433C2×C4C2×#13511yescharacteristic
434C2×C4#57611yescharacteristic
435C2×C4#57811yescharacteristic
436C8—22no—
437C8—41no—
438C8—41no—
439C8C2×C4×D411yescharacteristic
440C8#52011yescharacteristic
441C8#54811yescharacteristic
442C8#55311yescharacteristic
443C22—22no—
444C22—22no—
445C22—22no—
446C22—41no—
447C22—41no—
448C22—41no—
449C22—41no—
450C22—82no—
451C22—82no—
452C22—82no—
453C22—161no—
454C22—161no—
455C22—161no—
456C22—161no—
457C22C2×C4×D811yescharacteristic
458C22D4×D811yescharacteristic
459C22#333411yescharacteristic
460C22C2×#50911yescharacteristic
461C22#341411yescharacteristic
462C22#341611yescharacteristic
463C22#341711yescharacteristic
464C4—22no—
465C4—22no—
466C4—41no—
467C4—41no—
468C4—41no—
469C4—82no—
470C4—82no—
471C4—82no—
472C4—82no—
473C4—82no—
474C4—161no—
475C4—161no—
476C4#236111yescharacteristic
477C4#295111yescharacteristic
478C4#328511yescharacteristic
479C4#333911yescharacteristic
480C2—42no—
481C2—162no—
482C2#2302511yescharacteristic
483C2C4×#50911yescharacteristic
484C2#3345511yescharacteristic
485C2#3383811yescharacteristic
486C2#3384311yescharacteristic
487C2#3391011yescharacteristic
488C2#3391711yescharacteristic
subgroupG/H
center, socleC23C22×D8
commutatorC2×C8C23×C4
Frattini, cosocleC22×C8C24
prime radicalC2×#466C22
LCSG ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C4×C22⋊C4 ◁ C8×C22⋊C4 ◁ G
LECSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#2175C2semidirect
3. C2×#2159C2semidirect
4. #23026C2semidirect
6. #13237C2semidirect
7. #13168C2semidirect
8. #13132C2semidirect
9. #13130C2semidirect
10. #12993C2semidirect
12. #12937C2semidirect
13. #12936C2semidirect
14. C4×#366C2semidirect
15. #12612C2semidirect
26. #2220C4semidirect
74. C2×SD32C2×C4semidirect
77. C2×M32C2×C4semidirect
83. #467C2×C4semidirect
128. #366C2×C4semidirect
129. #366D4semidirect
132. #358C2×C4semidirect
135. #357C2×C4semidirect
363. C16C4×D4semidirect

Maximal subgroups

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

subgroupindex
C2×#22202
C2×#21752
C2×#21592
#230262
#230252
#132372
#131682
#131322
#131302
#129932
#129922
#129372
#129362
C4×#3662
#126122