Finite Group Databasebeta

← #498311

Subgroups — #498311

Order 512.

664 conjugacy classes of subgroups.

The subgroup lattice has 475 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2211C211yesmaximal, characteristic
2C2×#2176C211yesmaximal, characteristic, complemented
3C2×#2175C211yesmaximal, characteristic
4#23186C211yesmaximal, characteristic
5#22685C211yesmaximal, characteristic, complemented
6#13204C211yesmaximal, characteristic
7#13178C211yesmaximal, characteristic, complemented
8#13150C211yesmaximal, characteristic, complemented
9#13149C211yesmaximal, characteristic
10#13015C211yesmaximal, characteristic, complemented
11#13013C211yesmaximal, characteristic
12#12937C211yesmaximal, characteristic
13#12936C211yesmaximal, characteristic, complemented
14#12662C211yesmaximal, characteristic, complemented
15C4×#358C211yesmaximal, characteristic, complemented
16C22×Q32C2211yescharacteristic
17C22×SD32C2211yescharacteristic
18C2×#495C2211yescharacteristic
19C2×C4×Q16C2211yescharacteristic
20C2×C4×Q16C2211yescharacteristic
21C2×C4×Q16C2211yescharacteristic
22C2×C4×Q16C2211yescharacteristic
23C2×C4×D8C2211yescharacteristic
24#2367—21no—
25#2351—21no—
26#2211C418yescomplemented
27#2176—24no—
28#2175—24no—
29C2×#367C2211yescharacteristic
30C2×#366C2211yescharacteristic
31C2×C4⋊C16C2211yescharacteristic
32C2×#358C2211yescharacteristic
33C2×#358C2211yescharacteristic
34C2×#358C2211yescharacteristic
35C2×#358C2211yescharacteristic
36C2×#358C2211yescharacteristic
37C2×#357C2211yescharacteristic
38C2×#357C2211yescharacteristic
39C2×#357C2211yescharacteristic
40C2×#346C2211yescharacteristic
41C2×#346C2211yescharacteristic
42C2×C4×C16C2211yescharacteristic
43#1983—21no—
44#1970C2211yescharacteristic
45#1969C2211yescharacteristic
46#1942C2211yescharacteristic
47#1942C2211yescharacteristic
48#1880—21no—
49#1877—21no—
50#1865—21no—
51#1865—21no—
52#1846C2211yescharacteristic
53#1805—21no—
54C4×#107C2211yescharacteristic
55C8×C4⋊C4C2211yescharacteristic
56#1380C2211yescharacteristic
57#1380C2211yescharacteristic
58#1380C2211yescharacteristic
59#1380C2211yescharacteristic
60#1374C2211yescharacteristic
61#1374C2211yescharacteristic
62C22×Q16C2311yescharacteristic
63C22×Q16C2311yescharacteristic
64C22×SD16—21no—
65C22×D8C2311yescharacteristic
66C2×#124—21no—
67C2×C4×Q8—21no—
68C2×C4×Q8—21no—
69C2×C4×Q8—21no—
70C2×C4×Q8—21no—
71C2×C4×D4—21no—
72C2×Q32—22no—
73C2×Q32—22no—
74C2×Q32C2×C414yescomplemented
75C2×SD32—22no—
76C2×SD32—22no—
77C2×SD32C2×C414yes—
78C22×C16C2311yescharacteristic
79C22×C16C2311yescharacteristic
80C22×C16C2311yescharacteristic
81C22×C16C2311yescharacteristic
82#495—22no—
83#495C2×C414yes—
84C4×Q16—22no—
85C4×Q16—22no—
86C4×Q16—22no—
87C4×Q16—22no—
88C4×Q16—41no—
89C4×Q16—41no—
90C4×Q16—41no—
91C4×Q16C2×C414yescomplemented
92C4×D8—22no—
93C4×D8—41no—
94C2×#107C2311yescharacteristic
95C2×#107C2311yescharacteristic
96C2×#107C2311yescharacteristic
97C2×#107C2311yescharacteristic
98C2×#107C2311yescharacteristic
99C2×#106—21no—
100C2×C4⋊C8D411yescharacteristic
101C2×#103—21no—
102C2×#103—21no—
103C2×#103—21no—
104C2×#103—21no—
105C2×#103—21no—
106C2×#102—21no—
107C2×#102—21no—
108C2×C4×C8C2311yescharacteristic
109C2×C4×C8C2311yescharacteristic
110C2×C4×C8C2311yescharacteristic
111#391—21no—
112#390—21no—
113#389—21no—
114#389—21no—
115#389—41no—
116#389—41no—
117#388—41no—
118#388—41no—
119#385—41no—
120#382—21no—
121#382—41no—
122C4×C4⋊C4—21no—
123C4×C4⋊C4—41no—
124C4×C4⋊C4D411yescharacteristic
125C4×C22⋊C4—41no—
126#367—22no—
127#366—22no—
128C4⋊C16C2×C414yescomplemented
129#358—22no—
130#358—22no—
131#358—41no—
132#358C2×C414yescomplemented
133#358D414yes—
134#357—22no—
135#357—41no—
136#357C2×C414yes—
137#346—22no—
138#346—22no—
139C4×C16—22no—
140#340—21no—
141#340—21no—
142#340—21no—
143#336D411yescharacteristic
144#336D411yescharacteristic
145C22×Q8—21no—
146C22×Q8—21no—
147C22×D4—21no—
148C23×C4—41no—
149C2×Q16—21no—
150C2×Q16—21no—
151C2×Q16—21no—
152C2×Q16—21no—
153C2×Q16—21no—
154C2×Q16—21no—
155C2×Q16—21no—
156C2×Q16—21no—
157C2×Q16C22×C412yes—
158C2×Q16C22×C412yes—
159C2×Q16C2×D412yes—
160C2×Q16#4112yes—
161C2×SD16—24no—
162C2×SD16—81no—
163C2×SD16—81no—
164C2×D8—21no—
165C2×D8—21no—
166C2×D8—21no—
167C2×D8—21no—
168C2×D8—21no—
169C2×D8C22×C412yes—
170C22×C8C2×D411yescharacteristic
171C22×C8C2×D411yescharacteristic
172C22×C8C2×D411yescharacteristic
173C22×C8C2411yescharacteristic
174#124—24no—
175#124—42no—
176C4×Q8—24no—
177C4×Q8—42no—
178C4×Q8—42no—
179C4×Q8—42no—
180C4×Q8—42no—
181C4×Q8—81no—
182C4×Q8—81no—
183C4×Q8—81no—
184C4×D4—42no—
185C4×D4—81no—
186C2×C4⋊C4—21no—
187C2×C4⋊C4—21no—
188C2×C4⋊C4—21no—
189C2×C4⋊C4—21no—
190C2×C4⋊C4—21no—
191C2×C4⋊C4—41no—
192C2×C4⋊C4—41no—
193C2×C4⋊C4—41no—
194C2×C4⋊C4—41no—
195C2×C4⋊C4—41no—
196C2×C4⋊C4—41no—
197C2×C4⋊C4D811yescharacteristic
198C2×C4⋊C4D811yescharacteristic
199C2×C22⋊C4—41no—
200C2×C22⋊C4—41no—
201C2×C22⋊C4—41no—
202C2×C22⋊C4—41no—
203C2×C22⋊C4—41no—
204C2×C42—41no—
205C2×C42—41no—
206C2×C42—41no—
207C2×C42—41no—
208C2×C42—41no—
209C2×C42C2×D411yescharacteristic
210C2×C42C2×D411yescharacteristic
211C2×C42C2×D411yescharacteristic
212Q32—28no—
213SD32—28no—
214C2×C16—21no—
215C2×C16—21no—
216C2×C16—21no—
217C2×C16—21no—
218C2×C16—21no—
219C2×C16—21no—
220C2×C16—21no—
221C2×C16—21no—
222C2×C16—21no—
223C2×C16C22×C412yes—
224C2×C16C22×C412yes—
225C2×C16C2×D412yes—
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#107C22×C412yes—
235#107C2×D412yes—
236#106—81no—
237C4⋊C8—22no—
238#103—24no—
239#103—42no—
240#103—42no—
241#103—42no—
242#103—81no—
243#102—42no—
244#102—81no—
245C4×C8—21no—
246C4×C8—21no—
247C4×C8—21no—
248C4×C8—21no—
249C4×C8C22×C412yes—
250C4×C8#4112yes—
251#95—41no—
252#95—41no—
253#95—41no—
254#95—41no—
255#95—41no—
256#95—41no—
257#95—41no—
258#95D811yescharacteristic
259#95D811yescharacteristic
260C24—41no—
261C2×Q8—22no—
262C2×Q8—22no—
263C2×Q8—22no—
264C2×Q8—22no—
265C2×Q8—41no—
266C2×Q8—41no—
267C2×Q8—41no—
268C2×Q8—41no—
269C2×Q8—41no—
270C2×Q8—41no—
271C2×Q8—41no—
272C2×Q8—41no—
273C2×D4—22no—
274C2×D4—41no—
275C2×D4—41no—
276C2×D4—41no—
277C2×D4—41no—
278C2×D4—41no—
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—161no—
289C22×C4C2×D811yescharacteristic
290C22×C4C2×D811yescharacteristic
291C22×C4C2×D811yescharacteristic
292C22×C4C2×D811yescharacteristic
293C22×C4C2×D811yescharacteristic
294C22×C4C2×D811yescharacteristic
295C22×C4C22×D411yescharacteristic
296Q16—22no—
297Q16—22no—
298Q16—22no—
299Q16—22no—
300Q16—22no—
301Q16—41no—
302Q16—41no—
303Q16C4×D414yes—
304SD16—84no—
305D8—22no—
306D8—22no—
307D8—41no—
308D8—41no—
309C2×C8—21no—
310C2×C8—21no—
311C2×C8—21no—
312C2×C8—21no—
313C2×C8—21no—
314C2×C8—21no—
315C2×C8—21no—
316C2×C8—21no—
317C2×C8C4×D412yes—
318C2×C8C23×C411yescharacteristic
319C2×C8C22×D411yescharacteristic
320C2×C8C2×#4111yescharacteristic
321C2×C8C2×#4111yescharacteristic
322C2×C8H(5,𝔽2)11yescharacteristic
323C2×C82−1+411yescharacteristic
324C4⋊C4—22no—
325C4⋊C4—22no—
326C4⋊C4—41no—
327C4⋊C4—41no—
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—82no—
338C4⋊C4—82no—
339C4⋊C4—82no—
340C4⋊C4—82no—
341C4⋊C4—82no—
342C22⋊C4—44no—
343C22⋊C4—44no—
344C22⋊C4—82no—
345C22⋊C4—82no—
346C22⋊C4—161no—
347C42—21no—
348C42—21no—
349C42—21no—
350C42—21no—
351C42—44no—
352C42—82no—
353C42—82no—
354C42—82no—
355C42—161no—
356C42C4×D412yes—
357C42#13512yes—
358C16—22no—
359C16—22no—
360C16—22no—
361C16—41no—
362C23—42no—
363C23—81no—
364C23—81no—
365C23—81no—
366C23—81no—
367C23—81no—
368C23—81no—
369C23C22×D811yescharacteristic
370Q8—24no—
371Q8—42no—
372Q8—42no—
373Q8—42no—
374Q8—42no—
375Q8—42no—
376Q8—81no—
377Q8—81no—
378D4—42no—
379D4—42no—
380D4—81no—
381D4—81no—
382C2×C4—21no—
383C2×C4—21no—
384C2×C4—21no—
385C2×C4—21no—
386C2×C4—21no—
387C2×C4—21no—
388C2×C4—21no—
389C2×C4—21no—
390C2×C4—21no—
391C2×C4—21no—
392C2×C4—21no—
393C2×C4—21no—
394C2×C4—21no—
395C2×C4—21no—
396C2×C4—21no—
397C2×C4—21no—
398C2×C4—42no—
399C2×C4—42no—
400C2×C4—42no—
401C2×C4—42no—
402C2×C4—42no—
403C2×C4—42no—
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—164no—
420C2×C4C4×D812yes—
421C2×C4C4×D812yes—
422C2×C4C2×C4×D411yescharacteristic
423C2×C4D4211yescharacteristic
424C2×C4#54811yescharacteristic
425C2×C4C2×#13511yescharacteristic
426C2×C4#57611yescharacteristic
427C2×C4#57711yescharacteristic
428C8—22no—
429C8—41no—
430C8—41no—
431C8C2×C4×D411yescharacteristic
432C8#52011yescharacteristic
433C8#54711yescharacteristic
434C8#55311yescharacteristic
435C22—82no—
436C22—82no—
437C22—82no—
438C22—161no—
439C22—161no—
440C22—161no—
441C22—161no—
442C22C2×C4×D811yescharacteristic
443C22D4×D811yescharacteristic
444C22#335811yescharacteristic
445C22#341411yescharacteristic
446C22#341411yescharacteristic
447C22#341611yescharacteristic
448C22#341711yescharacteristic
449C4—22no—
450C4—22no—
451C4—41no—
452C4—41no—
453C4—41no—
454C4—41no—
455C4—44no—
456C4—82no—
457C4—82no—
458C4—82no—
459C4—161no—
460C4—161no—
461C4—161no—
462C4#236111yescharacteristic
463C4#294611yescharacteristic
464C4#328411yescharacteristic
465C4#336111yescharacteristic
466C2—162no—
467C2#2299511yescharacteristic
468C2#3345111yescharacteristic
469C2#3345511yescharacteristic
470C2#3384111yescharacteristic
471C2#3384311yescharacteristic
472C2#3393811yescharacteristic
473C2#3394411yescharacteristic
subgroupG/H
center, socleC23C22×D8
commutatorC2×C8C23×C4
Frattini, cosocleC22×C8C24
prime radicalC22×D8C23
LCSG ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C4×C4⋊C4 ◁ C8×C4⋊C4 ◁ G
LECSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#2176C2semidirect
5. #22685C2semidirect
7. #13178C2semidirect
8. #13150C2semidirect
10. #13015C2semidirect
13. #12936C2semidirect
14. #12662C2semidirect
15. C4×#358C2semidirect
26. #2211C4semidirect
74. C2×Q32C2×C4semidirect
91. C4×Q16C2×C4semidirect
128. C4⋊C16C2×C4semidirect
132. #358C2×C4semidirect

Maximal subgroups

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

subgroupindex
C2×#22112
C2×#21762
C2×#21752
#231862
#226852
#132042
#131782
#131502
#131492
#130152
#130132
#129372
#129362
#126622
C4×#3582