Finite Group Databasebeta

← #498566

Subgroups — #498566

Order 512.

696 conjugacy classes of subgroups.

The subgroup lattice has 502 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2251C211yesmaximal, characteristic
2C2×#2175C211yesmaximal, characteristic
3C2×C4×Q32C211yesmaximal, characteristic, complemented
4#23186C211yesmaximal, characteristic
5#23184C211yesmaximal, characteristic, complemented
6#13254C211yesmaximal, characteristic
7#13253C211yesmaximal, characteristic, complemented
8#13240C211yesmaximal, characteristic, complemented
9#13237C211yesmaximal, characteristic
10#13015C211yesmaximal, characteristic, complemented
11#13015C211yesmaximal, characteristic, complemented
12#13014C211yesmaximal, characteristic
13#13013C211yesmaximal, characteristic
14#12674C211yesmaximal, characteristic, complemented
15#12663C211yesmaximal, characteristic, complemented
16C22×Q32C2211yescharacteristic
17C22×Q32C2211yescharacteristic
18C22×SD32C2211yescharacteristic
19C22×SD32C2211yescharacteristic
20C2×#495C2211yescharacteristic
21C2×#494C2211yescharacteristic
22C2×C4×Q16C2211yescharacteristic
23C2×C4×Q16C2211yescharacteristic
24C2×C4×Q16C2211yescharacteristic
25C2×C4×D8C2211yescharacteristic
26#2367—21no—
27#2365—21no—
28#2251C418yescomplemented
29#2175—24no—
30C4×Q32—24no—
31C2×#367C2211yescharacteristic
32C2×#366C2211yescharacteristic
33C2×C4⋊C16C2211yescharacteristic
34C2×#358C2211yescharacteristic
35C2×#358C2211yescharacteristic
36C2×#358C2211yescharacteristic
37C2×#358C2211yescharacteristic
38C2×#358C2211yescharacteristic
39C2×#358C2211yescharacteristic
40C2×#357C2211yescharacteristic
41C2×#357C2211yescharacteristic
42C2×#346C2211yescharacteristic
43C2×#346C2211yescharacteristic
44C2×C4×C16C2211yescharacteristic
45#1983—21no—
46#1981—21no—
47#1970C2211yescharacteristic
48#1969C2211yescharacteristic
49#1969C2211yescharacteristic
50#1969C2211yescharacteristic
51#1880—21no—
52#1878—21no—
53#1878—21no—
54#1877—21no—
55C4×#107C2211yescharacteristic
56C4×#107C2211yescharacteristic
57C8×C4⋊C4C2211yescharacteristic
58#1380C2211yescharacteristic
59#1380C2211yescharacteristic
60#1374C2211yescharacteristic
61#1374C2211yescharacteristic
62C22×Q16—21no—
63C22×Q16C2311yescharacteristic
64C22×Q16C2311yescharacteristic
65C22×Q16C2311yescharacteristic
66C22×SD16—21no—
67C22×D8C2311yescharacteristic
68C2×C4⋊Q8—21no—
69C2×#124—21no—
70C2×C4×Q8—21no—
71C2×C4×Q8—21no—
72C2×C4×Q8—21no—
73C2×C4×D4—21no—
74C2×Q32—22no—
75C2×Q32—22no—
76C2×Q32—41no—
77C2×Q32—41no—
78C2×Q32C2×C414yescomplemented
79C2×Q32C2×C414yescomplemented
80C2×SD32—22no—
81C2×SD32—22no—
82C2×SD32—41no—
83C2×SD32—41no—
84C2×SD32C2×C414yes—
85C2×SD32C2×C414yes—
86C22×C16C2311yescharacteristic
87C22×C16C2311yescharacteristic
88C22×C16C2311yescharacteristic
89C22×C16C2311yescharacteristic
90#495—22no—
91#495C2×C414yes—
92#494—22no—
93#494C2×C414yescomplemented
94C4×Q16—22no—
95C4×Q16—22no—
96C4×Q16—22no—
97C4×Q16—41no—
98C4×Q16—41no—
99C4×Q16—41no—
100C4×D8—22no—
101C4×D8—41no—
102C2×#107C2311yescharacteristic
103C2×#107C2311yescharacteristic
104C2×#107C2311yescharacteristic
105C2×#107C2311yescharacteristic
106C2×C4⋊C8D411yescharacteristic
107C2×#103—21no—
108C2×#103—21no—
109C2×#103—21no—
110C2×#103—21no—
111C2×#103—21no—
112C2×#103—21no—
113C2×#102—21no—
114C2×#102—21no—
115C2×C4×C8C2311yescharacteristic
116C2×C4×C8C2311yescharacteristic
117C2×C4×C8C2311yescharacteristic
118#391—21no—
119#391—21no—
120#391—21no—
121#390—21no—
122#389—41no—
123#388—41no—
124#388—41no—
125#387—41no—
126#387—41no—
127#385—41no—
128C4×C4⋊C4—21no—
129C4×C4⋊C4—21no—
130C4×C4⋊C4—41no—
131C4×C4⋊C4D411yescharacteristic
132C4×C22⋊C4—41no—
133#367—22no—
134#366D414yes—
135C4⋊C16—22no—
136#358—22no—
137#358—22no—
138#358—22no—
139#358—41no—
140#358—41no—
141#358—41no—
142#357—22no—
143#357—41no—
144#346—22no—
145#346C2×C414yescomplemented
146C4×C16—22no—
147#340—21no—
148#340—21no—
149#336D411yescharacteristic
150#336D411yescharacteristic
151C22×Q8—21no—
152C22×Q8—21no—
153C22×Q8—21no—
154C22×D4—21no—
155C23×C4—41no—
156C2×Q16—24no—
157C2×Q16—21no—
158C2×Q16—21no—
159C2×Q16—21no—
160C2×Q16—21no—
161C2×Q16—21no—
162C2×Q16—21no—
163C2×Q16—21no—
164C2×Q16—21no—
165C2×Q16—21no—
166C2×Q16—21no—
167C2×Q16—21no—
168C2×Q16—21no—
169C2×Q16—21no—
170C2×Q16—21no—
171C2×Q16—21no—
172C2×Q16—81no—
173C2×Q16—81no—
174C2×Q16C22×C412yes—
175C2×Q16C22×C412yes—
176C2×Q16C22×C412yes—
177C2×SD16—24no—
178C2×SD16—81no—
179C2×SD16—81no—
180C2×D8—21no—
181C2×D8—21no—
182C2×D8—21no—
183C2×D8—21no—
184C2×D8—21no—
185C2×D8C22×C412yes—
186C22×C8C2×D411yescharacteristic
187C22×C8C2×D411yescharacteristic
188C22×C8C2×D411yescharacteristic
189C22×C8C2411yescharacteristic
190C4⋊Q8—24no—
191C4⋊Q8—42no—
192#124—24no—
193#124—42no—
194C4×Q8—42no—
195C4×Q8—42no—
196C4×Q8—42no—
197C4×Q8—81no—
198C4×Q8—81no—
199C4×Q8—81no—
200C4×D4—42no—
201C4×D4—81no—
202C2×C4⋊C4—21no—
203C2×C4⋊C4—21no—
204C2×C4⋊C4—21no—
205C2×C4⋊C4—21no—
206C2×C4⋊C4—41no—
207C2×C4⋊C4—41no—
208C2×C4⋊C4—41no—
209C2×C4⋊C4—41no—
210C2×C4⋊C4—41no—
211C2×C4⋊C4D811yescharacteristic
212C2×C4⋊C4D811yescharacteristic
213C2×C22⋊C4—41no—
214C2×C22⋊C4—41no—
215C2×C22⋊C4—41no—
216C2×C22⋊C4—41no—
217C2×C22⋊C4—41no—
218C2×C42—41no—
219C2×C42—41no—
220C2×C42—41no—
221C2×C42—41no—
222C2×C42—41no—
223C2×C42C2×D411yescharacteristic
224C2×C42C2×D411yescharacteristic
225C2×C42C2×D411yescharacteristic
226Q32—28no—
227Q32—44no—
228SD32—28no—
229SD32—44no—
230C2×C16—21no—
231C2×C16—21no—
232C2×C16—21no—
233C2×C16—21no—
234C2×C16—21no—
235C2×C16—21no—
236C2×C16—21no—
237C2×C16—21no—
238C2×C16C22×C412yes—
239C2×C16C22×C412yes—
240C2×C16C2×D412yes—
241C2×C16#4112yes—
242#107—21no—
243#107—21no—
244#107—21no—
245#107—21no—
246#107—21no—
247#107—21no—
248#107C2×D412yes—
249#107C2×D412yes—
250C4⋊C8—22no—
251#103—42no—
252#103—42no—
253#103—42no—
254#103—81no—
255#103—81no—
256#103—81no—
257#102—42no—
258#102—81no—
259C4×C8—21no—
260C4×C8—21no—
261C4×C8—21no—
262C4×C8—21no—
263C4×C8C22×C412yes—
264C4×C8#4112yes—
265#95—41no—
266#95—41no—
267#95—41no—
268#95—41no—
269#95—41no—
270#95—41no—
271#95—41no—
272#95—41no—
273#95D811yescharacteristic
274#95D811yescharacteristic
275C24—41no—
276C2×Q8—22no—
277C2×Q8—22no—
278C2×Q8—22no—
279C2×Q8—41no—
280C2×Q8—41no—
281C2×Q8—41no—
282C2×Q8—41no—
283C2×Q8—41no—
284C2×Q8—41no—
285C2×Q8—41no—
286C2×Q8—41no—
287C2×Q8—41no—
288C2×Q8—41no—
289C2×Q8—41no—
290C2×Q8—41no—
291C2×Q8—41no—
292C2×Q8—41no—
293C2×Q8—41no—
294C2×D4—22no—
295C2×D4—41no—
296C2×D4—41no—
297C2×D4—41no—
298C2×D4—41no—
299C2×D4—41no—
300C22×C4—41no—
301C22×C4—41no—
302C22×C4—41no—
303C22×C4—41no—
304C22×C4—41no—
305C22×C4—41no—
306C22×C4—41no—
307C22×C4—82no—
308C22×C4—82no—
309C22×C4—161no—
310C22×C4C2×D811yescharacteristic
311C22×C4C2×D811yescharacteristic
312C22×C4C2×D811yescharacteristic
313C22×C4C2×D811yescharacteristic
314C22×C4C2×D811yescharacteristic
315C22×C4C2×D811yescharacteristic
316C22×C4C22×D411yescharacteristic
317Q16—22no—
318Q16—22no—
319Q16—22no—
320Q16—22no—
321Q16—22no—
322Q16—22no—
323Q16—41no—
324Q16—41no—
325Q16—41no—
326Q16—41no—
327Q16—41no—
328Q16—41no—
329Q16—84no—
330SD16—84no—
331D8—22no—
332D8—22no—
333D8—41no—
334D8—41no—
335C2×C8—21no—
336C2×C8—21no—
337C2×C8—21no—
338C2×C8—21no—
339C2×C8—21no—
340C2×C8—21no—
341C2×C8—21no—
342C2×C8—21no—
343C2×C8C4×D412yes—
344C2×C8C23×C411yescharacteristic
345C2×C8C22×D411yescharacteristic
346C2×C8C2×#4111yescharacteristic
347C2×C8C2×#4111yescharacteristic
348C2×C8H(5,𝔽2)11yescharacteristic
349C2×C8H(5,𝔽2)11yescharacteristic
350C4⋊C4—22no—
351C4⋊C4—22no—
352C4⋊C4—22no—
353C4⋊C4—41no—
354C4⋊C4—41no—
355C4⋊C4—41no—
356C4⋊C4—44no—
357C4⋊C4—44no—
358C4⋊C4—41no—
359C4⋊C4—41no—
360C4⋊C4—41no—
361C4⋊C4—41no—
362C4⋊C4—82no—
363C4⋊C4—82no—
364C4⋊C4—82no—
365C22⋊C4—44no—
366C22⋊C4—44no—
367C22⋊C4—82no—
368C22⋊C4—82no—
369C22⋊C4—161no—
370C42—21no—
371C42—21no—
372C42—21no—
373C42—21no—
374C42—82no—
375C42—82no—
376C42—82no—
377C42—82no—
378C42—161no—
379C42C4×D412yes—
380C42#13512yes—
381C16—22no—
382C16—41no—
383C16—41no—
384C16C4×D414yes—
385C23—42no—
386C23—81no—
387C23—81no—
388C23—81no—
389C23—81no—
390C23—81no—
391C23—81no—
392C23C22×D811yescharacteristic
393Q8—42no—
394Q8—42no—
395Q8—42no—
396Q8—42no—
397Q8—42no—
398Q8—42no—
399Q8—81no—
400Q8—81no—
401Q8—81no—
402Q8—81no—
403Q8—81no—
404Q8—81no—
405D4—42no—
406D4—42no—
407D4—81no—
408D4—81no—
409C2×C4—21no—
410C2×C4—21no—
411C2×C4—21no—
412C2×C4—21no—
413C2×C4—21no—
414C2×C4—21no—
415C2×C4—21no—
416C2×C4—21no—
417C2×C4—21no—
418C2×C4—21no—
419C2×C4—21no—
420C2×C4—21no—
421C2×C4—21no—
422C2×C4—21no—
423C2×C4—21no—
424C2×C4—42no—
425C2×C4—42no—
426C2×C4—42no—
427C2×C4—42no—
428C2×C4—42no—
429C2×C4—81no—
430C2×C4—81no—
431C2×C4—81no—
432C2×C4—81no—
433C2×C4—81no—
434C2×C4—81no—
435C2×C4—81no—
436C2×C4—81no—
437C2×C4—81no—
438C2×C4—81no—
439C2×C4—81no—
440C2×C4—81no—
441C2×C4—81no—
442C2×C4—81no—
443C2×C4—81no—
444C2×C4—81no—
445C2×C4—164no—
446C2×C4C4×D812yes—
447C2×C4C4×D812yes—
448C2×C4#51012yes—
449C2×C4C2×C4×D411yescharacteristic
450C2×C4D4211yescharacteristic
451C2×C4#55011yescharacteristic
452C2×C4C2×#13511yescharacteristic
453C2×C4#57611yescharacteristic
454C2×C4#57611yescharacteristic
455C8—22no—
456C8—41no—
457C8—41no—
458C8C2×C4×D411yescharacteristic
459C8#51811yescharacteristic
460C8#55011yescharacteristic
461C8#55911yescharacteristic
462C22—82no—
463C22—82no—
464C22—82no—
465C22—161no—
466C22—161no—
467C22—161no—
468C22—161no—
469C22C2×C4×D811yescharacteristic
470C22D4×D811yescharacteristic
471C22#339111yescharacteristic
472C22C2×#51011yescharacteristic
473C22#341411yescharacteristic
474C22#341611yescharacteristic
475C22#341711yescharacteristic
476C4—22no—
477C4—22no—
478C4—22no—
479C4—41no—
480C4—41no—
481C4—41no—
482C4—82no—
483C4—82no—
484C4—82no—
485C4—82no—
486C4—82no—
487C4—161no—
488C4—161no—
489C4#236611yescharacteristic
490C4#294811yescharacteristic
491C4#328711yescharacteristic
492C4#339611yescharacteristic
493C2—162no—
494C2#2318511yescharacteristic
495C2C4×#51011yescharacteristic
496C2#3345511yescharacteristic
497C2#3383311yescharacteristic
498C2#3383611yescharacteristic
499C2#3396711yescharacteristic
500C2#3396911yescharacteristic
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
3. C2×C4×Q32C2semidirect
5. #23184C2semidirect
7. #13253C2semidirect
8. #13240C2semidirect
10. #13015C2semidirect
11. #13015C2semidirect
14. #12674C2semidirect
15. #12663C2semidirect
28. #2251C4semidirect
78. C2×Q32C2×C4semidirect
79. C2×Q32C2×C4semidirect
93. #494C2×C4semidirect
145. #346C2×C4semidirect

Maximal subgroups

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

subgroupindex
C2×#22512
C2×#21752
C2×C4×Q322
#231862
#231842
#132542
#132532
#132402
#132372
#130152
#130142
#130132
#126742
#126632