Finite Group Databasebeta

← #6434233

Subgroups — #6434233

Order 512.

1852 conjugacy classes of subgroups.

The subgroup lattice has 495 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35266C211yesmaximal, characteristic
2#34209C211yesmaximal, characteristic, complemented
3#30139C218yesmaximal, complemented
4#30130C214yesmaximal, complemented
5#30129C214yesmaximal, complemented
6#23744C211yesmaximal, characteristic, complemented
7#23731C211yesmaximal, characteristic
8#23615C212yesmaximal, complemented
9#23541C212yesmaximal
10#23507C211yesmaximal, characteristic, complemented
11C2×#2013C211yesmaximal, characteristic
12#20936C212yesmaximal
13#20877C212yesmaximal, complemented
14#20754C211yesmaximal, characteristic, complemented
15C23×SD16C2212yes—
16C22×C4⋊Q8C2211yescharacteristic
17C2×#519C2212yes—
18C2×#428C2212yes—
19C2×#424C2212yes—
20C2×#419C2212yes—
21C2×#419C2212yes—
22C2×#418C2212yes—
23C22×#103C2212yes—
24#2860C2214yes—
25#2644C2214yes—
26#2513—22no—
27#2512—22no—
28#2480—24no—
29#2365—22no—
30#2365—22no—
31#2344—24no—
32#2327C2214yescomplemented
33#2317C2211yescharacteristic
34#2316C2211yescharacteristic
35#2310C2212yes—
36#2309C2211yescharacteristic
37#2309C2211yescharacteristic
38C4×C4⋊Q8C2212yescomplemented
39C4×C4⋊Q8C2212yescomplemented
40#2296C2211yescharacteristic
41C2×#390C2211yescharacteristic
42#2285C2211yescharacteristic
43C4×#117C2211yescharacteristic
44#2101C2214yescomplemented
45#2100C2212yescomplemented
46#2099C2212yescomplemented
47#2094C2212yes—
48#2093C2212yes—
49#2092C2214yes—
50#2068C2218yescomplemented
51#2066C2218yescomplemented
52#2055C2218yes—
53#2055C2218yes—
54#2043C2214yescomplemented
55#2043C2214yescomplemented
56#2013C2214yes—
57#2013C2214yes—
58#1880C2218yes—
59#1880C2218yes—
60#1865C2218yescomplemented
61#1863C2218yescomplemented
62#1832C2214yescomplemented
63#1831C2214yescomplemented
64#1735C2212yes—
65#1735C2212yes—
66#1729C2212yes—
67#1728C2211yescharacteristic
68C2×#336C2211yescharacteristic
69C23×Q8C2312yes—
70C23×D4C2311yescharacteristic
71C22×SD16—28no—
72C22×SD16—44no—
73C22×SD16—44no—
74C22×SD16C2318yes—
75C22×M16C2312yes—
76C23×C8C2312yes—
77C2×C4⋊Q8—28no—
78C2×C4⋊Q8—22no—
79C2×C4⋊Q8C2312yes—
80C2×C4⋊Q8C2312yes—
81C2×#126—24no—
82#519—44no—
83#519—44no—
84C2×C4×Q8—22no—
85C2×C4×Q8—22no—
86C2×C4×Q8C2314yes—
87C2×C4×Q8C2314yes—
88C2×#117—24no—
89C2×#117C2311yescharacteristic
90C2×#117C2311yescharacteristic
91C2×#117C2312yes—
92C2×#117C2311yescharacteristic
93C2×#117C2311yescharacteristic
94C22×C4⋊C4—22no—
95C22×C4⋊C4C2311yescharacteristic
96C22×C22⋊C4—22no—
97C22×C42C2311yescharacteristic
98#428—44no—
99C2×#107C2314yes—
100C2×#106C2314yes—
101#424—44no—
102C2×C4⋊C8C2318yes—
103#419—28no—
104#419—44no—
105#418—44no—
106C2×#103—44no—
107C2×#103C2318yes—
108C2×#103C2318yes—
109C2×#103C2318yes—
110C2×#102C2314yes—
111C2×#102C2314yes—
112#397—24no—
113#397—24no—
114#397—24no—
115#396—24no—
116#391—24no—
117#391C2314yes—
118#391C2314yes—
119#390—21no—
120#390—22no—
121#390—21no—
122#390—22no—
123#390C2314yes—
124#390C2314yes—
125#390C2312yes—
126#390C2312yes—
127#389—22no—
128#389—24no—
129#389—24no—
130#389C2314yes—
131#389C2314yes—
132#389C2312yes—
133#389C2312yes—
134#388—24no—
135#388—24no—
136#388—24no—
137#387—28no—
138#387—28no—
139#385—22no—
140#385—22no—
141#384—21no—
142#384C2311yescharacteristic
143#384C2311yescharacteristic
144#382—21no—
145#382C2312yes—
146#381—24no—
147#380—22no—
148#380—22no—
149#379—22no—
150#379—22no—
151C4×C4⋊C4—24no—
152C4×C4⋊C4—24no—
153C4×C4⋊C4—22no—
154C4×C4⋊C4—22no—
155C4×C4⋊C4D412yes—
156C4×C4⋊C4D412yes—
157C4×C22⋊C4D414yes—
158#376C2312yes—
159C43C2312yes—
160#340C2314yes—
161#340C2314yes—
162#340C2314yes—
163#340C2314yes—
164#336C2312yes—
165#336C2312yes—
166#336C2318yes—
167#336C2312yes—
168#336C2312yes—
169C25—21no—
170C22×Q8—22no—
171C22×Q8—24no—
172C22×Q8—24no—
173C22×Q8—24no—
174C22×Q8—24no—
175C22×Q8—24no—
176C22×Q8#4118yes—
177C22×Q8C2414yes—
178C22×D4—21no—
179C22×D4—24no—
180C22×D4—22no—
181C22×D4—24no—
182C22×D4—22no—
183C22×D4C2412yes—
184C23×C4—22no—
185C23×C4—21no—
186C23×C4C2×D412yes—
187C23×C4C2411yescharacteristic
188C2×SD16—216no—
189C2×SD16—48no—
190C2×SD16—48no—
191C2×SD16—432no—
192C2×M16—24no—
193C2×M16—24no—
194C2×M16—24no—
195C22×C8—24no—
196C22×C8—24no—
197C22×C8—24no—
198C22×C8C2414yes—
199C22×C8C2414yes—
200C4⋊Q8—216no—
201C4⋊Q8—28no—
202C4⋊Q8—28no—
203#126—88no—
204C4×Q8—44no—
205C4×Q8—44no—
206C4×Q8—44no—
207C4×Q8—44no—
208C4×Q8—48no—
209C4×Q8—48no—
210#117—24no—
211#117—22no—
212#117—22no—
213#117—24no—
214#117—24no—
215#117—24no—
216#117—22no—
217#117—22no—
218#117—416no—
219C2×C4⋊C4—216no—
220C2×C4⋊C4—24no—
221C2×C4⋊C4—24no—
222C2×C4⋊C4—24no—
223C2×C4⋊C4—21no—
224C2×C4⋊C4—24no—
225C2×C4⋊C4—24no—
226C2×C4⋊C4—22no—
227C2×C4⋊C4—22no—
228C2×C4⋊C4—22no—
229C2×C4⋊C4—24no—
230C2×C4⋊C4—24no—
231C2×C4⋊C4—22no—
232C2×C4⋊C4C2×D411yescharacteristic
233C2×C4⋊C4C2×D411yescharacteristic
234C2×C4⋊C4C2×D412yes—
235C2×C4⋊C4C2412yes—
236C2×C4⋊C4C2412yes—
237C2×C4⋊C4C2412yes—
238C2×C4⋊C4C2412yes—
239C2×C4⋊C4C2412yes—
240C2×C22⋊C4—22no—
241C2×C22⋊C4—21no—
242C2×C22⋊C4—21no—
243C2×C22⋊C4—24no—
244C2×C22⋊C4—24no—
245C2×C22⋊C4—24no—
246C2×C22⋊C4—24no—
247C2×C22⋊C4—24no—
248C2×C22⋊C4—24no—
249C2×C22⋊C4—84no—
250C2×C22⋊C4C2×D412yes—
251C2×C22⋊C4C2×D412yes—
252C2×C42—24no—
253C2×C42—24no—
254C2×C42—24no—
255C2×C42—22no—
256C2×C42—21no—
257C2×C42—21no—
258C2×C42C2×D414yes—
259C2×C42C2×D412yes—
260C2×C42C2×D414yes—
261C2×C42C2412yes—
262C2×C42C2412yes—
263C2×C42C2411yescharacteristic
264C2×C42C2411yescharacteristic
265#107—44no—
266#106—44no—
267C4⋊C8—48no—
268#103—216no—
269#103—48no—
270#103—48no—
271#102—44no—
272#102—44no—
273#95—24no—
274#95—28no—
275#95—28no—
276#95—22no—
277#95—22no—
278#95—24no—
279#95—24no—
280#95C2×D414yes—
281#95C2×D414yes—
282C24—22no—
283C24—42no—
284C24—42no—
285C24—42no—
286C24—82no—
287C24—82no—
288C24C22×D411yescharacteristic
289C2×Q8—28no—
290C2×Q8—24no—
291C2×Q8—216no—
292C2×Q8—216no—
293C2×Q8—24no—
294C2×Q8—24no—
295C2×Q8—24no—
296C2×Q8—24no—
297C2×Q8—44no—
298C2×Q8—44no—
299C2×Q8—48no—
300C2×Q8—48no—
301C2×Q8C2×#4118yes—
302C2×D4—24no—
303C2×D4—24no—
304C2×D4—22no—
305C2×D4—22no—
306C2×D4—42no—
307C2×D4—42no—
308C2×D4—42no—
309C2×D4—48no—
310C2×D4—48no—
311C22×C4—24no—
312C22×C4—28no—
313C22×C4—24no—
314C22×C4—24no—
315C22×C4—22no—
316C22×C4—22no—
317C22×C4—22no—
318C22×C4—22no—
319C22×C4—22no—
320C22×C4—42no—
321C22×C4—44no—
322C22×C4—44no—
323C22×C4—42no—
324C22×C4—82no—
325C22×C4C22×D414yes—
326C22×C4C22×D412yes—
327C22×C4C22×D412yes—
328C22×C4C22×D412yes—
329C22×C4C22×D412yes—
330C22×C4C22×D411yescharacteristic
331C22×C4C2×#4114yes—
332C22×C4H(5,𝔽2)14yes—
333C22×C4H(5,𝔽2)12yes—
334C22×C42−1+412yes—
335C22×C4C2511yescharacteristic
336SD16—432no—
337M16—48no—
338C2×C8—24no—
339C2×C8—24no—
340C2×C8—24no—
341C2×C8—24no—
342C2×C8—24no—
343C2×C8—24no—
344C2×C8—48no—
345C4⋊C4—216no—
346C4⋊C4—216no—
347C4⋊C4—22no—
348C4⋊C4—22no—
349C4⋊C4—24no—
350C4⋊C4—24no—
351C4⋊C4—22no—
352C4⋊C4—22no—
353C4⋊C4—22no—
354C4⋊C4—24no—
355C4⋊C4—22no—
356C4⋊C4—22no—
357C4⋊C4—22no—
358C4⋊C4—44no—
359C4⋊C4—48no—
360C4⋊C4—48no—
361C4⋊C4—42no—
362C4⋊C4—44no—
363C4⋊C4—44no—
364C4⋊C4—82no—
365C4⋊C4—84no—
366C4⋊C4—84no—
367C22⋊C4—24no—
368C22⋊C4—24no—
369C22⋊C4—42no—
370C22⋊C4—44no—
371C22⋊C4—42no—
372C22⋊C4—48no—
373C22⋊C4—84no—
374C22⋊C4—84no—
375C22⋊C4—84no—
376C22⋊C4—84no—
377C22⋊C4—84no—
378C42—24no—
379C42—28no—
380C42—22no—
381C42—22no—
382C42—22no—
383C42—24no—
384C42—22no—
385C42—44no—
386C42—48no—
387C42—48no—
388C42—42no—
389C42—44no—
390C42—42no—
391C23—22no—
392C23—22no—
393C23—44no—
394C23—42no—
395C23—42no—
396C23—42no—
397C23—44no—
398C23—88no—
399C23—84no—
400C23—162no—
401C23—162no—
402C23—164no—
403C23#52512yes—
404C23#53512yes—
405C23#53712yes—
406C23C23×D411yescharacteristic
407Q8—28no—
408Q8—28no—
409Q8—416no—
410Q8—44no—
411Q8—44no—
412D4—44no—
413D4—44no—
414D4—84no—
415C2×C4—22no—
416C2×C4—22no—
417C2×C4—24no—
418C2×C4—24no—
419C2×C4—22no—
420C2×C4—22no—
421C2×C4—28no—
422C2×C4—22no—
423C2×C4—22no—
424C2×C4—24no—
425C2×C4—24no—
426C2×C4—24no—
427C2×C4—22no—
428C2×C4—48no—
429C2×C4—416no—
430C2×C4—44no—
431C2×C4—44no—
432C2×C4—42no—
433C2×C4—42no—
434C2×C4—44no—
435C2×C4—42no—
436C2×C4—42no—
437C2×C4—42no—
438C2×C4—42no—
439C2×C4—42no—
440C2×C4—44no—
441C2×C4—42no—
442C2×C4—42no—
443C2×C4—44no—
444C2×C4—162no—
445C2×C4#52512yes—
446C2×C4#53512yes—
447C2×C4#53712yes—
448C2×C4#53818yes—
449C2×C4#53918yes—
450C2×C4#57714yes—
451C2×C4C22×#4112yes—
452C2×C4C2×H(5,𝔽2)11yescharacteristic
453C2×C4C2×H(5,𝔽2)12yes—
454C2×C4C2×2−1+411yescharacteristic
455C8—44no—
456C8—44no—
457C22—22no—
458C22—22no—
459C22—24no—
460C22—24no—
461C22—82no—
462C22—82no—
463C22—82no—
464C22—84no—
465C22—84no—
466C22—168no—
467C22#263614yes—
468C22C2×#52511yescharacteristic
469C22C2×#53511yescharacteristic
470C22C2×#53711yescharacteristic
471C22C2×#57712yes—
472C22#358612yes—
473C4—24no—
474C4—48no—
475C4—42no—
476C4—42no—
477C4—42no—
478C4—44no—
479C4—82no—
480C4—82no—
481C4—82no—
482C4—82no—
483C4—82no—
484C4#263614yes—
485C4C2×#53812yes—
486C4C2×#53912yes—
487C2—24no—
488C2—162no—
489C2C2×#263611yescharacteristic
490C2#6058212yes—
491C2#6082011yescharacteristic
492C2#6082811yescharacteristic
493C2#6092312yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC23×D4C23
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C4×#117 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #34209C2semidirect
3. #30139C2semidirect
4. #30130C2semidirect
5. #30129C2semidirect
6. #23744C2semidirect
8. #23615C2semidirect
10. #23507C2semidirect
13. #20877C2semidirect
14. #20754C2semidirect
32. #2327C22semidirect
38. C4×C4⋊Q8C22semidirect
39. C4×C4⋊Q8C22semidirect
44. #2101C22semidirect
45. #2100C22semidirect
46. #2099C22semidirect
50. #2068C22semidirect
51. #2066C22semidirect
54. #2043C22semidirect
55. #2043C22semidirect
60. #1865C22semidirect
61. #1863C22semidirect
62. #1832C22semidirect
63. #1831C22semidirect

Maximal subgroups

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

subgroupindex
#352662
#342092
#301392
#301302
#301292
#237442
#237312
#236152
#235412
#235072
C2×#20132
#209362
#208772
#207542