Finite Group Databasebeta

← #949113

Subgroups — #949113

Order 512.

2606 conjugacy classes of subgroups.

The subgroup lattice has 519 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2438C214yesmaximal
2#17134C212yesmaximal, complemented
3#17132C212yesmaximal, complemented
4#16332C214yesmaximal, complemented
5#16281C214yesmaximal, complemented
6#15980C211yesmaximal, characteristic, complemented
7#15968C212yesmaximal, complemented
8#15964C211yesmaximal, characteristic, complemented
9#14935C214yesmaximal
10#14914C212yesmaximal
11#14275C212yesmaximal
12#13870C212yesmaximal
13C2×#1445C211yesmaximal, characteristic
14C22×C22⋊Q8C2212yes—
15C22×#117C2212yes—
16C23×C4⋊C4C2211yescharacteristic
17#2438—232no—
18C2×#400C2214yes—
19C2×#398C2214yes—
20C2×#398C2214yes—
21C2×#395C2214yes—
22C2×#391C2214yes—
23C2×#389C2214yes—
24C2×#387C2214yes—
25C2×#387C2214yes—
26C2×#385C2214yes—
27C2×#385C2214yes—
28C2×#381C2214yes—
29C2×#381C2212yes—
30C2×#380C2212yes—
31C2×#380C2212yes—
32C22×#95C2214yes—
33C22×#95C2211yescharacteristic
34C22×#95C2211yescharacteristic
35#1446C2212yes—
36#1446C2212yes—
37#1445C2211yescharacteristic
38#1445C2211yescharacteristic
39#1445C2214yes—
40#1445C2211yescharacteristic
41#1445C2211yescharacteristic
42#1444C2212yes—
43#1444C2212yes—
44#1444C2212yes—
45#1444C2212yes—
46#1443C2214yes—
47#1443C2214yes—
48#1442C2212yes—
49#1442C2212yes—
50#1441C2214yes—
51#1441C2214yes—
52#1441C2214yes—
53#1441C2214yes—
54#1441C2214yes—
55#1441C2214yes—
56#1441C2214yes—
57#1441C2214yes—
58#1441C2214yes—
59#1441C2214yes—
60#1441C2214yes—
61#1441C2214yes—
62#1436C2212yes—
63#1436C2212yes—
64#1436C2212yes—
65#1436C2212yes—
66C22⋊#95C2214yes—
67C22⋊#95C2212yes—
68C23×Q8C2312yes—
69C24×C4C2312yes—
70C24×C4C2312yes—
71C24×C4C2312yes—
72C24×C4C2311yescharacteristic
73C2×C22⋊Q8—216no—
74C2×C22⋊Q8—44no—
75C2×#117—216no—
76C2×#117—44no—
77C22×C4⋊C4—42no—
78C22×C4⋊C4—42no—
79C22×C4⋊C4Q818yescomplemented
80C22×C4⋊C4C2314yes—
81C22×C4⋊C4C2314yes—
82C22×C4⋊C4C2312yes—
83C22×C4⋊C4C2314yes—
84C22×C4⋊C4C2314yes—
85C22×C4⋊C4C2314yes—
86C22×C4⋊C4C2314yes—
87C22×C4⋊C4C2312yes—
88C22×C4⋊C4C2312yes—
89C22×C4⋊C4C2314yes—
90C22×C4⋊C4C2311yescharacteristic
91C22×C4⋊C4C2312yes—
92C22×C4⋊C4C2311yescharacteristic
93C22×C22⋊C4C2314yes—
94C22×C22⋊C4C2314yes—
95C22×C42C2312yes—
96C22×C42C2312yes—
97#400—216no—
98#398—216no—
99#398—216no—
100#395—216no—
101#391—216no—
102#389—216no—
103#387—216no—
104#387—216no—
105#385—216no—
106#385—216no—
107#381—216no—
108#381—44no—
109#380—44no—
110#380D4116yes—
111C2×#95—216no—
112C2×#95—42no—
113C2×#95—42no—
114C2×#95C2314yes—
115C2×#95C2314yes—
116C2×#95C2314yes—
117C2×#95C2314yes—
118C2×#95C2314yes—
119C2×#95C2314yes—
120C2×#95C2314yes—
121C2×#95C2314yes—
122C2×#95C2312yes—
123C2×#95C2314yes—
124C2×#95C2314yes—
125C2×#95C2314yes—
126C2×#95C2314yes—
127C2×#95C2314yes—
128C2×#95C2314yes—
129C2×#95C2314yes—
130C2×#95C2314yes—
131C2×#95C2314yes—
132C2×#95C2312yes—
133C2×#95C2312yes—
134C2×#95C2312yes—
135C2×#95C2314yes—
136C2×#95C2314yes—
137C2×#95C2311yescharacteristic
138C2×#95C2311yescharacteristic
139C2×#95C2312yes—
140C2×#95C2312yes—
141C2×#95C2311yescharacteristic
142C2×#95C2311yescharacteristic
143C2×#95C2312yes—
144C2×#95C2312yes—
145C25C2411yescharacteristic
146C22×Q8—28no—
147C22×Q8—24no—
148C22×Q8—42no—
149C22×Q8—44no—
150C22×Q8—42no—
151C23×C4—28no—
152C23×C4—24no—
153C23×C4—22no—
154C23×C4—22no—
155C23×C4—24no—
156C23×C4—28no—
157C23×C4—24no—
158C23×C4—24no—
159C23×C4—28no—
160C23×C4—24no—
161C23×C4—24no—
162C23×C4—44no—
163C23×C4—42no—
164C23×C4—44no—
165C23×C4—44no—
166C23×C4—44no—
167C23×C4C2×D418yes—
168C23×C4C2×Q818yes—
169C23×C4C2×Q814yes—
170C23×C4C2414yes—
171C23×C4C2414yes—
172C23×C4C2412yes—
173C23×C4C2414yes—
174C23×C4C2414yes—
175C23×C4C2412yes—
176C23×C4C2412yes—
177C23×C4C2412yes—
178C23×C4C2411yescharacteristic
179C23×C4C2411yescharacteristic
180C23×C4C2412yes—
181C23×C4C2412yes—
182C22⋊Q8—432no—
183#117—432no—
184C2×C4⋊C4—28no—
185C2×C4⋊C4—28no—
186C2×C4⋊C4—28no—
187C2×C4⋊C4—28no—
188C2×C4⋊C4—28no—
189C2×C4⋊C4—28no—
190C2×C4⋊C4—28no—
191C2×C4⋊C4—28no—
192C2×C4⋊C4—28no—
193C2×C4⋊C4—28no—
194C2×C4⋊C4—416no—
195C2×C4⋊C4—41no—
196C2×C4⋊C4—42no—
197C2×C4⋊C4—41no—
198C2×C4⋊C4—41no—
199C2×C4⋊C4—42no—
200C2×C4⋊C4—41no—
201C2×C4⋊C4—44no—
202C2×C4⋊C4—44no—
203C2×C4⋊C4—44no—
204C2×C4⋊C4—44no—
205C2×C4⋊C4—44no—
206C2×C4⋊C4—44no—
207C2×C4⋊C4—42no—
208C2×C4⋊C4—44no—
209C2×C4⋊C4—44no—
210C2×C4⋊C4—44no—
211C2×C4⋊C4—44no—
212C2×C4⋊C4—44no—
213C2×C4⋊C4—44no—
214C2×C4⋊C4—44no—
215C2×C4⋊C4—42no—
216C2×C4⋊C4—42no—
217C2×C4⋊C4C2×D4116yes—
218C2×C4⋊C4C2×Q814yes—
219C2×C4⋊C4C2×Q818yes—
220C2×C4⋊C4C2×Q814yes—
221C2×C22⋊C4—28no—
222C2×C22⋊C4—28no—
223C2×C22⋊C4—28no—
224C2×C22⋊C4—28no—
225C2×C22⋊C4—28no—
226C2×C22⋊C4C2×D4116yes—
227C2×C42—28no—
228C2×C42—28no—
229C2×C42—42no—
230C2×C42—42no—
231#95—28no—
232#95—28no—
233#95—28no—
234#95—28no—
235#95—28no—
236#95—28no—
237#95—28no—
238#95—28no—
239#95—28no—
240#95—28no—
241#95—28no—
242#95—28no—
243#95—24no—
244#95—41no—
245#95—42no—
246#95—41no—
247#95—41no—
248#95—42no—
249#95—41no—
250#95—44no—
251#95—44no—
252#95—44no—
253#95—44no—
254#95—42no—
255#95—44no—
256#95—44no—
257#95—42no—
258#95—42no—
259#95—44no—
260#95C2×D418yes—
261#95C2×D418yes—
262C24—22no—
263C24—24no—
264C24—22no—
265C24C22×D414yes—
266C24C22×Q812yes—
267C24H(5,𝔽2)12yes—
268C24H(5,𝔽2)12yes—
269C242−1+414yes—
270C24C2511yescharacteristic
271C2×Q8—48no—
272C2×Q8—416no—
273C2×Q8—88no—
274C2×Q8—84no—
275C2×Q8—84no—
276C22×C4—28no—
277C22×C4—216no—
278C22×C4—232no—
279C22×C4—24no—
280C22×C4—24no—
281C22×C4—24no—
282C22×C4—24no—
283C22×C4—24no—
284C22×C4—22no—
285C22×C4—22no—
286C22×C4—22no—
287C22×C4—22no—
288C22×C4—21no—
289C22×C4—21no—
290C22×C4—21no—
291C22×C4—21no—
292C22×C4—22no—
293C22×C4—22no—
294C22×C4—24no—
295C22×C4—24no—
296C22×C4—22no—
297C22×C4—24no—
298C22×C4—22no—
299C22×C4—22no—
300C22×C4—24no—
301C22×C4—24no—
302C22×C4—24no—
303C22×C4—24no—
304C22×C4—24no—
305C22×C4—24no—
306C22×C4—24no—
307C22×C4—24no—
308C22×C4—24no—
309C22×C4—22no—
310C22×C4—24no—
311C22×C4—24no—
312C22×C4—24no—
313C22×C4—24no—
314C22×C4—24no—
315C22×C4—24no—
316C22×C4—24no—
317C22×C4—216no—
318C22×C4—24no—
319C22×C4—24no—
320C22×C4—24no—
321C22×C4—24no—
322C22×C4—24no—
323C22×C4—22no—
324C22×C4—24no—
325C22×C4—24no—
326C22×C4—24no—
327C22×C4—24no—
328C22×C4—22no—
329C22×C4—24no—
330C22×C4—24no—
331C22×C4—24no—
332C22×C4—24no—
333C22×C4—24no—
334C22×C4—24no—
335C22×C4—24no—
336C22×C4—22no—
337C22×C4—48no—
338C22×C4—416no—
339C22×C4—416no—
340C22×C4—48no—
341C22×C4—48no—
342C22×C4—48no—
343C22×C4—416no—
344C22×C4—48no—
345C22×C4—48no—
346C22×C4—48no—
347C22×C4—88no—
348C22×C4—84no—
349C22×C4—84no—
350C22×C4—84no—
351C22×C4—88no—
352C22×C4C22×D414yes—
353C22×C4C22×D414yes—
354C22×C4C22×D418yes—
355C22×C4C22×D418yes—
356C22×C4C22×Q812yes—
357C22×C4C22×Q812yes—
358C22×C4C22×Q814yes—
359C22×C4C22×Q814yes—
360C22×C42−1+414yes—
361C4⋊C4—44no—
362C4⋊C4—48no—
363C4⋊C4—44no—
364C4⋊C4—416no—
365C4⋊C4—416no—
366C4⋊C4—416no—
367C4⋊C4—48no—
368C4⋊C4—416no—
369C4⋊C4—416no—
370C4⋊C4—88no—
371C4⋊C4—88no—
372C4⋊C4—84no—
373C4⋊C4—84no—
374C22⋊C4—232no—
375C22⋊C4—416no—
376C42—48no—
377C42—48no—
378C23—24no—
379C23—28no—
380C23—28no—
381C23—24no—
382C23—24no—
383C23—24no—
384C23—24no—
385C23—22no—
386C23—22no—
387C23—28no—
388C23—28no—
389C23#53718yes—
390C23#54314yes—
391C23D4214yes—
392C23D4×Q818yes—
393C23#56314yes—
394C23C23×D412yes—
395C23C23×Q811yescharacteristic
396C23C2×H(5,𝔽2)11yescharacteristic
397C23C2×H(5,𝔽2)11yescharacteristic
398C23C2×2−1+412yes—
399C23C2×2−1+412yes—
400C23#58514yes—
401C23#58512yes—
402Q8—816no—
403C2×C4—24no—
404C2×C4—24no—
405C2×C4—22no—
406C2×C4—22no—
407C2×C4—24no—
408C2×C4—24no—
409C2×C4—24no—
410C2×C4—24no—
411C2×C4—28no—
412C2×C4—28no—
413C2×C4—28no—
414C2×C4—28no—
415C2×C4—28no—
416C2×C4—28no—
417C2×C4—28no—
418C2×C4—28no—
419C2×C4—28no—
420C2×C4—28no—
421C2×C4—28no—
422C2×C4—24no—
423C2×C4—416no—
424C2×C4—44no—
425C2×C4—44no—
426C2×C4—432no—
427C2×C4—42no—
428C2×C4—42no—
429C2×C4—42no—
430C2×C4—44no—
431C2×C4—44no—
432C2×C4—44no—
433C2×C4—44no—
434C2×C4—44no—
435C2×C4—42no—
436C2×C4—42no—
437C2×C4—42no—
438C2×C4—44no—
439C2×C4—42no—
440C2×C4—42no—
441C2×C4—42no—
442C2×C4—42no—
443C2×C4—42no—
444C2×C4—42no—
445C2×C4—44no—
446C2×C4—44no—
447C2×C4—44no—
448C2×C4—42no—
449C2×C4—42no—
450C2×C4—44no—
451C2×C4—44no—
452C2×C4—44no—
453C2×C4—44no—
454C2×C4—42no—
455C2×C4—44no—
456C2×C4—44no—
457C2×C4—44no—
458C2×C4—44no—
459C2×C4—44no—
460C2×C4—44no—
461C2×C4—44no—
462C2×C4—44no—
463C2×C4—42no—
464C2×C4—42no—
465C2×C4—44no—
466C2×C4—44no—
467C2×C4—44no—
468C2×C4—44no—
469C2×C4—44no—
470C2×C4—44no—
471C2×C4—816no—
472C2×C4—816no—
473C22—28no—
474C22—28no—
475C22—216no—
476C22—24no—
477C22—24no—
478C22—28no—
479C22—28no—
480C22#268318yes—
481C22C2×#53712yes—
482C22C2×#54311yescharacteristic
483C22#345812yes—
484C22C2×D4211yescharacteristic
485C22C2×D4×Q812yes—
486C22#348514yes—
487C22#348914yes—
488C22#350212yes—
489C22C2×#56311yescharacteristic
490C22#353312yes—
491C22#353312yes—
492C22#353512yes—
493C22#354411yescharacteristic
494C22#354511yescharacteristic
495C22#356314yes—
496C22#356414yes—
497C4—48no—
498C4—44no—
499C4—48no—
500C4—42no—
501C4—42no—
502C4—44no—
503C4—44no—
504C4—44no—
505C4—44no—
506C4—82no—
507C4—84no—
508C4—84no—
509C2—28no—
510C2#3719411yescharacteristic
511C2#3747911yescharacteristic
512C2#3768914yes—
513C2#4140512yes—
514C2#4216712yes—
515C2#4352712yes—
516C2C2×#268311yescharacteristic
517C2#5275412yes—
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC25C24
LCSG ▷ C24 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C24 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #17134C2semidirect
3. #17132C2semidirect
4. #16332C2semidirect
5. #16281C2semidirect
6. #15980C2semidirect
7. #15968C2semidirect
8. #15964C2semidirect
79. C22×C4⋊C4Q8semidirect

Maximal subgroups

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

subgroupindex
C2×#24382
#171342
#171322
#163322
#162812
#159802
#159682
#159642
#149352
#149142
#142752
#138702
C2×#14452