Finite Group Databasebeta

← #2040756

Subgroups — #2040756

Order 512.

2474 conjugacy classes of subgroups.

The subgroup lattice has 496 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2722C214yesmaximal
2#17073C211yesmaximal, characteristic, complemented
3#17070C211yesmaximal, characteristic, complemented
4#16399C214yesmaximal
5#15911C214yesmaximal
6#15646C214yesmaximal, complemented
7#15629C214yesmaximal, complemented
8#15618C214yesmaximal, complemented
9#14697C212yesmaximal
10#14686C211yesmaximal, characteristic
11#14441C212yesmaximal, complemented
12C22×#126C2214yes—
13C23×C42C2211yescharacteristic
14#2722—232no—
15C2×#398C2214yes—
16C2×#398C2214yes—
17C2×#398C2214yes—
18C2×#398C2214yes—
19C2×#398C2214yes—
20C2×#397C2214yes—
21C2×#397C2214yes—
22C2×#389C2214yes—
23C2×#389C2214yes—
24C2×#389C2212yes—
25C2×#389C2212yes—
26C2×#387C2214yes—
27C2×#387C2214yes—
28C2×#385C2214yes—
29C2×#384C2212yes—
30C2×#380C2214yes—
31C2×#380C2212yes—
32C2×#380C2212yes—
33C2×#380C2212yes—
34#1445C2214yes—
35#1444C2214yes—
36#1444C2214yes—
37#1442C2214yes—
38#1442C2214yes—
39#1442C2214yes—
40#1441C2214yes—
41#1441C2214yes—
42#1441C2214yes—
43#1441C2214yes—
44#1441C2214yes—
45#1441C2214yes—
46#1441C2214yes—
47#1440C2214yes—
48#1436C2214yes—
49#1436C2214yes—
50#1436C2214yes—
51#1436C2214yes—
52C4⋊#95C2214yes—
53C4⋊#95C2212yes—
54C4⋊#95C2211yescharacteristic
55C4⋊#95C2211yescharacteristic
56C4⋊#95C2212yes—
57C22⋊#95C2214yes—
58C24×C4C2311yescharacteristic
59C24×C4C2312yes—
60C2×#126—216no—
61C2×#126—216no—
62C2×#126—48no—
63C22×C4⋊C4C2314yes—
64C22×C4⋊C4C2314yes—
65C22×C4⋊C4C2314yes—
66C22×C4⋊C4C2314yes—
67C22×C4⋊C4C2314yes—
68C22×C4⋊C4C2314yes—
69C22×C4⋊C4C2312yes—
70C22×C4⋊C4C2314yes—
71C22×C4⋊C4C2314yes—
72C22×C4⋊C4C2314yes—
73C22×C4⋊C4C2312yes—
74C22×C4⋊C4C2312yes—
75C22×C4⋊C4C2312yes—
76C22×C4⋊C4C2312yes—
77C22×C4⋊C4C2312yes—
78C22×C22⋊C4C2314yes—
79C22×C22⋊C4C2314yes—
80C22×C22⋊C4C2314yes—
81C22×C42—24no—
82C22×C42—44no—
83C22×C42C2311yescharacteristic
84C22×C42C2312yes—
85C22×C42C2314yes—
86C22×C42C2311yescharacteristic
87#398—216no—
88#398—216no—
89#398—216no—
90#398—216no—
91#398—48no—
92#397—216no—
93#397—216no—
94#389—216no—
95#389—28no—
96#389—216no—
97#389—44no—
98#387—216no—
99#387—216no—
100#385—216no—
101#384—44no—
102#380—216no—
103#380—44no—
104#380—44no—
105#380D4116yes—
106C2×#95C2314yes—
107C2×#95C2314yes—
108C2×#95C2314yes—
109C2×#95C2314yes—
110C2×#95C2314yes—
111C2×#95C2314yes—
112C2×#95C2314yes—
113C2×#95C2314yes—
114C2×#95C2314yes—
115C2×#95C2314yes—
116C2×#95C2314yes—
117C2×#95C2314yes—
118C2×#95C2314yes—
119C2×#95C2314yes—
120C2×#95C2314yes—
121C2×#95C2314yes—
122C2×#95C2314yes—
123C2×#95C2314yes—
124C2×#95C2314yes—
125C2×#95C2314yes—
126C2×#95C2314yes—
127C25C2411yescharacteristic
128C23×C4—22no—
129C23×C4—24no—
130C23×C4—22no—
131C23×C4—24no—
132C23×C4—22no—
133C23×C4—24no—
134C23×C4—24no—
135C23×C4—24no—
136C23×C4—42no—
137C23×C4—42no—
138C23×C4—42no—
139C23×C4C2×D418yes—
140C23×C4C2414yes—
141C23×C4C2414yes—
142C23×C4C2414yes—
143C23×C4C2414yes—
144C23×C4C2414yes—
145C23×C4C2414yes—
146C23×C4C2411yescharacteristic
147C23×C4C2412yes—
148C23×C4C2412yes—
149C23×C4C2411yescharacteristic
150#126—464no—
151C2×C4⋊C4—24no—
152C2×C4⋊C4—24no—
153C2×C4⋊C4—28no—
154C2×C4⋊C4—28no—
155C2×C4⋊C4—28no—
156C2×C4⋊C4—28no—
157C2×C4⋊C4—28no—
158C2×C4⋊C4—28no—
159C2×C4⋊C4—28no—
160C2×C4⋊C4—28no—
161C2×C4⋊C4—28no—
162C2×C4⋊C4—28no—
163C2×C4⋊C4—28no—
164C2×C4⋊C4—28no—
165C2×C4⋊C4—28no—
166C2×C4⋊C4—28no—
167C2×C4⋊C4—28no—
168C2×C4⋊C4—28no—
169C2×C4⋊C4—28no—
170C2×C4⋊C4—28no—
171C2×C4⋊C4—24no—
172C2×C4⋊C4—24no—
173C2×C4⋊C4—24no—
174C2×C4⋊C4—24no—
175C2×C4⋊C4—42no—
176C2×C4⋊C4—44no—
177C2×C4⋊C4—44no—
178C2×C4⋊C4—44no—
179C2×C4⋊C4—42no—
180C2×C4⋊C4—42no—
181C2×C4⋊C4—44no—
182C2×C4⋊C4—44no—
183C2×C4⋊C4—44no—
184C2×C4⋊C4—42no—
185C2×C4⋊C4—42no—
186C2×C4⋊C4—42no—
187C2×C4⋊C4—42no—
188C2×C4⋊C4—44no—
189C2×C4⋊C4—42no—
190C2×C4⋊C4—42no—
191C2×C4⋊C4—42no—
192C2×C4⋊C4—44no—
193C2×C4⋊C4—44no—
194C2×C4⋊C4C2×D418yes—
195C2×C4⋊C4C2×D418yes—
196C2×C22⋊C4—28no—
197C2×C22⋊C4—28no—
198C2×C22⋊C4—28no—
199C2×C22⋊C4—28no—
200C2×C22⋊C4—28no—
201C2×C22⋊C4—28no—
202C2×C22⋊C4—44no—
203C2×C22⋊C4—44no—
204C2×C22⋊C4C2×D4116yes—
205C2×C42—22no—
206C2×C42—24no—
207C2×C42—28no—
208C2×C42—28no—
209C2×C42—24no—
210C2×C42—24no—
211C2×C42—416no—
212C2×C42—44no—
213C2×C42—42no—
214C2×C42—42no—
215C2×C42#4114yes—
216#95—28no—
217#95—28no—
218#95—28no—
219#95—28no—
220#95—28no—
221#95—28no—
222#95—28no—
223#95—28no—
224#95—28no—
225#95—28no—
226#95—28no—
227#95—28no—
228#95—28no—
229#95—28no—
230#95—44no—
231#95—44no—
232#95—44no—
233#95—44no—
234#95—44no—
235#95—44no—
236#95C2×D4116yes—
237C24—21no—
238C24—22no—
239C24—22no—
240C24—22no—
241C24—21no—
242C24—22no—
243C24—22no—
244C24C22×D414yes—
245C24H(5,𝔽2)12yes—
246C24C2511yescharacteristic
247C22×C4—22no—
248C22×C4—22no—
249C22×C4—22no—
250C22×C4—22no—
251C22×C4—22no—
252C22×C4—21no—
253C22×C4—22no—
254C22×C4—22no—
255C22×C4—21no—
256C22×C4—22no—
257C22×C4—22no—
258C22×C4—28no—
259C22×C4—24no—
260C22×C4—24no—
261C22×C4—24no—
262C22×C4—21no—
263C22×C4—21no—
264C22×C4—22no—
265C22×C4—24no—
266C22×C4—24no—
267C22×C4—24no—
268C22×C4—24no—
269C22×C4—24no—
270C22×C4—24no—
271C22×C4—24no—
272C22×C4—24no—
273C22×C4—22no—
274C22×C4—24no—
275C22×C4—216no—
276C22×C4—216no—
277C22×C4—24no—
278C22×C4—24no—
279C22×C4—24no—
280C22×C4—24no—
281C22×C4—24no—
282C22×C4—24no—
283C22×C4—24no—
284C22×C4—24no—
285C22×C4—24no—
286C22×C4—24no—
287C22×C4—24no—
288C22×C4—24no—
289C22×C4—24no—
290C22×C4—24no—
291C22×C4—24no—
292C22×C4—24no—
293C22×C4—24no—
294C22×C4—24no—
295C22×C4—24no—
296C22×C4—24no—
297C22×C4—24no—
298C22×C4—24no—
299C22×C4—24no—
300C22×C4—24no—
301C22×C4—24no—
302C22×C4—24no—
303C22×C4—24no—
304C22×C4—24no—
305C22×C4—48no—
306C22×C4—48no—
307C22×C4—48no—
308C22×C4—48no—
309C22×C4—48no—
310C22×C4—48no—
311C22×C4—48no—
312C22×C4—44no—
313C22×C4—44no—
314C22×C4C22×D418yes—
315C22×C4C22×D418yes—
316C22×C4C22×D414yes—
317C22×C4C22×D414yes—
318C22×C4C2×#4112yes—
319C22×C4C2×#4114yes—
320C22×C4H(5,𝔽2)14yes—
321C22×C42−1+412yes—
322C22×C42−1+414yes—
323C22×C42−1+414yes—
324C22×C42−1+414yes—
325C4⋊C4—216no—
326C4⋊C4—48no—
327C4⋊C4—416no—
328C4⋊C4—416no—
329C4⋊C4—416no—
330C4⋊C4—416no—
331C4⋊C4—416no—
332C4⋊C4—416no—
333C4⋊C4—84no—
334C4⋊C4—88no—
335C4⋊C4—84no—
336C4⋊C4—84no—
337C4⋊C4—84no—
338C4⋊C4—88no—
339C4⋊C4—88no—
340C22⋊C4—232no—
341C22⋊C4—416no—
342C22⋊C4—88no—
343C42—28no—
344C42—44no—
345C42—48no—
346C42—416no—
347C23—24no—
348C23—24no—
349C23—22no—
350C23—24no—
351C23—24no—
352C23—24no—
353C23—24no—
354C23—24no—
355C23—22no—
356C23—24no—
357C23—41no—
358C23—41no—
359C23—42no—
360C23—41no—
361C23—42no—
362C23—42no—
363C23—42no—
364C23—42no—
365C23—41no—
366C23—42no—
367C23D4214yes—
368C23C23×D412yes—
369C23C22×#4111yescharacteristic
370C23C2×H(5,𝔽2)12yes—
371C23C2×H(5,𝔽2)11yescharacteristic
372C23C2×2−1+412yes—
373C23C2×2−1+412yes—
374C23C2×2−1+412yes—
375C23C2×2−1+411yescharacteristic
376C23#58512yes—
377C2×C4—24no—
378C2×C4—24no—
379C2×C4—22no—
380C2×C4—24no—
381C2×C4—24no—
382C2×C4—22no—
383C2×C4—24no—
384C2×C4—24no—
385C2×C4—24no—
386C2×C4—24no—
387C2×C4—24no—
388C2×C4—24no—
389C2×C4—24no—
390C2×C4—24no—
391C2×C4—24no—
392C2×C4—24no—
393C2×C4—28no—
394C2×C4—28no—
395C2×C4—28no—
396C2×C4—22no—
397C2×C4—28no—
398C2×C4—28no—
399C2×C4—28no—
400C2×C4—24no—
401C2×C4—432no—
402C2×C4—416no—
403C2×C4—44no—
404C2×C4—41no—
405C2×C4—44no—
406C2×C4—41no—
407C2×C4—44no—
408C2×C4—44no—
409C2×C4—44no—
410C2×C4—44no—
411C2×C4—42no—
412C2×C4—44no—
413C2×C4—44no—
414C2×C4—44no—
415C2×C4—44no—
416C2×C4—44no—
417C2×C4—44no—
418C2×C4—44no—
419C2×C4—44no—
420C2×C4—44no—
421C2×C4—44no—
422C2×C4—44no—
423C2×C4—44no—
424C2×C4—44no—
425C2×C4—44no—
426C2×C4—44no—
427C2×C4—44no—
428C2×C4—44no—
429C2×C4—44no—
430C2×C4—44no—
431C2×C4—44no—
432C2×C4—44no—
433C2×C4—44no—
434C2×C4—44no—
435C2×C4—44no—
436C2×C4—44no—
437C2×C4—44no—
438C2×C4—44no—
439C2×C4—44no—
440C2×C4—44no—
441C2×C4—44no—
442C2×C4#53718yes—
443C2×C4#54114yes—
444C2×C4#54718yes—
445C22—24no—
446C22—28no—
447C22—44no—
448C22—42no—
449C22—42no—
450C22—44no—
451C22—44no—
452C22—44no—
453C22—44no—
454C22C2×#53712yes—
455C22C2×#54111yescharacteristic
456C22C2×D4211yescharacteristic
457C22C2×#54712yes—
458C22#349011yescharacteristic
459C22#349211yescharacteristic
460C22#349512yes—
461C22#349612yes—
462C22#349712yes—
463C22#349912yes—
464C22#349912yes—
465C22#353111yescharacteristic
466C22#353512yes—
467C22#353612yes—
468C22#353812yes—
469C22#353812yes—
470C22#354411yescharacteristic
471C22#355512yes—
472C22#355611yescharacteristic
473C22#355612yes—
474C22#355912yes—
475C4—24no—
476C4—28no—
477C4—28no—
478C4—42no—
479C4—48no—
480C4—48no—
481C4—84no—
482C4—84no—
483C4—84no—
484C4—84no—
485C2—44no—
486C2#3672811yescharacteristic
487C2#3738712yes—
488C2#3786112yes—
489C2#3935312yes—
490C2#3935912yes—
491C2#4578712yes—
492C2#4623411yescharacteristic
493C2#5319612yes—
494C2#5931311yescharacteristic
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. #17073C2semidirect
3. #17070C2semidirect
6. #15646C2semidirect
7. #15629C2semidirect
8. #15618C2semidirect
11. #14441C2semidirect

Maximal subgroups

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

subgroupindex
C2×#27222
#170732
#170702
#163992
#159112
#156462
#156292
#156182
#146972
#146862
#144412