Finite Group Databasebeta

← #4414334

Subgroups — #4414334

Order 512.

3228 conjugacy classes of subgroups.

The subgroup lattice has 457 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2823C211yesmaximal, characteristic, complemented
2C2×#2852C212yesmaximal, complemented
3C22×#384C211yesmaximal, characteristic, complemented
4#16563C213yesmaximal, complemented
5#16515C216yesmaximal, complemented
6#16514C213yesmaximal, complemented
7#14821C213yesmaximal, complemented
8#14729C216yesmaximal, complemented
9#14719C213yesmaximal
10#14676C211yesmaximal, characteristic, complemented
11#14665C212yesmaximal, complemented
12C22×#126C2212yes—
13C22×#121C2211yescharacteristic
14C23×C4⋊C4C2213yes—
15C23×C4⋊C4C2213yes—
16C23×C42C2211yescharacteristic
17#2852—216no—
18#2823—28no—
19C2×#398C2213yes—
20C2×#398C2213yes—
21C2×#396C2216yes—
22C2×#396C2216yes—
23C2×#394C2213yes—
24C2×#394C2213yes—
25C2×#388C2216yes—
26C2×#387C2216yes—
27C2×#386C2216yes—
28C2×#386C2216yes—
29C2×#385C2216yes—
30C2×#384—28no—
31C2×#384C2211yescharacteristic
32C2×#384C2213yes—
33C2×#384C2213yes—
34C2×#384C2211yescharacteristic
35C2×#383C2212yes—
36C2×#380C2213yes—
37#1441C2216yes—
38#1441C2216yes—
39#1441C2213yes—
40#1441C2213yes—
41#1441C2216yes—
42#1441C2213yes—
43#1441C2216yes—
44#1441C2213yes—
45#1441C2216yes—
46#1441C2216yes—
47#1439C2216yes—
48#1439C2216yes—
49#1439C2216yes—
50#1439C2216yes—
51C22⋊#95C2213yes—
52C22⋊#95C2213yes—
53C23×D4C2312yes—
54C23×D4C2311yescharacteristic
55C24×C4C2313yes—
56C24×C4C2311yescharacteristic
57C24×C4C2313yes—
58C2×#126—28no—
59C2×#126—28no—
60C2×#126—28no—
61C2×#121—46no—
62C22×C4⋊C4—212no—
63C22×C4⋊C4—212no—
64C22×C4⋊C4—46no—
65C22×C4⋊C4—46no—
66C22×C4⋊C4—46no—
67C22×C4⋊C4—46no—
68C22×C4⋊C4C2316yes—
69C22×C4⋊C4C2316yes—
70C22×C4⋊C4C2313yes—
71C22×C4⋊C4C2316yes—
72C22×C4⋊C4C2313yes—
73C22×C4⋊C4C2313yes—
74C22×C4⋊C4C2313yes—
75C22×C4⋊C4C2313yes—
76C22×C4⋊C4C2313yes—
77C22×C4⋊C4C2311yescharacteristic
78C22×C4⋊C4C2313yes—
79C22×C4⋊C4C2313yes—
80C22×C22⋊C4C2316yes—
81C22×C22⋊C4C2316yes—
82C22×C22⋊C4C2316yes—
83C22×C22⋊C4C2316yes—
84C22×C22⋊C4C2316yes—
85C22×C22⋊C4C2312yes—
86C22×C42—24no—
87C22×C42—24no—
88C22×C42D418yescomplemented
89C22×C42C2313yes—
90C22×C42C2311yescharacteristic
91#398—212no—
92#398—46no—
93#396—224no—
94#396—412no—
95#394—212no—
96#394—46no—
97#388—224no—
98#387—224no—
99#386—224no—
100#386—412no—
101#385—224no—
102#384—212no—
103#384—212no—
104#384—24no—
105#384D418yescomplemented
106#383—28no—
107#380—46no—
108C2×#95C2313yes—
109C2×#95C2313yes—
110C2×#95C2316yes—
111C2×#95C2316yes—
112C2×#95C2316yes—
113C2×#95C2316yes—
114C2×#95C2313yes—
115C2×#95C2316yes—
116C2×#95C2316yes—
117C2×#95C2313yes—
118C2×#95C2316yes—
119C2×#95C2316yes—
120C2×#95C2316yes—
121C25C2412yes—
122C25C2411yescharacteristic
123C22×D4—24no—
124C22×D4—212no—
125C22×D4—212no—
126C22×D4—22no—
127C22×D4—43no—
128C22×D4—43no—
129C23×C4—26no—
130C23×C4—26no—
131C23×C4—26no—
132C23×C4—26no—
133C23×C4—26no—
134C23×C4—26no—
135C23×C4—26no—
136C23×C4—26no—
137C23×C4—212no—
138C23×C4—22no—
139C23×C4—26no—
140C23×C4—43no—
141C23×C4—43no—
142C23×C4—46no—
143C23×C4C2×D4112yes—
144C23×C4C2416yes—
145C23×C4C2416yes—
146C23×C4C2411yescharacteristic
147C23×C4C2413yes—
148C23×C4C2412yes—
149C23×C4C2413yes—
150C23×C4C2411yescharacteristic
151C23×C4C2413yes—
152C23×C4C2413yes—
153#126—432no—
154#121—416no—
155C2×C4⋊C4—26no—
156C2×C4⋊C4—212no—
157C2×C4⋊C4—26no—
158C2×C4⋊C4—26no—
159C2×C4⋊C4—26no—
160C2×C4⋊C4—212no—
161C2×C4⋊C4—212no—
162C2×C4⋊C4—212no—
163C2×C4⋊C4—212no—
164C2×C4⋊C4—26no—
165C2×C4⋊C4—212no—
166C2×C4⋊C4—26no—
167C2×C4⋊C4—212no—
168C2×C4⋊C4—212no—
169C2×C4⋊C4—26no—
170C2×C4⋊C4—43no—
171C2×C4⋊C4—43no—
172C2×C4⋊C4—43no—
173C2×C4⋊C4—43no—
174C2×C4⋊C4—46no—
175C2×C4⋊C4—43no—
176C2×C4⋊C4—43no—
177C2×C4⋊C4—43no—
178C2×C4⋊C4—43no—
179C2×C4⋊C4—43no—
180C2×C4⋊C4—43no—
181C2×C4⋊C4—43no—
182C2×C4⋊C4—448no—
183C2×C4⋊C4—448no—
184C2×C4⋊C4—43no—
185C2×C4⋊C4—43no—
186C2×C4⋊C4—46no—
187C2×C4⋊C4—43no—
188C2×C4⋊C4C2×D4112yes—
189C2×C4⋊C4C2×D4112yes—
190C2×C22⋊C4—212no—
191C2×C22⋊C4—212no—
192C2×C22⋊C4—212no—
193C2×C22⋊C4—212no—
194C2×C22⋊C4—212no—
195C2×C22⋊C4—212no—
196C2×C22⋊C4—212no—
197C2×C22⋊C4—212no—
198C2×C22⋊C4—212no—
199C2×C22⋊C4—212no—
200C2×C22⋊C4—46no—
201C2×C22⋊C4—46no—
202C2×C22⋊C4—46no—
203C2×C22⋊C4—46no—
204C2×C22⋊C4—46no—
205C2×C22⋊C4—46no—
206C2×C42—26no—
207C2×C42—26no—
208C2×C42—232no—
209C2×C42C2×D4112yes—
210C2×C42C2×D414yes—
211C2×C42C2×D414yes—
212C2×C42#4114yes—
213#95—212no—
214#95—212no—
215#95—212no—
216#95—212no—
217#95—26no—
218#95—26no—
219#95—46no—
220#95—46no—
221#95—46no—
222#95—43no—
223#95—46no—
224#95—43no—
225#95—46no—
226C24—21no—
227C24—21no—
228C24—23no—
229C24—23no—
230C24—22no—
231C24—22no—
232C24—22no—
233C24—26no—
234C24—26no—
235C24—26no—
236C24—26no—
237C24C22×D412yes—
238C24H(5,𝔽2)16yes—
239C242−1+416yes—
240C24C2511yescharacteristic
241C2×D4—44no—
242C2×D4—424no—
243C2×D4—48no—
244C2×D4—424no—
245C2×D4—86no—
246C2×D4—86no—
247C22×C4—224no—
248C22×C4—224no—
249C22×C4—224no—
250C22×C4—224no—
251C22×C4—224no—
252C22×C4—23no—
253C22×C4—23no—
254C22×C4—23no—
255C22×C4—23no—
256C22×C4—23no—
257C22×C4—23no—
258C22×C4—23no—
259C22×C4—23no—
260C22×C4—224no—
261C22×C4—224no—
262C22×C4—26no—
263C22×C4—23no—
264C22×C4—23no—
265C22×C4—26no—
266C22×C4—23no—
267C22×C4—23no—
268C22×C4—22no—
269C22×C4—26no—
270C22×C4—26no—
271C22×C4—26no—
272C22×C4—21no—
273C22×C4—23no—
274C22×C4—26no—
275C22×C4—26no—
276C22×C4—26no—
277C22×C4—26no—
278C22×C4—26no—
279C22×C4—26no—
280C22×C4—26no—
281C22×C4—23no—
282C22×C4—23no—
283C22×C4—26no—
284C22×C4—26no—
285C22×C4—23no—
286C22×C4—23no—
287C22×C4—26no—
288C22×C4—26no—
289C22×C4—26no—
290C22×C4—26no—
291C22×C4—26no—
292C22×C4—26no—
293C22×C4—424no—
294C22×C4—412no—
295C22×C4—412no—
296C22×C4—44no—
297C22×C4—412no—
298C22×C4—86no—
299C22×C4—812no—
300C22×C4—86no—
301C22×C4C22×D412yes—
302C22×C4C22×D416yes—
303C22×C4C22×D416yes—
304C22×C4C22×D416yes—
305C22×C4C22×D416yes—
306C22×C4C2×#4116yes—
307C22×C4H(5,𝔽2)16yes—
308C22×C42−1+416yes—
309C4⋊C4—412no—
310C4⋊C4—412no—
311C4⋊C4—424no—
312C4⋊C4—412no—
313C4⋊C4—412no—
314C4⋊C4—412no—
315C4⋊C4—412no—
316C4⋊C4—44no—
317C4⋊C4—424no—
318C4⋊C4—424no—
319C4⋊C4—86no—
320C4⋊C4—86no—
321C22⋊C4—48no—
322C22⋊C4—424no—
323C22⋊C4—424no—
324C22⋊C4—424no—
325C22⋊C4—812no—
326C22⋊C4—812no—
327C42—224no—
328C42#121116yescomplemented
329C23—22no—
330C23—26no—
331C23—26no—
332C23—26no—
333C23—26no—
334C23—26no—
335C23—26no—
336C23—26no—
337C23—212no—
338C23—42no—
339C23—46no—
340C23—46no—
341C23—46no—
342C23—46no—
343C23—46no—
344C23—46no—
345C23—42no—
346C23—46no—
347C23—46no—
348C23—46no—
349C23—412no—
350C23#537112yes—
351C23#56014yes—
352C23#563112yes—
353C23C23×D411yescharacteristic
354C23C23×D411yescharacteristic
355C23C22×#4111yescharacteristic
356C23C2×H(5,𝔽2)13yes—
357C23C2×H(5,𝔽2)13yes—
358C23C2×2−1+413yes—
359C23C2×2−1+413yes—
360D4—88no—
361D4—816no—
362C2×C4—296no—
363C2×C4—26no—
364C2×C4—26no—
365C2×C4—26no—
366C2×C4—26no—
367C2×C4—26no—
368C2×C4—26no—
369C2×C4—26no—
370C2×C4—26no—
371C2×C4—26no—
372C2×C4—26no—
373C2×C4—26no—
374C2×C4—26no—
375C2×C4—212no—
376C2×C4—26no—
377C2×C4—26no—
378C2×C4—41no—
379C2×C4—43no—
380C2×C4—43no—
381C2×C4—46no—
382C2×C4—43no—
383C2×C4—46no—
384C2×C4—46no—
385C2×C4—46no—
386C2×C4—46no—
387C2×C4—46no—
388C2×C4—46no—
389C2×C4—46no—
390C2×C4—46no—
391C2×C4—43no—
392C2×C4—43no—
393C2×C4—46no—
394C2×C4—46no—
395C2×C4—46no—
396C2×C4—46no—
397C2×C4—41no—
398C2×C4—42no—
399C2×C4—43no—
400C2×C4—46no—
401C2×C4—46no—
402C2×C4—46no—
403C2×C4—43no—
404C2×C4—43no—
405C2×C4—46no—
406C2×C4—46no—
407C2×C4—46no—
408C2×C4—824no—
409C2×C4—88no—
410C2×C4C2×#121112yes—
411C2×C4#537112yes—
412C22—212no—
413C22—212no—
414C22—24no—
415C22—212no—
416C22—212no—
417C22—24no—
418C22—82no—
419C22—86no—
420C22—86no—
421C22—82no—
422C22—86no—
423C22—86no—
424C22—82no—
425C22#282318yescomplemented
426C22C22×#12111yescharacteristic
427C22C2×#53713yes—
428C22C2×#53713yes—
429C22#348413yes—
430C22#348613yes—
431C22#349213yes—
432C22#349213yes—
433C22C2×#56011yescharacteristic
434C22C2×#56313yes—
435C22#354313yes—
436C22#354413yes—
437C22#355216yes—
438C4—212no—
439C4—212no—
440C4—46no—
441C4—42no—
442C4—82no—
443C4—86no—
444C4—86no—
445C4—81no—
446C4—83no—
447C2—28no—
448C2—162no—
449C2#3522313yes—
450C2C2×#282311yescharacteristic
451C2#3762013yes—
452C2#4632411yescharacteristic
453C2#4632513yes—
454C2#4632613yes—
455C2#4632711yescharacteristic
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC22×#121C22
LCSG ▷ C24 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C24 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#2823C2semidirect
2. C2×#2852C2semidirect
3. C22×#384C2semidirect
4. #16563C2semidirect
5. #16515C2semidirect
6. #16514C2semidirect
7. #14821C2semidirect
8. #14729C2semidirect
10. #14676C2semidirect
11. #14665C2semidirect
88. C22×C42D4semidirect
105. #384D4semidirect
328. C42#121semidirect
425. C22#2823semidirect

Maximal subgroups

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

subgroupindex
C2×#28232
C2×#28522
C22×#3842
#165632
#165152
#165142
#148212
#147292
#147192
#146762
#146652