Finite Group Databasebeta

← #382898

Subgroups — #382898

Order 512.

1074 conjugacy classes of subgroups.

The subgroup lattice has 441 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2330C211yesmaximal, characteristic
2C2×#1840C211yesmaximal, characteristic
3#14348C211yesmaximal, characteristic
4#11385C211yesmaximal, characteristic
5#11385C211yesmaximal, characteristic
6#11373C211yesmaximal, characteristic
7#11371C211yesmaximal, characteristic
8#11208C212yesmaximal
9#11207C212yesmaximal
10#10807C212yesmaximal
11#10806C212yesmaximal
12C22×#125C2211yescharacteristic
13#2330—24no—
14#2330C418yescomplemented
15C2×#389—22no—
16C2×#389C2212yes—
17C2×#387—22no—
18C2×#384C2211yescharacteristic
19C2×#384C2211yescharacteristic
20C2×#382C2211yescharacteristic
21C2×C4×C4⋊C4C2212yes—
22C2×#376C2211yescharacteristic
23#1840—24no—
24C2×#340C2212yes—
25C2×#340C2212yes—
26C2×#340C2212yes—
27C2×#340C2212yes—
28C2×#340C2212yes—
29C2×#340C2212yes—
30C2×#340C2212yes—
31C2×#340C2212yes—
32C2×#336C2212yes—
33C2×#336C2211yescharacteristic
34C2×#336C2211yescharacteristic
35C2×#336C2212yes—
36#1442—22no—
37#1442—22no—
38C4⋊#95C2211yescharacteristic
39C4⋊#95C2211yescharacteristic
40C4⋊#95C2211yescharacteristic
41C4⋊#95C2211yescharacteristic
42C23×C8C2312yes—
43C23×C8C2312yes—
44C2×#125—24no—
45C2×#125—24no—
46C2×#125—22no—
47C2×#125C2×C414yes—
48C22×C4⋊C4—22no—
49C22×C4⋊C4—22no—
50C22×C4⋊C4—22no—
51C22×C4⋊C4—21no—
52C22×C4⋊C4—21no—
53C22×C4⋊C4C2311yescharacteristic
54C22×C4⋊C4C2311yescharacteristic
55C22×C4⋊C4C2311yescharacteristic
56C22×C4⋊C4C2311yescharacteristic
57C22×C4⋊C4C2312yes—
58C22×C4⋊C4C2312yes—
59C22×C42—22no—
60C22×C42C2311yescharacteristic
61C22×C42C2311yescharacteristic
62C22×C42C2311yescharacteristic
63#389—24no—
64#389—216no—
65#389C2×C418yes—
66#387—216no—
67#384—22no—
68#384—42no—
69#384C2×C414yes—
70#382—42no—
71C4×C4⋊C4—24no—
72C4×C4⋊C4C2×C418yes—
73#376C2×C414yes—
74#376D414yes—
75C2×#95—22no—
76C2×#95—22no—
77C2×#95—21no—
78C2×#95—21no—
79C2×#95—22no—
80C2×#95—22no—
81C2×#95—22no—
82C2×#95—22no—
83C2×#95D412yes—
84C2×#95D412yes—
85#340—24no—
86#340—24no—
87#340—24no—
88#340—24no—
89#340—42no—
90#340—42no—
91#340—42no—
92#340—42no—
93#336—22no—
94#336—22no—
95#336—24no—
96#336—24no—
97C23×C4—22no—
98C23×C4—22no—
99C23×C4—21no—
100C23×C4—21no—
101C23×C4—21no—
102C23×C4—21no—
103C23×C4C2×D411yescharacteristic
104C23×C4C2×D412yes—
105C23×C4C2×D411yescharacteristic
106C23×C4C2×D412yes—
107C23×C4C2411yescharacteristic
108C22×C8—22no—
109C22×C8—22no—
110C22×C8—22no—
111C22×C8—22no—
112C22×C8—22no—
113C22×C8—22no—
114C22×C8—22no—
115C22×C8—22no—
116C22×C8—22no—
117C22×C8—22no—
118C22×C8—22no—
119C22×C8—22no—
120C22×C8—22no—
121C22×C8—22no—
122#125—216no—
123#125—216no—
124C2×C4⋊C4—28no—
125C2×C4⋊C4—28no—
126C2×C4⋊C4—28no—
127C2×C4⋊C4—22no—
128C2×C4⋊C4—22no—
129C2×C4⋊C4—22no—
130C2×C4⋊C4—22no—
131C2×C4⋊C4—22no—
132C2×C4⋊C4—22no—
133C2×C4⋊C4—22no—
134C2×C4⋊C4—22no—
135C2×C4⋊C4—22no—
136C2×C4⋊C4—22no—
137C2×C4⋊C4—22no—
138C2×C4⋊C4—22no—
139C2×C4⋊C4—22no—
140C2×C4⋊C4—22no—
141C2×C4⋊C4—22no—
142C2×C4⋊C4—22no—
143C2×C4⋊C4—21no—
144C2×C4⋊C4—21no—
145C2×C4⋊C4—21no—
146C2×C4⋊C4—21no—
147C2×C4⋊C4—21no—
148C2×C4⋊C4—21no—
149C2×C4⋊C4—21no—
150C2×C4⋊C4—21no—
151C2×C4⋊C4—28no—
152C2×C4⋊C4—28no—
153C2×C4⋊C4—42no—
154C2×C4⋊C4—42no—
155C2×C4⋊C4—82no—
156C2×C4⋊C4—82no—
157C2×C4⋊C4—82no—
158C2×C4⋊C4—82no—
159C2×C4⋊C4—82no—
160C2×C4⋊C4—82no—
161C2×C4⋊C4C22×C414yes—
162C2×C4⋊C4C22×C414yes—
163C2×C4⋊C4#4114yes—
164C2×C4⋊C4#4114yes—
165C2×C42—22no—
166C2×C42—22no—
167C2×C42—22no—
168C2×C42—21no—
169C2×C42—21no—
170C2×C42—21no—
171C2×C42—21no—
172C2×C42—22no—
173C2×C42—28no—
174C2×C42—82no—
175C2×C42—82no—
176C2×C42C22×C412yes—
177C2×C42C22×C412yes—
178C2×C42C22×C412yes—
179C2×C42C2×D412yes—
180C2×C42C2×D412yes—
181C2×C42C2×D412yes—
182#95—28no—
183#95—28no—
184#95—42no—
185#95—42no—
186#95—82no—
187#95—82no—
188#95—82no—
189#95—82no—
190#95C22⋊C418yes—
191#95C22⋊C418yes—
192C24C22×D411yescharacteristic
193C22×C4—24no—
194C22×C4—22no—
195C22×C4—22no—
196C22×C4—24no—
197C22×C4—24no—
198C22×C4—21no—
199C22×C4—21no—
200C22×C4—22no—
201C22×C4—22no—
202C22×C4—21no—
203C22×C4—21no—
204C22×C4—22no—
205C22×C4—22no—
206C22×C4—22no—
207C22×C4—22no—
208C22×C4—22no—
209C22×C4—22no—
210C22×C4—22no—
211C22×C4—22no—
212C22×C4—22no—
213C22×C4—22no—
214C22×C4—22no—
215C22×C4—22no—
216C22×C4—24no—
217C22×C4—42no—
218C22×C4—42no—
219C22×C4—42no—
220C22×C4—42no—
221C22×C4—42no—
222C22×C4—42no—
223C22×C4—42no—
224C22×C4—42no—
225C22×C4—42no—
226C22×C4—42no—
227C22×C4—41no—
228C22×C4—41no—
229C22×C4—41no—
230C22×C4—41no—
231C22×C4—41no—
232C22×C4—41no—
233C22×C4—41no—
234C22×C4—41no—
235C22×C4—41no—
236C22×C4—41no—
237C22×C4—41no—
238C22×C4—41no—
239C22×C4—42no—
240C22×C4—42no—
241C22×C4—42no—
242C22×C4—42no—
243C22×C4—42no—
244C22×C4—42no—
245C22×C4—42no—
246C22×C4—42no—
247C22×C4C2×C22⋊C412yes—
248C22×C4C2×C22⋊C414yes—
249C22×C4C2×C22⋊C412yes—
250C22×C4C2×C22⋊C414yes—
251C22×C4C23×C411yescharacteristic
252C22×C4C22×D411yescharacteristic
253C22×C4C2×#4112yes—
254C22×C4C2×#4112yes—
255C22×C4H(5,𝔽2)12yes—
256C22×C42−1+411yescharacteristic
257C22×C42−1+411yescharacteristic
258C22×C42−1+411yescharacteristic
259C22×C42−1+411yescharacteristic
260C22×C42−1+412yes—
261C2×C8—42no—
262C2×C8—42no—
263C2×C8—42no—
264C2×C8—42no—
265C2×C8—42no—
266C2×C8—42no—
267C2×C8—42no—
268C2×C8—42no—
269C2×C8—42no—
270C2×C8—42no—
271C2×C8—42no—
272C2×C8—42no—
273C2×C8—42no—
274C2×C8—42no—
275C4⋊C4—232no—
276C4⋊C4—24no—
277C4⋊C4—24no—
278C4⋊C4—24no—
279C4⋊C4—24no—
280C4⋊C4—24no—
281C4⋊C4—42no—
282C4⋊C4—42no—
283C4⋊C4—42no—
284C4⋊C4—42no—
285C4⋊C4—42no—
286C4⋊C4—42no—
287C4⋊C4—42no—
288C4⋊C4—42no—
289C4⋊C4—42no—
290C4⋊C4—42no—
291C4⋊C4—88no—
292C4⋊C4—88no—
293C4⋊C4—84no—
294C4⋊C4—84no—
295C4⋊C4#11718yes—
296C42—24no—
297C42—22no—
298C42—22no—
299C42—24no—
300C42—24no—
301C42—22no—
302C42—22no—
303C42—24no—
304C42—88no—
305C23C22×C22⋊C411yescharacteristic
306C23#52511yescharacteristic
307C23#52511yescharacteristic
308C23#53511yescharacteristic
309C23#53611yescharacteristic
310C23#53611yescharacteristic
311C23#53611yescharacteristic
312C23#57612yes—
313C23#57712yes—
314C23#57712yes—
315C23#57812yes—
316C2×C4—24no—
317C2×C4—22no—
318C2×C4—22no—
319C2×C4—24no—
320C2×C4—24no—
321C2×C4—24no—
322C2×C4—24no—
323C2×C4—24no—
324C2×C4—24no—
325C2×C4—24no—
326C2×C4—28no—
327C2×C4—42no—
328C2×C4—44no—
329C2×C4—42no—
330C2×C4—44no—
331C2×C4—42no—
332C2×C4—44no—
333C2×C4—42no—
334C2×C4—44no—
335C2×C4—44no—
336C2×C4—44no—
337C2×C4—41no—
338C2×C4—41no—
339C2×C4—41no—
340C2×C4—41no—
341C2×C4—42no—
342C2×C4—42no—
343C2×C4—42no—
344C2×C4—42no—
345C2×C4—42no—
346C2×C4—42no—
347C2×C4—44no—
348C2×C4—44no—
349C2×C4—44no—
350C2×C4—82no—
351C2×C4—82no—
352C2×C4—82no—
353C2×C4—82no—
354C2×C4—82no—
355C2×C4—82no—
356C2×C4—82no—
357C2×C4—82no—
358C2×C4—81no—
359C2×C4—81no—
360C2×C4—81no—
361C2×C4—81no—
362C2×C4—81no—
363C2×C4—81no—
364C2×C4—81no—
365C2×C4—81no—
366C2×C4—81no—
367C2×C4—81no—
368C2×C4—81no—
369C2×C4—81no—
370C2×C4—82no—
371C2×C4—82no—
372C2×C4—82no—
373C2×C4—82no—
374C2×C4C2×#11712yes—
375C2×C4#51911yescharacteristic
376C2×C4#51911yescharacteristic
377C2×C4#52012yes—
378C2×C4#52512yes—
379C2×C4#53611yescharacteristic
380C2×C4#53611yescharacteristic
381C2×C4#53712yes—
382C2×C4#54112yes—
383C2×C4#54112yes—
384C2×C4#54112yes—
385C2×C4#54112yes—
386C2×C4#54212yes—
387C2×C4#54212yes—
388C2×C4#54212yes—
389C2×C4#54212yes—
390C8—82no—
391C8—82no—
392C22#228011yescharacteristic
393C22#231011yescharacteristic
394C22#231111yescharacteristic
395C22#263211yescharacteristic
396C22#263211yescharacteristic
397C22#263411yescharacteristic
398C22#263411yescharacteristic
399C22#289712yes—
400C22#289812yes—
401C22#311912yes—
402C22#312012yes—
403C22#312412yes—
404C22#312512yes—
405C22#322011yescharacteristic
406C22#322111yescharacteristic
407C22#322211yescharacteristic
408C22#322311yescharacteristic
409C22#324112yes—
410C22#324112yes—
411C22#324212yes—
412C22#324212yes—
413C22#324312yes—
414C22#324412yes—
415C4—44no—
416C4—48no—
417C4—44no—
418C4—42no—
419C4—42no—
420C4—84no—
421C4—161no—
422C4—161no—
423C4—161no—
424C4—161no—
425C4#232012yes—
426C4#232012yes—
427C4#263712yes—
428C4#263712yes—
429C2#1434611yescharacteristic
430C2#2074612yes—
431C2#2184312yes—
432C2#2187211yescharacteristic
433C2#2187311yescharacteristic
434C2#3007111yescharacteristic
435C2#3007211yescharacteristic
436C2#3007311yescharacteristic
437C2#3007411yescharacteristic
438C2#3010312yes—
439C2#3010412yes—
subgroupG/H
center, socle, prime radicalC24C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C2×#376 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
14. #2330C4semidirect

Maximal subgroups

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

subgroupindex
C2×#23302
C2×#18402
#143482
#113852
#113732
#113712
#112082
#112072
#108072
#108062