Finite Group Databasebeta

← #391927

Subgroups — #391927

Order 512.

958 conjugacy classes of subgroups.

The subgroup lattice has 437 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2647C211yesmaximal, characteristic
2C2×#1832C211yesmaximal, characteristic
3#14348C211yesmaximal, characteristic
4#11373C211yesmaximal, characteristic
5#11371C211yesmaximal, characteristic
6#11358C211yesmaximal, characteristic
7#11358C211yesmaximal, characteristic
8#11228C212yesmaximal
9#11227C212yesmaximal
10#11208C212yesmaximal
11#11207C212yesmaximal
12C22×C4⋊C8C2212yes—
13#2647—24no—
14#2647C418yescomplemented
15C2×#400—22no—
16C2×#398—22no—
17C2×#389C2212yes—
18C2×#389C2212yes—
19C2×#384C2211yescharacteristic
20C2×#384C2211yescharacteristic
21C2×#382C2211yescharacteristic
22C2×#382C2211yescharacteristic
23C2×#376C2211yescharacteristic
24#1832—24no—
25C2×#340C2212yes—
26C2×#340C2212yes—
27C2×#340C2212yes—
28C2×#340C2212yes—
29C2×#340C2212yes—
30C2×#340C2212yes—
31C2×#340C2212yes—
32C2×#340C2212yes—
33C2×#336C2211yescharacteristic
34C2×#336C2211yescharacteristic
35C2×#336C2212yes—
36#1442—22no—
37#1442—22no—
38C4⋊#95C2211yescharacteristic
39C4⋊#95C2211yescharacteristic
40C4⋊#95C2211yescharacteristic
41C4⋊#95C2211yescharacteristic
42C23×C8C2312yes—
43C23×C8C2312yes—
44C22×C4⋊C4—22no—
45C22×C4⋊C4—22no—
46C22×C4⋊C4—22no—
47C22×C4⋊C4—21no—
48C22×C4⋊C4—21no—
49C22×C4⋊C4C2311yescharacteristic
50C22×C4⋊C4C2311yescharacteristic
51C22×C4⋊C4C2311yescharacteristic
52C22×C4⋊C4C2311yescharacteristic
53C22×C4⋊C4C2312yes—
54C22×C4⋊C4C2312yes—
55C22×C42C2311yescharacteristic
56C22×C42C2311yescharacteristic
57C22×C42C2311yescharacteristic
58C2×C4⋊C8—24no—
59C2×C4⋊C8—42no—
60C2×C4⋊C8—42no—
61#400—216no—
62#398—216no—
63#389—24no—
64#389—24no—
65#389C2×C418yes—
66#389C2×C418yes—
67#384—22no—
68#384—42no—
69#384C2×C414yes—
70#382—22no—
71#382—42no—
72#382C2×C414yes—
73#376C2×C414yes—
74#376D414yes—
75C2×#95—22no—
76C2×#95—22no—
77C2×#95—22no—
78C2×#95—21no—
79C2×#95—21no—
80C2×#95—22no—
81C2×#95—22no—
82C2×#95—22no—
83C2×#95—22no—
84C2×#95D412yes—
85C2×#95D412yes—
86#340—42no—
87#340—42no—
88#340—42no—
89#340—42no—
90#340—42no—
91#340—42no—
92#340—42no—
93#340—42no—
94#336—22no—
95#336—22no—
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×D411yescharacteristic
105C23×C4C2×D412yes—
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—
122C2×C4⋊C4—22no—
123C2×C4⋊C4—22no—
124C2×C4⋊C4—22no—
125C2×C4⋊C4—22no—
126C2×C4⋊C4—22no—
127C2×C4⋊C4—22no—
128C2×C4⋊C4—22no—
129C2×C4⋊C4—22no—
130C2×C4⋊C4—22no—
131C2×C4⋊C4—22no—
132C2×C4⋊C4—21no—
133C2×C4⋊C4—21no—
134C2×C4⋊C4—21no—
135C2×C4⋊C4—21no—
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—22no—
144C2×C4⋊C4—21no—
145C2×C4⋊C4—21no—
146C2×C4⋊C4—21no—
147C2×C4⋊C4—21no—
148C2×C4⋊C4—28no—
149C2×C4⋊C4—28no—
150C2×C4⋊C4—28no—
151C2×C4⋊C4—42no—
152C2×C4⋊C4—42no—
153C2×C4⋊C4—82no—
154C2×C4⋊C4—82no—
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×C42—22no—
164C2×C42—21no—
165C2×C42—21no—
166C2×C42—22no—
167C2×C42—22no—
168C2×C42—21no—
169C2×C42—21no—
170C2×C42—22no—
171C2×C42C22×C412yes—
172C2×C42C22×C412yes—
173C2×C42C22×C412yes—
174C2×C42C2×D412yes—
175C2×C42C2×D412yes—
176C2×C42C2×D412yes—
177C4⋊C8—48no—
178#95—28no—
179#95—28no—
180#95—28no—
181#95—42no—
182#95—42no—
183#95—82no—
184#95—82no—
185#95—82no—
186#95—82no—
187#95C22⋊C418yes—
188#95C22⋊C418yes—
189C24C22×D411yescharacteristic
190C22×C4—21no—
191C22×C4—21no—
192C22×C4—22no—
193C22×C4—22no—
194C22×C4—21no—
195C22×C4—21no—
196C22×C4—22no—
197C22×C4—22no—
198C22×C4—22no—
199C22×C4—22no—
200C22×C4—22no—
201C22×C4—22no—
202C22×C4—22no—
203C22×C4—22no—
204C22×C4—22no—
205C22×C4—22no—
206C22×C4—22no—
207C22×C4—22no—
208C22×C4—22no—
209C22×C4—22no—
210C22×C4—24no—
211C22×C4—24no—
212C22×C4—42no—
213C22×C4—42no—
214C22×C4—42no—
215C22×C4—42no—
216C22×C4—42no—
217C22×C4—42no—
218C22×C4—42no—
219C22×C4—42no—
220C22×C4—42no—
221C22×C4—42no—
222C22×C4—42no—
223C22×C4—42no—
224C22×C4—41no—
225C22×C4—41no—
226C22×C4—41no—
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—42no—
237C22×C4—42no—
238C22×C4—42no—
239C22×C4—42no—
240C22×C4—42no—
241C22×C4—42no—
242C22×C4—42no—
243C22×C4—42no—
244C22×C4C2×C22⋊C412yes—
245C22×C4C2×C22⋊C414yes—
246C22×C4C2×C22⋊C414yes—
247C22×C4C2×C22⋊C412yes—
248C22×C4C23×C411yescharacteristic
249C22×C4C22×D411yescharacteristic
250C22×C4H(5,𝔽2)11yescharacteristic
251C22×C4H(5,𝔽2)11yescharacteristic
252C22×C4H(5,𝔽2)12yes—
253C22×C42−1+412yes—
254C22×C42−1+412yes—
255C22×C42−1+411yescharacteristic
256C22×C42−1+411yescharacteristic
257C22×C42−1+412yes—
258C2×C8—42no—
259C2×C8—42no—
260C2×C8—42no—
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—
272C4⋊C4—24no—
273C4⋊C4—24no—
274C4⋊C4—24no—
275C4⋊C4—24no—
276C4⋊C4—42no—
277C4⋊C4—42no—
278C4⋊C4—42no—
279C4⋊C4—42no—
280C4⋊C4—42no—
281C4⋊C4—42no—
282C4⋊C4—42no—
283C4⋊C4—42no—
284C4⋊C4—42no—
285C4⋊C4—42no—
286C4⋊C4—42no—
287C4⋊C4—42no—
288C4⋊C4—88no—
289C4⋊C4—88no—
290C4⋊C4—88no—
291C4⋊C4—84no—
292C4⋊C4—84no—
293C42—24no—
294C42—24no—
295C42—22no—
296C42—22no—
297C42—22no—
298C42—24no—
299C42—22no—
300C42—24no—
301C23C22×C22⋊C411yescharacteristic
302C23#53511yescharacteristic
303C23#53611yescharacteristic
304C23#53611yescharacteristic
305C23#53611yescharacteristic
306C23#53711yescharacteristic
307C23#53711yescharacteristic
308C23#57612yes—
309C23#57712yes—
310C23#57712yes—
311C23#57812yes—
312C2×C4—24no—
313C2×C4—22no—
314C2×C4—22no—
315C2×C4—24no—
316C2×C4—24no—
317C2×C4—24no—
318C2×C4—24no—
319C2×C4—24no—
320C2×C4—24no—
321C2×C4—24no—
322C2×C4—41no—
323C2×C4—41no—
324C2×C4—41no—
325C2×C4—41no—
326C2×C4—42no—
327C2×C4—42no—
328C2×C4—42no—
329C2×C4—42no—
330C2×C4—42no—
331C2×C4—42no—
332C2×C4—42no—
333C2×C4—42no—
334C2×C4—42no—
335C2×C4—42no—
336C2×C4—44no—
337C2×C4—44no—
338C2×C4—44no—
339C2×C4—44no—
340C2×C4—44no—
341C2×C4—44no—
342C2×C4—82no—
343C2×C4—82no—
344C2×C4—82no—
345C2×C4—82no—
346C2×C4—82no—
347C2×C4—82no—
348C2×C4—82no—
349C2×C4—82no—
350C2×C4—81no—
351C2×C4—81no—
352C2×C4—81no—
353C2×C4—81no—
354C2×C4—81no—
355C2×C4—81no—
356C2×C4—81no—
357C2×C4—81no—
358C2×C4—81no—
359C2×C4—81no—
360C2×C4—81no—
361C2×C4—81no—
362C2×C4—82no—
363C2×C4—82no—
364C2×C4—82no—
365C2×C4—82no—
366C2×C4—82no—
367C2×C4—82no—
368C2×C4—82no—
369C2×C4—82no—
370C2×C4#51912yes—
371C2×C4#52012yes—
372C2×C4#52011yescharacteristic
373C2×C4#52011yescharacteristic
374C2×C4#53511yescharacteristic
375C2×C4#53612yes—
376C2×C4#53611yescharacteristic
377C2×C4#53712yes—
378C2×C4#56212yes—
379C2×C4#56212yes—
380C2×C4#56312yes—
381C2×C4#56312yes—
382C2×C4#56312yes—
383C2×C4#56312yes—
384C2×C4#56412yes—
385C2×C4#56412yes—
386C8—82no—
387C8—82no—
388C22#231011yescharacteristic
389C22#231111yescharacteristic
390C22#231111yescharacteristic
391C22#282011yescharacteristic
392C22#282011yescharacteristic
393C22#282211yescharacteristic
394C22#282211yescharacteristic
395C22#289712yes—
396C22#289812yes—
397C22#322011yescharacteristic
398C22#322111yescharacteristic
399C22#322211yescharacteristic
400C22#322311yescharacteristic
401C22#324112yes—
402C22#324112yes—
403C22#324212yes—
404C22#324212yes—
405C22#324312yes—
406C22#324412yes—
407C22#325712yes—
408C22#325812yes—
409C22#326212yes—
410C22#326312yes—
411C4—44no—
412C4—42no—
413C4—42no—
414C4—44no—
415C4—84no—
416C4—84no—
417C4—161no—
418C4—161no—
419C4—161no—
420C4—161no—
421C4#238212yes—
422C4#238212yes—
423C4#282012yes—
424C4#282212yes—
425C2#1466411yescharacteristic
426C2#2184312yes—
427C2#2187211yescharacteristic
428C2#2187311yescharacteristic
429C2#2189612yes—
430C2#3282911yescharacteristic
431C2#3283011yescharacteristic
432C2#3283111yescharacteristic
433C2#3283211yescharacteristic
434C2#3285412yes—
435C2#3285512yes—
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. #2647C4semidirect

Maximal subgroups

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

subgroupindex
C2×#26472
C2×#18322
#143482
#113732
#113712
#113582
#112282
#112272
#112082
#112072