Finite Group Databasebeta

← #515147

Subgroups — #515147

Order 512.

2630 conjugacy classes of subgroups.

The subgroup lattice has 429 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2399C211yesmaximal, characteristic
2C2×#2295C211yesmaximal, characteristic
3C2×C42×Q8C211yesmaximal, characteristic
4#14320C214yesmaximal
5#14314C214yesmaximal
6#14308C214yesmaximal
7#14302C214yesmaximal
8#13854C211yesmaximal, characteristic
9#13850C211yesmaximal, characteristic
10#13850C211yesmaximal, characteristic
11#13846C214yesmaximal
12C4×#387C214yesmaximal
13#13757C211yesmaximal, characteristic
14C22×C4×Q8C2212yes—
15C22×C4×Q8C2211yescharacteristic
16C22×C4×Q8C2211yescharacteristic
17#2399C4116yescomplemented
18C2×#398C2212yes—
19C2×#398C2212yes—
20C2×#395C2212yes—
21C2×#395C2212yes—
22#2295—28no—
23C2×#391C2212yes—
24C2×#389C2212yes—
25C2×#387C2218yes—
26C2×#387C2214yes—
27C42×Q8—28no—
28C2×C4×C4⋊C4C2212yes—
29C2×C4×C4⋊C4C2211yescharacteristic
30C2×C4×C4⋊C4C2211yescharacteristic
31C2×C4×C4⋊C4C2212yes—
32C2×C4×C4⋊C4C2212yes—
33C2×C4×C4⋊C4C2214yes—
34C2×C4×C4⋊C4C2214yes—
35C2×#376C2212yes—
36C2×#376C2211yescharacteristic
37C2×#376C2211yescharacteristic
38C2×C43C2212yes—
39C2×C43C2211yescharacteristic
40#1446C2212yes—
41#1446C2212yes—
42#1445C2218yes—
43#1445C2212yes—
44#1445C2212yes—
45#1443C2218yes—
46#1442C2218yes—
47#1442C2214yes—
48#1442C2214yes—
49#1441C2218yes—
50#1441C2214yes—
51#1441C2214yes—
52#1436C2218yes—
53#1436C2218yes—
54#1434C2214yes—
55#1434C2214yes—
56#1434C2218yes—
57C4×#95C2212yes—
58C4×#95C2214yes—
59C4×#95C2214yes—
60C4×#95C2214yes—
61C4×#95C2212yes—
62C23×Q8C2311yescharacteristic
63C23×Q8C2311yescharacteristic
64C2×C4×Q8—28no—
65C2×C4×Q8—24no—
66C2×C4×Q8—24no—
67C2×C4×Q8—216no—
68C2×C4×Q8—24no—
69C2×C4×Q8—24no—
70C2×C4×Q8C2×C418yes—
71C2×C4×Q8C2×C418yes—
72C22×C4⋊C4C2314yes—
73C22×C4⋊C4C2312yes—
74C22×C4⋊C4C2314yes—
75C22×C4⋊C4C2312yes—
76C22×C4⋊C4C2312yes—
77C22×C4⋊C4C2314yes—
78C22×C4⋊C4C2311yescharacteristic
79C22×C4⋊C4C2311yescharacteristic
80C22×C4⋊C4C2312yes—
81C22×C4⋊C4C2314yes—
82C22×C42C2312yes—
83C22×C42C2312yes—
84C22×C42C2312yes—
85C22×C42C2312yes—
86C22×C42C2312yes—
87C22×C42C2312yes—
88C22×C42C2312yes—
89C22×C42C2311yescharacteristic
90C22×C42C2311yescharacteristic
91C22×C42C2314yes—
92C22×C42C2314yes—
93C22×C42C2312yes—
94C22×C42C2311yescharacteristic
95#398C2×C4116yes—
96#398C2×C4116yes—
97#395C2×C4116yes—
98#395C2×C4116yes—
99#391—28no—
100#389—28no—
101#387—216no—
102#387—232no—
103C4×C4⋊C4—28no—
104C4×C4⋊C4—216no—
105C4×C4⋊C4—28no—
106C4×C4⋊C4—24no—
107C4×C4⋊C4—24no—
108C4×C4⋊C4—28no—
109C4×C4⋊C4C2×C4132yes—
110#376—28no—
111#376—24no—
112#376C2×C418yes—
113C2×#95C2318yes—
114C2×#95C2318yes—
115C2×#95C2318yes—
116C2×#95C2318yes—
117C2×#95C2314yes—
118C2×#95C2312yes—
119C2×#95C2312yes—
120C2×#95C2314yes—
121C2×#95C2314yes—
122C2×#95C2312yes—
123C2×#95C2312yes—
124C2×#95C2314yes—
125C2×#95C2318yes—
126C2×#95C2318yes—
127C2×#95C2318yes—
128C2×#95C2318yes—
129C2×#95C2314yes—
130C2×#95C2312yes—
131C2×#95C2312yes—
132C2×#95C2314yes—
133C43—28no—
134C43D418yes—
135C22×Q8—22no—
136C22×Q8—22no—
137C22×Q8—22no—
138C22×Q8—28no—
139C22×Q8—22no—
140C22×Q8—44no—
141C22×Q8C22×C414yes—
142C22×Q8C22×C414yes—
143C23×C4C2414yes—
144C23×C4C2414yes—
145C23×C4C2412yes—
146C23×C4C2412yes—
147C23×C4C2412yes—
148C23×C4C2414yes—
149C23×C4C2412yes—
150C23×C4C2414yes—
151C23×C4C2411yescharacteristic
152C23×C4C2411yescharacteristic
153C23×C4C2412yes—
154C23×C4C2412yes—
155C23×C4C2411yescharacteristic
156C4×Q8—264no—
157C4×Q8—232no—
158C4×Q8—232no—
159C2×C4⋊C4—28no—
160C2×C4⋊C4—24no—
161C2×C4⋊C4—24no—
162C2×C4⋊C4—22no—
163C2×C4⋊C4—24no—
164C2×C4⋊C4—22no—
165C2×C4⋊C4—28no—
166C2×C4⋊C4—28no—
167C2×C4⋊C4—28no—
168C2×C4⋊C4—24no—
169C2×C4⋊C4—22no—
170C2×C4⋊C4—22no—
171C2×C4⋊C4—24no—
172C2×C4⋊C4—216no—
173C2×C4⋊C4—28no—
174C2×C4⋊C4—24no—
175C2×C4⋊C4—216no—
176C2×C4⋊C4—48no—
177C2×C4⋊C4—44no—
178C2×C4⋊C4C22×C418yes—
179C2×C4⋊C4C22×C418yes—
180C2×C4⋊C4C22×C414yes—
181C2×C4⋊C4C22×C414yes—
182C2×C4⋊C4C22×C4116yes—
183C2×C4⋊C4C22×C4116yes—
184C2×C42—24no—
185C2×C42—28no—
186C2×C42—24no—
187C2×C42—24no—
188C2×C42—24no—
189C2×C42—28no—
190C2×C42—28no—
191C2×C42—24no—
192C2×C42—24no—
193C2×C42—24no—
194C2×C42—24no—
195C2×C42—28no—
196C2×C42—24no—
197C2×C42—24no—
198C2×C42—24no—
199C2×C42—28no—
200C2×C42—28no—
201C2×C42—24no—
202C2×C42—24no—
203C2×C42—44no—
204C2×C42—48no—
205C2×C42C22×C418yes—
206C2×C42C22×C418yes—
207C2×C42C22×C418yes—
208C2×C42C22×C414yes—
209C2×C42C2×D418yes—
210C2×C42C2×D414yes—
211C2×C42C2×D414yes—
212C2×C42C2×D418yes—
213C2×C42C2×D414yes—
214C2×C42#41116yes—
215C2×C42#4114yes—
216#95—28no—
217#95—28no—
218#95—28no—
219#95—24no—
220#95—24no—
221#95—28no—
222#95—24no—
223#95—24no—
224#95—216no—
225#95—216no—
226#95—216no—
227#95—216no—
228#95—48no—
229#95—48no—
230#95—48no—
231#95—48no—
232#95C22×C418yes—
233#95C22×C418yes—
234#95C22×C4116yes—
235#95C22×C4116yes—
236C24C2511yescharacteristic
237C2×Q8—216no—
238C2×Q8—216no—
239C2×Q8—28no—
240C2×Q8—28no—
241C2×Q8—216no—
242C2×Q8—48no—
243C2×Q8—48no—
244C2×Q8—48no—
245C22×C4—22no—
246C22×C4—22no—
247C22×C4—24no—
248C22×C4—24no—
249C22×C4—24no—
250C22×C4—24no—
251C22×C4—24no—
252C22×C4—22no—
253C22×C4—22no—
254C22×C4—24no—
255C22×C4—22no—
256C22×C4—28no—
257C22×C4—28no—
258C22×C4—22no—
259C22×C4—22no—
260C22×C4—24no—
261C22×C4—24no—
262C22×C4—22no—
263C22×C4—24no—
264C22×C4—24no—
265C22×C4—24no—
266C22×C4—28no—
267C22×C4—24no—
268C22×C4—28no—
269C22×C4—28no—
270C22×C4—24no—
271C22×C4—28no—
272C22×C4—24no—
273C22×C4—24no—
274C22×C4—28no—
275C22×C4—24no—
276C22×C4—28no—
277C22×C4—28no—
278C22×C4—24no—
279C22×C4—24no—
280C22×C4—28no—
281C22×C4—24no—
282C22×C4—24no—
283C22×C4C23×C418yes—
284C22×C4C23×C414yes—
285C22×C4C23×C414yes—
286C22×C4C23×C414yes—
287C22×C4C23×C412yes—
288C22×C4C23×C412yes—
289C22×C4C23×C414yes—
290C22×C4C23×C412yes—
291C22×C4C22×D414yes—
292C22×C4C22×D412yes—
293C22×C4C22×D412yes—
294C22×C4C22×D414yes—
295C22×C4C22×D412yes—
296C22×C4C2×#4118yes—
297C22×C4C2×#4112yes—
298C22×C4C2×#4112yes—
299C22×C4C2×#4118yes—
300C22×C4C2×#4118yes—
301C22×C4C2×#4112yes—
302C22×C42−1+414yes—
303C22×C42−1+414yes—
304C4⋊C4—216no—
305C4⋊C4—232no—
306C4⋊C4—216no—
307C4⋊C4—28no—
308C4⋊C4—28no—
309C4⋊C4—216no—
310C4⋊C4—232no—
311C4⋊C4—416no—
312C4⋊C4—48no—
313C4⋊C4—416no—
314C42—216no—
315C42—216no—
316C42—216no—
317C42—216no—
318C42—232no—
319C42—28no—
320C42—28no—
321C42—48no—
322C42—416no—
323C42—48no—
324C42—48no—
325C42—48no—
326C42C4×D4116yes—
327C23C24×C411yescharacteristic
328C23C23×D411yescharacteristic
329C23C22×#4111yescharacteristic
330C23C22×#4114yes—
331C23C2×2−1+412yes—
332C23C2×2−1+412yes—
333C23#58514yes—
334Q8—232no—
335Q8—416no—
336C2×C4—24no—
337C2×C4—24no—
338C2×C4—24no—
339C2×C4—24no—
340C2×C4—24no—
341C2×C4—28no—
342C2×C4—24no—
343C2×C4—28no—
344C2×C4—24no—
345C2×C4—24no—
346C2×C4—24no—
347C2×C4—24no—
348C2×C4—216no—
349C2×C4—24no—
350C2×C4—24no—
351C2×C4—24no—
352C2×C4—28no—
353C2×C4—28no—
354C2×C4—28no—
355C2×C4—24no—
356C2×C4—28no—
357C2×C4—28no—
358C2×C4—24no—
359C2×C4—24no—
360C2×C4—28no—
361C2×C4—24no—
362C2×C4—24no—
363C2×C4—216no—
364C2×C4—44no—
365C2×C4—44no—
366C2×C4—44no—
367C2×C4—44no—
368C2×C4—44no—
369C2×C4—44no—
370C2×C4—44no—
371C2×C4—44no—
372C2×C4—48no—
373C2×C4—48no—
374C2×C4—48no—
375C2×C4—48no—
376C2×C4—44no—
377C2×C4—44no—
378C2×C4—44no—
379C2×C4—44no—
380C2×C4—44no—
381C2×C4—44no—
382C2×C4—48no—
383C2×C4—48no—
384C2×C4—48no—
385C2×C4—48no—
386C2×C4C2×C4×D414yes—
387C2×C4C2×C4×D414yes—
388C2×C4C2×C4×D414yes—
389C2×C4C4×#4118yes—
390C2×C4#51918yes—
391C2×C4#52518yes—
392C2×C4#53614yes—
393C2×C4#53614yes—
394C2×C4#54118yes—
395C22C22×C4×D411yescharacteristic
396C22C2×C4×#4112yes—
397C22C2×#51912yes—
398C22#342812yes—
399C22C2×#52512yes—
400C22#344412yes—
401C22C2×#53611yescharacteristic
402C22C2×#53611yescharacteristic
403C22#344912yes—
404C22C2×#54112yes—
405C22#345512yes—
406C22#346118yes—
407C22#346118yes—
408C4—24no—
409C4—28no—
410C4—24no—
411C4—28no—
412C4—44no—
413C4—44no—
414C4—48no—
415C4—44no—
416C4—48no—
417C4—44no—
418C4—84no—
419C4—84no—
420C4#232718yes—
421C2C4×#52511yescharacteristic
422C2#3406511yescharacteristic
423C2C2×#232711yescharacteristic
424C2#3408514yes—
425C2#3460212yes—
426C2#3460414yes—
427C2#3462712yes—
subgroupG/H
centerC23×C4C24
commutatorC23C24×C4
Frattini, socle, prime radical, cosocleC24C25
LCSG ▷ C23 ▷ C1
UCSC1 ◁ C23×C4 ◁ G
LECSG ▷ C24 ▷ C1

Complements

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

NH ≅ G/Ntype
17. #2399C4semidirect

Maximal subgroups

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

subgroupindex
C2×#23992
C2×#22952
C2×C42×Q82
#143202
#143142
#143082
#143022
#138542
#138502
#138462
C4×#3872
#137572