Finite Group Databasebeta

← #11805259

Subgroups — #11805259

Order 1152.

1305 conjugacy classes of subgroups.

The subgroup lattice has 390 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C4⋊C4×C3⋊D3C211yesmaximal, characteristic
2C3×#4984C212yesmaximal, complemented
3#10591325C211yesmaximal, characteristic
4#10591325C212yesmaximal
5C4×#3722C211yesmaximal, characteristic
6Dic3×C4⋊Dic3C212yesmaximal, complemented
7#10591263C214yesmaximal, complemented
8#10591260C212yesmaximal
9#76201—32nomaximal
10C22×C4×C3⋊D3C2211yescharacteristic
11C22×C4×C3⋊D3C2212yes—
12C2×C32×C4⋊C4C2211yescharacteristic
13C2×C4⋊(C3⋊Dic3)C2211yescharacteristic
14C2×#3750C2212yes—
15C4⋊C4×C3⋊D3C418yescomplemented
16C2×C3×C4⋊Dic3C2212yes—
17C2×C3×C4⋊Dic3C2214yes—
18C2×C3×C4×Dic3C2212yes—
19C2×#3722C2212yes—
20C2×#3722C2211yescharacteristic
21C2×#3722C2211yescharacteristic
22C2×Dic32C2214yes—
23C3×C3⋊#95C2214yes—
24#63579C2212yes—
25#63579C2212yes—
26#63579C2214yes—
27C2×D3×C4⋊C4—32no—
28C2×D3×C4⋊C4—61no—
29C3×#389—32no—
30#4987—34no—
31#4987—32no—
32#4984D312yescomplemented
33#4978—34no—
34Dic3×C4⋊C4—32no—
35C4×(C2×D3)⋊C4—32no—
36C4×C4⋊Dic3—32no—
37#4666—34no—
38#4655—34no—
39C23×C3⋊D3C2311yescharacteristic
40C22×C32×C4C2312yes—
41C22×C32×C4C2311yescharacteristic
42C22×C3⋊Dic3C2311yescharacteristic
43C22×C3⋊Dic3C2312yes—
44C2×C4×C3⋊D3—24no—
45C2×C4×C3⋊D3—44no—
46C2×C4×C3⋊D3C2×C414yes—
47C2×C4×C3⋊D3C2×C418yes—
48C22×C3×Dic3C2312yes—
49C22×C3×Dic3C2314yes—
50C22×C3×Dic3C2312yes—
51C32×C4⋊C4C2×C414yes—
52C4⋊(C3⋊Dic3)C2×C414yes—
53#3750C2×C418yes—
54C3×C4⋊Dic3—24no—
55C3×C4⋊Dic3—44no—
56C3×C4×Dic3—24no—
57#3722—24no—
58#3722—22no—
59#3722Q814yes—
60Dic32—44no—
61#2345—91noSylow
62C22×C4×D3—32no—
63C22×C4×D3—34no—
64C22×C4×D3—61no—
65C22×C4×D3—62no—
66C2×C3×C4⋊C4—21no—
67C2×C3×C4⋊C4—32no—
68C2×C3×C4⋊C4—34no—
69C2×C3×C4⋊C4C2×D312yes—
70C2×C3×C42—32no—
71C2×(C2×D3)⋊C4—32no—
72C2×(C2×D3)⋊C4—34no—
73C2×(C2×D3)⋊C4—32no—
74C2×C4⋊Dic3—32no—
75C2×C4⋊Dic3—61no—
76C2×C4⋊Dic3C2×D312yes—
77C2×C4⋊Dic3C2×D314yes—
78C2×Dic3⋊C4—34no—
79C2×Dic3⋊C4—62no—
80C2×C4×Dic3—34no—
81C2×C4×Dic3—34no—
82C2×C4×Dic3C2×D312yes—
83D3×C4⋊C4—316no—
84D3×C4⋊C4—68no—
85C3×#95—34no—
86C3⋊#95—32no—
87C3⋊#95—34no—
88C3⋊#95—34no—
89C3⋊#95—34no—
90C3⋊#95—32no—
91C3⋊#95C2×D314yes—
92C23×C32C2411yescharacteristic
93C22×C3⋊D3—22no—
94C22×C3⋊D3—22no—
95C22×C3⋊D3C22×C412yes—
96C22×C3⋊D3C2×Q812yes—
97C22×C3⋊D3#4112yes—
98C2×C32×C4—24no—
99C2×C32×C4C22×C414yes—
100C2×C32×C4C22×C412yes—
101C2×C32×C4#4114yes—
102C2×C3⋊Dic3—22no—
103C2×C3⋊Dic3—24no—
104C2×C3⋊Dic3C22×C412yes—
105C2×C3⋊Dic3C22×C414yes—
106C4×C3⋊D3—28no—
107C4×C3⋊D3—48no—
108C2×C3×Dic3—22no—
109C2×C3×Dic3—22no—
110C2×C3×Dic3—22no—
111C2×C3×Dic3—24no—
112C2×C3×Dic3—22no—
113C2×C3×Dic3—24no—
114C2×C3×Dic3—24no—
115C2×C3×Dic3—22no—
116C2×C3×Dic3C2×Q814yes—
117C22×C4⋊C4—91no—
118#389—32no—
119#387—94no—
120#380—92no—
121#380—91no—
122C4×C4⋊C4—92no—
123C4×C22⋊C4—92no—
124C4×C22⋊C4—91no—
125C23×D3—32no—
126C23×D3—61no—
127C22×C3×C4—21no—
128C22×C3×C4—22no—
129C22×C3×C4—32no—
130C22×C3×C4—32no—
131C22×C3×C4—34no—
132C22×C3×C4C22×D314yes—
133C22×C3×C4C22×D312yes—
134C22×Dic3—32no—
135C22×Dic3—34no—
136C22×Dic3—61no—
137C22×Dic3—62no—
138C22×Dic3C22×D312yes—
139C22×Dic3C22×D314yes—
140C22×Dic3C22×D312yes—
141C2×C4×D3—38no—
142C2×C4×D3—316no—
143C2×C4×D3—64no—
144C2×C4×D3—68no—
145C2×C4×D3—68no—
146C2×C4×D3—64no—
147C2×C4×D3—64no—
148C2×C4×D3—124no—
149C2×C4×D3—124no—
150C2×C4×D3—128no—
151C3×C4⋊C4—24no—
152C3×C4⋊C4—64no—
153C3×C4⋊C4—124no—
154C3×C4⋊C4C4×D318yes—
155C3×C42—64no—
156(C2×D3)⋊C4—38no—
157(C2×D3)⋊C4—68no—
158(C2×D3)⋊C4—64no—
159C4⋊Dic3—24no—
160C4⋊Dic3—38no—
161C4⋊Dic3—44no—
162C4⋊Dic3—64no—
163Dic3⋊C4—316no—
164Dic3⋊C4—68no—
165C4×Dic3—24no—
166C4×Dic3—124no—
167C4×Dic3—124no—
168C22×C32C23×C411yescharacteristic
169C22×C32C22×Q811yescharacteristic
170C22×C32C2×#4112yes—
171C22×C32C2×#4111yescharacteristic
172C22×C32H(5,𝔽2)12yes—
173C2×C3⋊D3—22no—
174C2×C3⋊D3—22no—
175C2×C3⋊D3—24no—
176C2×C3⋊D3—24no—
177C2×C3⋊D3C4×Q814yes—
178C32×C4—24no—
179C32×C4#11714yes—
180C3⋊Dic3—22no—
181C3⋊Dic3—24no—
182C3×Dic3—24no—
183C3×Dic3—42no—
184C3×Dic3—44no—
185C23×C4—91no—
186C23×C4—92no—
187C2×C4⋊C4—32no—
188C2×C4⋊C4—34no—
189C2×C4⋊C4—91no—
190C2×C4⋊C4—92no—
191C2×C4⋊C4—98no—
192C2×C4⋊C4D3211yescharacteristic
193C2×C22⋊C4—91no—
194C2×C22⋊C4—92no—
195C2×C22⋊C4—91no—
196C2×C42—32no—
197C2×C42—94no—
198#95—34no—
199#95—92no—
200#95—94no—
201#95—92no—
202C23×C3—21no—
203C23×C3C23×D312yes—
204C22×D3—34no—
205C22×D3—34no—
206C22×D3—34no—
207C22×D3—62no—
208C22×D3—62no—
209C22×D3—62no—
210C22×D3—64no—
211C22×D3—64no—
212C22×D3—62no—
213C22×D3—62no—
214C22×D3—62no—
215C22×D3—62no—
216C2×C3×C4—24no—
217C2×C3×C4—24no—
218C2×C3×C4—24no—
219C2×C3×C4—22no—
220C2×C3×C4—24no—
221C2×C3×C4—34no—
222C2×C3×C4—44no—
223C2×C3×C4—62no—
224C2×C3×C4—62no—
225C2×C3×C4—62no—
226C2×C3×C4—64no—
227C2×C3×C4—62no—
228C2×C3×C4—64no—
229C2×C3×C4—64no—
230C2×C3×C4—62no—
231C2×C3×C4C2×C4×D318yes—
232C2×C3×C4C2×C4×D314yes—
233C2×C3×C4#23714yes—
234C2×C3×C4#23914yes—
235C2×Dic3—22no—
236C2×Dic3—22no—
237C2×Dic3—22no—
238C2×Dic3—24no—
239C2×Dic3—24no—
240C2×Dic3—22no—
241C2×Dic3—22no—
242C2×Dic3—24no—
243C2×Dic3—34no—
244C2×Dic3—38no—
245C2×Dic3—62no—
246C2×Dic3—64no—
247C2×Dic3—64no—
248C2×Dic3—62no—
249C2×Dic3—64no—
250C2×Dic3—62no—
251C2×Dic3—64no—
252C2×Dic3—124no—
253C2×Dic3D3×Q814yes—
254C4×D3—616no—
255C4×D3—616no—
256C4×D3—1216no—
257C4×D3—1216no—
258C2×C32C2×#11711yescharacteristic
259C2×C32C2×C4×Q811yescharacteristic
260C2×C32#51811yescharacteristic
261C2×C32#55312yes—
262C2×C32#55412yes—
263C3⋊D3—24no—
264C24—91no—
265C22×C4—32no—
266C22×C4—34no—
267C22×C4—32no—
268C22×C4—91no—
269C22×C4—92no—
270C22×C4—94no—
271C22×C4—98no—
272C22×C4—184no—
273C22×C4—364no—
274C22×C4C2×D3212yes—
275C22×C4C2×D3211yescharacteristic
276C4⋊C4—64no—
277C4⋊C4—98no—
278C4⋊C4—94no—
279C4⋊C4—124no—
280C4⋊C4#62414yes—
281C22⋊C4—94no—
282C22⋊C4—184no—
283C22⋊C4—182no—
284C42—64no—
285C42—364no—
286C22×C3—22no—
287C22×C3—22no—
288C22×C3—21no—
289C22×C3—21no—
290C22×C3—21no—
291C22×C3C22×C4×D312yes—
292C22×C3C2×#23712yes—
293C22×C3#100412yes—
294C22×C3C2×D3×Q812yes—
295C22×C3C2×#23912yes—
296C22×C3D3×#4112yes—
297C22×C3#100912yes—
298C2×D3—38no—
299C2×D3—64no—
300C2×D3—64no—
301C2×D3—64no—
302C2×D3—68no—
303C2×D3—68no—
304C2×D3—64no—
305C2×D3—64no—
306C2×D3—64no—
307C2×D3—64no—
308C2×D3—64no—
309C2×D3—64no—
310C3×C4—28no—
311C3×C4—24no—
312C3×C4—44no—
313C3×C4—64no—
314C3×C4—122no—
315C3×C4—124no—
316C3×C4#89218yes—
317Dic3—24no—
318Dic3—42no—
319Dic3—44no—
320Dic3—64no—
321Dic3—64no—
322Dic3—68no—
323Dic3—124no—
324C32#234511yescharacteristic, complemented, Sylow
325C23—92no—
326C23—92no—
327C23—92no—
328C23—182no—
329C23—182no—
330C23C22×D3211yescharacteristic
331C2×C4—24no—
332C2×C4—34no—
333C2×C4—62no—
334C2×C4—62no—
335C2×C4—62no—
336C2×C4—64no—
337C2×C4—64no—
338C2×C4—62no—
339C2×C4—64no—
340C2×C4—62no—
341C2×C4—92no—
342C2×C4—94no—
343C2×C4—182no—
344C2×C4—184no—
345C2×C4—188no—
346C2×C4—368no—
347C2×C4#379614yes—
348C2×C4C2×#62414yes—
349C2×C4C2×#62412yes—
350C2×C3—21no—
351C2×C3—21no—
352C2×C3—22no—
353C2×C3—21no—
354C2×C3—22no—
355C2×C3C2×#89212yes—
356C2×C3#554512yes—
357C2×C3C4×D3×Q812yes—
358C2×C3#561412yes—
359C2×C3#563512yes—
360C2×C3#569512yes—
361C2×C3#573012yes—
362D3—68no—
363D3—68no—
364C22—94no—
365C22—182no—
366C22—184no—
367C22—182no—
368C22—184no—
369C22C2×#379612yes—
370C22#6431412yes—
371C22D32×Q811yescharacteristic
372C22#6431911yescharacteristic
373C22C22×#62411yescharacteristic
374C4—24no—
375C4—64no—
376C4—122no—
377C4—124no—
378C4—182no—
379C4—184no—
380C4#6384914yes—
381C3—21no—
382C3#7620112yescomplemented
383C2—184no—
384C2#1059439312yes—
385C2#1059448912yes—
386C2C2×#6384911yescharacteristic
387C2#1059522311yescharacteristic
388C2Q8×#62411yescharacteristic
subgroupG/H
center, FrattiniC23C22×D32
commutatorC22×C32C23×C4
FittingC2×C32×C4⋊C4C22
socle, cosocleC23×C32C24
prime radicalC23×C3⋊D3C23
hypercenterC2×C4⋊C4D32
hypocenterC32#2345
LCSG ▷ C22×C32 ▷ C32
UCSC1 ◁ C23 ◁ C2×C4⋊C4

Complements

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

NH ≅ G/Ntype
2. C3×#4984C2semidirect
6. Dic3×C4⋊Dic3C2semidirect
7. #10591263C2semidirect
15. C4⋊C4×C3⋊D3C4semidirect
32. #4984D3semidirect
324. C32#2345semidirect
382. C3#76201semidirect

Maximal subgroups

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

subgroupindex
C2×C4⋊C4×C3⋊D32
C3×#49842
#105913252
C4×#37222
Dic3×C4⋊Dic32
#105912632
#105912602
#762013