Finite Group Databasebeta

← #13149569

Subgroups — #13149569

Order 1728.

1135 conjugacy classes of subgroups; only the 223 normal subgroups are shown.

normal subgroups only

This group has 1135 conjugacy classes of subgroups — too many to draw. Only its 223 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#88651C211yesmaximal, characteristic
2C4⋊C4×#272C211yesmaximal, characteristic
3#64096⋊C3C212yesmaximal, complemented
4#11721147C212yesmaximal
5C4×#6199C211yesmaximal, characteristic
6C4×#6199C212yesmaximal
7#11721082C212yesmaximal, complemented
8#11721077C214yesmaximal, complemented
9D3×#887—32nomaximal
10C2×C4×#272C2212yes—
11C2×C4×#272C2211yescharacteristic
12C2×#634⋊C3C2212yes—
13C2×#6199C2211yescharacteristic
14C2×#6199C2212yes—
15C2×#6199C2211yescharacteristic
16C2×#6196C2212yes—
17#88651C418yescomplemented
18C4⋊C4×H(3,𝔽3)C2211yescharacteristic
19C4⋊#1084C2211yescharacteristic
20#88540C2212yes—
21#3750⋊C3C2214yes—
22C4×(C3⋊Dic3)⋊C3C2214yes—
23C4×(C3⋊Dic3)⋊C3C2212yes—
24#88448C2214yes—
25#88447C2212yes—
26#88447C2214yes—
27C2×D3×Dic6—32no—
28#64096D312yescomplemented
29C3×D3×C4⋊C4—32no—
30C3×D3×C4⋊C4—61no—
31C3×#887—32no—
32D3×Dic3⋊C4—34no—
33C4×D3×Dic3—32no—
34C4×D3×Dic3—34no—
35#63844—32no—
36#63843—32no—
37#63838—34no—
38#63837—34no—
39C22×#272C2311yescharacteristic
40C2×C4×H(3,𝔽3)C2312yes—
41C2×C4×H(3,𝔽3)C2311yescharacteristic
42C2×#1084C2312yes—
43C2×#1084C2311yescharacteristic
44C4×#272—24no—
45C4×#272C2×C414yes—
46C2×(C3⋊Dic3)⋊C3C2312yes—
47C2×(C3⋊Dic3)⋊C3C2314yes—
48C2×(C3⋊Dic3)⋊C3C2312yes—
49#634⋊C3C2×C418yescomplemented
50#6199—22no—
51#6199C2×C418yes—
52#6199Q814yescomplemented
53#6196C2×C418yescomplemented
54C4×D3×Q8—92no—
55C2×#887—91no—
56C2×#634C2×D312yes—
57C2×C3×C4×D3—32no—
58C2×C3×C4×D3—34no—
59C2×C3×C4×D3—61no—
60C2×C3×C4×D3—62no—
61C2×C3×Dic6—32no—
62C2×#627—34no—
63C2×D3×Dic3—34no—
64C2×D3×Dic3—32no—
65C2×D3×Dic3—32no—
66D3×Dic6—316no—
67C32×C4⋊C4—21no—
68C32×C4⋊C4C2×D312yes—
69#3750C2×D314yes—
70C4×C3⋊Dic3C2×D314yes—
71C4×C3⋊Dic3C2×D312yes—
72C3×C4⋊Dic3—32no—
73C3×C4⋊Dic3—61no—
74C3×Dic3⋊C4—34no—
75C3×Dic3⋊C4—34no—
76C3×Dic3⋊C4—62no—
77C3×C4×Dic3—34no—
78C3×C4×Dic3—32no—
79#3724—34no—
80Dic3⋊Dic3—34no—
81Dic32—32no—
82Dic32—34no—
83Dic32—32no—
84Dic32—34no—
85C22×H(3,𝔽3)C2411yescharacteristic
86C2×#272C22×C412yes—
87C2×#272C2×Q812yes—
88C2×#272#4112yes—
89C4×H(3,𝔽3)—22no—
90C4×H(3,𝔽3)C22×C412yes—
91#1084—22no—
92#1084C22×C412yes—
93(C3⋊Dic3)⋊C3—22no—
94(C3⋊Dic3)⋊C3C22×C418yes—
95(C3⋊Dic3)⋊C3C2×Q814yescomplemented
96C2×D3×Q8—92no—
97C22×Dic6—91no—
98C3×C4×Q8—92no—
99C2×C3×C4⋊C4—91no—
100Q8×Dic3—92no—
101C2×Dic3⋊C4—92no—
102C2×C4×Dic3—92no—
103C2×C4×Dic3—91no—
104D3×C4⋊C4—32no—
105D3×C4⋊C4—61no—
106D3×C4⋊C4—94no—
107#887—32no—
108#887—92no—
109#887—94no—
110#887—92no—
111#887—94no—
112C42×D3—94no—
113C42×D3—92no—
114C4×Dic6—94no—
115C4×Dic6—92no—
116C22×C3×D3—32no—
117C22×C3×D3—61no—
118C2×C32×C4—21no—
119C2×C32×C4—22no—
120C2×C32×C4C22×D314yes—
121C2×C32×C4C22×D312yes—
122C2×C3⋊Dic3C22×D312yes—
123C2×C3⋊Dic3C22×D314yes—
124C2×C3⋊Dic3C22×D312yes—
125#634C4×D318yescomplemented
126C2×C3×Dic3—32no—
127C2×C3×Dic3—34no—
128C2×C3×Dic3—32no—
129C2×C3×Dic3—32no—
130C2×C3×Dic3—34no—
131C2×C3×Dic3—61no—
132C2×C3×Dic3—62no—
133C3×C4×D3—38no—
134C3×C4×D3—64no—
135C3×C4×D3—68no—
136C3×C4×D3—124no—
137C3×Dic6—38no—
138#627—316no—
139D3×Dic3—316no—
140D3×Dic3—38no—
141D3×Dic3—64no—
142C2×C4×Q8—271noSylow
143C2×H(3,𝔽3)C23×C411yescharacteristic
144C2×H(3,𝔽3)C22×Q811yescharacteristic
145C2×H(3,𝔽3)C2×#4111yescharacteristic
146#272C4×Q814yescomplemented
147C2×C3×Q8—92no—
148C22×C3×C4—91no—
149C22×C3×C4—92no—
150C22×Dic3—91no—
151C22×Dic3—92no—
152C22×Dic3—91no—
153D3×Q8—916no—
154C2×C4×D3—32no—
155C2×C4×D3—34no—
156C2×C4×D3—61no—
157C2×C4×D3—62no—
158C2×C4×D3—92no—
159C2×C4×D3—94no—
160C2×C4×D3—92no—
161C2×Dic6—32no—
162C2×Dic6—92no—
163C2×Dic6—98no—
164C2×Dic6—92no—
165C2×Dic6—94no—
166C3×C4⋊C4—32no—
167C3×C4⋊C4—61no—
168C3×C4⋊C4—94no—
169C3×C4⋊C4—92no—
170C3×C4⋊C4—91no—
171C3×C4⋊C4D3211yescharacteristic
172C3×C42—92no—
173C3×C42—94no—
174C4⋊Dic3—32no—
175C4⋊Dic3—61no—
176C4⋊Dic3—94no—
177Dic3⋊C4—34no—
178Dic3⋊C4—34no—
179Dic3⋊C4—62no—
180Dic3⋊C4—94no—
181Dic3⋊C4—94no—
182Dic3⋊C4—94no—
183C4×Dic3—34no—
184C4×Dic3—32no—
185C4×Dic3—94no—
186C4×Dic3—92no—
187C4×Dic3—94no—
188C4×Dic3—92no—
189C4×Dic3—92no—
190C4×Dic3—94no—
191C4×Dic3—92no—
192C4×Dic3—94no—
193C22×C32—21no—
194C22×C32C23×D312yes—
195C2×C3×D3—34no—
196C2×C3×D3—34no—
197C2×C3×D3—34no—
198C2×C3×D3—62no—
199C2×C3×D3—62no—
200C2×C3×D3—62no—
201C32×C4—24no—
202C32×C4—22no—
203C32×C4—42no—
204C32×C4C2×C4×D314yes—
205C3⋊Dic3—22no—
206C3⋊Dic3C2×C4×D318yes—
207C3⋊Dic3D3×Q814yescomplemented
208C3×Dic3—38no—
209C3×Dic3—34no—
210C3×Dic3—34no—
211C3×Dic3—62no—
212C3×Dic3—62no—
213C3×Dic3—64no—
214C3×Dic3—122no—
215C22×Q8—271no—
216C4×Q8—92no—
217C4×Q8—274no—
218C4×Q8—272no—
219C2×C4⋊C4—91no—
220C2×C4⋊C4—272no—
221C2×C42—271no—
222C2×C42—272no—
223H(3,𝔽3)C2×C4×Q811yescharacteristic, complemented, Sylow
224C23×C3—91no—
225C22×D3—32no—
226C22×D3—61no—
227C3×Q8—98no—
228C2×C3×C4—32no—
229C2×C3×C4—34no—
230C2×C3×C4—61no—
231C2×C3×C4—62no—
232C2×C3×C4—92no—
233C2×C3×C4—94no—
234C2×C3×C4—91no—
235C2×C3×C4—94no—
236C2×C3×C4—92no—
237C2×C3×C4—92no—
238C2×C3×C4—184no—
239C2×C3×C4C2×D3212yes—
240C2×C3×C4C2×D3211yescharacteristic
241C2×Dic3—32no—
242C2×Dic3—32no—
243C2×Dic3—34no—
244C2×Dic3—34no—
245C2×Dic3—32no—
246C2×Dic3—61no—
247C2×Dic3—62no—
248C2×Dic3—92no—
249C2×Dic3—94no—
250C2×Dic3—92no—
251C2×Dic3—98no—
252C2×Dic3—94no—
253C2×Dic3—182no—
254C4×D3—38no—
255C4×D3—64no—
256C4×D3—68no—
257C4×D3—98no—
258C4×D3—916no—
259C4×D3—124no—
260C4×D3—184no—
261Dic6—38no—
262Dic6—98no—
263Dic6—98no—
264Dic6—916no—
265C2×C32—21no—
266C2×C32—21no—
267C2×C32—21no—
268C2×C32C22×C4×D312yes—
269C2×C32C2×D3×Q812yes—
270C2×C32D3×#4112yes—
271C3×D3—38no—
272C3×D3—64no—
273C2×Q8—92no—
274C2×Q8—272no—
275C2×Q8—278no—
276C22×C4—91no—
277C22×C4—92no—
278C22×C4—271no—
279C22×C4—272no—
280C22×C4—271no—
281C4⋊C4—92no—
282C4⋊C4—91no—
283C4⋊C4—94no—
284C4⋊C4—274no—
285C4⋊C4((C3⋊D3)⋊C3)⋊C211yescharacteristic
286C42—94no—
287C42—92no—
288C42—274no—
289C42—272no—
290C22×C3—32no—
291C22×C3—61no—
292C22×C3—92no—
293C22×C3—92no—
294C22×C3—92no—
295C22×C3C22×D3211yescharacteristic
296C2×D3—34no—
297C2×D3—34no—
298C2×D3—34no—
299C2×D3—62no—
300C2×D3—62no—
301C2×D3—62no—
302C3×C4—22no—
303C3×C4—34no—
304C3×C4—64no—
305C3×C4—62no—
306C3×C4—98no—
307C3×C4—94no—
308C3×C4—92no—
309C3×C4—122no—
310C3×C4—182no—
311C3×C4—182no—
312C3×C4C4×D3212yes—
313Dic3—34no—
314Dic3—38no—
315Dic3—34no—
316Dic3—62no—
317Dic3—64no—
318Dic3—62no—
319Dic3—98no—
320Dic3—94no—
321Dic3—122no—
322Dic3—182no—
323C32—21no—
324C32C4×D3×Q812yescomplemented
325C23—91no—
326Q8—98no—
327Q8—278no—
328C2×C4—92no—
329C2×C4—94no—
330C2×C4—91no—
331C2×C4—94no—
332C2×C4—92no—
333C2×C4—92no—
334C2×C4—184no—
335C2×C4—278no—
336C2×C4—274no—
337C2×C4—542no—
338C2×C4C2×((C3⋊D3)⋊C3)⋊C212yes—
339C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
340C2×C3—32no—
341C2×C3—32no—
342C2×C3—32no—
343C2×C3—61no—
344C2×C3—61no—
345C2×C3—61no—
346C2×C3—94no—
347C2×C3C2×C4×D3211yescharacteristic
348C2×C3#6431211yescharacteristic
349C2×C3D32×Q811yescharacteristic
350D3—38no—
351D3—64no—
352C22—92no—
353C22—92no—
354C22—92no—
355C22C22×((C3⋊D3)⋊C3)⋊C211yescharacteristic
356C4—22no—
357C4—94no—
358C4—92no—
359C4—98no—
360C4—182no—
361C4—182no—
362C4C4×((C3⋊D3)⋊C3)⋊C212yes—
363C3—32no—
364C3—61no—
365C3#1059436311yescharacteristic
366C2—94no—
367C2C2×C4×((C3⋊D3)⋊C3)⋊C211yescharacteristic
368C2#1172250511yescharacteristic
369C2Q8×((C3⋊D3)⋊C3)⋊C211yescharacteristic
subgroupG/H
centerC22C22×((C3⋊D3)⋊C3)⋊C2
commutatorC2×H(3,𝔽3)C23×C4
Frattini, socleC22×C3C22×D32
FittingC4⋊C4×H(3,𝔽3)C22
prime radicalC22×#272C23
cosocleC22×H(3,𝔽3)C24
hypercenterC4⋊C4((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)C2×C4×Q8
LCSG ▷ C2×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C22 ◁ C4⋊C4

Complements

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

NH ≅ G/Ntype
3. #64096⋊C3C2semidirect
7. #11721082C2semidirect
8. #11721077C2semidirect
17. #88651C4semidirect
28. #64096D3semidirect
49. #634⋊C3C2×C4semidirect
52. #6199Q8semidirect
53. #6196C2×C4semidirect
95. (C3⋊Dic3)⋊C3C2×Q8semidirect
125. #634C4×D3semidirect
146. #272C4×Q8semidirect
207. C3⋊Dic3D3×Q8semidirect
223. H(3,𝔽3)C2×C4×Q8semidirect
324. C32C4×D3×Q8semidirect

Maximal subgroups

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

subgroupindex
C2×#886512
C4⋊C4×#2722
#64096⋊C32
#117211472
C4×#61992
#117210822
#117210772
D3×#8873