Finite Group Databasebeta

← #13147876

Subgroups — #13147876

Order 1728.

1193 conjugacy classes of subgroups; only the 175 normal subgroups are shown.

normal subgroups only

This group has 1193 conjugacy classes of subgroups — too many to draw. Only its 175 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×#88652C212yesmaximal, complemented
2#11721720C211yesmaximal, characteristic, complemented
3C42×(C3⋊D3)⋊C3C211yesmaximal, characteristic, complemented
4#64078⋊C3C211yesmaximal, characteristic, complemented
5#11721156C214yesmaximal, complemented
6#11721149C214yesmaximal, complemented
7#11721114C212yesmaximal, complemented
8#10593907—31nomaximal
9#10593897—31nomaximal
10C2×C4⋊#272C2212yes—
11C2×C4⋊#272C2214yes—
12C2×C4×(C3⋊D3)⋊C3C2212yes—
13C2×C4×(C3⋊D3)⋊C3C2211yescharacteristic
14C2×#634⋊C3C2212yes—
15C2×#6200C2214yes—
16C2×#6199C2214yes—
17#88652—28no—
18C42×H(3,𝔽3)C2211yescharacteristic
19C4⋊((C3⋊Dic3)⋊C3)C2214yes—
20C4×(C3⋊Dic3)⋊C3C2211yescharacteristic
21C4×(C3⋊Dic3)⋊C3C2212yes—
22#88451C2218yes—
23C2×#3796—32no—
24C2×#3795—32no—
25C42×C3⋊D3D311yescharacteristic, complemented
26#64078D311yescharacteristic, complemented
27C3×#874—31no—
28C3×#874—31no—
29C3×#874—61no—
30C3×C42×D3—31no—
31C3×#869—31no—
32#63916—34no—
33#63908—34no—
34#63898—34no—
35Dic3×D12—34no—
36#63863—32no—
37#63860—32no—
38C22×#272C2314yes—
39C22×(C3⋊D3)⋊C3C2311yescharacteristic
40C2×C4×H(3,𝔽3)C2311yescharacteristic
41C2×C4×H(3,𝔽3)C2312yes—
42C4⋊#272—24no—
43C4⋊#272—28no—
44C2×(C3⋊Dic3)⋊C3C2314yes—
45C2×(C3⋊Dic3)⋊C3C2312yes—
46C2×(C3⋊Dic3)⋊C3C2311yescharacteristic
47C4×(C3⋊D3)⋊C3—24no—
48C4×(C3⋊D3)⋊C3D414yescomplemented
49#634⋊C3—24no—
50#6200—28no—
51#6199—28no—
52#5725—91no—
53#5716—91no—
54#5477—91no—
55C2×C4×C3⋊D3C2×D312yes—
56C2×C4×C3⋊D3C2×D311yescharacteristic
57C2×#634C2×D312yes—
58C2×C3×D12—32no—
59C2×C3×D12—32no—
60C2×C3×D12—34no—
61C2×C3×D12—34no—
62C2×C3×D12—64no—
63C2×C3×D12—62no—
64C2×C3×C4×D3—31no—
65C2×C3×C4×D3—32no—
66C2×C3×Dic6—32no—
67C2×#626—34no—
68C2×#625—34no—
69C2×D3×Dic3—34no—
70C2×D3×Dic3—34no—
71#3796—68no—
72#3795—68no—
73C32×C42—21no—
74C32×C42C2×D311yescharacteristic
75C32×C42C2×D311yescharacteristic
76C4⋊(C3⋊Dic3)C2×D314yes—
77C4×C3⋊Dic3C2×D311yescharacteristic
78C4×C3⋊Dic3C2×D312yes—
79C3×C4⋊Dic3—34no—
80C3×C4×Dic3—31no—
81C3×C4×Dic3—32no—
82(C2×D3)⋊Dic3—38no—
83(C2×D3)⋊Dic3—38no—
84C22×H(3,𝔽3)C2411yescharacteristic
85C2×#272—24no—
86C2×#272—28no—
87C2×(C3⋊D3)⋊C3—22no—
88C2×(C3⋊D3)⋊C3C2×D412yes—
89C4×H(3,𝔽3)C2×D412yes—
90C4×H(3,𝔽3)#4114yescomplemented
91(C3⋊Dic3)⋊C3—24no—
92(C3⋊Dic3)⋊C3—22no—
93(C3⋊Dic3)⋊C3C2×D412yes—
94C2×#239—92no—
95C2×#237—92no—
96C2×#235—92no—
97C3×C4⋊Q8—91no—
98C3×#127—91no—
99C2×C3×C42—91no—
100#947—92no—
101#938—94no—
102#936—92no—
103D4×Dic3—94no—
104#930—94no—
105#895—94no—
106#893—94no—
107#875—92no—
108#874—31no—
109#874—31no—
110#874—61no—
111C4×D12—94no—
112C42×D3—31no—
113C42×D3—91no—
114#869—31no—
115#869—91no—
116C22×C3⋊D3C22×D311yescharacteristic
117C22×C3×D3—31no—
118C22×C3×D3—34no—
119C22×C3×D3—34no—
120C22×C3×D3—64no—
121C2×C32×C4—22no—
122C2×C32×C4—21no—
123C2×C32×C4C22×D311yescharacteristic
124C2×C32×C4C22×D312yes—
125C2×C32×C4C22×D311yescharacteristic
126C2×C32×C4C22×D312yes—
127C2×C3⋊Dic3C22×D312yes—
128C2×C3⋊Dic3C22×D314yes—
129C2×C3⋊Dic3C22×D311yescharacteristic
130C4×C3⋊D3—24no—
131C4×C3⋊D3D1214yescomplemented
132#634—24no—
133C2×C3×Dic3—31no—
134C2×C3×Dic3—32no—
135C2×C3×Dic3—34no—
136C3×D12—64no—
137C3×D12—68no—
138C3×D12—64no—
139C3×D12—68no—
140C3×D12—128no—
141C3×D12—124no—
142C3×C4×D3—34no—
143C3×C4×D3—64no—
144C3×Dic6—64no—
145#626—68no—
146#625—68no—
147D3×Dic3—68no—
148D3×Dic3—68no—
149#532—271noSylow
150C2×H(3,𝔽3)C22×D411yescharacteristic
151C2×H(3,𝔽3)C2×#4112yes—
152#272—44no—
153(C3⋊D3)⋊C3—22no—
154C2×C3×Q8—92no—
155C2×C3×D4—94no—
156C2×C3×D4—92no—
157C22×C3×C4—91no—
158C22×C3×C4—92no—
159C2×#67—94no—
160C2×#67—94no—
161C22×Dic3—94no—
162#239—188no—
163#237—188no—
164#235—188no—
165C2×D12—34no—
166C2×D12—32no—
167C2×D12—34no—
168C2×D12—32no—
169C2×D12—64no—
170C2×D12—62no—
171C2×D12—94no—
172C2×C4×D3—31no—
173C2×C4×D3—32no—
174C2×C4×D3—91no—
175C2×C4×D3—92no—
176C2×C4×D3—94no—
177C2×C4×D3—94no—
178C2×Dic6—32no—
179C2×Dic6—92no—
180C3×C4⋊C4—94no—
181C3×C42—31no—
182C3×C42—31no—
183C3×C42—61no—
184C3×C42—92no—
185C3×C42—91no—
186C3×C42D3211yescharacteristic
187#217—98no—
188(C2×D3)⋊C4—98no—
189(C2×D3)⋊C4—98no—
190C4⋊Dic3—34no—
191C4⋊Dic3—94no—
192C4×Dic3—32no—
193C4×Dic3—31no—
194C4×Dic3—92no—
195C4×Dic3—91no—
196C22×C32—21no—
197C22×C32C23×D311yescharacteristic
198C22×C32C23×D311yescharacteristic
199C2×C3⋊D3—22no—
200C2×C3⋊D3C2×D1212yes—
201C2×C3×D3—32no—
202C2×C3×D3—62no—
203C2×C3×D3—64no—
204C2×C3×D3—68no—
205C2×C3×D3—64no—
206C2×C3×D3—68no—
207C2×C3×D3—128no—
208C2×C3×D3—124no—
209C32×C4—24no—
210C32×C4—22no—
211C32×C4C2×D1212yes—
212C32×C4#23514yescomplemented
213C32×C4D3×D412yes—
214C32×C4#23914yescomplemented
215C3⋊Dic3—22no—
216C3⋊Dic3—24no—
217C3⋊Dic3C2×D1212yes—
218C3×Dic3—32no—
219C3×Dic3—64no—
220C3×Dic3—62no—
221C2×#41—272no—
222C4⋊Q8—91no—
223#127—91no—
224#124—272no—
225#121—274no—
226C4×D4—274no—
227C2×C42—91no—
228H(3,𝔽3)#53211yescharacteristic, complemented, Sylow
229C23×C3—91no—
230C23×C3—94no—
231C22×D3—31no—
232C22×D3—34no—
233C22×D3—34no—
234C22×D3—64no—
235C22×D3—91no—
236C3×Q8—184no—
237C3×D4—188no—
238C3×D4—184no—
239C2×C3×C4—32no—
240C2×C3×C4—31no—
241C2×C3×C4—32no—
242C2×C3×C4—31no—
243C2×C3×C4—62no—
244C2×C3×C4—61no—
245C2×C3×C4—94no—
246C2×C3×C4—91no—
247C2×C3×C4—92no—
248C2×C3×C4—94no—
249C2×C3×C4—184no—
250C2×C3×C4C2×D3212yes—
251C2×C3×C4C2×D3211yescharacteristic
252#67—188no—
253#67—188no—
254C2×Dic3—31no—
255C2×Dic3—32no—
256C2×Dic3—34no—
257C2×Dic3—91no—
258C2×Dic3—92no—
259C2×Dic3—94no—
260C2×Dic3—188no—
261D12—68no—
262D12—68no—
263D12—64no—
264D12—64no—
265D12—128no—
266D12—124no—
267D12—188no—
268C4×D3—34no—
269C4×D3—64no—
270C4×D3—94no—
271C4×D3—184no—
272C4×D3—188no—
273C4×D3—188no—
274Dic6—64no—
275Dic6—184no—
276C2×C32—22no—
277C2×C32—21no—
278C2×C32C22×D1211yescharacteristic
279C2×C32C2×#23512yes—
280C2×C32C2×D3×D411yescharacteristic
281C2×C32C2×#23912yes—
282C3⋊D3—22no—
283C3×D3—62no—
284C3×D3—124no—
285C3×D3—124no—
286C3×D3—244no—
287#41—548no—
288C2×Q8—92no—
289C2×D4—94no—
290C2×D4—92no—
291C2×D4—274no—
292C22×C4—91no—
293C22×C4—92no—
294C22×C4—274no—
295C4⋊C4—94no—
296C22⋊C4—278no—
297C42—92no—
298C42—91no—
299C42((C3⋊D3)⋊C3)⋊C211yescharacteristic
300C22×C3—31no—
301C22×C3—31no—
302C22×C3—61no—
303C22×C3—92no—
304C22×C3—182no—
305C22×C3—188no—
306C22×C3—184no—
307C22×C3C22×D3211yescharacteristic
308C2×D3—32no—
309C2×D3—62no—
310C2×D3—64no—
311C2×D3—64no—
312C2×D3—68no—
313C2×D3—68no—
314C2×D3—92no—
315C2×D3—128no—
316C2×D3—124no—
317C2×D3—182no—
318C3×C4—34no—
319C3×C4—34no—
320C3×C4—32no—
321C3×C4—32no—
322C3×C4—64no—
323C3×C4—62no—
324C3×C4—92no—
325C3×C4—184no—
326C3×C4—182no—
327C3×C4#379914yes—
328C3×C4D3×D1212yes—
329Dic3—32no—
330Dic3—62no—
331Dic3—64no—
332Dic3—92no—
333Dic3—184no—
334Dic3—182no—
335C32—21no—
336C32#547711yescharacteristic, complemented
337C32#572511yescharacteristic, complemented
338C23—91no—
339C23—94no—
340Q8—184no—
341D4—188no—
342D4—184no—
343D4—548no—
344C2×C4—91no—
345C2×C4—94no—
346C2×C4—92no—
347C2×C4—94no—
348C2×C4—184no—
349C2×C4—548no—
350C2×C4C2×((C3⋊D3)⋊C3)⋊C212yes—
351C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
352C2×C3—32no—
353C2×C3—32no—
354C2×C3—31no—
355C2×C3—31no—
356C2×C3—62no—
357C2×C3—61no—
358C2×C3—182no—
359C2×C3—364no—
360C2×C3C2×#379912yes—
361C2×C3C2×D3×D1211yescharacteristic
362D3—62no—
363D3—124no—
364D3—124no—
365D3—182no—
366D3—244no—
367C22—92no—
368C22—182no—
369C22—188no—
370C22—184no—
371C22C22×((C3⋊D3)⋊C3)⋊C211yescharacteristic
372C4—92no—
373C4—182no—
374C4—184no—
375C4#8865214yescomplemented
376C4#8865412yes—
377C3—31no—
378C3—31no—
379C3—61no—
380C3#1059397011yescharacteristic
381C2—182no—
382C2—364no—
383C2C2×#8865212yes—
384C2C2×#8865411yescharacteristic
subgroupG/H
centerC22C22×((C3⋊D3)⋊C3)⋊C2
commutator, cosocleC22×H(3,𝔽3)C24
Frattini, socleC22×C3C22×D32
FittingC42×H(3,𝔽3)C22
hypercenterC42((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)#532
LCSG ▷ C22×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C22 ◁ C42

Complements

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

NH ≅ G/Ntype
1. C2×#88652C2semidirect
2. #11721720C2semidirect
3. C42×(C3⋊D3)⋊C3C2semidirect
4. #64078⋊C3C2semidirect
5. #11721156C2semidirect
6. #11721149C2semidirect
7. #11721114C2semidirect
25. C42×C3⋊D3D3semidirect
26. #64078D3semidirect
48. C4×(C3⋊D3)⋊C3D4semidirect
90. C4×H(3,𝔽3)#41semidirect
131. C4×C3⋊D3D12semidirect
212. C32×C4#235semidirect
214. C32×C4#239semidirect
228. H(3,𝔽3)#532semidirect
336. C32#5477semidirect
337. C32#5725semidirect
375. C4#88652semidirect

Maximal subgroups

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

subgroupindex
C2×#886522
#117217202
C42×(C3⋊D3)⋊C32
#64078⋊C32
#117211562
#117211492
#117211142
#105939073
#105938973