Finite Group Databasebeta

← #13147846

Subgroups — #13147846

Order 1728.

1153 conjugacy classes of subgroups; only the 175 normal subgroups are shown.

normal subgroups only

This group has 1153 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×#88648C212yesmaximal, complemented
2#11721715C211yesmaximal, characteristic, complemented
3#11721385C211yesmaximal, characteristic, complemented
4C42×(C3⋊D3)⋊C3C211yesmaximal, characteristic, complemented
5#11721141C214yesmaximal, complemented
6#11721132C214yesmaximal, complemented
7#11721110C212yesmaximal, complemented
8#10593970—31nomaximal
9#10593907—31nomaximal
10C2×#6229C2212yes—
11C2×C4⋊((C3⋊D3)⋊C3)C2212yes—
12C2×C4⋊((C3⋊D3)⋊C3)C2214yes—
13C2×C4×(C3⋊D3)⋊C3C2212yes—
14C2×C4×(C3⋊D3)⋊C3C2211yescharacteristic
15C2×#6198C2214yes—
16C2×((C3⋊D3)⋊C3)⋊C4C2214yes—
17#88648—28no—
18C42×H(3,𝔽3)C2211yescharacteristic
19C4⋊#1084C2214yes—
20C4×(C3⋊Dic3)⋊C3C2211yescharacteristic
21C4×(C3⋊Dic3)⋊C3C2212yes—
22#88450C2218yes—
23C2×#3799—32no—
24C2×#3796—32no—
25#64083D311yescharacteristic, complemented
26C42×C3⋊D3D311yescharacteristic, complemented
27C3×#874—31no—
28C3×C42×D3—31no—
29C3×#869—31no—
30C3×#869—31no—
31C3×#869—61no—
32#63916—34no—
33#63909—34no—
34#63898—34no—
35#63894—34no—
36#63864—32no—
37#63863—32no—
38C22×(C3⋊D3)⋊C3C2314yes—
39C22×(C3⋊D3)⋊C3C2311yescharacteristic
40C2×C4×H(3,𝔽3)C2311yescharacteristic
41C2×C4×H(3,𝔽3)C2312yes—
42C2×#1084C2314yes—
43#6229—24no—
44C2×(C3⋊Dic3)⋊C3C2312yes—
45C2×(C3⋊Dic3)⋊C3C2311yescharacteristic
46C4⋊((C3⋊D3)⋊C3)—24no—
47C4⋊((C3⋊D3)⋊C3)—28no—
48C4×(C3⋊D3)⋊C3—24no—
49C4×(C3⋊D3)⋊C3D414yescomplemented
50#6198—28no—
51((C3⋊D3)⋊C3)⋊C4—28no—
52#5725—91no—
53#5716—91no—
54#5477—91no—
55C2×C4⋊(C3⋊D3)C2×D312yes—
56C2×C4⋊(C3⋊D3)C2×D314yes—
57C2×C4×C3⋊D3C2×D312yes—
58C2×C4×C3⋊D3C2×D311yescharacteristic
59C2×C3×D12—32no—
60C2×C3×D12—34no—
61C2×C3×C4×D3—31no—
62C2×C3×C4×D3—32no—
63C2×C3×Dic6—32no—
64C2×C3×Dic6—32no—
65C2×C3×Dic6—62no—
66C2×#626—34no—
67C2×#626—34no—
68C2×#624—34no—
69C2×D3×Dic3—34no—
70#3799—68no—
71#3796—68no—
72C32×C42—21no—
73C32×C42C2×D311yescharacteristic
74C32×C42C2×D311yescharacteristic
75C4×C3⋊Dic3C2×D311yescharacteristic
76C4×C3⋊Dic3C2×D312yes—
77C3×C4⋊Dic3—34no—
78C3×C4⋊Dic3—34no—
79C3×C4⋊Dic3—64no—
80C3×C4×Dic3—31no—
81C3×C4×Dic3—32no—
82#3722—38no—
83(C2×D3)⋊Dic3—38no—
84C22×H(3,𝔽3)C2411yescharacteristic
85C2×(C3⋊D3)⋊C3—22no—
86C2×(C3⋊D3)⋊C3—24no—
87C2×(C3⋊D3)⋊C3—28no—
88C2×(C3⋊D3)⋊C3C2×D412yes—
89C4×H(3,𝔽3)C2×D412yes—
90C4×H(3,𝔽3)#4114yescomplemented
91#1084—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—91no—
110C4×D12—94no—
111C42×D3—31no—
112C42×D3—91no—
113#869—31no—
114#869—31no—
115#869—61no—
116C22×C3⋊D3C22×D314yes—
117C22×C3⋊D3C22×D311yescharacteristic
118C22×C3×D3—31no—
119C22×C3×D3—34no—
120C2×C32×C4—22no—
121C2×C32×C4—21no—
122C2×C32×C4C22×D311yescharacteristic
123C2×C32×C4C22×D312yes—
124C2×C32×C4C22×D311yescharacteristic
125C2×C32×C4C22×D312yes—
126C2×C3⋊Dic3C22×D312yes—
127C2×C3⋊Dic3C22×D311yescharacteristic
128C4⋊(C3⋊D3)—24no—
129C4⋊(C3⋊D3)—28no—
130C4×C3⋊D3—24no—
131C4×C3⋊D3D1214yescomplemented
132C2×C3×Dic3—31no—
133C2×C3×Dic3—32no—
134C2×C3×Dic3—34no—
135C2×C3×Dic3—34no—
136C2×C3×Dic3—64no—
137C3×D12—64no—
138C3×D12—68no—
139C3×C4×D3—34no—
140C3×C4×D3—64no—
141C3×Dic6—64no—
142C3×Dic6—64no—
143C3×Dic6—124no—
144#626—68no—
145#626—68no—
146#624—68no—
147D3×Dic3—68no—
148#532—271noSylow
149C2×H(3,𝔽3)C22×D411yescharacteristic
150C2×H(3,𝔽3)C2×#4112yes—
151(C3⋊D3)⋊C3—22no—
152(C3⋊D3)⋊C3—44no—
153C2×C3×Q8—92no—
154C2×C3×D4—94no—
155C2×C3×D4—92no—
156C22×C3×C4—91no—
157C22×C3×C4—92no—
158C2×#67—94no—
159C2×#67—94no—
160C22×Dic3—94no—
161#239—188no—
162#237—188no—
163#235—188no—
164C2×D12—34no—
165C2×D12—32no—
166C2×D12—94no—
167C2×D12—92no—
168C2×D12—94no—
169C2×C4×D3—31no—
170C2×C4×D3—32no—
171C2×C4×D3—91no—
172C2×C4×D3—92no—
173C2×C4×D3—94no—
174C2×C4×D3—94no—
175C2×Dic6—32no—
176C2×Dic6—32no—
177C2×Dic6—62no—
178C3×C4⋊C4—94no—
179C3×C42—31no—
180C3×C42—31no—
181C3×C42—61no—
182C3×C42—92no—
183C3×C42—91no—
184C3×C42D3211yescharacteristic
185#217—98no—
186(C2×D3)⋊C4—98no—
187(C2×D3)⋊C4—98no—
188C4⋊Dic3—34no—
189C4⋊Dic3—34no—
190C4⋊Dic3—64no—
191C4×Dic3—32no—
192C4×Dic3—31no—
193C4×Dic3—92no—
194C4×Dic3—91no—
195C22×C32—21no—
196C22×C32C23×D311yescharacteristic
197C22×C32C23×D311yescharacteristic
198C2×C3⋊D3—22no—
199C2×C3⋊D3—24no—
200C2×C3⋊D3—28no—
201C2×C3⋊D3C2×D1212yes—
202C2×C3×D3—32no—
203C2×C3×D3—62no—
204C2×C3×D3—64no—
205C2×C3×D3—68no—
206C32×C4—24no—
207C32×C4—22no—
208C32×C4C2×D1212yes—
209C32×C4#23514yescomplemented
210C32×C4D3×D412yes—
211C32×C4#23714yescomplemented
212C3⋊Dic3—22no—
213C3⋊Dic3C2×D1212yes—
214C3×Dic3—32no—
215C3×Dic3—64no—
216C3×Dic3—64no—
217C3×Dic3—62no—
218C3×Dic3—124no—
219C2×#41—272no—
220C4⋊Q8—91no—
221#127—91no—
222#124—272no—
223#121—274no—
224C4×D4—274no—
225C2×C42—91no—
226H(3,𝔽3)#53211yescharacteristic, complemented, Sylow
227C23×C3—91no—
228C23×C3—94no—
229C22×D3—31no—
230C22×D3—34no—
231C22×D3—91no—
232C22×D3—94no—
233C3×Q8—184no—
234C3×D4—184no—
235C3×D4—188no—
236C2×C3×C4—32no—
237C2×C3×C4—31no—
238C2×C3×C4—32no—
239C2×C3×C4—31no—
240C2×C3×C4—62no—
241C2×C3×C4—61no—
242C2×C3×C4—94no—
243C2×C3×C4—91no—
244C2×C3×C4—92no—
245C2×C3×C4—94no—
246C2×C3×C4—184no—
247C2×C3×C4C2×D3212yes—
248C2×C3×C4C2×D3211yescharacteristic
249#67—188no—
250#67—188no—
251C2×Dic3—31no—
252C2×Dic3—34no—
253C2×Dic3—34no—
254C2×Dic3—32no—
255C2×Dic3—64no—
256C2×Dic3—91no—
257C2×Dic3—92no—
258C2×Dic3—188no—
259D12—68no—
260D12—64no—
261D12—188no—
262D12—184no—
263D12—188no—
264C4×D3—34no—
265C4×D3—64no—
266C4×D3—94no—
267C4×D3—184no—
268C4×D3—188no—
269C4×D3—188no—
270Dic6—64no—
271Dic6—64no—
272Dic6—124no—
273C2×C32—22no—
274C2×C32—21no—
275C2×C32C22×D1211yescharacteristic
276C2×C32C2×#23512yes—
277C2×C32C2×D3×D411yescharacteristic
278C2×C32C2×#23712yes—
279C3⋊D3—22no—
280C3⋊D3—44no—
281C3×D3—62no—
282C3×D3—124no—
283#41—548no—
284C2×Q8—92no—
285C2×D4—94no—
286C2×D4—92no—
287C2×D4—274no—
288C22×C4—91no—
289C22×C4—92no—
290C22×C4—274no—
291C4⋊C4—94no—
292C22⋊C4—278no—
293C42—92no—
294C42—91no—
295C42((C3⋊D3)⋊C3)⋊C211yescharacteristic
296C22×C3—31no—
297C22×C3—31no—
298C22×C3—61no—
299C22×C3—92no—
300C22×C3—182no—
301C22×C3—184no—
302C22×C3—188no—
303C22×C3C22×D3211yescharacteristic
304C2×D3—32no—
305C2×D3—62no—
306C2×D3—68no—
307C2×D3—64no—
308C2×D3—92no—
309C2×D3—182no—
310C2×D3—184no—
311C2×D3—188no—
312C3×C4—34no—
313C3×C4—34no—
314C3×C4—32no—
315C3×C4—32no—
316C3×C4—64no—
317C3×C4—62no—
318C3×C4—92no—
319C3×C4—182no—
320C3×C4—184no—
321C3×C4#379514yes—
322C3×C4D3×D1212yes—
323Dic3—32no—
324Dic3—64no—
325Dic3—64no—
326Dic3—62no—
327Dic3—92no—
328Dic3—124no—
329Dic3—182no—
330C32—21no—
331C32#547711yescharacteristic, complemented
332C32#571611yescharacteristic, complemented
333C23—91no—
334C23—94no—
335Q8—184no—
336D4—188no—
337D4—184no—
338D4—548no—
339C2×C4—91no—
340C2×C4—94no—
341C2×C4—92no—
342C2×C4—94no—
343C2×C4—184no—
344C2×C4—548no—
345C2×C4C2×((C3⋊D3)⋊C3)⋊C212yes—
346C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
347C2×C3—32no—
348C2×C3—32no—
349C2×C3—31no—
350C2×C3—31no—
351C2×C3—62no—
352C2×C3—61no—
353C2×C3—182no—
354C2×C3—364no—
355C2×C3C2×#379512yes—
356C2×C3C2×D3×D1211yescharacteristic
357D3—62no—
358D3—124no—
359D3—182no—
360D3—364no—
361C22—92no—
362C22—182no—
363C22—188no—
364C22—184no—
365C22C22×((C3⋊D3)⋊C3)⋊C211yescharacteristic
366C4—92no—
367C4—182no—
368C4—184no—
369C4#8864814yescomplemented
370C4#8865412yes—
371C3—31no—
372C3—31no—
373C3—61no—
374C3#1059389711yescharacteristic
375C2—182no—
376C2—364no—
377C2C2×#8864812yes—
378C2C2×#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×#88648C2semidirect
2. #11721715C2semidirect
3. #11721385C2semidirect
4. C42×(C3⋊D3)⋊C3C2semidirect
5. #11721141C2semidirect
6. #11721132C2semidirect
7. #11721110C2semidirect
25. #64083D3semidirect
26. C42×C3⋊D3D3semidirect
49. C4×(C3⋊D3)⋊C3D4semidirect
90. C4×H(3,𝔽3)#41semidirect
131. C4×C3⋊D3D12semidirect
209. C32×C4#235semidirect
211. C32×C4#237semidirect
226. H(3,𝔽3)#532semidirect
331. C32#5477semidirect
332. C32#5716semidirect
369. C4#88648semidirect

Maximal subgroups

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

subgroupindex
C2×#886482
#117217152
#117213852
C42×(C3⋊D3)⋊C32
#117211412
#117211322
#117211102
#105939703
#105939073