Finite Group Databasebeta

← #13138023

Subgroups — #13138023

Order 1728.

420 conjugacy classes of subgroups; only the 77 normal subgroups are shown.

normal subgroups only

This group has 420 conjugacy classes of subgroups — too many to draw. Only its 77 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
1#64096⋊C3C211yesmaximal, characteristic
2#11721090C211yesmaximal, characteristic
3C2×#88439C211yesmaximal, characteristic
4C2×#88433C211yesmaximal, characteristic
5#11720532C211yesmaximal, characteristic
6#63637⋊C3C211yesmaximal, characteristic
7#11720311C211yesmaximal, characteristic
8#10590566—31nomaximal
9#10590561—31nomaximal
10C2×#6229C2211yescharacteristic
11C2×#634⋊C3C2211yescharacteristic
12C2×#634⋊C3C2211yescharacteristic
13C4⋊C4×H(3,𝔽3)C2211yescharacteristic
14C2×#6180C2211yescharacteristic
15#3750⋊C3—21no—
16C4×(C3⋊Dic3)⋊C3—21no—
17C4×(C3⋊Dic3)⋊C3C2211yescharacteristic
18C2×(C3⋊(C3⋊C8))⋊C3C2211yescharacteristic
19#88448—21no—
20#88448—21no—
21#88439—22no—
22#88433—22no—
23#64096D311yescharacteristic
24C3×#887—31no—
25#63873—31no—
26#63866—31no—
27C2×#3719—31no—
28C2×#3719—31no—
29C2×#3719—31no—
30C2×#3718—31no—
31#63637D311yescharacteristic
32C3×#810—31no—
33C3×#810—31no—
34C3×#810—31no—
35C3×#810—61no—
36#63572—31no—
37Dic3⋊(C3⋊C8)—31no—
38C2×C4×H(3,𝔽3)—21no—
39C2×C4×H(3,𝔽3)C2311yescharacteristic
40C2×#1084—21no—
41#6229—21no—
42#6229—21no—
43C2×(C3⋊Dic3)⋊C3—21no—
44C2×(C3⋊Dic3)⋊C3—21no—
45C2×(C3⋊Dic3)⋊C3D411yescharacteristic
46C2×(C3⋊Dic3)⋊C3D411yescharacteristic
47#634⋊C3—21no—
48#634⋊C3—21no—
49#634⋊C3—21no—
50#634⋊C3D412yes—
51#6180—21no—
52(C3⋊(C3⋊C8))⋊C3—21no—
53#5031—91no—
54#4795—91no—
55#4795—91no—
56C2×#634C2×D311yescharacteristic
57C2×#634C2×D311yescharacteristic
58C2×C3×Dic6—31no—
59C2×C3×Dic6—31no—
60C2×C3×Dic6—31no—
61C2×C3×Dic6—31no—
62C2×C3×Dic6—61no—
63C32×C4⋊C4—21no—
64C32×C4⋊C4C2×D311yescharacteristic
65C32×C4⋊C4C2×D311yescharacteristic
66#3750—21no—
67C4×C3⋊Dic3—21no—
68C4×C3⋊Dic3C2×D311yescharacteristic
69C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
70C3×Dic3⋊C4—61no—
71C3×C4×Dic3—31no—
72C3×C4×Dic3—61no—
73C2×C3×C3⋊C8—31no—
74C2×C3×C3⋊C8—31no—
75C2×C3×C3⋊C8—31no—
76C2×C3×C3⋊C8—61no—
77#3724—61no—
78#3724—61no—
79Dic3⋊Dic3—61no—
80Dic3⋊Dic3—61no—
81#3719—62no—
82#3719—62no—
83#3719—62no—
84#3718—62no—
85C22×H(3,𝔽3)C2×D411yescharacteristic
86C4×H(3,𝔽3)—41no—
87C4×H(3,𝔽3)C2×D411yescharacteristic
88C4×H(3,𝔽3)#4111yescharacteristic
89#1084—41no—
90(C3⋊Dic3)⋊C3—22no—
91(C3⋊Dic3)⋊C3—21no—
92(C3⋊Dic3)⋊C3—41no—
93(C3⋊Dic3)⋊C3Q1612yescomplemented
94C3×C4×Q8—91no—
95#944—91no—
96#944—91no—
97C2×#216—91no—
98C2×#216—91no—
99C2×#216—91no—
100C2×#216—91no—
101C2×Dic12—91no—
102C2×Dic12—91no—
103#887—31no—
104#887—91no—
105#869—91no—
106C3×#103—91no—
107C3×#103—91no—
108Dic3⋊C8—91no—
109Dic3⋊C8—91no—
110#810—31no—
111#810—31no—
112#810—31no—
113#810—61no—
114#810—91no—
115C4⋊(C3⋊C8)—91no—
116C2×C32×C4—21no—
117C2×C32×C4—21no—
118C2×C32×C4—21no—
119C2×C32×C4—41no—
120C2×C32×C4C22×D311yescharacteristic
121C2×C32×C4C22×D311yescharacteristic
122C2×C3⋊Dic3—21no—
123C2×C3⋊Dic3—21no—
124C2×C3⋊Dic3#6711yescharacteristic
125C2×C3⋊Dic3#6711yescharacteristic
126#634—21no—
127#634—21no—
128#634—21no—
129#634D1212yes—
130C2×C3×Dic3—31no—
131C2×C3×Dic3—31no—
132C2×C3×Dic3—61no—
133C2×C3×Dic3—61no—
134C2×C3×Dic3—61no—
135C2×C3×Dic3—61no—
136C2×C3×Dic3—121no—
137C3×Dic6—32no—
138C3×Dic6—61no—
139C3×Dic6—62no—
140C3×Dic6—61no—
141C3×Dic6—61no—
142C3×Dic6—61no—
143C3×Dic6—61no—
144C3×Dic6—61no—
145C3×Dic6—61no—
146C3×Dic6—121no—
147C3⋊(C3⋊C8)—21no—
148C3×C3⋊C8—61no—
149C3×C3⋊C8—61no—
150C3×C3⋊C8—61no—
151C3×C3⋊C8—121no—
152#462—271noSylow
153C2×H(3,𝔽3)#12111yescharacteristic
154C2×H(3,𝔽3)C2×Q1611yescharacteristic
155C2×H(3,𝔽3)#13711yescharacteristic
156C2×C3×Q8—91no—
157C2×C3×Q8—91no—
158C2×C3×Q8—91no—
159C2×Dic6—31no—
160C2×Dic6—31no—
161C2×Dic6—31no—
162C2×Dic6—31no—
163C2×Dic6—61no—
164C2×Dic6—91no—
165C2×Dic6—91no—
166C2×C3×C8—91no—
167C2×C3×C8—91no—
168C3×C4⋊C4—31no—
169C3×C4⋊C4—31no—
170C3×C4⋊C4—61no—
171C3×C4⋊C4—181no—
172C3×C4⋊C4D3211yescharacteristic
173C3×C42—91no—
174C3×C42—181no—
175#216—182no—
176#216—182no—
177#216—182no—
178#216—182no—
179C4⋊Dic3—181no—
180C4⋊Dic3—181no—
181Dic3⋊C4—61no—
182Dic3⋊C4—181no—
183Dic3⋊C4—181no—
184Dic3⋊C4—181no—
185Dic3⋊C4—181no—
186Dic3⋊C4—181no—
187C4×Dic3—31no—
188C4×Dic3—61no—
189C4×Dic3—91no—
190C4×Dic3—181no—
191C2×C3⋊C8—31no—
192C2×C3⋊C8—31no—
193C2×C3⋊C8—31no—
194C2×C3⋊C8—61no—
195C2×C3⋊C8—91no—
196Dic12—182no—
197Dic12—182no—
198C22×C32—21no—
199C22×C32D3×D411yescharacteristic
200C22×C32C2×#6711yescharacteristic
201C32×C4—21no—
202C32×C4—21no—
203C32×C4—41no—
204C32×C4—41no—
205C32×C4—81no—
206C32×C4C2×D1211yescharacteristic
207C32×C4#23511yescharacteristic
208C32×C4#23511yescharacteristic
209C32×C4D3×D411yescharacteristic
210C3⋊Dic3—22no—
211C3⋊Dic3—21no—
212C3⋊Dic3—41no—
213C3⋊Dic3#21612yescomplemented
214C3×Dic3—32no—
215C3×Dic3—62no—
216C3×Dic3—61no—
217C3×Dic3—122no—
218C3×Dic3—121no—
219C3×Dic3—121no—
220C3×Dic3—121no—
221C2×Q16—271no—
222C2×Q16—271no—
223C4⋊Q8—271no—
224C4×Q8—91no—
225C4⋊C8—271no—
226#103—91no—
227#103—91no—
228H(3,𝔽3)#46211yescharacteristic, complemented, Sylow
229C3×Q8—92no—
230C3×Q8—181no—
231C3×Q8—181no—
232C3×Q8—181no—
233C3×Q8—181no—
234C3×Q8—181no—
235C2×C3×C4—21no—
236C2×C3×C4—31no—
237C2×C3×C4—31no—
238C2×C3×C4—61no—
239C2×C3×C4—61no—
240C2×C3×C4—61no—
241C2×C3×C4—91no—
242C2×C3×C4—91no—
243C2×C3×C4—121no—
244C2×C3×C4—181no—
245C2×C3×C4—181no—
246C2×C3×C4—181no—
247C2×C3×C4C2×D3211yescharacteristic
248C2×Dic3—31no—
249C2×Dic3—31no—
250C2×Dic3—61no—
251C2×Dic3—61no—
252C2×Dic3—61no—
253C2×Dic3—61no—
254C2×Dic3—91no—
255C2×Dic3—91no—
256C2×Dic3—121no—
257C2×Dic3—181no—
258C2×Dic3—181no—
259Dic6—32no—
260Dic6—61no—
261Dic6—61no—
262Dic6—61no—
263Dic6—61no—
264Dic6—61no—
265Dic6—62no—
266Dic6—61no—
267Dic6—61no—
268Dic6—92no—
269Dic6—121no—
270Dic6—181no—
271Dic6—181no—
272Dic6—181no—
273C3×C8—181no—
274C3×C8—181no—
275C3⋊C8—61no—
276C3⋊C8—61no—
277C3⋊C8—61no—
278C3⋊C8—121no—
279C3⋊C8—181no—
280C2×C32—21no—
281C2×C32—21no—
282C2×C32—21no—
283C2×C32#88411yescharacteristic
284C2×C32#91511yescharacteristic
285C2×C32D3×Q1611yescharacteristic
286C2×C32#93011yescharacteristic
287C2×C32C2×#21611yescharacteristic
288C2×C32#95111yescharacteristic
289C2×Q8—91no—
290C2×Q8—91no—
291C2×Q8—91no—
292Q16—542no—
293Q16—542no—
294C2×C8—91no—
295C2×C8—91no—
296C4⋊C4—181no—
297C4⋊C4—541no—
298C4⋊C4—541no—
299C4⋊C4((C3⋊D3)⋊C3)⋊C211yescharacteristic
300C42—91no—
301C42—181no—
302C22×C3—31no—
303C22×C3—31no—
304C22×C3—61no—
305C22×C3D3×#6711yescharacteristic
306C3×C4—31no—
307C3×C4—31no—
308C3×C4—31no—
309C3×C4—31no—
310C3×C4—41no—
311C3×C4—61no—
312C3×C4—61no—
313C3×C4—92no—
314C3×C4—121no—
315C3×C4—121no—
316C3×C4—181no—
317C3×C4—182no—
318C3×C4—241no—
319C3×C4—361no—
320C3×C4—361no—
321C3×C4#379811yescharacteristic
322C3×C4D3×D1211yescharacteristic
323Dic3—32no—
324Dic3—62no—
325Dic3—61no—
326Dic3—92no—
327Dic3—121no—
328Dic3—121no—
329Dic3—122no—
330Dic3—121no—
331Dic3—182no—
332Dic3—181no—
333Dic3—361no—
334C32—21no—
335C32#479511yescharacteristic, complemented
336C32#503111yescharacteristic, complemented
337Q8—92no—
338Q8—181no—
339Q8—181no—
340Q8—181no—
341Q8—181no—
342Q8—181no—
343C2×C4—21no—
344C2×C4—91no—
345C2×C4—91no—
346C2×C4—181no—
347C2×C4—181no—
348C2×C4—181no—
349C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
350C8—181no—
351C8—181no—
352C2×C3—31no—
353C2×C3—31no—
354C2×C3—31no—
355C2×C3—31no—
356C2×C3—31no—
357C2×C3—31no—
358C2×C3—61no—
359C2×C3—61no—
360C2×C3—61no—
361C2×C3#6390611yescharacteristic
362C2×C3#6392911yescharacteristic
363C2×C3D3×#21611yescharacteristic
364C22#8867611yescharacteristic
365C4—41no—
366C4—92no—
367C4—182no—
368C4—181no—
369C4—361no—
370C4—361no—
371C4#8865011yescharacteristic
372C4#8865411yescharacteristic
373C3—31no—
374C3—31no—
375C3—61no—
376C3#1059056111yescharacteristic
377C2#1172114011yescharacteristic
378C2#1172119911yescharacteristic
379C2#1172123611yescharacteristic
subgroupG/H
centerC22#88676
commutator, cosocleC2×C4×H(3,𝔽3)C23
FrattiniC2×C3×C4C2×D32
FittingC4⋊C4×H(3,𝔽3)C22
socleC22×C3D3×#67
prime radicalC22×H(3,𝔽3)C2×D4
hypercenterC4⋊C4((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)#462
LCSG ▷ C2×C4×H(3,𝔽3) ▷ C2×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C22 ◁ C2×C4 ◁ C4⋊C4

Complements

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

NH ≅ G/Ntype
93. (C3⋊Dic3)⋊C3Q16semidirect
213. C3⋊Dic3#216semidirect
228. H(3,𝔽3)#462semidirect
335. C32#4795semidirect
336. C32#5031semidirect

Maximal subgroups

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

subgroupindex
#64096⋊C32
#117210902
C2×#884392
C2×#884332
#117205322
#63637⋊C32
#117203112
#105905663
#105905613