Finite Group Databasebeta

← #13140894

Subgroups — #13140894

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#64153⋊C3C211yesmaximal, characteristic
2C2×#3756⋊C3C211yesmaximal, characteristic
3#11721082C211yesmaximal, characteristic
4C2×#88433C211yesmaximal, characteristic
5#11720583C211yesmaximal, characteristic
6#11720311C211yesmaximal, characteristic
7#11720309C211yesmaximal, characteristic
8#10591580—31nomaximal
9#10591575—31nomaximal
10C2×Q8×H(3,𝔽3)C2211yescharacteristic
11C2×#634⋊C3C2211yescharacteristic
12C2×#634⋊C3C2211yescharacteristic
13C4⋊#1084C2211yescharacteristic
14C2×#6180C2211yescharacteristic
15#3756⋊C3—22no—
16#3750⋊C3—21no—
17#3750⋊C3—21no—
18C4×(C3⋊Dic3)⋊C3C2211yescharacteristic
19C2×(C3⋊(C3⋊C8))⋊C3C2211yescharacteristic
20#88448—21no—
21#88447—21no—
22#88433—22no—
23#64153D311yescharacteristic
24C2×#3756D311yescharacteristic
25C3×#944—31no—
26C2×C3×#216—31no—
27#63844—31no—
28#63843—31no—
29C2×#3719—31no—
30C2×#3719—31no—
31C3×#835—31no—
32C3×#835—31no—
33C3×#835—61no—
34#63572—31no—
35Dic3⋊(C3⋊C8)—31no—
36#63567—31no—
37#63565—31no—
38Q8×H(3,𝔽3)—21no—
39Q8×H(3,𝔽3)—21no—
40C2×C4×H(3,𝔽3)—21no—
41C2×C4×H(3,𝔽3)C2311yescharacteristic
42C2×#1084—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#5182—91no—
54#5090—91no—
55#4795—91no—
56C2×C32×Q8—21no—
57C2×C32×Q8C2×D311yescharacteristic
58C2×C32×Q8C2×D311yescharacteristic
59C2×#634C2×D311yescharacteristic
60C2×#634C2×D311yescharacteristic
61C2×C3×Dic6—31no—
62C2×C3×Dic6—31no—
63#3756—22no—
64#3750—21no—
65#3750—21no—
66C4×C3⋊Dic3C2×D311yescharacteristic
67C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
68C3×#216—62no—
69C3×C4⋊Dic3—31no—
70C3×C4⋊Dic3—31no—
71C3×C4⋊Dic3—61no—
72C3×Dic3⋊C4—61no—
73C3×Dic3⋊C4—61no—
74C3×C4×Dic3—31no—
75C2×C3×C3⋊C8—31no—
76C2×C3×C3⋊C8—31no—
77C2×C3×C3⋊C8—31no—
78C2×C3×C3⋊C8—61no—
79#3724—61no—
80Dic3⋊Dic3—61no—
81Dic32—61no—
82Dic32—61no—
83#3719—62no—
84#3719—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
94C2×C3×Q16—91no—
95C3×C4⋊Q8—91no—
96Q8×Dic3—91no—
97#944—31no—
98#944—91no—
99C2×#216—31no—
100C2×#216—91no—
101C2×#216—91no—
102C2×#216—91no—
103C2×Dic12—91no—
104#887—91no—
105C4×Dic6—91no—
106C3×#103—91no—
107#835—31no—
108#835—31no—
109#835—61no—
110#835—91no—
111#816—91no—
112Dic3⋊C8—91no—
113Dic3⋊C8—91no—
114#810—91no—
115C4⋊(C3⋊C8)—91no—
116C32×Q8—22no—
117C32×Q8—21no—
118C32×Q8—21no—
119C32×Q8—21no—
120C32×Q8—21no—
121C32×Q8—41no—
122C2×C32×C4—21no—
123C2×C32×C4—21no—
124C2×C32×C4—21no—
125C2×C32×C4—41no—
126C2×C32×C4C22×D311yescharacteristic
127C2×C32×C4C22×D311yescharacteristic
128C2×C3⋊Dic3—21no—
129C2×C3⋊Dic3—21no—
130C2×C3⋊Dic3#6711yescharacteristic
131C2×C3⋊Dic3#6711yescharacteristic
132#634—21no—
133#634—21no—
134#634—21no—
135#634#6712yes—
136C2×C3×Dic3—31no—
137C2×C3×Dic3—31no—
138C2×C3×Dic3—61no—
139C2×C3×Dic3—61no—
140C2×C3×Dic3—61no—
141C2×C3×Dic3—61no—
142C2×C3×Dic3—121no—
143C3×Dic6—32no—
144C3×Dic6—61no—
145C3×Dic6—61no—
146C3×Dic6—61no—
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—31no—
157C2×C3×Q8—31no—
158C2×C3×Q8—61no—
159C2×C3×Q8—91no—
160C2×C3×Q8—91no—
161C2×C3×Q8D3211yescharacteristic
162C2×Dic6—31no—
163C2×Dic6—31no—
164C2×Dic6—91no—
165C2×Dic6—91no—
166C3×Q16—182no—
167C2×C3×C8—91no—
168C2×C3×C8—91no—
169C3×C4⋊C4—91no—
170C3×C4⋊C4—181no—
171C3×C4⋊C4—181no—
172C3×C42—91no—
173#216—62no—
174#216—182no—
175#216—182no—
176#216—182no—
177C4⋊Dic3—31no—
178C4⋊Dic3—31no—
179C4⋊Dic3—61no—
180C4⋊Dic3—181no—
181Dic3⋊C4—61no—
182Dic3⋊C4—61no—
183Dic3⋊C4—181no—
184Dic3⋊C4—181no—
185Dic3⋊C4—181no—
186Dic3⋊C4—181no—
187C4×Dic3—31no—
188C4×Dic3—91no—
189C4×Dic3—181no—
190C4×Dic3—181no—
191C4×Dic3—181no—
192C2×C3⋊C8—31no—
193C2×C3⋊C8—31no—
194C2×C3⋊C8—31no—
195C2×C3⋊C8—61no—
196C2×C3⋊C8—91no—
197Dic12—182no—
198C22×C32—21no—
199C22×C32D3×D411yescharacteristic
200C22×C32C2×#6711yescharacteristic
201C32×C4—21no—
202C32×C4—21no—
203C32×C4—42no—
204C32×C4—41no—
205C32×C4—41no—
206C32×C4D3×D411yescharacteristic
207C32×C4#23711yescharacteristic
208C32×C4#23711yescharacteristic
209C32×C4C2×#6711yescharacteristic
210C3⋊Dic3—22no—
211C3⋊Dic3—21no—
212C3⋊Dic3—41no—
213C3⋊Dic3#21612yescomplemented
214C3×Dic3—32no—
215C3×Dic3—61no—
216C3×Dic3—62no—
217C3×Dic3—121no—
218C3×Dic3—121no—
219C3×Dic3—121no—
220C3×Dic3—241no—
221C2×Q16—91no—
222C2×Q16—271no—
223C4⋊Q8—91no—
224C4×Q8—271no—
225C4⋊C8—271no—
226#103—91no—
227#103—271no—
228H(3,𝔽3)#46211yescharacteristic, complemented, Sylow
229C3×Q8—21no—
230C3×Q8—21no—
231C3×Q8—61no—
232C3×Q8—61no—
233C3×Q8—61no—
234C3×Q8—61no—
235C3×Q8—62no—
236C3×Q8—92no—
237C3×Q8—121no—
238C3×Q8—181no—
239C3×Q8—181no—
240C3×Q8—181no—
241C2×C3×C4—21no—
242C2×C3×C4—31no—
243C2×C3×C4—31no—
244C2×C3×C4—61no—
245C2×C3×C4—61no—
246C2×C3×C4—61no—
247C2×C3×C4—91no—
248C2×C3×C4—91no—
249C2×C3×C4—121no—
250C2×C3×C4—181no—
251C2×C3×C4—181no—
252C2×C3×C4—181no—
253C2×C3×C4C2×D3211yescharacteristic
254C2×Dic3—31no—
255C2×Dic3—31no—
256C2×Dic3—61no—
257C2×Dic3—61no—
258C2×Dic3—61no—
259C2×Dic3—61no—
260C2×Dic3—91no—
261C2×Dic3—91no—
262C2×Dic3—121no—
263C2×Dic3—181no—
264C2×Dic3—181no—
265Dic6—32no—
266Dic6—61no—
267Dic6—61no—
268Dic6—61no—
269Dic6—92no—
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×C32D3×Q1611yescharacteristic
284C2×C32#91811yescharacteristic
285C2×C32#93811yescharacteristic
286C2×C32#93911yescharacteristic
287C2×C32#94211yescharacteristic
288C2×C32C2×#21611yescharacteristic
289C2×Q8—91no—
290C2×Q8—91no—
291C2×Q8((C3⋊D3)⋊C3)⋊C211yescharacteristic
292Q16—182no—
293Q16—542no—
294C2×C8—91no—
295C2×C8—91no—
296C4⋊C4—91no—
297C4⋊C4—181no—
298C4⋊C4—181no—
299C4⋊C4—541no—
300C42—91no—
301C42—541no—
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—122no—
317C3×C4—181no—
318C3×C4—182no—
319C3×C4—361no—
320C3×C4—361no—
321C3×C4#380511yescharacteristic
322C3×C4D3×#6711yescharacteristic
323Dic3—32no—
324Dic3—61no—
325Dic3—62no—
326Dic3—92no—
327Dic3—121no—
328Dic3—121no—
329Dic3—121no—
330Dic3—182no—
331Dic3—181no—
332Dic3—241no—
333Dic3—361no—
334C32—21no—
335C32#509011yescharacteristic, complemented
336C32#518211yescharacteristic, complemented
337Q8—21no—
338Q8—21no—
339Q8—92no—
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×C3D3×#21611yescharacteristic
362C2×C3#6394811yescharacteristic
363C2×C3#6397011yescharacteristic
364C22#8867611yescharacteristic
365C4—41no—
366C4—92no—
367C4—181no—
368C4—182no—
369C4—361no—
370C4—361no—
371C4#8867211yescharacteristic
372C4#8867611yescharacteristic
373C3—31no—
374C3—31no—
375C3—61no—
376C3#1059157411yescharacteristic
377C2#1172123411yescharacteristic
378C2#1172123611yescharacteristic
379C2#1172131311yescharacteristic
subgroupG/H
centerC22#88676
commutator, cosocleC2×C4×H(3,𝔽3)C23
FrattiniC2×C3×C4C2×D32
FittingC2×Q8×H(3,𝔽3)C22
socleC22×C3D3×#67
prime radicalC22×H(3,𝔽3)C2×D4
hypercenterC2×Q8((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 ◁ C2×Q8

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#5090semidirect
336. C32#5182semidirect

Maximal subgroups

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

subgroupindex
#64153⋊C32
C2×#3756⋊C32
#117210822
C2×#884332
#117205832
#117203112
#117203092
#105915803
#105915753