Finite Group Databasebeta

← #13140981

Subgroups — #13140981

Order 1728.

438 conjugacy classes of subgroups; only the 75 normal subgroups are shown.

normal subgroups only

This group has 438 conjugacy classes of subgroups — too many to draw. Only its 75 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×#88550C211yesmaximal, characteristic
2#64153⋊C3C211yesmaximal, characteristic
3#11721083C211yesmaximal, characteristic
4C2×#88439C211yesmaximal, characteristic
5#63662⋊C3C211yesmaximal, characteristic
6#11720313C211yesmaximal, characteristic
7#11720311C211yesmaximal, characteristic
8#10591580—31nomaximal
9#10591574—31nomaximal
10C2×Q8×H(3,𝔽3)C2211yescharacteristic
11C2×#6229C2211yescharacteristic
12C2×#634⋊C3C2211yescharacteristic
13#88550—22no—
14C2×#6180C2211yescharacteristic
15C4⋊((C3⋊Dic3)⋊C3)C2211yescharacteristic
16#3750⋊C3—21no—
17#3750⋊C3—21no—
18C4×(C3⋊Dic3)⋊C3C2211yescharacteristic
19C2×(C3⋊(C3⋊C8))⋊C3C2211yescharacteristic
20#88449—21no—
21#88447—21no—
22#88439—22no—
23#64153D311yescharacteristic
24C3×#944—31no—
25C2×C3×#216—31no—
26C2×C3×#216—31no—
27C2×C3×#216—61no—
28#63843—31no—
29Dic3×Dic6—31no—
30C2×#3719—31no—
31C2×#3718—31no—
32#63662D311yescharacteristic
33C3×#835—31no—
34#63572—31no—
35Dic3⋊(C3⋊C8)—31no—
36#63567—31no—
37#63566—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—
43#6229—21no—
44#6229D412yes—
45C2×(C3⋊Dic3)⋊C3—21no—
46C2×(C3⋊Dic3)⋊C3—21no—
47C2×(C3⋊Dic3)⋊C3D411yescharacteristic
48C2×(C3⋊Dic3)⋊C3D411yescharacteristic
49#634⋊C3—21no—
50#634⋊C3—21no—
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×C3×Dic6—31no—
61C2×C3×Dic6—31no—
62C2×C3×Dic6—31no—
63C2×C3×Dic6—61no—
64C4⋊(C3⋊Dic3)C2×D311yescharacteristic
65#3750—21no—
66#3750—21no—
67C4×C3⋊Dic3C2×D311yescharacteristic
68C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
69C3×#216—62no—
70C3×#216—62no—
71C3×#216—64no—
72C3×C4⋊Dic3—31no—
73C3×Dic3⋊C4—61no—
74C3×Dic3⋊C4—61no—
75C3×C4×Dic3—31no—
76C2×C3×C3⋊C8—31no—
77C2×C3×C3⋊C8—31no—
78C2×C3×C3⋊C8—31no—
79C2×C3×C3⋊C8—61no—
80Dic3⋊Dic3—61no—
81Dic3⋊Dic3—61no—
82Dic32—61no—
83Dic32—61no—
84#3719—62no—
85#3718—62no—
86C22×H(3,𝔽3)C2×D411yescharacteristic
87C4×H(3,𝔽3)—41no—
88C4×H(3,𝔽3)C2×D411yescharacteristic
89C4×H(3,𝔽3)#4111yescharacteristic
90#1084—22no—
91(C3⋊Dic3)⋊C3—21no—
92(C3⋊Dic3)⋊C3—41no—
93(C3⋊Dic3)⋊C3—41no—
94(C3⋊Dic3)⋊C3Q1612yescomplemented
95C2×C3×Q16—91no—
96C3×C4⋊Q8—91no—
97Q8×Dic3—91no—
98#944—31no—
99#944—91no—
100C2×#216—31no—
101C2×#216—31no—
102C2×#216—61no—
103C2×#216—91no—
104C2×#216—91no—
105C2×Dic12—91no—
106#887—91no—
107C4×Dic6—91no—
108C3×#103—91no—
109#835—31no—
110#835—91no—
111#835—91no—
112#816—91no—
113Dic3⋊C8—91no—
114Dic3⋊C8—91no—
115#810—91no—
116C4⋊(C3⋊C8)—91no—
117C32×Q8—22no—
118C32×Q8—21no—
119C32×Q8—21no—
120C32×Q8—21no—
121C32×Q8—21no—
122C32×Q8—41no—
123C2×C32×C4—21no—
124C2×C32×C4—21no—
125C2×C32×C4—21no—
126C2×C32×C4—41no—
127C2×C32×C4C22×D311yescharacteristic
128C2×C32×C4C22×D311yescharacteristic
129C2×C3⋊Dic3—21no—
130C2×C3⋊Dic3—21no—
131C2×C3⋊Dic3#6711yescharacteristic
132C2×C3⋊Dic3#6711yescharacteristic
133#634—21no—
134#634—21no—
135C2×C3×Dic3—31no—
136C2×C3×Dic3—31no—
137C2×C3×Dic3—61no—
138C2×C3×Dic3—61no—
139C2×C3×Dic3—61no—
140C2×C3×Dic3—61no—
141C2×C3×Dic3—121no—
142C3×Dic6—32no—
143C3×Dic6—32no—
144C3×Dic6—62no—
145C3×Dic6—61no—
146C3×Dic6—61no—
147C3×Dic6—61no—
148C3×Dic6—61no—
149C3×Dic6—121no—
150C3⋊(C3⋊C8)—21no—
151C3×C3⋊C8—62no—
152C3×C3⋊C8—61no—
153C3×C3⋊C8—61no—
154C3×C3⋊C8—61no—
155#462—271noSylow
156C2×H(3,𝔽3)#12111yescharacteristic
157C2×H(3,𝔽3)C2×Q1611yescharacteristic
158C2×H(3,𝔽3)#13711yescharacteristic
159C2×C3×Q8—31no—
160C2×C3×Q8—31no—
161C2×C3×Q8—61no—
162C2×C3×Q8—91no—
163C2×C3×Q8—91no—
164C2×C3×Q8D3211yescharacteristic
165C2×Dic6—31no—
166C2×Dic6—31no—
167C2×Dic6—31no—
168C2×Dic6—61no—
169C2×Dic6—91no—
170C3×Q16—182no—
171C2×C3×C8—91no—
172C2×C3×C8—91no—
173C3×C4⋊C4—91no—
174C3×C4⋊C4—181no—
175C3×C4⋊C4—181no—
176C3×C42—91no—
177#216—62no—
178#216—62no—
179#216—64no—
180#216—182no—
181#216—182no—
182C4⋊Dic3—31no—
183C4⋊Dic3—91no—
184C4⋊Dic3—181no—
185Dic3⋊C4—61no—
186Dic3⋊C4—61no—
187Dic3⋊C4—181no—
188Dic3⋊C4—181no—
189Dic3⋊C4—181no—
190Dic3⋊C4—181no—
191C4×Dic3—31no—
192C4×Dic3—91no—
193C4×Dic3—181no—
194C4×Dic3—181no—
195C4×Dic3—181no—
196C2×C3⋊C8—31no—
197C2×C3⋊C8—31no—
198C2×C3⋊C8—31no—
199C2×C3⋊C8—61no—
200C2×C3⋊C8—91no—
201Dic12—182no—
202C22×C32—21no—
203C22×C32D3×D411yescharacteristic
204C22×C32C2×#6711yescharacteristic
205C32×C4—21no—
206C32×C4—21no—
207C32×C4—42no—
208C32×C4—41no—
209C32×C4—41no—
210C32×C4#23511yescharacteristic
211C32×C4D3×D411yescharacteristic
212C32×C4D3×D411yescharacteristic
213C32×C4#23711yescharacteristic
214C3⋊Dic3—21no—
215C3⋊Dic3—41no—
216C3⋊Dic3—41no—
217C3⋊Dic3#21612yescomplemented
218C3×Dic3—32no—
219C3×Dic3—62no—
220C3×Dic3—61no—
221C3×Dic3—62no—
222C3×Dic3—122no—
223C3×Dic3—121no—
224C3×Dic3—121no—
225C2×Q16—91no—
226C2×Q16—271no—
227C4⋊Q8—91no—
228C4×Q8—271no—
229C4⋊C8—271no—
230#103—91no—
231#103—271no—
232H(3,𝔽3)#46211yescharacteristic, complemented, Sylow
233C3×Q8—21no—
234C3×Q8—21no—
235C3×Q8—61no—
236C3×Q8—61no—
237C3×Q8—61no—
238C3×Q8—61no—
239C3×Q8—62no—
240C3×Q8—92no—
241C3×Q8—121no—
242C3×Q8—181no—
243C3×Q8—181no—
244C3×Q8—181no—
245C2×C3×C4—21no—
246C2×C3×C4—31no—
247C2×C3×C4—31no—
248C2×C3×C4—61no—
249C2×C3×C4—61no—
250C2×C3×C4—61no—
251C2×C3×C4—91no—
252C2×C3×C4—91no—
253C2×C3×C4—121no—
254C2×C3×C4—181no—
255C2×C3×C4—181no—
256C2×C3×C4—181no—
257C2×C3×C4C2×D3211yescharacteristic
258C2×Dic3—31no—
259C2×Dic3—31no—
260C2×Dic3—61no—
261C2×Dic3—61no—
262C2×Dic3—61no—
263C2×Dic3—61no—
264C2×Dic3—91no—
265C2×Dic3—91no—
266C2×Dic3—121no—
267C2×Dic3—181no—
268C2×Dic3—181no—
269Dic6—32no—
270Dic6—32no—
271Dic6—62no—
272Dic6—61no—
273Dic6—61no—
274Dic6—61no—
275Dic6—61no—
276Dic6—121no—
277Dic6—181no—
278Dic6—181no—
279C3×C8—181no—
280C3×C8—181no—
281C3⋊C8—61no—
282C3⋊C8—61no—
283C3⋊C8—62no—
284C3⋊C8—61no—
285C3⋊C8—181no—
286C2×C32—21no—
287C2×C32—21no—
288C2×C32—21no—
289C2×C32#88411yescharacteristic
290C2×C32#91511yescharacteristic
291C2×C32D3×Q1611yescharacteristic
292C2×C32#93811yescharacteristic
293C2×C32#94211yescharacteristic
294C2×C32C2×#21611yescharacteristic
295C2×Q8—91no—
296C2×Q8—91no—
297C2×Q8((C3⋊D3)⋊C3)⋊C211yescharacteristic
298Q16—182no—
299Q16—542no—
300C2×C8—91no—
301C2×C8—91no—
302C4⋊C4—91no—
303C4⋊C4—181no—
304C4⋊C4—181no—
305C4⋊C4—541no—
306C42—91no—
307C42—541no—
308C22×C3—31no—
309C22×C3—31no—
310C22×C3—61no—
311C22×C3D3×#6711yescharacteristic
312C3×C4—31no—
313C3×C4—31no—
314C3×C4—31no—
315C3×C4—31no—
316C3×C4—41no—
317C3×C4—61no—
318C3×C4—61no—
319C3×C4—92no—
320C3×C4—121no—
321C3×C4—121no—
322C3×C4—122no—
323C3×C4—181no—
324C3×C4—182no—
325C3×C4—361no—
326C3×C4—361no—
327C3×C4#380411yescharacteristic
328C3×C4#381111yescharacteristic
329Dic3—32no—
330Dic3—62no—
331Dic3—62no—
332Dic3—61no—
333Dic3—92no—
334Dic3—122no—
335Dic3—121no—
336Dic3—121no—
337Dic3—181no—
338Dic3—361no—
339Dic3—361no—
340C32—21no—
341C32#479511yescharacteristic, complemented
342C32#509011yescharacteristic, complemented
343Q8—21no—
344Q8—21no—
345Q8—92no—
346Q8—181no—
347Q8—181no—
348Q8—181no—
349C2×C4—21no—
350C2×C4—91no—
351C2×C4—91no—
352C2×C4—181no—
353C2×C4—181no—
354C2×C4—181no—
355C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
356C8—181no—
357C8—181no—
358C2×C3—31no—
359C2×C3—31no—
360C2×C3—31no—
361C2×C3—31no—
362C2×C3—31no—
363C2×C3—31no—
364C2×C3—61no—
365C2×C3—61no—
366C2×C3—61no—
367C2×C3D3×#21611yescharacteristic
368C2×C3#6394611yescharacteristic
369C2×C3#6395811yescharacteristic
370C22#8867611yescharacteristic
371C4—41no—
372C4—92no—
373C4—181no—
374C4—182no—
375C4—361no—
376C4—361no—
377C4#8867111yescharacteristic
378C4#8867711yescharacteristic
379C3—31no—
380C3—31no—
381C3—61no—
382C3#1059157511yescharacteristic
383C2#1172123611yescharacteristic
384C2#1172123811yescharacteristic
385C2#1172130311yescharacteristic
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
94. (C3⋊Dic3)⋊C3Q16semidirect
217. C3⋊Dic3#216semidirect
232. H(3,𝔽3)#462semidirect
341. C32#4795semidirect
342. C32#5090semidirect

Maximal subgroups

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

subgroupindex
C2×#885502
#64153⋊C32
#117210832
C2×#884392
#63662⋊C32
#117203132
#117203112
#105915803
#105915743