Finite Group Databasebeta

← #13140977

Subgroups — #13140977

Order 1728.

487 conjugacy classes of subgroups; only the 83 normal subgroups are shown.

normal subgroups only

This group has 487 conjugacy classes of subgroups — too many to draw. Only its 83 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
3C2×#3756⋊C3C211yesmaximal, characteristic
4#11721090C211yesmaximal, characteristic
5C2×#88439C211yesmaximal, characteristic
6C2×#88433C211yesmaximal, characteristic
7#11720310C211yesmaximal, characteristic
8#10591628—31nomaximal
9#10591626—31nomaximal
10C2×Q8×H(3,𝔽3)C2211yescharacteristic
11C2×#6229C2211yescharacteristic
12C2×#634⋊C3C2211yescharacteristic
13C2×#634⋊C3C2211yescharacteristic
14#88550—22no—
15C2×#6180C2211yescharacteristic
16#3756⋊C3—22no—
17#3750⋊C3—22no—
18C4×(C3⋊Dic3)⋊C3C2211yescharacteristic
19C2×(C3⋊(C3⋊C8))⋊C3C2211yescharacteristic
20#88448—22no—
21#88439—22no—
22#88433—22no—
23#64153D311yescharacteristic
24C2×#3756D311yescharacteristic
25C3×#944—31no—
26C2×C3×#216—31no—
27C2×C3×#216—31no—
28C2×C3×#216—31no—
29C2×C3×#216—61no—
30#63873—31no—
31#63866—31no—
32C2×#3719—31no—
33C2×#3719—31no—
34C2×#3719—31no—
35C2×#3718—31no—
36#63553—31no—
37Dic3×C3⋊C8—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#6229—21no—
45C2×(C3⋊Dic3)⋊C3—21no—
46C2×(C3⋊Dic3)⋊C3—21no—
47C2×(C3⋊Dic3)⋊C3D412yes—
48#634⋊C3—21no—
49#634⋊C3—21no—
50#634⋊C3—21no—
51#634⋊C3—21no—
52#6180D412yes—
53(C3⋊(C3⋊C8))⋊C3D412yes—
54#5183—91no—
55#5183—91no—
56#5092—91no—
57C2×C32×Q8—21no—
58C2×C32×Q8C2×D311yescharacteristic
59C2×C32×Q8C2×D311yescharacteristic
60C2×#634C2×D311yescharacteristic
61C2×#634C2×D311yescharacteristic
62C2×C3×Dic6—31no—
63C2×C3×Dic6—31no—
64C2×C3×Dic6—31no—
65C2×C3×Dic6—31no—
66C2×C3×Dic6—61no—
67#3756—22no—
68#3750—22no—
69C4×C3⋊Dic3C2×D311yescharacteristic
70C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
71C3×#216—62no—
72C3×#216—62no—
73C3×#216—62no—
74C3×#216—64no—
75C3×Dic3⋊C4—62no—
76C3×C4×Dic3—31no—
77C2×C3×C3⋊C8—31no—
78C2×C3×C3⋊C8—31no—
79C2×C3×C3⋊C8—31no—
80C2×C3×C3⋊C8—61no—
81#3724—62no—
82Dic3⋊Dic3—62no—
83#3719—62no—
84#3719—62no—
85#3719—62no—
86#3718—62no—
87C22×H(3,𝔽3)C2×D411yescharacteristic
88C4×H(3,𝔽3)—41no—
89C4×H(3,𝔽3)C2×D411yescharacteristic
90C4×H(3,𝔽3)C2×D411yescharacteristic
91#1084—41no—
92(C3⋊Dic3)⋊C3—41no—
93(C3⋊Dic3)⋊C3—41no—
94(C3⋊Dic3)⋊C3Q1614yescomplemented
95C2×C3×Q16—91no—
96C2×C3×Q16—91no—
97C3×C4⋊Q8—91no—
98#944—31no—
99#944—91no—
100#944—91no—
101#944—91no—
102C2×#216—31no—
103C2×#216—31no—
104C2×#216—31no—
105C2×#216—61no—
106C2×#216—91no—
107C2×#216—91no—
108C2×#216—91no—
109C2×#216—91no—
110C2×#216—91no—
111C2×Dic12—91no—
112C2×Dic12—91no—
113#869—91no—
114C8×Dic3—91no—
115C8×Dic3—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#6712yes—
132#634—21no—
133#634—21no—
134#634—21no—
135#634—21no—
136C2×C3×Dic3—32no—
137C2×C3×Dic3—61no—
138C2×C3×Dic3—61no—
139C2×C3×Dic3—61no—
140C2×C3×Dic3—61no—
141C2×C3×Dic3—121no—
142C3×Dic6—62no—
143C3×Dic6—61no—
144C3×Dic6—61no—
145C3×Dic6—61no—
146C3×Dic6—61no—
147C3×Dic6—61no—
148C3×Dic6—61no—
149C3×Dic6—61no—
150C3×Dic6—61no—
151C3×Dic6—121no—
152C3⋊(C3⋊C8)#6712yes—
153C3×C3⋊C8—32no—
154C3×C3⋊C8—32no—
155C3×C3⋊C8—32no—
156C3×C3⋊C8—62no—
157#494—271noSylow
158C2×H(3,𝔽3)#12711yescharacteristic
159C2×H(3,𝔽3)C2×Q1612yes—
160C2×C3×Q8—31no—
161C2×C3×Q8—31no—
162C2×C3×Q8—61no—
163C2×C3×Q8—91no—
164C2×C3×Q8—91no—
165C2×C3×Q8—91no—
166C2×C3×Q8D3211yescharacteristic
167C2×Dic6—31no—
168C2×Dic6—31no—
169C2×Dic6—31no—
170C2×Dic6—31no—
171C2×Dic6—61no—
172C2×Dic6—91no—
173C2×Dic6—91no—
174C3×Q16—182no—
175C3×Q16—182no—
176C2×C3×C8—91no—
177C2×C3×C8—91no—
178C3×C4⋊C4—182no—
179C3×C42—91no—
180#216—62no—
181#216—62no—
182#216—62no—
183#216—64no—
184#216—182no—
185#216—182no—
186#216—182no—
187#216—182no—
188#216—182no—
189C4⋊Dic3—182no—
190Dic3⋊C4—62no—
191Dic3⋊C4—182no—
192Dic3⋊C4—182no—
193Dic3⋊C4—182no—
194C4×Dic3—31no—
195C4×Dic3—91no—
196C2×C3⋊C8—31no—
197C2×C3⋊C8—31no—
198C2×C3⋊C8—31no—
199C2×C3⋊C8—61no—
200C2×C3⋊C8—91no—
201Dic12—182no—
202Dic12—182no—
203C22×C32—21no—
204C22×C32D3×D411yescharacteristic
205C22×C32C2×#6711yescharacteristic
206C32×C4—21no—
207C32×C4—21no—
208C32×C4—42no—
209C32×C4—41no—
210C32×C4—41no—
211C32×C4D3×D411yescharacteristic
212C32×C4D3×D411yescharacteristic
213C32×C4D3×D411yescharacteristic
214C32×C4C2×#6711yescharacteristic
215C3⋊Dic3—41no—
216C3⋊Dic3—41no—
217C3⋊Dic3#21614yescomplemented
218C3×Dic3—34no—
219C3×Dic3—122no—
220C3×Dic3—121no—
221C3×Dic3—121no—
222C3×Dic3—121no—
223C3×Dic3—121no—
224C2×Q16—91no—
225C2×Q16—91no—
226C2×Q16—271no—
227C2×Q16—271no—
228C4⋊Q8—91no—
229C4⋊Q8—271no—
230C4×C8—271no—
231H(3,𝔽3)#49411yescharacteristic, complemented, Sylow
232C3×Q8—21no—
233C3×Q8—21no—
234C3×Q8—61no—
235C3×Q8—61no—
236C3×Q8—61no—
237C3×Q8—61no—
238C3×Q8—62no—
239C3×Q8—121no—
240C3×Q8—181no—
241C3×Q8—181no—
242C3×Q8—181no—
243C3×Q8—181no—
244C3×Q8—181no—
245C3×Q8—181no—
246C2×C3×C4—21no—
247C2×C3×C4—31no—
248C2×C3×C4—31no—
249C2×C3×C4—61no—
250C2×C3×C4—61no—
251C2×C3×C4—61no—
252C2×C3×C4—92no—
253C2×C3×C4—121no—
254C2×C3×C4—181no—
255C2×C3×C4—181no—
256C2×C3×C4—181no—
257C2×C3×C4C2×D3211yescharacteristic
258C2×Dic3—32no—
259C2×Dic3—61no—
260C2×Dic3—61no—
261C2×Dic3—61no—
262C2×Dic3—61no—
263C2×Dic3—92no—
264C2×Dic3—121no—
265C2×Dic3—181no—
266C2×Dic3—181no—
267Dic6—61no—
268Dic6—61no—
269Dic6—61no—
270Dic6—61no—
271Dic6—62no—
272Dic6—61no—
273Dic6—61no—
274Dic6—61no—
275Dic6—61no—
276Dic6—121no—
277Dic6—181no—
278Dic6—181no—
279Dic6—181no—
280Dic6—181no—
281C3×C8—92no—
282C3×C8—92no—
283C3⋊C8—32no—
284C3⋊C8—32no—
285C3⋊C8—32no—
286C3⋊C8—62no—
287C3⋊C8—92no—
288C2×C32—22no—
289C2×C32—21no—
290C2×C32D3×Q1612yes—
291C2×C32#94011yescharacteristic
292C2×C32#94011yescharacteristic
293C2×C32C2×#21612yes—
294C2×Q8—91no—
295C2×Q8—91no—
296C2×Q8—91no—
297C2×Q8((C3⋊D3)⋊C3)⋊C211yescharacteristic
298Q16—182no—
299Q16—182no—
300Q16—542no—
301Q16—542no—
302C2×C8—91no—
303C2×C8—91no—
304C4⋊C4—182no—
305C4⋊C4—542no—
306C42—91no—
307C22×C3—31no—
308C22×C3—31no—
309C22×C3—61no—
310C22×C3D3×#6711yescharacteristic
311C3×C4—31no—
312C3×C4—31no—
313C3×C4—31no—
314C3×C4—31no—
315C3×C4—41no—
316C3×C4—61no—
317C3×C4—61no—
318C3×C4—94no—
319C3×C4—121no—
320C3×C4—121no—
321C3×C4—122no—
322C3×C4—361no—
323C3×C4—361no—
324C3×C4—361no—
325C3×C4D3×#6711yescharacteristic
326C3×C4#381111yescharacteristic
327Dic3—34no—
328Dic3—94no—
329Dic3—121no—
330Dic3—121no—
331Dic3—122no—
332Dic3—121no—
333Dic3—121no—
334Dic3—361no—
335Dic3—361no—
336C32—21no—
337C32#509211yescharacteristic, complemented
338C32#518311yescharacteristic, complemented
339Q8—21no—
340Q8—21no—
341Q8—181no—
342Q8—181no—
343Q8—181no—
344Q8—181no—
345Q8—181no—
346Q8—181no—
347C2×C4—21no—
348C2×C4—92no—
349C2×C4—181no—
350C2×C4—181no—
351C2×C4—181no—
352C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
353C8—92no—
354C8—92no—
355C2×C3—32no—
356C2×C3—32no—
357C2×C3—31no—
358C2×C3—31no—
359C2×C3—62no—
360C2×C3—61no—
361C2×C3D3×#21612yes—
362C2×C3#6397911yescharacteristic
363C22#8867611yescharacteristic
364C4—41no—
365C4—94no—
366C4—361no—
367C4—361no—
368C4—361no—
369C4#8867611yescharacteristic
370C4#8867711yescharacteristic
371C3—31no—
372C3—31no—
373C3—61no—
374C3#1059162611yescharacteristic
375C2#1172123612yes—
376C2#1172131711yescharacteristic
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)#494
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
231. H(3,𝔽3)#494semidirect
337. C32#5092semidirect
338. C32#5183semidirect

Maximal subgroups

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

subgroupindex
C2×#885502
#64153⋊C32
C2×#3756⋊C32
#117210902
C2×#884392
C2×#884332
#117203102
#105916283
#105916263