Finite Group Databasebeta

← #13140979

Subgroups — #13140979

Order 1728.

445 conjugacy classes of subgroups; only the 91 normal subgroups are shown.

normal subgroups only

This group has 445 conjugacy classes of subgroups — too many to draw. Only its 91 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
2Q8×(C3⋊Dic3)⋊C3C211yesmaximal, characteristic
3#11721083C211yesmaximal, characteristic
4#63662⋊C3C211yesmaximal, characteristic
5#11720326C211yesmaximal, characteristic
6#11720315C211yesmaximal, characteristic
7#11720310C211yesmaximal, characteristic
8#10591581—31nomaximal
9Dic3×#216—31nomaximal
10C2×Q8×H(3,𝔽3)C2211yescharacteristic
11C2×#6229C2211yescharacteristic
12#88550C414yescomplemented
13C2×#6180C2211yescharacteristic
14C4⋊((C3⋊Dic3)⋊C3)—21no—
15C4⋊((C3⋊Dic3)⋊C3)C2211yescharacteristic
16C4⋊((C3⋊Dic3)⋊C3)C2211yescharacteristic
17C4×(C3⋊Dic3)⋊C3—21no—
18C4×(C3⋊Dic3)⋊C3C2211yescharacteristic
19C2×(C3⋊(C3⋊C8))⋊C3C2211yescharacteristic
20#88449—21no—
21#88447—21no—
22Q8×C3⋊Dic3D311yescharacteristic
23C3×Q8×Dic3—31no—
24C2×C3×#216—31no—
25C2×C3×#216—31no—
26C2×C3×#216—61no—
27#63843—31no—
28Dic3×Dic6—31no—
29#63662D311yescharacteristic
30C3×#835—31no—
31#63575—31no—
32#63575—31no—
33#63566—31no—
34#63565—31no—
35#63553—31no—
36Dic3×C3⋊C8—31no—
37Q8×H(3,𝔽3)—21no—
38Q8×H(3,𝔽3)C2×C412yes—
39C2×C4×H(3,𝔽3)—21no—
40C2×C4×H(3,𝔽3)C2311yescharacteristic
41C2×#1084—21no—
42#6229—21no—
43#6229C2×C412yes—
44C2×(C3⋊Dic3)⋊C3—21no—
45C2×(C3⋊Dic3)⋊C3—21no—
46C2×(C3⋊Dic3)⋊C3D411yescharacteristic
47C2×(C3⋊Dic3)⋊C3D411yescharacteristic
48#6180C2×C412yes—
49(C3⋊(C3⋊C8))⋊C3—21no—
50Dic3×Q16—91no—
51C4×#216—91no—
52#4790—91no—
53C2×C32×Q8—21no—
54C2×C32×Q8C2×D311yescharacteristic
55C2×C32×Q8C2×D311yescharacteristic
56C2×C3×Dic6—31no—
57C2×C3×Dic6—31no—
58C2×C3×Dic6—61no—
59C4⋊(C3⋊Dic3)—21no—
60C4⋊(C3⋊Dic3)C2×D311yescharacteristic
61C4⋊(C3⋊Dic3)C2×D311yescharacteristic
62C4×C3⋊Dic3—21no—
63C4×C3⋊Dic3C2×D311yescharacteristic
64C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
65C3×#216—34no—
66C3×#216—34no—
67C3×#216—64no—
68C3×C4⋊Dic3—31no—
69C3×C4⋊Dic3—31no—
70C3×C4⋊Dic3—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—
77Dic3⋊Dic3—61no—
78Dic3⋊Dic3—61no—
79Dic32—61no—
80Dic32—61no—
81C22×H(3,𝔽3)C2×D411yescharacteristic
82C4×H(3,𝔽3)—22no—
83C4×H(3,𝔽3)C22×C411yescharacteristic
84C4×H(3,𝔽3)#4111yescharacteristic
85#1084—22no—
86(C3⋊Dic3)⋊C3—21no—
87(C3⋊Dic3)⋊C3—41no—
88(C3⋊Dic3)⋊C3—41no—
89(C3⋊Dic3)⋊C3Q1612yescomplemented
90C2×C3×Q16—91no—
91C3×C4×Q8—91no—
92Q8×Dic3—31no—
93Q8×Dic3—91no—
94Q8×Dic3—91no—
95C2×#216—31no—
96C2×#216—31no—
97C2×#216—61no—
98#887—91no—
99C4×Dic6—91no—
100C3×#103—91no—
101#835—31no—
102#835—91no—
103#835—91no—
104#818—91no—
105#816—91no—
106C8×Dic3—91no—
107C8×Dic3—91no—
108#810—91no—
109#807—91no—
110#807—91no—
111C4×C3⋊C8—91no—
112C32×Q8—21no—
113C32×Q8—21no—
114C32×Q8—22no—
115C32×Q8—41no—
116C32×Q8C4×D312yes—
117C32×Q8C4×D312yes—
118C2×C32×C4—21no—
119C2×C32×C4—21no—
120C2×C32×C4—21no—
121C2×C32×C4—41no—
122C2×C32×C4C22×D311yescharacteristic
123C2×C32×C4C22×D311yescharacteristic
124C2×C3⋊Dic3—21no—
125C2×C3⋊Dic3—21no—
126C2×C3⋊Dic3#6711yescharacteristic
127C2×C3⋊Dic3#6711yescharacteristic
128C2×C3×Dic3—31no—
129C2×C3×Dic3—31no—
130C2×C3×Dic3—61no—
131C2×C3×Dic3—61no—
132C2×C3×Dic3—61no—
133C2×C3×Dic3—61no—
134C2×C3×Dic3—121no—
135C3×Dic6—32no—
136C3×Dic6—32no—
137C3×Dic6—62no—
138C3×Dic6—61no—
139C3×Dic6—61no—
140C3×Dic6—121no—
141C3⋊(C3⋊C8)—21no—
142C3×C3⋊C8—32no—
143C3×C3⋊C8—32no—
144C3×C3⋊C8—62no—
145C3×C3⋊C8—61no—
146C4×Q16—271noSylow
147C2×H(3,𝔽3)C4×D411yescharacteristic
148C2×H(3,𝔽3)C2×Q1611yescharacteristic
149C2×H(3,𝔽3)#13511yescharacteristic
150C2×C3×Q8—31no—
151C2×C3×Q8—31no—
152C2×C3×Q8—61no—
153C2×C3×Q8—91no—
154C2×C3×Q8D3211yescharacteristic
155C2×Dic6—31no—
156C2×Dic6—31no—
157C2×Dic6—61no—
158C3×Q16—94no—
159C2×C3×C8—91no—
160C2×C3×C8—91no—
161C3×C4⋊C4—91no—
162C3×C4⋊C4—91no—
163C3×C4⋊C4—181no—
164C3×C42—91no—
165C3×C42—181no—
166#216—34no—
167#216—34no—
168#216—64no—
169C4⋊Dic3—31no—
170C4⋊Dic3—31no—
171C4⋊Dic3—61no—
172C4⋊Dic3—91no—
173C4⋊Dic3—91no—
174C4⋊Dic3—181no—
175C4⋊Dic3—181no—
176Dic3⋊C4—181no—
177Dic3⋊C4—181no—
178C4×Dic3—31no—
179C4×Dic3—61no—
180C4×Dic3—91no—
181C4×Dic3—181no—
182C4×Dic3—181no—
183C4×Dic3—181no—
184C4×Dic3—181no—
185C2×C3⋊C8—31no—
186C2×C3⋊C8—31no—
187C2×C3⋊C8—31no—
188C2×C3⋊C8—61no—
189C2×C3⋊C8—91no—
190C22×C32—21no—
191C22×C32D3×D411yescharacteristic
192C22×C32C2×#6711yescharacteristic
193C32×C4—22no—
194C32×C4—22no—
195C32×C4—21no—
196C32×C4—21no—
197C32×C4—42no—
198C32×C4C2×C4×D311yescharacteristic
199C32×C4C2×C4×D311yescharacteristic
200C32×C4#23511yescharacteristic
201C32×C4#23711yescharacteristic
202C3⋊Dic3—21no—
203C3⋊Dic3—41no—
204C3⋊Dic3—41no—
205C3⋊Dic3#21612yescomplemented
206C3×Dic3—32no—
207C3×Dic3—62no—
208C3×Dic3—61no—
209C3×Dic3—62no—
210C3×Dic3—122no—
211C3×Dic3—121no—
212C3×Dic3—121no—
213C2×Q16—91no—
214C4×Q8—91no—
215C4×Q8—271no—
216#107—271no—
217#103—91no—
218#103—271no—
219C4×C8—271no—
220H(3,𝔽3)C4×Q1611yescharacteristic, complemented, Sylow
221C3×Q8—21no—
222C3×Q8—32no—
223C3×Q8—32no—
224C3×Q8—61no—
225C3×Q8—61no—
226C3×Q8—62no—
227C3×Q8—92no—
228C3×Q8—121no—
229C3×Q8—181no—
230C3×Q8#62412yes—
231C2×C3×C4—21no—
232C2×C3×C4—31no—
233C2×C3×C4—31no—
234C2×C3×C4—61no—
235C2×C3×C4—61no—
236C2×C3×C4—61no—
237C2×C3×C4—91no—
238C2×C3×C4—91no—
239C2×C3×C4—121no—
240C2×C3×C4—181no—
241C2×C3×C4—181no—
242C2×C3×C4—181no—
243C2×C3×C4C2×D3211yescharacteristic
244C2×Dic3—31no—
245C2×Dic3—31no—
246C2×Dic3—61no—
247C2×Dic3—61no—
248C2×Dic3—61no—
249C2×Dic3—61no—
250C2×Dic3—91no—
251C2×Dic3—91no—
252C2×Dic3—121no—
253C2×Dic3—181no—
254C2×Dic3—181no—
255Dic6—32no—
256Dic6—32no—
257Dic6—62no—
258Dic6—61no—
259Dic6—61no—
260Dic6—121no—
261C3×C8—92no—
262C3×C8—181no—
263C3⋊C8—32no—
264C3⋊C8—32no—
265C3⋊C8—62no—
266C3⋊C8—61no—
267C3⋊C8—181no—
268C2×C32—21no—
269C2×C32—21no—
270C2×C32—21no—
271C2×C32#67⋊C411yescharacteristic
272C2×C32#91611yescharacteristic
273C2×C32D3×Q1611yescharacteristic
274C2×C32C4×#6711yescharacteristic
275C2×C32C2×#21611yescharacteristic
276C2×C32#95011yescharacteristic
277C2×Q8—91no—
278C2×Q8((C3⋊D3)⋊C3)⋊C211yescharacteristic
279Q16—94no—
280C2×C8—91no—
281C2×C8—91no—
282C4⋊C4—91no—
283C4⋊C4—91no—
284C4⋊C4—181no—
285C4⋊C4—541no—
286C42—91no—
287C42—181no—
288C42—541no—
289C22×C3—31no—
290C22×C3—31no—
291C22×C3—61no—
292C22×C3D3×#6711yescharacteristic
293C3×C4—22no—
294C3×C4—31no—
295C3×C4—31no—
296C3×C4—31no—
297C3×C4—31no—
298C3×C4—62no—
299C3×C4—62no—
300C3×C4—61no—
301C3×C4—61no—
302C3×C4—92no—
303C3×C4—122no—
304C3×C4—181no—
305C3×C4—182no—
306C3×C4—361no—
307C3×C4—361no—
308C3×C4#380411yescharacteristic
309C3×C4C2×#62411yescharacteristic
310Dic3—32no—
311Dic3—62no—
312Dic3—62no—
313Dic3—61no—
314Dic3—92no—
315Dic3—122no—
316Dic3—121no—
317Dic3—121no—
318Dic3—181no—
319Dic3—361no—
320Dic3—361no—
321C32—21no—
322C32#479011yescharacteristic, complemented
323C32C4×#21611yescharacteristic, complemented
324Q8—21no—
325Q8—92no—
326Q8—181no—
327Q8#619912yes—
328C2×C4—21no—
329C2×C4—91no—
330C2×C4—91no—
331C2×C4—181no—
332C2×C4—181no—
333C2×C4—181no—
334C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
335C8—92no—
336C8—181no—
337C2×C3—31no—
338C2×C3—31no—
339C2×C3—31no—
340C2×C3—31no—
341C2×C3—31no—
342C2×C3—31no—
343C2×C3—61no—
344C2×C3—61no—
345C2×C3—61no—
346C2×C3D3×#21611yescharacteristic
347C2×C3#6394711yescharacteristic
348C2×C3#6395211yescharacteristic
349C22#8867611yescharacteristic
350C4—22no—
351C4—92no—
352C4—181no—
353C4—182no—
354C4—361no—
355C4—361no—
356C4#8867111yescharacteristic
357C4C2×#619911yescharacteristic
358C3—31no—
359C3—31no—
360C3—61no—
361C3#1059153711yescharacteristic
362C2#1172123611yescharacteristic
363C2#1172123911yescharacteristic
364C2#1172129311yescharacteristic
subgroupG/H
centerC22#88676
commutatorC4×H(3,𝔽3)C22×C4
FrattiniC2×C3×C4C2×D32
FittingC2×Q8×H(3,𝔽3)C22
socleC22×C3D3×#67
prime radicalC22×H(3,𝔽3)C2×D4
cosocleC2×C4×H(3,𝔽3)C23
hypercenterC2×Q8((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)C4×Q16
LCSG ▷ 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
12. #88550C4semidirect
89. (C3⋊Dic3)⋊C3Q16semidirect
205. C3⋊Dic3#216semidirect
220. H(3,𝔽3)C4×Q16semidirect
322. C32#4790semidirect
323. C32C4×#216semidirect

Maximal subgroups

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

subgroupindex
C2×#885502
Q8×(C3⋊Dic3)⋊C32
#117210832
#63662⋊C32
#117203262
#117203152
#117203102
#105915813
Dic3×#2163