Finite Group Databasebeta

← #13140893

Subgroups — #13140893

Order 1728.

427 conjugacy classes of subgroups; only the 99 normal subgroups are shown.

normal subgroups only

This group has 427 conjugacy classes of subgroups — too many to draw. Only its 99 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
1Q8×(C3⋊Dic3)⋊C3C211yesmaximal, characteristic
2C2×#3756⋊C3C211yesmaximal, characteristic
3#11721082C211yesmaximal, characteristic
4#11720583C211yesmaximal, characteristic
5#11720313C211yesmaximal, characteristic
6#11720310C211yesmaximal, characteristic
7#11720306C211yesmaximal, characteristic
8#10591581—31nomaximal
9#10591537—31nomaximal
10C2×Q8×H(3,𝔽3)C2211yescharacteristic
11C2×#634⋊C3C2211yescharacteristic
12C4⋊#1084C2211yescharacteristic
13C2×#6180C2211yescharacteristic
14#3756⋊C3C414yescomplemented
15C4⋊((C3⋊Dic3)⋊C3)—21no—
16C4⋊((C3⋊Dic3)⋊C3)C2211yescharacteristic
17C4×(C3⋊Dic3)⋊C3—21no—
18C4×(C3⋊Dic3)⋊C3C2211yescharacteristic
19C2×(C3⋊(C3⋊C8))⋊C3C2211yescharacteristic
20#88448—21no—
21#88447—21no—
22Q8×C3⋊Dic3D311yescharacteristic
23C2×#3756D311yescharacteristic
24C3×Q8×Dic3—31no—
25C2×C3×#216—31no—
26#63844—31no—
27#63843—31no—
28C3×#835—31no—
29C3×#835—31no—
30C3×#835—61no—
31#63576—31no—
32#63575—31no—
33#63567—31no—
34#63566—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—
42C2×(C3⋊Dic3)⋊C3—21no—
43C2×(C3⋊Dic3)⋊C3—21no—
44C2×(C3⋊Dic3)⋊C3D411yescharacteristic
45C2×(C3⋊Dic3)⋊C3D411yescharacteristic
46#634⋊C3—21no—
47#634⋊C3C2×C412yes—
48#6180—21no—
49(C3⋊(C3⋊C8))⋊C3C2×C412yes—
50Dic3×Q16—91no—
51C4×#216—91no—
52#4790—91no—
53C2×C32×Q8—21no—
54C2×C32×Q8C2×D311yescharacteristic
55C2×C32×Q8C2×D311yescharacteristic
56C2×#634C2×D311yescharacteristic
57C2×C3×Dic6—31no—
58#3756Dic314yescomplemented
59C4⋊(C3⋊Dic3)—21no—
60C4⋊(C3⋊Dic3)C2×D311yescharacteristic
61C4×C3⋊Dic3—21no—
62C4×C3⋊Dic3C2×D311yescharacteristic
63C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
64C3×#216—34no—
65C3×C4⋊Dic3—31no—
66C3×C4⋊Dic3—31no—
67C3×C4⋊Dic3—31no—
68C3×C4⋊Dic3—61no—
69C3×C4⋊Dic3—61no—
70C3×C4×Dic3—31no—
71C3×C4×Dic3—61no—
72C2×C3×C3⋊C8—31no—
73C2×C3×C3⋊C8—31no—
74C2×C3×C3⋊C8—31no—
75C2×C3×C3⋊C8—61no—
76#3724—61no—
77Dic3⋊Dic3—61no—
78Dic32—61no—
79Dic32—61no—
80C22×H(3,𝔽3)C2×D411yescharacteristic
81C4×H(3,𝔽3)—22no—
82C4×H(3,𝔽3)C22×C411yescharacteristic
83C4×H(3,𝔽3)#4111yescharacteristic
84#1084—41no—
85(C3⋊Dic3)⋊C3—22no—
86(C3⋊Dic3)⋊C3—21no—
87(C3⋊Dic3)⋊C3—41no—
88(C3⋊Dic3)⋊C3Q1612yescomplemented
89C2×C3×Q16—91no—
90C3×C4×Q8—91no—
91Q8×Dic3—31no—
92Q8×Dic3—91no—
93Q8×Dic3—91no—
94C2×#216—31no—
95C2×#216—91no—
96#887—91no—
97C4×Dic6—91no—
98C3×#103—91no—
99#835—31no—
100#835—31no—
101#835—61no—
102#835—91no—
103#818—91no—
104#816—91no—
105C8×Dic3—91no—
106C8×Dic3—91no—
107#810—91no—
108#807—91no—
109#807—91no—
110C4×C3⋊C8—91no—
111C32×Q8—21no—
112C32×Q8—21no—
113C32×Q8—22no—
114C32×Q8—41no—
115C32×Q8C4×D312yes—
116C32×Q8C2×Dic312yes—
117C2×C32×C4—21no—
118C2×C32×C4—21no—
119C2×C32×C4—21no—
120C2×C32×C4—41no—
121C2×C32×C4C22×D311yescharacteristic
122C2×C32×C4C22×D311yescharacteristic
123C2×C3⋊Dic3—21no—
124C2×C3⋊Dic3—21no—
125C2×C3⋊Dic3#6711yescharacteristic
126C2×C3⋊Dic3#6711yescharacteristic
127#634—21no—
128#634C2×Dic312yes—
129C2×C3×Dic3—31no—
130C2×C3×Dic3—31no—
131C2×C3×Dic3—61no—
132C2×C3×Dic3—61no—
133C2×C3×Dic3—61no—
134C2×C3×Dic3—61no—
135C2×C3×Dic3—121no—
136C3×Dic6—32no—
137C3×Dic6—61no—
138C3⋊(C3⋊C8)C2×Dic312yes—
139C3×C3⋊C8—32no—
140C3×C3⋊C8—61no—
141C3×C3⋊C8—61no—
142C3×C3⋊C8—121no—
143C4×Q16—271noSylow
144C2×H(3,𝔽3)C4×D411yescharacteristic
145C2×H(3,𝔽3)C2×Q1611yescharacteristic
146C2×H(3,𝔽3)#13511yescharacteristic
147C2×C3×Q8—31no—
148C2×C3×Q8—31no—
149C2×C3×Q8—61no—
150C2×C3×Q8—91no—
151C2×C3×Q8D3211yescharacteristic
152C2×Dic6—31no—
153C2×Dic6—91no—
154C3×Q16—94no—
155C2×C3×C8—91no—
156C2×C3×C8—91no—
157C3×C4⋊C4—91no—
158C3×C4⋊C4—91no—
159C3×C4⋊C4—181no—
160C3×C42—91no—
161C3×C42—181no—
162#216—34no—
163#216—94no—
164C4⋊Dic3—31no—
165C4⋊Dic3—31no—
166C4⋊Dic3—31no—
167C4⋊Dic3—61no—
168C4⋊Dic3—61no—
169C4⋊Dic3—91no—
170C4⋊Dic3—181no—
171C4⋊Dic3—181no—
172Dic3⋊C4—181no—
173Dic3⋊C4—181no—
174C4×Dic3—31no—
175C4×Dic3—61no—
176C4×Dic3—91no—
177C4×Dic3—181no—
178C4×Dic3—181no—
179C4×Dic3—181no—
180C4×Dic3—181no—
181C2×C3⋊C8—31no—
182C2×C3⋊C8—31no—
183C2×C3⋊C8—31no—
184C2×C3⋊C8—61no—
185C2×C3⋊C8—91no—
186C22×C32—21no—
187C22×C32D3×D411yescharacteristic
188C22×C32C2×#6711yescharacteristic
189C32×C4—22no—
190C32×C4—22no—
191C32×C4—21no—
192C32×C4—21no—
193C32×C4—42no—
194C32×C4C2×C4×D311yescharacteristic
195C32×C4#23511yescharacteristic
196C32×C4#23711yescharacteristic
197C32×C4C22×Dic311yescharacteristic
198C3⋊Dic3—22no—
199C3⋊Dic3—21no—
200C3⋊Dic3—41no—
201C3⋊Dic3#21612yescomplemented
202C3×Dic3—32no—
203C3×Dic3—61no—
204C3×Dic3—62no—
205C3×Dic3—121no—
206C3×Dic3—121no—
207C3×Dic3—121no—
208C3×Dic3—241no—
209C2×Q16—91no—
210C4×Q8—91no—
211C4×Q8—271no—
212#107—271no—
213#103—91no—
214#103—271no—
215C4×C8—271no—
216H(3,𝔽3)C4×Q1611yescharacteristic, complemented, Sylow
217C3×Q8—21no—
218C3×Q8—32no—
219C3×Q8—32no—
220C3×Q8—61no—
221C3×Q8—61no—
222C3×Q8—62no—
223C3×Q8—92no—
224C3×Q8—121no—
225C3×Q8—181no—
226C3×Q8D3×Dic312yes—
227C2×C3×C4—21no—
228C2×C3×C4—31no—
229C2×C3×C4—31no—
230C2×C3×C4—61no—
231C2×C3×C4—61no—
232C2×C3×C4—61no—
233C2×C3×C4—91no—
234C2×C3×C4—91no—
235C2×C3×C4—121no—
236C2×C3×C4—181no—
237C2×C3×C4—181no—
238C2×C3×C4—181no—
239C2×C3×C4C2×D3211yescharacteristic
240C2×Dic3—31no—
241C2×Dic3—31no—
242C2×Dic3—61no—
243C2×Dic3—61no—
244C2×Dic3—61no—
245C2×Dic3—61no—
246C2×Dic3—91no—
247C2×Dic3—91no—
248C2×Dic3—121no—
249C2×Dic3—181no—
250C2×Dic3—181no—
251Dic6—32no—
252Dic6—61no—
253Dic6—92no—
254Dic6—181no—
255C3×C8—92no—
256C3×C8—181no—
257C3⋊C8—32no—
258C3⋊C8—61no—
259C3⋊C8—61no—
260C3⋊C8—92no—
261C3⋊C8—121no—
262C2×C32—21no—
263C2×C32—21no—
264C2×C32—21no—
265C2×C32D3×Q1611yescharacteristic
266C2×C32#91911yescharacteristic
267C2×C32C4×#6711yescharacteristic
268C2×C32D4×Dic311yescharacteristic
269C2×C32C2×#21611yescharacteristic
270C2×C32#95011yescharacteristic
271C2×Q8—91no—
272C2×Q8((C3⋊D3)⋊C3)⋊C211yescharacteristic
273Q16—94no—
274C2×C8—91no—
275C2×C8—91no—
276C4⋊C4—91no—
277C4⋊C4—91no—
278C4⋊C4—181no—
279C4⋊C4—541no—
280C42—91no—
281C42—181no—
282C42—541no—
283C22×C3—31no—
284C22×C3—31no—
285C22×C3—61no—
286C22×C3D3×#6711yescharacteristic
287C3×C4—22no—
288C3×C4—31no—
289C3×C4—31no—
290C3×C4—31no—
291C3×C4—31no—
292C3×C4—62no—
293C3×C4—62no—
294C3×C4—61no—
295C3×C4—61no—
296C3×C4—92no—
297C3×C4—122no—
298C3×C4—181no—
299C3×C4—182no—
300C3×C4—361no—
301C3×C4—361no—
302C3×C4C2×D3×Dic311yescharacteristic
303C3×C4#380411yescharacteristic
304Dic3—32no—
305Dic3—61no—
306Dic3—62no—
307Dic3—92no—
308Dic3—121no—
309Dic3—121no—
310Dic3—121no—
311Dic3—182no—
312Dic3—181no—
313Dic3—241no—
314Dic3—361no—
315C32—21no—
316C32C4×#21611yescharacteristic, complemented
317C32Dic3×Q1611yescharacteristic, complemented
318Q8—21no—
319Q8—92no—
320Q8—181no—
321Q8((C3⋊D3)⋊C3)⋊C412yes—
322C2×C4—21no—
323C2×C4—91no—
324C2×C4—91no—
325C2×C4—181no—
326C2×C4—181no—
327C2×C4—181no—
328C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
329C8—92no—
330C8—181no—
331C2×C3—31no—
332C2×C3—31no—
333C2×C3—31no—
334C2×C3—31no—
335C2×C3—31no—
336C2×C3—31no—
337C2×C3—61no—
338C2×C3—61no—
339C2×C3—61no—
340C2×C3D3×#21611yescharacteristic
341C2×C3#6394911yescharacteristic
342C2×C3Dic3×#6711yescharacteristic
343C22#8867611yescharacteristic
344C4—22no—
345C4—92no—
346C4—181no—
347C4—182no—
348C4—361no—
349C4—361no—
350C4C2×((C3⋊D3)⋊C3)⋊C411yescharacteristic
351C4#8867111yescharacteristic
352C3—31no—
353C3—31no—
354C3—61no—
355C3Dic3×#21611yescharacteristic
356C2#1172123511yescharacteristic
357C2#1172123611yescharacteristic
358C2#1172131411yescharacteristic
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
14. #3756⋊C3C4semidirect
58. #3756Dic3semidirect
88. (C3⋊Dic3)⋊C3Q16semidirect
201. C3⋊Dic3#216semidirect
216. H(3,𝔽3)C4×Q16semidirect
316. C32C4×#216semidirect
317. C32Dic3×Q16semidirect

Maximal subgroups

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

subgroupindex
Q8×(C3⋊Dic3)⋊C32
C2×#3756⋊C32
#117210822
#117205832
#117203132
#117203102
#117203062
#105915813
#105915373