Finite Group Databasebeta

← #10592489⋊C3 #13141913

Subgroups — #10592489⋊C3 #13141913

Order 1728.

427 conjugacy classes of subgroups; only the 116 normal subgroups are shown.

normal subgroups only

This group has 427 conjugacy classes of subgroups — too many to draw. Only its 116 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#64096⋊C3C211yesmaximal, characteristic
4H(3,𝔽3)×#103C211yesmaximal, characteristic
5#63643⋊C3C211yesmaximal, characteristic
6C8×(C3⋊Dic3)⋊C3C211yesmaximal, characteristic
7#63634⋊C3C211yesmaximal, characteristic
8#10592489C311yesmaximal, characteristic, complemented
9C3×#4790—31nomaximal
10C2×Q8×H(3,𝔽3)C2211yescharacteristic
11C2×#634⋊C3C2211yescharacteristic
12C2×C8×H(3,𝔽3)C2211yescharacteristic
13C4⋊C4×H(3,𝔽3)C2211yescharacteristic
14#3756⋊C3C414yescomplemented
15C4⋊((C3⋊Dic3)⋊C3)—21no—
16C4⋊((C3⋊Dic3)⋊C3)C2211yescharacteristic
17#3750⋊C3—21no—
18C4×(C3⋊Dic3)⋊C3—21no—
19C4×(C3⋊Dic3)⋊C3—21no—
20C4×(C3⋊Dic3)⋊C3C2211yescharacteristic
21C2×(C3⋊(C3⋊C8))⋊C3C2211yescharacteristic
22Q8×C3⋊Dic3C2×C311yescharacteristic
23C2×#3756C2×C311yescharacteristic
24#64096C2×C311yescharacteristic
25C3×Q8×Dic3—31no—
26C2×C3×#216—31no—
27C3×#887—31no—
28C32×#103—21no—
29C32×#103D311yescharacteristic
30C32×#103C2×C311yescharacteristic
31#63643C2×C311yescharacteristic
32C8×C3⋊Dic3C2×C311yescharacteristic
33#63634C2×C311yescharacteristic
34C3×#816—31no—
35C3×C8×Dic3—31no—
36C3×#807—31no—
37Q8×H(3,𝔽3)—21no—
38Q8×H(3,𝔽3)C2×C412yes—
39C2×C4×H(3,𝔽3)—21no—
40C2×C4×H(3,𝔽3)—21no—
41C2×C4×H(3,𝔽3)C2311yescharacteristic
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—
48C8×H(3,𝔽3)—21no—
49(C3⋊(C3⋊C8))⋊C3C2×C412yes—
50C3×C4×Q16—91no—
51#4790—31no—
52#4790—91no—
53C2×C32×Q8—21no—
54C2×C32×Q8C2×D311yescharacteristic
55C2×C32×Q8C22×C311yescharacteristic
56C2×#634C22×C311yescharacteristic
57C2×C3×Dic6—31no—
58C2×C32×C8—21no—
59C2×C32×C8C2×D311yescharacteristic
60C2×C32×C8C22×C311yescharacteristic
61C32×C4⋊C4—21no—
62C32×C4⋊C4C2×D311yescharacteristic
63C32×C4⋊C4C22×C311yescharacteristic
64#3756C3×C414yescomplemented
65C4⋊(C3⋊Dic3)—21no—
66C4⋊(C3⋊Dic3)C22×C311yescharacteristic
67#3750—21no—
68C4×C3⋊Dic3—21no—
69C4×C3⋊Dic3—21no—
70C4×C3⋊Dic3C22×C311yescharacteristic
71C2×C3⋊(C3⋊C8)C22×C311yescharacteristic
72C3×#216—34no—
73C3×C4⋊Dic3—31no—
74C3×C4⋊Dic3—61no—
75C3×Dic3⋊C4—61no—
76C3×C4×Dic3—31no—
77C3×C4×Dic3—61no—
78C3×C4×Dic3—61no—
79C2×C3×C3⋊C8—31no—
80C22×H(3,𝔽3)C2×D411yescharacteristic
81C4×H(3,𝔽3)—22no—
82C4×H(3,𝔽3)—41no—
83C4×H(3,𝔽3)C22×C411yescharacteristic
84C4×H(3,𝔽3)#4111yescharacteristic
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—
91C3×C4×Q8—91no—
92Q8×Dic3—31no—
93Q8×Dic3—91no—
94C2×#216—31no—
95C2×#216—91no—
96#887—31no—
97#887—91no—
98C3×#107—91no—
99C3×#103—31no—
100C3×#103—31no—
101C3×#103—61no—
102C3×#103—91no—
103C3×#103C3×D311yescharacteristic
104C3×C4×C8—91no—
105#816—31no—
106#816—91no—
107C8×Dic3—31no—
108C8×Dic3—91no—
109#807—31no—
110#807—91no—
111C32×Q8—21no—
112C32×Q8—21no—
113C32×Q8—22no—
114C32×Q8—41no—
115C32×Q8C4×D312yes—
116C32×Q8C2×C3×C412yes—
117C2×C32×C4—21no—
118C2×C32×C4—21no—
119C2×C32×C4—21no—
120C2×C32×C4—21no—
121C2×C32×C4—21no—
122C2×C32×C4—41no—
123C2×C32×C4—41no—
124C2×C32×C4C22×D311yescharacteristic
125C2×C32×C4C23×C311yescharacteristic
126C2×C3⋊Dic3—21no—
127C2×C3⋊Dic3—21no—
128C2×C3⋊Dic3C3×D411yescharacteristic
129C2×C3⋊Dic3C3×D411yescharacteristic
130#634—21no—
131#634C2×C3×C412yes—
132C2×C3×Dic3—31no—
133C2×C3×Dic3—31no—
134C2×C3×Dic3—61no—
135C2×C3×Dic3—61no—
136C3×Dic6—32no—
137C3×Dic6—61no—
138C32×C8—21no—
139C32×C8—21no—
140C32×C8—41no—
141C3⋊(C3⋊C8)C2×C3×C412yes—
142C3×C3⋊C8—32no—
143C4×Q16—91noSylow
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×Q8C2×C3×D311yescharacteristic
152C2×Dic6—31no—
153C2×Dic6—91no—
154C3×Q16—94no—
155C2×C3×C8—31no—
156C2×C3×C8—31no—
157C2×C3×C8—61no—
158C2×C3×C8—91no—
159C2×C3×C8C2×C3×D311yescharacteristic
160C3×C4⋊C4—31no—
161C3×C4⋊C4—31no—
162C3×C4⋊C4—61no—
163C3×C4⋊C4—91no—
164C3×C4⋊C4—181no—
165C3×C4⋊C4—181no—
166C3×C4⋊C4C2×C3×D311yescharacteristic
167C3×C42—91no—
168C3×C42—181no—
169C3×C42—181no—
170#216—34no—
171#216—94no—
172C4⋊Dic3—31no—
173C4⋊Dic3—61no—
174C4⋊Dic3—91no—
175C4⋊Dic3—181no—
176Dic3⋊C4—61no—
177Dic3⋊C4—181no—
178C4×Dic3—31no—
179C4×Dic3—61no—
180C4×Dic3—61no—
181C4×Dic3—91no—
182C4×Dic3—181no—
183C4×Dic3—181no—
184C2×C3⋊C8—31no—
185C2×C3⋊C8—91no—
186C22×C32—21no—
187C22×C32D3×D411yescharacteristic
188C22×C32C2×C3×D411yescharacteristic
189C32×C4—22no—
190C32×C4—22no—
191C32×C4—21no—
192C32×C4—21no—
193C32×C4—42no—
194C32×C4—41no—
195C32×C4—41no—
196C32×C4—81no—
197C32×C4C2×C4×D311yescharacteristic
198C32×C4#23711yescharacteristic
199C32×C4C22×C3×C411yescharacteristic
200C32×C4C3×#4111yescharacteristic
201C3⋊Dic3—22no—
202C3⋊Dic3—21no—
203C3⋊Dic3—41no—
204C3⋊Dic3C3×Q1612yescomplemented
205C3×Dic3—32no—
206C3×Dic3—62no—
207C3×Dic3—61no—
208C3×Dic3—121no—
209C2×Q16—91no—
210C4×Q8—91no—
211C4×Q8—91no—
212#107—91no—
213#103—91no—
214#103(C3⋊D3)⋊C311yescharacteristic
215C4×C8—91no—
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×Q8C3×C4×D312yes—
227C2×C3×C4—21no—
228C2×C3×C4—21no—
229C2×C3×C4—31no—
230C2×C3×C4—31no—
231C2×C3×C4—61no—
232C2×C3×C4—61no—
233C2×C3×C4—61no—
234C2×C3×C4—61no—
235C2×C3×C4—61no—
236C2×C3×C4—91no—
237C2×C3×C4—91no—
238C2×C3×C4—121no—
239C2×C3×C4—121no—
240C2×C3×C4—181no—
241C2×C3×C4—181no—
242C2×C3×C4C22×C3×D311yescharacteristic
243C2×Dic3—31no—
244C2×Dic3—31no—
245C2×Dic3—61no—
246C2×Dic3—61no—
247C2×Dic3—91no—
248C2×Dic3—91no—
249C2×Dic3—181no—
250C2×Dic3—181no—
251Dic6—32no—
252Dic6—61no—
253Dic6—92no—
254Dic6—181no—
255C3×C8—21no—
256C3×C8—61no—
257C3×C8—61no—
258C3×C8—92no—
259C3×C8—121no—
260C3⋊C8—32no—
261C3⋊C8—92no—
262C2×C32—21no—
263C2×C32—21no—
264C2×C32—21no—
265C2×C32#67⋊C411yescharacteristic
266C2×C32#91611yescharacteristic
267C2×C32D3×Q1611yescharacteristic
268C2×C32C3×C4×D411yescharacteristic
269C2×C32C2×C3×Q1611yescharacteristic
270C2×C32C3×#13511yescharacteristic
271C2×Q8—91no—
272C2×Q8C2×(C3⋊D3)⋊C311yescharacteristic
273Q16—94no—
274C2×C8—91no—
275C2×C8C2×(C3⋊D3)⋊C311yescharacteristic
276C4⋊C4—91no—
277C4⋊C4—181no—
278C4⋊C4—181no—
279C4⋊C4C2×(C3⋊D3)⋊C311yescharacteristic
280C42—91no—
281C42—181no—
282C42—181no—
283C22×C3—31no—
284C22×C3—31no—
285C22×C3—61no—
286C22×C3C3×D3×D411yescharacteristic
287C3×C4—22no—
288C3×C4—31no—
289C3×C4—31no—
290C3×C4—31no—
291C3×C4—31no—
292C3×C4—41no—
293C3×C4—62no—
294C3×C4—62no—
295C3×C4—61no—
296C3×C4—61no—
297C3×C4—92no—
298C3×C4—122no—
299C3×C4—121no—
300C3×C4—121no—
301C3×C4—181no—
302C3×C4—182no—
303C3×C4—241no—
304C3×C4—361no—
305C3×C4C2×C3×C4×D311yescharacteristic
306C3×C4C3×#23711yescharacteristic
307Dic3—32no—
308Dic3—62no—
309Dic3—61no—
310Dic3—92no—
311Dic3—121no—
312Dic3—182no—
313Dic3—181no—
314Dic3—361no—
315C32—21no—
316C32#479011yescharacteristic, complemented
317C32C3×C4×Q1611yescharacteristic, complemented
318Q8—21no—
319Q8—92no—
320Q8—181no—
321Q8C4×(C3⋊D3)⋊C312yes—
322C2×C4—21no—
323C2×C4—21no—
324C2×C4—91no—
325C2×C4—91no—
326C2×C4—181no—
327C2×C4—181no—
328C2×C4C22×(C3⋊D3)⋊C311yescharacteristic
329C8—21no—
330C8—92no—
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×C3C3×#67⋊C411yescharacteristic
341C2×C3C3×#91611yescharacteristic
342C2×C3C3×D3×Q1611yescharacteristic
343C22D4×(C3⋊D3)⋊C311yescharacteristic
344C4—22no—
345C4—41no—
346C4—92no—
347C4—181no—
348C4—182no—
349C4—361no—
350C4C2×C4×(C3⋊D3)⋊C311yescharacteristic
351C4#3830⋊C311yescharacteristic
352C3—31no—
353C3—31no—
354C3—61no—
355C3C3×#479011yescharacteristic
356C2#64090⋊C311yescharacteristic
357C2#64125⋊C311yescharacteristic
358C2Q16×(C3⋊D3)⋊C311yescharacteristic
subgroupG/H
centerC22D4×(C3⋊D3)⋊C3
commutatorC32×C4C22×C3×C4
FrattiniC2×C3×C4C22×C3×D3
FittingH(3,𝔽3)×#103C2
socleC22×C3C3×D3×D4
prime radicalC22×H(3,𝔽3)C2×D4
cosocleC2×C32×C4C23×C3
hypercenter#103(C3⋊D3)⋊C3
hypocenterC32C3×C4×Q16
LCSG ▷ C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C22 ◁ C2×C4 ◁ #103

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
8. #10592489C3semidirect
14. #3756⋊C3C4semidirect
64. #3756C3×C4semidirect
88. (C3⋊Dic3)⋊C3Q16semidirect
204. C3⋊Dic3C3×Q16semidirect
216. H(3,𝔽3)C4×Q16semidirect
316. C32#4790semidirect
317. C32C3×C4×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
#64096⋊C32
H(3,𝔽3)×#1032
#63643⋊C32
C8×(C3⋊Dic3)⋊C32
#63634⋊C32
#105924893
C3×#47903