Finite Group Databasebeta

← #13160886

Subgroups — #13160886

Order 1728.

884 conjugacy classes of subgroups; only the 164 normal subgroups are shown.

normal subgroups only

This group has 884 conjugacy classes of subgroups — too many to draw. Only its 164 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
1C32×#938C211yesmaximal, characteristic
2D3×C4⋊(C3⋊Dic3)C211yesmaximal, characteristic, complemented
3#11722202C211yesmaximal, characteristic, complemented
4C3×#63659C211yesmaximal, characteristic, complemented
5#11722057C211yesmaximal, characteristic, complemented
6#11722056C211yesmaximal, characteristic, complemented
7(C2×D3)⋊(C3⋊(C3⋊C8))C211yesmaximal, characteristic, complemented
8#10592732—31nomaximal
9#10591413—34nomaximal
10C2×C33×D4C2211yescharacteristic
11C2×C32×#67—21no—
12C2×C32×C4×D3C2211yescharacteristic
13C2×D3×C3⋊Dic3—21no—
14C4⋊(C3⋊(C3⋊Dic3))C2211yescharacteristic
15C2×C3⋊(C3⋊(C3⋊C8))C2211yescharacteristic
16C3×C4⋊(C3⋊Dic3)C2211yescharacteristic
17C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
18C32×#217—21no—
19C32×C4⋊Dic3C2211yescharacteristic
20Dic3⋊(C3⋊Dic3)—21no—
21C32×#121—31no—
22C2×C4⋊(C3⋊Dic3)—31no—
23C3×#938D314yes—
24D3×C4⋊Dic3—34no—
25#63659—31no—
26#63659—34no—
27#63659—64no—
28#63659D311yescharacteristic, complemented
29#63657—31no—
30#63634—31no—
31#63634—31no—
32C3×#832—34no—
33#63576—34no—
34#63575—34no—
35(C2×D3)⋊(C3⋊C8)—34no—
36C23×C33—21no—
37C22×C32×D3D411yescharacteristic
38C33×D4—21no—
39C33×D4—21no—
40C2×C33×C4C2311yescharacteristic
41C2×C3⋊(C3⋊Dic3)—21no—
42C2×C3×C3⋊Dic3—21no—
43C32×#67—42no—
44C2×C32×Dic3—21no—
45C2×C32×Dic3D411yescharacteristic
46C32×C4×D3—21no—
47C32×C4×D3—21no—
48D3×C3⋊Dic3—42no—
49C3⋊(C3⋊(C3⋊C8))—21no—
50C3×C3⋊(C3⋊C8)—21no—
51#5033—94no—
52#4774—91no—
53C2×C32×D4—24no—
54C2×C32×D4—61no—
55C2×C32×D4C2×D314yes—
56C2×C32×D4C2×D311yescharacteristic
57C22×C32×C4—31no—
58C22×C3⋊Dic3—61no—
59C2×C3×#67—24no—
60C2×C3×C4×D3C2×D314yes—
61C2×D3×Dic3—64no—
62C32×C4⋊C4—31no—
63C32×C22⋊C4—61no—
64C4⋊(C3⋊Dic3)—31no—
65C4⋊(C3⋊Dic3)—34no—
66C4⋊(C3⋊Dic3)—61no—
67C4⋊(C3⋊Dic3)—64no—
68C4⋊(C3⋊Dic3)C2×D311yescharacteristic
69C2×C3⋊(C3⋊C8)—31no—
70C2×C3⋊(C3⋊C8)—34no—
71C2×C3⋊(C3⋊C8)—64no—
72C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
73C3×#217—24no—
74C3×C4⋊Dic3—34no—
75C3×C4⋊Dic3C2×D314yes—
76C2×C3×C3⋊C8—34no—
77Dic3⋊Dic3—64no—
78C22×C33—41no—
79C22×C33—41no—
80C22×C33—41no—
81C22×C33C2×D411yescharacteristic
82C2×C32×D3—21no—
83C2×C32×D3—21no—
84C2×C32×D3D812yes—
85C33×C4#4111yescharacteristic
86C33×C4#4111yescharacteristic
87C3⋊(C3⋊Dic3)—41no—
88C3×C3⋊Dic3—41no—
89C32×Dic3—21no—
90C32×Dic3—41no—
91C3×#121—34no—
92#938C3⋊D311yescharacteristic
93C2×C4⋊Dic3—94no—
94D3×C4⋊C4—91no—
95C3×#102—91no—
96#832—34no—
97#832—94no—
98#832—91no—
99#832—184no—
100#830—94no—
101(C2×D3)⋊C8—91no—
102#818—91no—
103#807—94no—
104#807—94no—
105#807—91no—
106C23×C32—21no—
107C23×C32—24no—
108C23×C32—31no—
109C23×C32—44no—
110C22×C3×D3#6714yes—
111C32×D4—28no—
112C32×D4—21no—
113C32×D4—24no—
114C32×D4—21no—
115C32×D4—24no—
116C32×D4—44no—
117C32×D4—122no—
118C2×C32×C4—24no—
119C2×C32×C4—31no—
120C2×C32×C4—61no—
121C2×C32×C4—61no—
122C2×C32×C4—61no—
123C2×C32×C4C22×D314yes—
124C2×C32×C4C22×D311yescharacteristic
125C2×C3⋊Dic3—21no—
126C2×C3⋊Dic3—61no—
127C2×C3⋊Dic3—64no—
128C2×C3⋊Dic3—124no—
129C2×C3⋊Dic3—122no—
130C3×#67—48no—
131C2×C3×Dic3—24no—
132C2×C3×Dic3—64no—
133C2×C3×Dic3#6714yes—
134C3×C4×D3—24no—
135C3×C4×D3—24no—
136D3×Dic3—128no—
137C3⋊(C3⋊C8)—21no—
138C3⋊(C3⋊C8)—61no—
139C3⋊(C3⋊C8)—64no—
140C3⋊(C3⋊C8)—124no—
141C3×C3⋊C8—64no—
142#480—271noSylow
143C2×C33—81no—
144C2×C33#12311yescharacteristic
145C2×C33C2×D811yescharacteristic
146C2×C33#13711yescharacteristic
147C32×D3—22no—
148C2×C3×D4—24no—
149C2×C3×D4—64no—
150C2×C3×D4D3214yes—
151C2×C3×D4C2×C3⋊D311yescharacteristic
152C22×C3×C4—34no—
153C2×#67—21no—
154C22×Dic3—184no—
155C2×C4×D3—181no—
156C2×C4×D3C2×C3⋊D311yescharacteristic
157C2×C3×C8—91no—
158C3×C4⋊C4—34no—
159C3×C4⋊C4—91no—
160C3×C22⋊C4—64no—
161#217—21no—
162C4⋊Dic3—34no—
163C4⋊Dic3—94no—
164C4⋊Dic3—91no—
165C4⋊Dic3—184no—
166C4⋊Dic3—184no—
167C4⋊Dic3C2×C3⋊D311yescharacteristic
168Dic3⋊C4—181no—
169C2×C3⋊C8—34no—
170C2×C3⋊C8—94no—
171C2×C3⋊C8—91no—
172C2×C3⋊C8—184no—
173C22×C32—24no—
174C22×C32—32no—
175C22×C32—48no—
176C22×C32—41no—
177C22×C32—41no—
178C22×C32—41no—
179C22×C32—44no—
180C22×C32—44no—
181C22×C32—44no—
182C22×C32—61no—
183C22×C32—61no—
184C22×C32—84no—
185C22×C32—84no—
186C22×C32D3×D411yescharacteristic
187C22×C32C2×#6714yes—
188C2×C3×D3—24no—
189C2×C3×D3—24no—
190C2×C3×D3#21318yes—
191C32×C4—24no—
192C32×C4—24no—
193C32×C4—61no—
194C32×C4—121no—
195C32×C4#23511yescharacteristic
196C32×C4#23714yes—
197C32×C4#23714yes—
198C32×C4#23711yescharacteristic
199C3⋊Dic3—41no—
200C3⋊Dic3—121no—
201C3⋊Dic3—124no—
202C3⋊Dic3—244no—
203C3×Dic3—24no—
204C3×Dic3—44no—
205C3×Dic3—124no—
206#121—31no—
207C2×C4⋊C4—271no—
208#107—271no—
209#107—271no—
210#102—91no—
211#102—271no—
212C22⋊C8—271no—
213C33#48011yescharacteristic, complemented, Sylow
214C23×C3—24no—
215C23×C3—21no—
216C23×C3—34no—
217C23×C3—44no—
218C22×D3#63811yescharacteristic
219C3×D4—28no—
220C3×D4—24no—
221C3×D4—24no—
222C3×D4—21no—
223C3×D4—21no—
224C3×D4—44no—
225C3×D4—128no—
226C2×C3×C4—24no—
227C2×C3×C4—34no—
228C2×C3×C4—64no—
229C2×C3×C4—64no—
230C2×C3×C4—64no—
231C2×C3×C4—181no—
232C2×C3×C4C2×D3214yes—
233C2×C3×C4C22×C3⋊D311yescharacteristic
234#67—42no—
235C2×Dic3—21no—
236C2×Dic3—64no—
237C2×Dic3—184no—
238C2×Dic3—181no—
239C2×Dic3—364no—
240C2×Dic3—368no—
241C2×Dic3#63811yescharacteristic
242C4×D3—21no—
243C4×D3—21no—
244C4×D3—362no—
245C3×C8—181no—
246C3⋊C8—64no—
247C3⋊C8—184no—
248C3⋊C8—181no—
249C3⋊C8—364no—
250C2×C32—24no—
251C2×C32—24no—
252C2×C32—24no—
253C2×C32—62no—
254C2×C32—88no—
255C2×C32—81no—
256C2×C32—84no—
257C2×C32#88311yescharacteristic
258C2×C32D3×D811yescharacteristic
259C2×C32#91511yescharacteristic
260C2×C32C2×#21314yes—
261C2×C32#93514yes—
262C2×C32#95114yes—
263C3×D3—28no—
264C2×D4—61no—
265C2×D4D3×C3⋊D311yescharacteristic
266C22×C4—31no—
267C22×C4—541no—
268C2×C8—91no—
269C2×C8—271no—
270C4⋊C4—31no—
271C4⋊C4—91no—
272C4⋊C4—271no—
273C4⋊C4—541no—
274C22⋊C4—61no—
275C22×C3—24no—
276C22×C3—38no—
277C22×C3—48no—
278C22×C3—41no—
279C22×C3—41no—
280C22×C3—41no—
281C22×C3—44no—
282C22×C3—44no—
283C22×C3—44no—
284C22×C3—64no—
285C22×C3—64no—
286C22×C3—84no—
287C22×C3—84no—
288C22×C3D3×#6714yes—
289C22×C3C2×#63811yescharacteristic
290C2×D3—21no—
291C2×D3—21no—
292C2×D3#375312yes—
293C3×C4—24no—
294C3×C4—24no—
295C3×C4—64no—
296C3×C4—124no—
297C3×C4—361no—
298C3×C4#380414yes—
299C3×C4#380514yes—
300C3×C4#383011yescharacteristic
301C3×C4#383011yescharacteristic
302Dic3—21no—
303Dic3—41no—
304Dic3—124no—
305Dic3—361no—
306Dic3—364no—
307Dic3—724no—
308C32—24no—
309C32#477411yescharacteristic, complemented
310C32#503314yescomplemented
311C23—21no—
312C23—31no—
313D4—21no—
314D4—21no—
315D4—122no—
316C2×C4—31no—
317C2×C4—61no—
318C2×C4—61no—
319C2×C4—61no—
320C2×C4—181no—
321C2×C4—541no—
322C2×C4—1082no—
323C2×C4C2×D3×C3⋊D311yescharacteristic
324C8—181no—
325C8—541no—
326C2×C3—24no—
327C2×C3—24no—
328C2×C3—24no—
329C2×C3—68no—
330C2×C3—88no—
331C2×C3—84no—
332C2×C3—81no—
333C2×C3D3×#21314yes—
334C2×C3#6392914yes—
335C2×C3#6396914yes—
336C2×C3C2×#375311yescharacteristic
337C2×C3#6414411yescharacteristic
338C2×C3#6416011yescharacteristic
339D3—22no—
340C22—32no—
341C22—41no—
342C22—41no—
343C22—41no—
344C22—61no—
345C22—61no—
346C22D3×#63811yescharacteristic
347C4—61no—
348C4—121no—
349C4—361no—
350C4—1081no—
351C4#8902811yescharacteristic
352C4#8902911yescharacteristic
353C3—24no—
354C3#1059141314yescomplemented
355C3#1059273211yescharacteristic, complemented
356C2—62no—
357C2—81no—
358C2D3×#375311yescharacteristic
359C2#1172355011yescharacteristic
360C2#1172361411yescharacteristic
subgroupG/H
centerC22D3×#638
commutator, cosocleC2×C33×C4C23
FrattiniC2×C4C2×D3×C3⋊D3
FittingC2×C33×D4C22
socleC22×C33C2×D4
prime radicalC32×#938C2
hypercenterC2×D4D3×C3⋊D3
hypocenterC33#480
LCSG ▷ C2×C33×C4 ▷ C2×C33 ▷ C33
UCSC1 ◁ C22 ◁ C2×C4 ◁ C2×D4

Complements

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

NH ≅ G/Ntype
2. D3×C4⋊(C3⋊Dic3)C2semidirect
3. #11722202C2semidirect
4. C3×#63659C2semidirect
5. #11722057C2semidirect
6. #11722056C2semidirect
7. (C2×D3)⋊(C3⋊(C3⋊C8))C2semidirect
28. #63659D3semidirect
213. C33#480semidirect
309. C32#4774semidirect
310. C32#5033semidirect
354. C3#10591413semidirect
355. C3#10592732semidirect

Maximal subgroups

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

subgroupindex
C32×#9382
D3×C4⋊(C3⋊Dic3)2
#117222022
C3×#636592
#117220572
#117220562
(C2×D3)⋊(C3⋊(C3⋊C8))2
#105927323
#105914133