Finite Group Databasebeta

← #13166372

Subgroups — #13166372

Order 1728.

688 conjugacy classes of subgroups; only the 78 normal subgroups are shown.

normal subgroups only

This group has 688 conjugacy classes of subgroups — too many to draw. Only its 78 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×C3⋊D3×Dic6C211yesmaximal, characteristic
2#11724514C211yesmaximal, characteristic
3C3×#64283C211yesmaximal, characteristic
4#11722709C214yesmaximal, complemented
5#10593829—31nomaximal
6C2×#6308C411yescharacteristic, complemented
7C2×C3×C4×C3⋊D3C2211yescharacteristic
8C2×C32×Dic6C411yescharacteristic, complemented
9C2×#6293—21no—
10C2×Dic3×C3⋊D3—21no—
11C3⋊D3×Dic6—21no—
12C3⋊D3×Dic6—21no—
13C3⋊D3×Dic6C412yescomplemented
14C3⋊D3×Dic6C2212yes—
15#88994—21no—
16C4⋊(C3⋊#157)C2212yes—
17C4×C3⋊#157—21no—
18C3×#3792—21no—
19C3×#3788—21no—
20C3×#3787C2212yes—
21C2×Q8×C3⋊D3—31no—
22C2×D3×Dic6—62no—
23#64286—31no—
24#64283D311yescharacteristic
25#63784—34no—
26C22×C3×C3⋊D3D411yescharacteristic
27C2×C3⋊#157—22no—
28C2×C33×C4C2×C411yescharacteristic
29C2×C3⋊(C3⋊Dic3)—21no—
30#6308—21no—
31#6308C2×C412yes—
32C2×C3×C3⋊Dic3D411yescharacteristic
33C3×C4×C3⋊D3C2×C411yescharacteristic
34C3×C4×C3⋊D3D411yescharacteristic
35C3×C4×C3⋊D3D411yescharacteristic
36C3×C4×C3⋊D3C2311yescharacteristic
37C2×C32×Dic3—21no—
38C32×Dic6—21no—
39C32×Dic6C2×C412yes—
40#6293—22no—
41#6293—22no—
42Dic3×C3⋊D3—22no—
43Dic3×C3⋊D3—22no—
44C3×#622—22no—
45#5120—91nomaximal
46C2×C32×Q8—31no—
47Q8×C3⋊D3—32no—
48Q8×C3⋊D3—32no—
49Q8×C3⋊D3—61no—
50Q8×C3⋊D3—61no—
51C2×C4×C3⋊D3—61no—
52C2×C4×C3⋊D3C2×D311yescharacteristic
53C2×#634—31no—
54C2×#634—62no—
55C2×#634—61no—
56C2×#634—122no—
57C2×C3×C4×D3—62no—
58C2×C3×Dic6—22no—
59C2×#627—64no—
60C2×D3×Dic3—64no—
61D3×Dic6—64no—
62D3×Dic6—64no—
63D3×Dic6—64no—
64D3×Dic6—64no—
65C22⋊#157—61no—
66#3792—21no—
67C4⋊#157—32no—
68C4×#157—61no—
69#3788—21no—
70#3787C2×D312yes—
71C22×C33C22⋊C411yescharacteristic
72C2×C3×C3⋊D3—22no—
73C2×C3×C3⋊D3C22⋊C411yescharacteristic
74C2×C3×C3⋊D3C2×D411yescharacteristic
75C3⋊#157—42no—
76C33×C4C22⋊C411yescharacteristic
77C33×C4C22×C411yescharacteristic
78C3⋊(C3⋊Dic3)—22no—
79C3×C3⋊Dic3C22⋊C411yescharacteristic
80C3×C3⋊Dic3C2×D411yescharacteristic
81C32×Dic3—22no—
82C2×D3×Q8—182no—
83C22×Dic6—91no—
84C2×C3×M16—91no—
85#926—91no—
86#816—94no—
87C22×C3⋊D3D1211yescharacteristic
88C22×C3×D3—62no—
89C2×#157—62no—
90C32×Q8—32no—
91C32×Q8—61no—
92C2×C32×C4—22no—
93C2×C32×C4—42no—
94C2×C32×C4—61no—
95C2×C32×C4C4×D311yescharacteristic
96C2×C3⋊Dic3—61no—
97C2×C3⋊Dic3—64no—
98C2×C3⋊Dic3—124no—
99C2×C3⋊Dic3#6711yescharacteristic
100C4×C3⋊D3—62no—
101C4×C3⋊D3—62no—
102C4×C3⋊D3C4×D311yescharacteristic
103C4×C3⋊D3D1211yescharacteristic
104C4×C3⋊D3#6711yescharacteristic
105C4×C3⋊D3C22×D311yescharacteristic
106#634—32no—
107#634—64no—
108#634—61no—
109#634—62no—
110#634—62no—
111#634—64no—
112#634—124no—
113#634—124no—
114C2×C3×Dic3—24no—
115C2×C3×Dic3—62no—
116C3×C4×D3—62no—
117C3×C4×D3—62no—
118C3×C4×D3—62no—
119C3×C4×D3—62no—
120C3×Dic6—24no—
121C3×Dic6—24no—
122#627—68no—
123#627—68no—
124D3×Dic3—68no—
125D3×Dic3—68no—
126#622—22no—
127#419—271noSylow
128C2×C33—21no—
129C2×C33C2×C22⋊C411yescharacteristic
130C3×C3⋊D3—21no—
131C3×C3⋊D3#13712yes—
132C2×C3×Q8—62no—
133C22×C3×C4—91no—
134C22×Dic3—181no—
135D3×Q8—184no—
136D3×Q8—184no—
137D3×Q8—184no—
138D3×Q8—184no—
139C2×C4×D3—62no—
140C2×C4×D3—184no—
141C2×Dic6—91no—
142C2×Dic6—92no—
143C2×Dic6—92no—
144C2×Dic6—182no—
145C2×Dic6—181no—
146C2×Dic6—181no—
147C2×Dic6—181no—
148C2×Dic6—184no—
149C2×Dic6—362no—
150C2×Dic6#15711yescharacteristic, complemented
151C3×M16—181no—
152C3×M16—181no—
153C2×C3×C8—92no—
154#217—181no—
155C4⋊Dic3—92no—
156C4×Dic3—181no—
157C22×C32—22no—
158C22×C32—42no—
159C22×C32(C2×D3)⋊C411yescharacteristic
160C2×C3⋊D3—22no—
161C2×C3⋊D3(C2×D3)⋊C411yescharacteristic
162C2×C3⋊D3C2×D1211yescharacteristic
163C2×C3×D3—62no—
164C2×C3×D3—62no—
165C2×C3×D3—68no—
166#157—122no—
167C32×C4—22no—
168C32×C4—22no—
169C32×C4—42no—
170C32×C4—42no—
171C32×C4—62no—
172C32×C4(C2×D3)⋊C411yescharacteristic
173C32×C4C2×C4×D311yescharacteristic
174C3⋊Dic3—62no—
175C3⋊Dic3—68no—
176C3⋊Dic3—128no—
177C3⋊Dic3(C2×D3)⋊C411yescharacteristic
178C3⋊Dic3C2×#6711yescharacteristic
179C3×Dic3—28no—
180C3×Dic3—62no—
181C3×Dic3—62no—
182C22×Q8—271no—
183C2×M16—91no—
184#117—271no—
185#103—274no—
186C33#41911yescharacteristic, complemented, Sylow
187C23×C3—91no—
188C22×D3—62no—
189C3×Q8—64no—
190C3×Q8—64no—
191C2×C3×C4—22no—
192C2×C3×C4—42no—
193C2×C3×C4—64no—
194C2×C3×C4—91no—
195C2×C3×C4—91no—
196C2×C3×C4—91no—
197C2×C3×C4—91no—
198C2×C3×C4—91no—
199C2×C3×C4C2×#15711yescharacteristic
200C2×Dic3—21no—
201C2×Dic3—62no—
202C2×Dic3—182no—
203C2×Dic3—182no—
204C2×Dic3—184no—
205C2×Dic3—181no—
206C2×Dic3—182no—
207C2×Dic3—364no—
208C4×D3—62no—
209C4×D3—62no—
210C4×D3—62no—
211C4×D3—62no—
212C4×D3—188no—
213C4×D3—188no—
214Dic6—21no—
215Dic6—92no—
216Dic6—184no—
217Dic6—182no—
218Dic6—182no—
219Dic6—188no—
220Dic6—188no—
221Dic6—184no—
222Dic6—181no—
223Dic6—364no—
224Dic6—364no—
225Dic6C2×#15712yes—
226C3×C8—182no—
227C2×C32—21no—
228C2×C32—24no—
229C2×C32—22no—
230C2×C32—44no—
231C2×C32—42no—
232C2×C32C2×(C2×D3)⋊C411yescharacteristic
233C3⋊D3—21no—
234C3⋊D3#90912yes—
235C3×D3—64no—
236C3×D3—64no—
237C2×Q8—31no—
238C2×Q8—272no—
239C2×Q8—272no—
240C2×Q8—271no—
241C2×Q8—541no—
242C2×Q8—541no—
243C2×Q8—541no—
244C22×C4—91no—
245C22×C4—541no—
246M16—181no—
247M16—181no—
248C2×C8—92no—
249C4⋊C4—272no—
250C22⋊C4—541no—
251C42—541no—
252C22×C3—22no—
253C22×C3—42no—
254C22×C3—91no—
255C22×C3—91no—
256C22×C3—182no—
257C22×C3C22⋊#15711yescharacteristic
258C2×D3—62no—
259C2×D3—62no—
260C2×D3—68no—
261C3×C4—22no—
262C3×C4—22no—
263C3×C4—42no—
264C3×C4—42no—
265C3×C4—68no—
266C3×C4—91no—
267C3×C4—91no—
268C3×C4C22⋊#15711yescharacteristic
269C3×C4C22×#15711yescharacteristic
270Dic3—22no—
271Dic3—62no—
272Dic3—62no—
273Dic3—188no—
274Dic3—182no—
275Dic3—368no—
276Dic3—362no—
277C32—22no—
278C32—42no—
279C32#512011yescharacteristic, complemented
280C23—91no—
281Q8—32no—
282Q8—61no—
283Q8—272no—
284Q8—542no—
285Q8—542no—
286Q8—541no—
287C2×C4—61no—
288C2×C4—91no—
289C2×C4—91no—
290C2×C4—91no—
291C2×C4—91no—
292C2×C4—91no—
293C2×C4—542no—
294C2×C4—542no—
295C2×C4—541no—
296C2×C4—542no—
297C2×C4D3×#15711yescharacteristic
298C8—182no—
299C2×C3—24no—
300C2×C3—21no—
301C2×C3—22no—
302C2×C3—44no—
303C2×C3—42no—
304C2×C3—92no—
305C2×C3—181no—
306C2×C3C2×C22⋊#15711yescharacteristic
307D3—64no—
308D3—64no—
309C22—91no—
310C22—91no—
311C22—182no—
312C22(C2×D3)⋊#15711yescharacteristic
313C4—62no—
314C4—91no—
315C4—91no—
316C4—542no—
317C4—1082no—
318C4(C2×D3)⋊#15711yescharacteristic
319C4C2×D3×#15711yescharacteristic
320C3—22no—
321C3—42no—
322C3#1059382911yescharacteristic, complemented
323C2—21no—
324C2—92no—
325C2—181no—
326C2C2×(C2×D3)⋊#15711yescharacteristic
subgroupG/H
centerC2C2×(C2×D3)⋊#157
commutatorC33×C4C22×C4
FrattiniC4C2×D3×#157
FittingC2×C33×C4C2×C4
socleC2×C33C2×C22⋊C4
prime radicalC22×C3×C3⋊D3D4
cosocleC3×C4×C3⋊D3C23
hypercenterC2×C4D3×#157
hypocenterC33#419
LCSG ▷ C33×C4 ▷ C2×C33 ▷ C33
UCSC1 ◁ C2 ◁ C2×C4

Complements

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

NH ≅ G/Ntype
4. #11722709C2semidirect
6. C2×#6308C4semidirect
8. C2×C32×Dic6C4semidirect
13. C3⋊D3×Dic6C4semidirect
150. C2×Dic6#157semidirect
186. C33#419semidirect
279. C32#5120semidirect
322. C3#10593829semidirect

Maximal subgroups

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

subgroupindex
C2×C3⋊D3×Dic62
#117245142
C3×#642832
#117227092
#105938293
#51209