Finite Group Databasebeta

← #13180951

Subgroups — #13180951

Order 1728.

920 conjugacy classes of subgroups; only the 174 normal subgroups are shown.

normal subgroups only

This group has 920 conjugacy classes of subgroups — too many to draw. Only its 174 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×#64286C211yesmaximal, characteristic
4#11724261C214yesmaximal
5#11724245C214yesmaximal, complemented
6Dic6×#157C214yesmaximal, complemented
7#10596646—31nomaximal
8C2×#6308C411yescharacteristic, complemented
9C2×C3×C4×C3⋊D3C2211yescharacteristic
10C2×C32×Dic6C411yescharacteristic, complemented
11C2×#6293C412yescomplemented
12C2×Dic3×C3⋊D3C2212yes—
13C3⋊D3×Dic6C412yescomplemented
14C3⋊D3×Dic6C412yescomplemented
15C3⋊D3×Dic6C2212yes—
16C3⋊D3×Dic6C2212yes—
17#88994C2212yes—
18C4⋊(C3⋊#157)C2212yes—
19C4×C3⋊#157C2212yes—
20C3×C22⋊#157C2212yes—
21C3×C4⋊#157C2212yes—
22C3×C4×#157C2212yes—
23Dic3⋊#157C2218yes—
24Dic3×#157C2218yes—
25C2×Q8×C3⋊D3—31no—
26C2×D3×Dic6—62no—
27Q8×#157—34no—
28Q8×#157—34no—
29#64286—31no—
30#64286—34no—
31#64286D311yescharacteristic
32C22×C3×C3⋊D3C2311yescharacteristic
33C2×C3⋊#157C2314yes—
34C2×C3×#157C2314yes—
35C2×C33×C4C2×C411yescharacteristic
36C2×C3⋊(C3⋊Dic3)C2×C412yes—
37#6308C2×C412yes—
38#6308C2×C412yes—
39C2×C3×C3⋊Dic3C2×C411yescharacteristic
40C3×C4×C3⋊D3C2×C411yescharacteristic
41C3×C4×C3⋊D3C2×C411yescharacteristic
42C3×C4×C3⋊D3C2311yescharacteristic
43C3×C4×C3⋊D3C2311yescharacteristic
44C2×C32×Dic3C2×C412yes—
45C32×Dic6C2×C412yes—
46C32×Dic6C2×C412yes—
47#6293C2×C414yes—
48#6293C2×C414yes—
49Dic3×C3⋊D3C2×C414yes—
50Dic3×C3⋊D3C2314yes—
51#5516—91nomaximal
52C2×C32×Q8—31no—
53Q8×C3⋊D3—32no—
54Q8×C3⋊D3—32no—
55Q8×C3⋊D3—32no—
56Q8×C3⋊D3—32no—
57C2×C4×C3⋊D3—32no—
58C2×C4×C3⋊D3C2×D311yescharacteristic
59C2×#634—31no—
60C2×#634—32no—
61C2×#634—62no—
62C2×#634—122no—
63C2×C3×C4×D3—62no—
64C2×C3×Dic6—22no—
65C2×#627—64no—
66C2×D3×Dic3—64no—
67D3×Dic6—64no—
68D3×Dic6—64no—
69D3×Dic6—64no—
70D3×Dic6—64no—
71C22⋊#157—32no—
72C22⋊#157C2×D312yes—
73C4⋊#157—38no—
74C4⋊#157—32no—
75C4⋊#157C2×D312yes—
76C4×#157—38no—
77C4×#157—32no—
78C4×#157C2×D312yes—
79C22×C33C22×C411yescharacteristic
80C2×C3×C3⋊D3—22no—
81C2×C3×C3⋊D3C22×C411yescharacteristic
82C2×C3×C3⋊D3C2411yescharacteristic
83C3⋊#157—24no—
84C3×#157—24no—
85C33×C4C22×C411yescharacteristic
86C33×C4C22×C411yescharacteristic
87C3⋊(C3⋊Dic3)C22×C414yes—
88C3×C3⋊Dic3C22×C411yescharacteristic
89C3×C3⋊Dic3C22×C411yescharacteristic
90C32×Dic3C22×C414yes—
91C2×D3×Q8—182no—
92C22×Dic6—91no—
93C3×#117—91no—
94#926—91no—
95#887—94no—
96#877—94no—
97C4×Dic6—94no—
98C22×C3⋊D3C22×D311yescharacteristic
99C22×C3×D3—62no—
100C2×#157—34no—
101C2×#157C22×D314yes—
102C32×Q8—32no—
103C32×Q8—32no—
104C2×C32×C4—22no—
105C2×C32×C4—32no—
106C2×C32×C4—42no—
107C2×C32×C4C4×D311yescharacteristic
108C2×C3⋊Dic3—32no—
109C2×C3⋊Dic3—64no—
110C2×C3⋊Dic3—124no—
111C2×C3⋊Dic3C4×D311yescharacteristic
112C4×C3⋊D3—34no—
113C4×C3⋊D3—34no—
114C4×C3⋊D3C4×D311yescharacteristic
115C4×C3⋊D3C4×D311yescharacteristic
116C4×C3⋊D3C22×D311yescharacteristic
117C4×C3⋊D3C22×D311yescharacteristic
118#634—32no—
119#634—32no—
120#634—34no—
121#634—34no—
122#634—64no—
123#634—64no—
124#634—124no—
125#634—124no—
126C2×C3×Dic3—24no—
127C2×C3×Dic3—62no—
128C3×C4×D3—62no—
129C3×C4×D3—62no—
130C3×C4×D3—62no—
131C3×C4×D3—62no—
132C3×Dic6—24no—
133C3×Dic6—24no—
134#627—68no—
135#627—68no—
136D3×Dic3—68no—
137D3×Dic3—68no—
138#519—271noSylow
139C2×C33—21no—
140C2×C33C23×C411yescharacteristic
141C3×C3⋊D3—21no—
142C3×C3⋊D32−1+412yes—
143C2×C3×Q8—62no—
144C22×C3×C4—91no—
145C22×Dic3—92no—
146D3×Q8—184no—
147D3×Q8—184no—
148D3×Q8—184no—
149D3×Q8—184no—
150C2×C4×D3—62no—
151C2×C4×D3—184no—
152C2×Dic6—91no—
153C2×Dic6—92no—
154C2×Dic6—92no—
155C2×Dic6—92no—
156C2×Dic6—92no—
157C2×Dic6—92no—
158C2×Dic6—184no—
159C2×Dic6—182no—
160C2×Dic6—362no—
161C2×Dic6#15711yescharacteristic, complemented
162C3×C4⋊C4—92no—
163C3×C22⋊C4—92no—
164C3×C42—92no—
165#217—92no—
166C4⋊Dic3—92no—
167Dic3⋊C4—98no—
168C4×Dic3—98no—
169C4×Dic3—92no—
170C22×C32—22no—
171C22×C32—42no—
172C22×C32C2×C4×D311yescharacteristic
173C2×C3⋊D3—22no—
174C2×C3⋊D3C2×C4×D311yescharacteristic
175C2×C3⋊D3C23×D311yescharacteristic
176C2×C3×D3—62no—
177C2×C3×D3—62no—
178C2×C3×D3—68no—
179#157—24no—
180#157—64no—
181C32×C4—22no—
182C32×C4—22no—
183C32×C4—34no—
184C32×C4—42no—
185C32×C4—42no—
186C32×C4C2×C4×D311yescharacteristic
187C32×C4C2×C4×D311yescharacteristic
188C3⋊Dic3—34no—
189C3⋊Dic3—68no—
190C3⋊Dic3—128no—
191C3⋊Dic3C2×C4×D311yescharacteristic
192C3⋊Dic3C2×C4×D311yescharacteristic
193C3×Dic3—28no—
194C3×Dic3—62no—
195C3×Dic3—62no—
196C22×Q8—271no—
197C4×Q8—274no—
198C4×Q8—274no—
199#117—91no—
200#117—274no—
201#117—271no—
202C33#51911yescharacteristic, complemented, Sylow
203C23×C3—91no—
204C22×D3—62no—
205C3×Q8—64no—
206C3×Q8—64no—
207C2×C3×C4—22no—
208C2×C3×C4—42no—
209C2×C3×C4—64no—
210C2×C3×C4—91no—
211C2×C3×C4—91no—
212C2×C3×C4—91no—
213C2×C3×C4—91no—
214C2×C3×C4—91no—
215C2×C3×C4—94no—
216C2×C3×C4C2×#15711yescharacteristic
217C2×Dic3—62no—
218C2×Dic3—92no—
219C2×Dic3—94no—
220C2×Dic3—94no—
221C2×Dic3—94no—
222C2×Dic3—184no—
223C2×Dic3—364no—
224C2×Dic3C2×#15712yes—
225C4×D3—62no—
226C4×D3—62no—
227C4×D3—62no—
228C4×D3—62no—
229C4×D3—188no—
230C4×D3—188no—
231Dic6—92no—
232Dic6—92no—
233Dic6—94no—
234Dic6—94no—
235Dic6—188no—
236Dic6—188no—
237Dic6—184no—
238Dic6—184no—
239Dic6—364no—
240Dic6—364no—
241Dic6C2×#15712yes—
242Dic6C2×#15712yes—
243C2×C32—21no—
244C2×C32—24no—
245C2×C32—22no—
246C2×C32—44no—
247C2×C32—42no—
248C2×C32C22×C4×D311yescharacteristic
249C3⋊D3—21no—
250C3⋊D3#101012yes—
251C3×D3—64no—
252C3×D3—64no—
253C2×Q8—31no—
254C2×Q8—272no—
255C2×Q8—272no—
256C2×Q8—272no—
257C2×Q8—272no—
258C2×Q8—272no—
259C2×Q8—271no—
260C22×C4—91no—
261C22×C4—272no—
262C4⋊C4—92no—
263C4⋊C4—278no—
264C4⋊C4—272no—
265C22⋊C4—92no—
266C22⋊C4—272no—
267C42—92no—
268C42—272no—
269C42—278no—
270C22×C3—22no—
271C22×C3—42no—
272C22×C3—91no—
273C22×C3—91no—
274C22×C3—182no—
275C22×C3C22×#15711yescharacteristic
276C2×D3—62no—
277C2×D3—62no—
278C2×D3—68no—
279C3×C4—22no—
280C3×C4—22no—
281C3×C4—42no—
282C3×C4—42no—
283C3×C4—68no—
284C3×C4—91no—
285C3×C4—91no—
286C3×C4—184no—
287C3×C4C22×#15711yescharacteristic
288C3×C4C22×#15711yescharacteristic
289Dic3—62no—
290Dic3—62no—
291Dic3—94no—
292Dic3—188no—
293Dic3—184no—
294Dic3—368no—
295Dic3C22×#15714yes—
296C32—22no—
297C32—42no—
298C32#551611yescharacteristic, complemented
299C23—91no—
300Q8—32no—
301Q8—32no—
302Q8—274no—
303Q8—274no—
304Q8—272no—
305Q8—272no—
306C2×C4—32no—
307C2×C4—91no—
308C2×C4—91no—
309C2×C4—91no—
310C2×C4—91no—
311C2×C4—91no—
312C2×C4—94no—
313C2×C4—274no—
314C2×C4—274no—
315C2×C4—272no—
316C2×C4—274no—
317C2×C4D3×#15711yescharacteristic
318C2×C3—24no—
319C2×C3—21no—
320C2×C3—22no—
321C2×C3—44no—
322C2×C3—42no—
323C2×C3—92no—
324C2×C3—181no—
325C2×C3C23×#15711yescharacteristic
326D3—64no—
327D3—64no—
328C22—91no—
329C22—91no—
330C22—182no—
331C22C2×D3×#15711yescharacteristic
332C4—34no—
333C4—91no—
334C4—91no—
335C4—184no—
336C4—274no—
337C4—544no—
338C4C2×D3×#15711yescharacteristic
339C4C2×D3×#15711yescharacteristic
340C3—22no—
341C3—42no—
342C3#1059664611yescharacteristic, complemented
343C2—21no—
344C2—92no—
345C2—181no—
346C2C22×D3×#15711yescharacteristic
subgroupG/H
center, FrattiniC2C22×D3×#157
commutator, socleC2×C33C23×C4
FittingC2×C33×C4C2×C4
prime radicalC22×C3×C3⋊D3C23
cosocleC2×C3×C3⋊D3C24
hypercenterC2×C4D3×#157
hypocenterC33#519
LCSG ▷ C2×C33 ▷ C33
UCSC1 ◁ C2 ◁ C2×C4

Complements

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

NH ≅ G/Ntype
5. #11724245C2semidirect
6. Dic6×#157C2semidirect
8. C2×#6308C4semidirect
10. C2×C32×Dic6C4semidirect
11. C2×#6293C4semidirect
13. C3⋊D3×Dic6C4semidirect
14. C3⋊D3×Dic6C4semidirect
161. C2×Dic6#157semidirect
202. C33#519semidirect
298. C32#5516semidirect
342. C3#10596646semidirect

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×#642862
#117242612
#117242452
Dic6×#1572
#105966463
#55169