Finite Group Databasebeta

← #13166382

Subgroups — #13166382

Order 1728.

712 conjugacy classes of subgroups; only the 74 normal subgroups are shown.

normal subgroups only

This group has 712 conjugacy classes of subgroups — too many to draw. Only its 74 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×#89014C211yesmaximal, characteristic
2#11724514C211yesmaximal, characteristic, complemented
3C3×#64283C211yesmaximal, characteristic, complemented
4#11722710C214yesmaximal, complemented
5#10593816—31nomaximal
6C2×#6308C411yescharacteristic, complemented
7C2×C3×C4×C3⋊D3C2211yescharacteristic
8C2×C32×D12C411yescharacteristic, complemented
9C2×#6290—21no—
10C2×D3×C3⋊Dic3—21no—
11#89014—21no—
12#89014—21no—
13#89014C412yescomplemented
14#89014C2212yes—
15#88994—21no—
16C4⋊(C3⋊#157)—21no—
17C4×C3⋊#157C2212yescomplemented
18C3×#3792—21no—
19C3×#3788C2212yescomplemented
20C3×#3787—21no—
21C2×#3830—31no—
22C2×#3795—62no—
23#64286—31no—
24#64283D311yescharacteristic, complemented
25#63783—34no—
26C22×C3×C3⋊D3D411yescharacteristic
27C22×C32×D3—21no—
28C2×C3⋊#157—22no—
29C2×C33×C4C2×C411yescharacteristic
30C2×C3⋊(C3⋊Dic3)—21no—
31#6308—21no—
32#6308C2×C412yes—
33C2×C3×C3⋊Dic3D411yescharacteristic
34C3×C4×C3⋊D3C2×C411yescharacteristic
35C3×C4×C3⋊D3D411yescharacteristic
36C3×C4×C3⋊D3D411yescharacteristic
37C3×C4×C3⋊D3C2311yescharacteristic
38C32×D12—21no—
39C32×D12C2×C412yes—
40#6290—22no—
41#6290—22no—
42D3×C3⋊Dic3—22no—
43D3×C3⋊Dic3—22no—
44C3×#622—22no—
45#5139—91nomaximal
46C2×C32×D4—31no—
47C2×#638—61no—
48C22×C3⋊Dic3—61no—
49#3830—32no—
50#3830—32no—
51#3830—61no—
52#3830—61no—
53C2×C4×C3⋊D3C2×D311yescharacteristic
54C2×#634—31no—
55C2×#634—62no—
56C2×#634—122no—
57C2×C3×D12—22no—
58C2×C3×C4×D3—62no—
59C2×#625—64no—
60C2×D3×Dic3—64no—
61#3795—64no—
62#3795—64no—
63#3795—64no—
64#3795—64no—
65C22⋊#157—61no—
66#3792—21no—
67C4⋊#157—61no—
68C4×#157—32no—
69#3788C2×D312yescomplemented
70#3787—21no—
71C22×C33C22⋊C411yescharacteristic
72C2×C3×C3⋊D3—41no—
73C2×C3×C3⋊D3C22⋊C411yescharacteristic
74C2×C3×C3⋊D3C2×D411yescharacteristic
75C2×C32×D3—22no—
76C2×C32×D3—42no—
77C3⋊#157—42no—
78C33×C4C22⋊C411yescharacteristic
79C33×C4C22×C411yescharacteristic
80C3⋊(C3⋊Dic3)—22no—
81C3×C3⋊Dic3C22⋊C411yescharacteristic
82C3×C3⋊Dic3C2×D411yescharacteristic
83C2×#237—182no—
84C2×#235—91no—
85C2×C3×M16—91no—
86#926—91no—
87#825—94no—
88C23×C32—61no—
89C22×C3⋊D3D1211yescharacteristic
90C22×C3×D3—24no—
91C22×C3×D3—62no—
92C2×#157—62no—
93C32×D4—32no—
94C32×D4—61no—
95C2×C32×C4—22no—
96C2×C32×C4—42no—
97C2×C32×C4C4×D311yescharacteristic
98#638—62no—
99#638—62no—
100C2×C3⋊Dic3—61no—
101C2×C3⋊Dic3—62no—
102C2×C3⋊Dic3—62no—
103C2×C3⋊Dic3—64no—
104C2×C3⋊Dic3—124no—
105C2×C3⋊Dic3#6711yescharacteristic
106C4×C3⋊D3C4×D311yescharacteristic
107C4×C3⋊D3D1211yescharacteristic
108C4×C3⋊D3#6711yescharacteristic
109C4×C3⋊D3C22×D311yescharacteristic
110#634—32no—
111#634—64no—
112#634—61no—
113#634—64no—
114#634—124no—
115#634—124no—
116C2×C3×Dic3—62no—
117C3×D12—24no—
118C3×D12—24no—
119C3×C4×D3—62no—
120C3×C4×D3—62no—
121C3×C4×D3—62no—
122C3×C4×D3—62no—
123#625—68no—
124#625—68no—
125D3×Dic3—68no—
126D3×Dic3—68no—
127#622—22no—
128#421—271noSylow
129C2×C33—21no—
130C2×C33C2×C22⋊C411yescharacteristic
131C3×C3⋊D3—21no—
132C3×C3⋊D3—21no—
133C32×D3—42no—
134C2×C3×D4—62no—
135C22×C3×C4—91no—
136C2×#67—181no—
137C2×#67—184no—
138C22×Dic3—184no—
139#237—184no—
140#237—184no—
141#237—184no—
142#237—184no—
143#235—92no—
144#235—92no—
145#235—181no—
146#235—181no—
147C2×D12#15711yescharacteristic, complemented
148C2×C4×D3—62no—
149C2×C4×D3—181no—
150C2×Dic6—91no—
151C2×Dic6—182no—
152C2×Dic6—362no—
153C3×M16—92no—
154C3×M16—181no—
155C2×C3×C8—181no—
156#217—181no—
157C4⋊Dic3—181no—
158C4×Dic3—92no—
159C22×C32—22no—
160C22×C32—42no—
161C22×C32—62no—
162C22×C32—122no—
163C22×C32(C2×D3)⋊C411yescharacteristic
164C2×C3⋊D3—41no—
165C2×C3⋊D3(C2×D3)⋊C411yescharacteristic
166C2×C3⋊D3C2×D1211yescharacteristic
167C2×C3×D3—28no—
168C2×C3×D3—48no—
169C2×C3×D3—62no—
170C2×C3×D3—62no—
171C2×C3×D3—124no—
172#157—122no—
173C32×C4—22no—
174C32×C4—22no—
175C32×C4—42no—
176C32×C4—42no—
177C32×C4(C2×D3)⋊C411yescharacteristic
178C32×C4C2×C4×D311yescharacteristic
179C3⋊Dic3—62no—
180C3⋊Dic3—68no—
181C3⋊Dic3—128no—
182C3⋊Dic3(C2×D3)⋊C411yescharacteristic
183C3⋊Dic3C2×#6711yescharacteristic
184C3×Dic3—62no—
185C3×Dic3—62no—
186C2×#41—271no—
187C2×M16—91no—
188#117—271no—
189#104—274no—
190C33#42111yescharacteristic, complemented, Sylow
191C23×C3—64no—
192C23×C3—91no—
193C22×D3—21no—
194C22×D3—62no—
195C3×D4—64no—
196C3×D4—64no—
197C2×C3×C4—22no—
198C2×C3×C4—42no—
199C2×C3×C4—91no—
200C2×C3×C4—91no—
201C2×C3×C4—91no—
202C2×C3×C4—91no—
203C2×C3×C4—91no—
204C2×C3×C4C2×#15711yescharacteristic
205#67—182no—
206#67—182no—
207#67—188no—
208#67—188no—
209C2×Dic3—62no—
210C2×Dic3—188no—
211C2×Dic3—188no—
212C2×Dic3—184no—
213C2×Dic3—181no—
214C2×Dic3—182no—
215C2×Dic3—364no—
216D12—21no—
217D12C2×#15712yes—
218C4×D3—62no—
219C4×D3—62no—
220C4×D3—62no—
221C4×D3—62no—
222C4×D3—182no—
223C4×D3—182no—
224Dic6—92no—
225Dic6—184no—
226Dic6—184no—
227Dic6—181no—
228Dic6—364no—
229Dic6—364no—
230C3×C8—182no—
231C2×C32—21no—
232C2×C32—24no—
233C2×C32—22no—
234C2×C32—44no—
235C2×C32—42no—
236C2×C32—122no—
237C2×C32C2×(C2×D3)⋊C411yescharacteristic
238C3⋊D3—21no—
239C3⋊D3—21no—
240C3×D3—48no—
241C3×D3—122no—
242C3×D3—122no—
243#41—272no—
244#41—272no—
245#41—541no—
246#41—541no—
247C2×Q8—271no—
248C2×D4—31no—
249C2×D4—541no—
250C22×C4—91no—
251C22×C4—541no—
252M16—92no—
253M16—181no—
254C2×C8—181no—
255C4⋊C4—541no—
256C22⋊C4—541no—
257C42—272no—
258C22×C3—22no—
259C22×C3—42no—
260C22×C3—68no—
261C22×C3—91no—
262C22×C3—91no—
263C22×C3—128no—
264C22×C3—361no—
265C22×C3C22⋊#15711yescharacteristic
266C2×D3—22no—
267C2×D3—42no—
268C2×D3—62no—
269C2×D3—62no—
270C2×D3—124no—
271C3×C4—22no—
272C3×C4—22no—
273C3×C4—42no—
274C3×C4—42no—
275C3×C4—91no—
276C3×C4—91no—
277C3×C4C22⋊#15711yescharacteristic
278C3×C4C22×#15711yescharacteristic
279Dic3—62no—
280Dic3—62no—
281Dic3—188no—
282Dic3—182no—
283Dic3—368no—
284Dic3—362no—
285C32—22no—
286C32—42no—
287C32#513911yescharacteristic, complemented
288C23—61no—
289C23—91no—
290Q8—272no—
291Q8—541no—
292D4—32no—
293D4—61no—
294D4—542no—
295D4—542no—
296C2×C4—91no—
297C2×C4—91no—
298C2×C4—91no—
299C2×C4—91no—
300C2×C4—91no—
301C2×C4—542no—
302C2×C4—542no—
303C2×C4—541no—
304C2×C4—542no—
305C2×C4D3×#15711yescharacteristic
306C8—182no—
307C2×C3—24no—
308C2×C3—21no—
309C2×C3—22no—
310C2×C3—44no—
311C2×C3—42no—
312C2×C3—128no—
313C2×C3—181no—
314C2×C3—181no—
315C2×C3C2×C22⋊#15711yescharacteristic
316D3—42no—
317D3—122no—
318D3—122no—
319C22—62no—
320C22—91no—
321C22—91no—
322C22—122no—
323C22—361no—
324C22(C2×D3)⋊#15711yescharacteristic
325C4—91no—
326C4—91no—
327C4—542no—
328C4—1082no—
329C4(C2×D3)⋊#15711yescharacteristic
330C4C2×D3×#15711yescharacteristic
331C3—22no—
332C3—42no—
333C3#1059381611yescharacteristic, complemented
334C2—21no—
335C2—122no—
336C2—181no—
337C2—181no—
338C2C2×(C2×D3)⋊#15711yescharacteristic
subgroupG/H
centerC2C2×(C2×D3)⋊#157
commutatorC33×C4C22×C4
FrattiniC4C2×D3×#157
FittingC2×C33×C4C2×C4
socleC2×C33C2×C22⋊C4
prime radicalC2×#89014C2
cosocleC3×C4×C3⋊D3C23
hypercenterC2×C4D3×#157
hypocenterC33#421
LCSG ▷ C33×C4 ▷ C2×C33 ▷ C33
UCSC1 ◁ C2 ◁ C2×C4

Complements

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

NH ≅ G/Ntype
2. #11724514C2semidirect
3. C3×#64283C2semidirect
4. #11722710C2semidirect
6. C2×#6308C4semidirect
8. C2×C32×D12C4semidirect
13. #89014C4semidirect
17. C4×C3⋊#157C22semidirect
19. C3×#3788C22semidirect
24. #64283D3semidirect
69. #3788C2×D3semidirect
147. C2×D12#157semidirect
190. C33#421semidirect
287. C32#5139semidirect
333. C3#10593816semidirect

Maximal subgroups

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

subgroupindex
C2×#890142
#117245142
C3×#642832
#117227102
#105938163
#51399