Finite Group Databasebeta

← #13155622

Subgroups — #13155622

Order 1728.

798 conjugacy classes of subgroups; only the 114 normal subgroups are shown.

normal subgroups only

This group has 798 conjugacy classes of subgroups — too many to draw. Only its 114 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×#88810C211yesmaximal, characteristic
2#11723501C212yesmaximal, complemented
3#11722018C211yesmaximal, characteristic, complemented
4C3×#63576C211yesmaximal, characteristic, complemented
5C3×#63564C212yesmaximal, complemented
6#10590740—31nomaximal
7#10590459—32nomaximal
8C2×C3×C4⋊(C3⋊D3)C2212yes—
9C2×Dic3×C3⋊D3—22no—
10C4×C3⋊(C3⋊Dic3)C2211yescharacteristic
11C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
12C32×C4⋊Dic3C2212yes—
13C2×C32×C3⋊C8C2211yescharacteristic
14#88810C414yescomplemented
15#88803—22no—
16#64102—32no—
17#63898—32no—
18#63898—32no—
19#63898—62no—
20C2×#3714—32no—
21C2×#3713—31no—
22C8×C3⋊Dic3—31no—
23C3×#821—32no—
24C3×#809—32no—
25C3×#807—32no—
26#63576D311yescharacteristic, complemented
27#63564D312yescomplemented
28#63553—32no—
29#63553—61no—
30C22×C3×C3⋊D3—22no—
31C2×C33×C4C2311yescharacteristic
32C2×C3⋊(C3⋊Dic3)D411yescharacteristic
33C2×C3⋊(C3⋊Dic3)D411yescharacteristic
34C3×C4⋊(C3⋊D3)—22no—
35C3×C4⋊(C3⋊D3)C2×C414yescomplemented
36C2×C32×Dic3—22no—
37Dic3×C3⋊D3—44no—
38C3×C3⋊(C3⋊C8)C2×C412yescomplemented
39C32×C3⋊C8—21no—
40#4872—91no—
41#4756—92no—
42C2×C4⋊(C3⋊D3)C2×D312yes—
43C2×C4×C3⋊D3—62no—
44C2×C3×D12—32no—
45C2×C3×D12—32no—
46C2×C3×D12—62no—
47C2×D3×Dic3—62no—
48C2×D3×Dic3—62no—
49C2×D3×Dic3—64no—
50C2×C32×C8—31no—
51C32×C4⋊C4—32no—
52#3752—62no—
53C4×C3⋊Dic3—32no—
54C4×C3⋊Dic3—31no—
55C4×C3⋊Dic3—61no—
56C4×C3⋊Dic3—62no—
57C4×C3⋊Dic3—121no—
58C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
59C3×C4⋊Dic3—22no—
60C3×C4⋊Dic3C2×D312yes—
61C3×C4⋊Dic3C2×D312yes—
62C2×C3×C3⋊C8—21no—
63C2×C3×C3⋊C8—32no—
64C2×C3×C3⋊C8—61no—
65C2×C3×C3⋊C8C2×D312yes—
66(C2×D3)⋊Dic3—62no—
67(C2×D3)⋊Dic3—62no—
68(C2×D3)⋊Dic3—64no—
69#3714—38no—
70#3713—34no—
71C22×C33C2×D411yescharacteristic
72C2×C3×C3⋊D3—24no—
73C2×C3×C3⋊D3—42no—
74C2×C3×C3⋊D3—42no—
75C33×C4C22×C411yescharacteristic
76C33×C4#4111yescharacteristic
77C3⋊(C3⋊Dic3)—21no—
78C3⋊(C3⋊Dic3)D812yescomplemented
79C32×Dic3—42no—
80D4×Dic3—92no—
81C2×#213—92no—
82C2×D24—91no—
83#893—92no—
84#893—92no—
85#893—182no—
86C3×#107—91no—
87C3×#102—92no—
88#821—32no—
89C8×Dic3—92no—
90C8×Dic3—91no—
91C8×Dic3—181no—
92#809—32no—
93#807—32no—
94C22×C3⋊D3—22no—
95C22×C3×D3—64no—
96C22×C3×D3—62no—
97C22×C3×D3—62no—
98C2×C32×C4—21no—
99C2×C32×C4—22no—
100C2×C32×C4—41no—
101C2×C32×C4—62no—
102C2×C32×C4C22×D312yes—
103C2×C32×C4C22×D311yescharacteristic
104C2×C3⋊Dic3—31no—
105C2×C3⋊Dic3—32no—
106C2×C3⋊Dic3—31no—
107C2×C3⋊Dic3—32no—
108C2×C3⋊Dic3—61no—
109C2×C3⋊Dic3—61no—
110C2×C3⋊Dic3—62no—
111C2×C3⋊Dic3—62no—
112C2×C3⋊Dic3—121no—
113C2×C3⋊Dic3—121no—
114C4⋊(C3⋊D3)—22no—
115C4⋊(C3⋊D3)C4×D314yescomplemented
116C4×C3⋊D3—124no—
117C2×C3×Dic3—24no—
118C2×C3×Dic3—22no—
119C2×C3×Dic3—22no—
120C3×D12—34no—
121C3×D12—34no—
122C3×D12—64no—
123C3×D12—64no—
124C3×D12—62no—
125C3×D12—62no—
126D3×Dic3—124no—
127D3×Dic3—124no—
128D3×Dic3—128no—
129C32×C8—61no—
130C3⋊(C3⋊C8)C4×D312yescomplemented
131C3×C3⋊C8—22no—
132C3×C3⋊C8—22no—
133C3×C3⋊C8—34no—
134C3×C3⋊C8—62no—
135C4×D8—271noSylow
136C2×C33C4×D411yescharacteristic
137C2×C33C2×D811yescharacteristic
138C2×C33#13511yescharacteristic
139C3×C3⋊D3—44no—
140C2×C3×D4—92no—
141C22×Dic3—182no—
142C2×D12—32no—
143C2×D12—32no—
144C2×D12—62no—
145C2×C4×D3—182no—
146C2×C4×D3—182no—
147C2×C4×D3—184no—
148C2×C3×C8—32no—
149C2×C3×C8—61no—
150C2×C3×C8—91no—
151C3×C4⋊C4—32no—
152C3×C4⋊C4—32no—
153C3×C4⋊C4—62no—
154#217—182no—
155#213—98no—
156(C2×D3)⋊C4—182no—
157(C2×D3)⋊C4—182no—
158(C2×D3)⋊C4—184no—
159C4⋊Dic3D3212yes—
160C4×Dic3—91no—
161C4×Dic3—92no—
162C4×Dic3—181no—
163C4×Dic3—182no—
164C4×Dic3—361no—
165C2×C3⋊C8—32no—
166C2×C3⋊C8—61no—
167C2×C3⋊C8D3211yescharacteristic
168D24—94no—
169C22×C32—21no—
170C22×C32—22no—
171C22×C32—41no—
172C22×C32D3×D412yes—
173C22×C32D3×D411yescharacteristic
174C2×C3⋊D3—24no—
175C2×C3⋊D3—42no—
176C2×C3⋊D3—42no—
177C2×C3×D3—68no—
178C2×C3×D3—64no—
179C2×C3×D3—64no—
180C2×C3×D3—124no—
181C2×C3×D3—124no—
182C2×C3×D3—122no—
183C2×C3×D3—122no—
184C2×C3×D3—122no—
185C2×C3×D3—122no—
186C32×C4—22no—
187C32×C4—21no—
188C32×C4—22no—
189C32×C4—21no—
190C32×C4—41no—
191C32×C4—41no—
192C32×C4—122no—
193C32×C4C2×C4×D312yes—
194C32×C4C2×C4×D311yescharacteristic
195C32×C4#23712yes—
196C32×C4#23911yescharacteristic
197C3⋊Dic3—32no—
198C3⋊Dic3—34no—
199C3⋊Dic3—64no—
200C3⋊Dic3—62no—
201C3⋊Dic3—61no—
202C3⋊Dic3—62no—
203C3⋊Dic3—62no—
204C3⋊Dic3—122no—
205C3⋊Dic3—122no—
206C3⋊Dic3—122no—
207C3×Dic3—44no—
208C3×Dic3—42no—
209C3×Dic3—42no—
210C2×D8—271no—
211C4×D4—272no—
212#107—91no—
213#102—92no—
214C4×C8—271no—
215C33C4×D811yescharacteristic, complemented, Sylow
216C23×C3—182no—
217C22×D3—64no—
218C22×D3—62no—
219C22×D3—62no—
220C3×D4—94no—
221C3×D4—182no—
222C2×C3×C4—21no—
223C2×C3×C4—22no—
224C2×C3×C4—41no—
225C2×C3×C4—64no—
226C2×C3×C4—62no—
227C2×C3×C4—62no—
228C2×C3×C4C2×D3212yes—
229C2×C3×C4C2×D3211yescharacteristic
230C2×Dic3—22no—
231C2×Dic3—91no—
232C2×Dic3—92no—
233C2×Dic3—91no—
234C2×Dic3—92no—
235C2×Dic3—181no—
236C2×Dic3—181no—
237C2×Dic3—182no—
238C2×Dic3—182no—
239C2×Dic3—361no—
240C2×Dic3—361no—
241C2×Dic3—364no—
242D12—34no—
243D12—34no—
244D12—64no—
245D12—64no—
246D12—62no—
247D12—62no—
248C4×D3—364no—
249C4×D3—364no—
250C4×D3—368no—
251C3×C8—62no—
252C3×C8—62no—
253C3×C8—92no—
254C3⋊C8—21no—
255C3⋊C8—34no—
256C3⋊C8—62no—
257C2×C32—21no—
258C2×C32—21no—
259C2×C32—22no—
260C2×C32—21no—
261C2×C32—22no—
262C2×C32—22no—
263C2×C32—41no—
264C2×C32—41no—
265C2×C32—41no—
266C2×C32#67⋊C412yes—
267C2×C32#89311yescharacteristic
268C2×C32D3×D812yes—
269C2×C32D3×D811yescharacteristic
270C2×C32#91612yes—
271C2×C32#91911yescharacteristic
272C3⋊D3—44no—
273C3×D3—128no—
274C3×D3—124no—
275C3×D3—124no—
276C2×D4—92no—
277C22×C4—542no—
278D8—274no—
279C2×C8—31no—
280C2×C8—91no—
281C4⋊C4—32no—
282C22⋊C4—542no—
283C42—271no—
284C22×C3—22no—
285C22×C3—21no—
286C22×C3—41no—
287C22×C3—184no—
288C22×C3—362no—
289C22×C3—362no—
290C22×C3#380211yescharacteristic
291C22×C3#381112yes—
292C2×D3—68no—
293C2×D3—64no—
294C2×D3—64no—
295C2×D3—124no—
296C2×D3—124no—
297C2×D3—122no—
298C2×D3—122no—
299C2×D3—122no—
300C2×D3—122no—
301C3×C4—21no—
302C3×C4—22no—
303C3×C4—21no—
304C3×C4—22no—
305C3×C4—41no—
306C3×C4—41no—
307C3×C4—124no—
308C3×C4—122no—
309C3×C4—122no—
310C3×C4#379612yes—
311C3×C4C4×D3212yes—
312C3×C4C4×D3211yescharacteristic
313C3×C4#380511yescharacteristic
314Dic3—42no—
315Dic3—92no—
316Dic3—94no—
317Dic3—184no—
318Dic3—182no—
319Dic3—182no—
320Dic3—181no—
321Dic3—182no—
322Dic3—362no—
323Dic3—362no—
324Dic3—362no—
325C32—21no—
326C32—22no—
327C32—41no—
328C32#475612yescomplemented
329C32#487211yescharacteristic, complemented
330C23—182no—
331D4—94no—
332D4—182no—
333C2×C4—62no—
334C2×C4—271no—
335C2×C4—271no—
336C2×C4—1084no—
337C2×C4D3311yescharacteristic
338C8—61no—
339C8—92no—
340C2×C3—22no—
341C2×C3—21no—
342C2×C3—21no—
343C2×C3—22no—
344C2×C3—22no—
345C2×C3—21no—
346C2×C3—41no—
347C2×C3—41no—
348C2×C3—41no—
349C2×C3—364no—
350C2×C3#6379711yescharacteristic
351C2×C3#6380911yescharacteristic
352C2×C3#6388112yes—
353C2×C3#6390211yescharacteristic
354C2×C3#6392612yes—
355C2×C3#6395012yes—
356D3—128no—
357D3—124no—
358D3—124no—
359C22—184no—
360C22—362no—
361C22—362no—
362C22#8895411yescharacteristic
363C4—122no—
364C4—272no—
365C4—541no—
366C4#8894911yescharacteristic
367C4#8896411yescharacteristic
368C3—22no—
369C3—21no—
370C3—41no—
371C3#1059045912yescomplemented
372C3#1059074011yescharacteristic, complemented
373C2—364no—
374C2#1172283211yescharacteristic
375C2#1172286311yescharacteristic
376C2#1172293811yescharacteristic
subgroupG/H
centerC22#88954
commutatorC33×C4C22×C4
Frattini, hypercenterC2×C4D33
Fitting, cosocleC2×C33×C4C23
socleC22×C33C2×D4
prime radicalC2×#88810C2
hypocenterC33C4×D8
LCSG ▷ C33×C4 ▷ C2×C33 ▷ C33
UCSC1 ◁ C22 ◁ C2×C4

Complements

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

NH ≅ G/Ntype
2. #11723501C2semidirect
3. #11722018C2semidirect
4. C3×#63576C2semidirect
5. C3×#63564C2semidirect
14. #88810C4semidirect
26. #63576D3semidirect
27. #63564D3semidirect
35. C3×C4⋊(C3⋊D3)C2×C4semidirect
38. C3×C3⋊(C3⋊C8)C2×C4semidirect
78. C3⋊(C3⋊Dic3)D8semidirect
115. C4⋊(C3⋊D3)C4×D3semidirect
130. C3⋊(C3⋊C8)C4×D3semidirect
215. C33C4×D8semidirect
328. C32#4756semidirect
329. C32#4872semidirect
371. C3#10590459semidirect
372. C3#10590740semidirect

Maximal subgroups

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

subgroupindex
C2×#888102
#117235012
#117220182
C3×#635762
C3×#635642
#105907403
#105904593