Finite Group Databasebeta

← #13155470

Subgroups — #13155470

Order 1728.

710 conjugacy classes of subgroups; only the 142 normal subgroups are shown.

normal subgroups only

This group has 710 conjugacy classes of subgroups — too many to draw. Only its 142 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×#88808C211yesmaximal, characteristic
2C3⋊D3×C4⋊Dic3C211yesmaximal, characteristic
3C3×#63900C211yesmaximal, characteristic
4#11722057C212yesmaximal, complemented
5C3×#63576C212yesmaximal, complemented
6#10590573—31nomaximal
7D3×#807—32nomaximal
8C2×C3×C4×C3⋊D3C2211yescharacteristic
9C2×Dic3×C3⋊D3—21no—
10C2×C3×#624—21no—
11C4⋊(C3⋊(C3⋊Dic3))C2211yescharacteristic
12C2×C3×C3⋊(C3⋊C8)C2212yes—
13C32×C4⋊Dic3C2211yescharacteristic
14C32×C4⋊Dic3C2212yes—
15#88808C414yescomplemented
16#88807—21no—
17C3×#3724—21no—
18C4⋊C4×C3⋊D3—31no—
19C3×D3×C4⋊C4—32no—
20#63900D311yescharacteristic
21D3×C4⋊Dic3—32no—
22D3×C4⋊Dic3—61no—
23C2×#3710—31no—
24C2×D3×C3⋊C8—32no—
25#63634—32no—
26C3×#807—32no—
27C3×#807—32no—
28#63576D312yescomplemented
29#63575—32no—
30#63575—32no—
31#63575—62no—
32C22×C3×C3⋊D3D411yescharacteristic
33C2×C33×C4C2311yescharacteristic
34C2×C3⋊(C3⋊Dic3)—21no—
35C2×C3×C3⋊Dic3D411yescharacteristic
36C3×C4×C3⋊D3C2×C412yes—
37C3×C4×C3⋊D3Q812yes—
38C2×C32×Dic3—21no—
39C2×C32×Dic3—22no—
40Dic3×C3⋊D3—42no—
41C3×#624—42no—
42C3×C3⋊(C3⋊C8)C2×C414yescomplemented
43C2×#807—91no—
44D3×#107—92no—
45C2×C4×C3⋊D3—61no—
46C2×C4×C3⋊D3C2×D311yescharacteristic
47C2×C3×C4×D3—32no—
48C2×C3×C4×D3—61no—
49C2×C3×C4×D3—62no—
50C2×#624—21no—
51C2×D3×Dic3—62no—
52C2×D3×Dic3—62no—
53C32×C4⋊C4—32no—
54C32×C4⋊C4—31no—
55C4⋊(C3⋊Dic3)—31no—
56C4⋊(C3⋊Dic3)—32no—
57C4⋊(C3⋊Dic3)—61no—
58C4⋊(C3⋊Dic3)—62no—
59C4⋊(C3⋊Dic3)—121no—
60#3750—61no—
61C2×C3⋊(C3⋊C8)C2×D312yes—
62C3×C4⋊Dic3—21no—
63C3×C4⋊Dic3—22no—
64C3×C4⋊Dic3C2×D312yes—
65C3×C4⋊Dic3C2×D312yes—
66C3×C4⋊Dic3C2×D312yes—
67C3×Dic3⋊C4—62no—
68C2×C3×C3⋊C8—32no—
69C2×C3×C3⋊C8—32no—
70C2×C3×C3⋊C8—62no—
71#3724—21no—
72Dic3⋊Dic3—62no—
73Dic3⋊Dic3—62no—
74#3710—34no—
75D3×C3⋊C8—38no—
76C22×C33C2×D411yescharacteristic
77C2×C3×C3⋊D3C4⋊C412yes—
78C2×C3×C3⋊D3D812yes—
79C2×C3×C3⋊D3Q1612yes—
80C33×C4C22×C411yescharacteristic
81C33×C4C2×Q811yescharacteristic
82C3⋊(C3⋊Dic3)—41no—
83C3×C3⋊Dic3C4⋊C412yes—
84C32×Dic3—41no—
85C32×Dic3—42no—
86C2×C3×C4⋊C4—91no—
87C2×C4⋊Dic3—91no—
88C22×C3⋊C8—91no—
89C2×D3×C8—92no—
90D3×C4⋊C4—32no—
91D3×C4⋊C4—92no—
92D3×C4⋊C4—181no—
93C3×#107—92no—
94#818—92no—
95#807—32no—
96#807—32no—
97#807—92no—
98#807—92no—
99#807—182no—
100C22×C3⋊D3#6711yescharacteristic
101C22×C3×D3—32no—
102C22×C3×D3—61no—
103C2×C32×C4—22no—
104C2×C32×C4—21no—
105C2×C32×C4—41no—
106C2×C32×C4—61no—
107C2×C32×C4—62no—
108C2×C32×C4C22×D311yescharacteristic
109C2×C32×C4C22×D312yes—
110C2×C3⋊Dic3—61no—
111C2×C3⋊Dic3—62no—
112C2×C3⋊Dic3—62no—
113C2×C3⋊Dic3—122no—
114C2×C3⋊Dic3—122no—
115C2×C3⋊Dic3#6711yescharacteristic
116C4×C3⋊D3—122no—
117C4×C3⋊D3Dic612yes—
118C4×C3⋊D3C4×D312yes—
119C2×C3×Dic3—22no—
120C2×C3×Dic3—22no—
121C2×C3×Dic3—22no—
122C2×C3×Dic3—22no—
123C2×C3×Dic3—32no—
124C2×C3×Dic3—42no—
125C2×C3×Dic3—61no—
126C3×C4×D3—34no—
127C3×C4×D3—34no—
128C3×C4×D3—62no—
129C3×C4×D3—62no—
130C3×C4×D3—124no—
131#624—42no—
132D3×Dic3—124no—
133D3×Dic3—124no—
134C3⋊(C3⋊C8)C4×D314yescomplemented
135C3×C3⋊C8—34no—
136C3×C3⋊C8—34no—
137C3×C3⋊C8—64no—
138C2×#107—271noSylow
139C2×C33C2×C4⋊C411yescharacteristic
140C2×C33C2×D811yescharacteristic
141C2×C33C2×Q1611yescharacteristic
142C3×C3⋊D3#10714yescomplemented
143C22×C3×C4—91no—
144C22×C3×C4—181no—
145C22×Dic3—181no—
146C2×C4×D3—32no—
147C2×C4×D3—61no—
148C2×C4×D3—62no—
149C2×C4×D3—182no—
150C2×C4×D3—182no—
151C2×C3×C8—92no—
152C3×C4⋊C4—32no—
153C3×C4⋊C4—32no—
154C3×C4⋊C4—32no—
155C3×C4⋊C4—61no—
156C3×C4⋊C4—62no—
157C3×C4⋊C4—181no—
158C4⋊Dic3—91no—
159C4⋊Dic3—92no—
160C4⋊Dic3—181no—
161C4⋊Dic3—182no—
162C4⋊Dic3—181no—
163C4⋊Dic3—361no—
164C4⋊Dic3D3211yescharacteristic
165C4⋊Dic3D3212yes—
166Dic3⋊C4—62no—
167Dic3⋊C4—182no—
168Dic3⋊C4—182no—
169C2×C3⋊C8—32no—
170C2×C3⋊C8—32no—
171C2×C3⋊C8—62no—
172C2×C3⋊C8—94no—
173D3×C8—98no—
174C22×C32—22no—
175C22×C32—21no—
176C22×C32—41no—
177C22×C32D3×D412yes—
178C22×C32C2×#6711yescharacteristic
179C2×C3⋊D3Dic3⋊C412yes—
180C2×C3⋊D3#21312yes—
181C2×C3⋊D3#21612yes—
182C2×C3×D3—34no—
183C2×C3×D3—34no—
184C2×C3×D3—34no—
185C2×C3×D3—62no—
186C2×C3×D3—62no—
187C2×C3×D3—62no—
188C32×C4—22no—
189C32×C4—21no—
190C32×C4—22no—
191C32×C4—21no—
192C32×C4—41no—
193C32×C4—41no—
194C32×C4—121no—
195C32×C4—122no—
196C32×C4C2×Dic611yescharacteristic
197C32×C4C2×C4×D312yes—
198C32×C4C2×C4×D311yescharacteristic
199C32×C4D3×Q812yes—
200C3⋊Dic3—121no—
201C3⋊Dic3—122no—
202C3⋊Dic3—122no—
203C3⋊Dic3—242no—
204C3⋊Dic3—242no—
205C3⋊Dic3Dic3⋊C412yes—
206C3×Dic3—34no—
207C3×Dic3—42no—
208C3×Dic3—42no—
209C3×Dic3—42no—
210C3×Dic3—42no—
211C3×Dic3—62no—
212C3×Dic3—82no—
213C22×C8—271no—
214C2×C4⋊C4—91no—
215C2×C4⋊C4—271no—
216#107—92no—
217#107—272no—
218C33C2×#10711yescharacteristic, complemented, Sylow
219C23×C3—91no—
220C22×D3—32no—
221C22×D3—61no—
222C2×C3×C4—22no—
223C2×C3×C4—21no—
224C2×C3×C4—41no—
225C2×C3×C4—62no—
226C2×C3×C4—62no—
227C2×C3×C4—62no—
228C2×C3×C4—62no—
229C2×C3×C4—91no—
230C2×C3×C4—92no—
231C2×C3×C4—92no—
232C2×C3×C4—122no—
233C2×C3×C4—362no—
234C2×C3×C4C2×D3212yes—
235C2×C3×C4C2×D3211yescharacteristic
236C2×Dic3—22no—
237C2×Dic3—21no—
238C2×Dic3—32no—
239C2×Dic3—61no—
240C2×Dic3—182no—
241C2×Dic3—182no—
242C2×Dic3—181no—
243C2×Dic3—362no—
244C2×Dic3—362no—
245C2×Dic3—362no—
246C4×D3—34no—
247C4×D3—34no—
248C4×D3—62no—
249C4×D3—62no—
250C4×D3—124no—
251C4×D3—364no—
252C4×D3—364no—
253C3×C8—94no—
254C3⋊C8—34no—
255C3⋊C8—34no—
256C3⋊C8—64no—
257C2×C32—22no—
258C2×C32—21no—
259C2×C32—22no—
260C2×C32—21no—
261C2×C32—22no—
262C2×C32—21no—
263C2×C32—41no—
264C2×C32—41no—
265C2×C32—41no—
266C2×C32D3×C4⋊C412yes—
267C2×C32D3×D812yes—
268C2×C32D3×Q1612yes—
269C2×C32C2×Dic3⋊C411yescharacteristic
270C2×C32C2×#21311yescharacteristic
271C2×C32C2×#21611yescharacteristic
272C3⋊D3#80714yescomplemented
273C3×D3—38no—
274C3×D3—64no—
275C22×C4—91no—
276C22×C4—181no—
277C22×C4—541no—
278C2×C8—92no—
279C2×C8—274no—
280C4⋊C4—31no—
281C4⋊C4—32no—
282C4⋊C4—181no—
283C4⋊C4—271no—
284C4⋊C4—541no—
285C22×C3—21no—
286C22×C3—22no—
287C22×C3—41no—
288C22×C3—92no—
289C22×C3—92no—
290C22×C3—92no—
291C22×C3D3×#6712yes—
292C22×C3#381111yescharacteristic
293C2×D3—34no—
294C2×D3—34no—
295C2×D3—34no—
296C2×D3—62no—
297C2×D3—62no—
298C2×D3—62no—
299C3×C4—22no—
300C3×C4—21no—
301C3×C4—21no—
302C3×C4—22no—
303C3×C4—41no—
304C3×C4—41no—
305C3×C4—92no—
306C3×C4—122no—
307C3×C4—122no—
308C3×C4—122no—
309C3×C4—122no—
310C3×C4—242no—
311C3×C4D3×Dic612yes—
312C3×C4#379711yescharacteristic
313C3×C4C4×D3212yes—
314C3×C4C4×D3211yescharacteristic
315Dic3—34no—
316Dic3—42no—
317Dic3—41no—
318Dic3—62no—
319Dic3—362no—
320Dic3—362no—
321Dic3—361no—
322Dic3—722no—
323Dic3—722no—
324C32—22no—
325C32—21no—
326C32—41no—
327C32D3×#10712yescomplemented
328C32C2×#80711yescharacteristic, complemented
329C23—91no—
330C2×C4—62no—
331C2×C4—61no—
332C2×C4—91no—
333C2×C4—92no—
334C2×C4—92no—
335C2×C4—362no—
336C2×C4—541no—
337C2×C4—1082no—
338C2×C4D3311yescharacteristic
339C8—94no—
340C2×C3—21no—
341C2×C3—22no—
342C2×C3—21no—
343C2×C3—22no—
344C2×C3—21no—
345C2×C3—22no—
346C2×C3—41no—
347C2×C3—41no—
348C2×C3—41no—
349C2×C3—94no—
350C2×C3D3×Dic3⋊C412yes—
351C2×C3#6388311yescharacteristic
352C2×C3D3×#21312yes—
353C2×C3#6392611yescharacteristic
354C2×C3D3×#21612yes—
355C2×C3#6394411yescharacteristic
356D3—38no—
357D3—64no—
358C22—92no—
359C22—92no—
360C22—92no—
361C22#8895211yescharacteristic
362C4—92no—
363C4—122no—
364C4—121no—
365C4—1081no—
366C4#8894911yescharacteristic
367C4#8895711yescharacteristic
368C3—21no—
369C3—22no—
370C3—41no—
371C3D3×#80712yescomplemented
372C3#1059057311yescharacteristic, complemented
373C2—94no—
374C2#1172283011yescharacteristic
375C2#1172284611yescharacteristic
376C2#1172291011yescharacteristic
subgroupG/H
centerC22#88952
commutatorC33×C4C22×C4
Frattini, hypercenterC2×C4D33
Fitting, cosocleC2×C33×C4C23
socleC22×C33C2×D4
prime radicalC22×C3×C3⋊D3D4
hypocenterC33C2×#107
LCSG ▷ C33×C4 ▷ C2×C33 ▷ C33
UCSC1 ◁ C22 ◁ C2×C4

Complements

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

NH ≅ G/Ntype
4. #11722057C2semidirect
5. C3×#63576C2semidirect
15. #88808C4semidirect
28. #63576D3semidirect
42. C3×C3⋊(C3⋊C8)C2×C4semidirect
134. C3⋊(C3⋊C8)C4×D3semidirect
142. C3×C3⋊D3#107semidirect
218. C33C2×#107semidirect
272. C3⋊D3#807semidirect
327. C32D3×#107semidirect
328. C32C2×#807semidirect
371. C3D3×#807semidirect
372. C3#10590573semidirect

Maximal subgroups

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

subgroupindex
C2×#888082
C3⋊D3×C4⋊Dic32
C3×#639002
#117220572
C3×#635762
#105905733
D3×#8073