Finite Group Databasebeta

← #13155457

Subgroups — #13155457

Order 1728.

630 conjugacy classes of subgroups; only the 126 normal subgroups are shown.

normal subgroups only

This group has 630 conjugacy classes of subgroups — too many to draw. Only its 126 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
1#11723673C211yesmaximal, characteristic
2C2×C3⋊D3×C3⋊C8C211yesmaximal, characteristic
3C2×C3×#3712C211yesmaximal, characteristic
4#11722051C212yesmaximal, complemented
5C3×#63572C212yesmaximal, complemented
6#10590701—31nomaximal
7#10590695—32nomaximal
8C2×C3×C4×C3⋊D3C2211yescharacteristic
9C2×C3⋊(C3⋊(C3⋊C8))C2211yescharacteristic
10C3×C4×C3⋊Dic3C2212yes—
11C2×C32×C3⋊C8C2211yescharacteristic
12C2×C32×C3⋊C8C2212yes—
13#88815C411yescharacteristic
14#88815C411yescharacteristic
15#88814C412yes—
16C3⋊D3×C3⋊C8—22no—
17C3×#3712—22no—
18C2×C8×C3⋊D3—31no—
19C2×C3×ΓL(1,𝔽25)—32no—
20#63874—31no—
21#63855—32no—
22C2×#3712D311yescharacteristic
23C2×D3×C3⋊C8—32no—
24C2×D3×C3⋊C8—61no—
25#63641—32no—
26C3×Dic3⋊C8—32no—
27C3×Dic3⋊C8—32no—
28#63572D312yescomplemented
29Dic3⋊(C3⋊C8)—32no—
30Dic3⋊(C3⋊C8)—32no—
31Dic3⋊(C3⋊C8)—62no—
32C22×C3×C3⋊D3C2×C411yescharacteristic
33C2×C33×C4C2311yescharacteristic
34C2×C3×C3⋊Dic3C2×C412yes—
35C2×C3×C3⋊Dic3C2×C412yes—
36C2×C3×C3⋊Dic3C2×C411yescharacteristic
37C3×C4×C3⋊D3D412yes—
38C3×C4×C3⋊D3Q812yes—
39C3⋊(C3⋊(C3⋊C8))—21no—
40C32×C3⋊C8—21no—
41C32×C3⋊C8—22no—
42#5119—91no—
43#4839—92no—
44C2×C4×C3⋊D3C2×D311yescharacteristic
45C2×C3×C4×D3—32no—
46C2×C3×C4×D3—61no—
47C2×C32×C8—32no—
48C2×C32×C8—31no—
49C4×C3⋊Dic3C2×D312yes—
50C2×C3⋊(C3⋊C8)—32no—
51C2×C3⋊(C3⋊C8)—31no—
52C2×C3⋊(C3⋊C8)—61no—
53C2×C3⋊(C3⋊C8)—62no—
54C2×C3⋊(C3⋊C8)—121no—
55C8×C3⋊D3—62no—
56C3×C4×Dic3—32no—
57C3×C4×Dic3—32no—
58C3×C4×Dic3—62no—
59C2×C3×C3⋊C8—21no—
60C2×C3×C3⋊C8—22no—
61C2×C3×C3⋊C8C2×D312yes—
62C2×C3×C3⋊C8C2×D312yes—
63C2×C3×C3⋊C8C2×D312yes—
64C3×ΓL(1,𝔽25)—64no—
65#3724—31no—
66#3724—31no—
67Dic3⋊Dic3—32no—
68Dic3⋊Dic3—32no—
69#3722—32no—
70(C2×D3)⋊Dic3—32no—
71(C2×D3)⋊Dic3—32no—
72#3712—22no—
73D3×C3⋊C8—64no—
74D3×C3⋊C8—64no—
75C22×C33C22×C411yescharacteristic
76C2×C3×C3⋊D3—21no—
77C2×C3×C3⋊D3—21no—
78C2×C3×C3⋊D3C4⋊C412yes—
79C33×C4C2×D411yescharacteristic
80C33×C4C2×Q811yescharacteristic
81C3×C3⋊Dic3—22no—
82C3×C3⋊Dic3—22no—
83C3×C3⋊Dic3C4⋊C412yes—
84C2×C3×M16—91no—
85#926—91no—
86C22×C3⋊C8—91no—
87C2×ΓL(1,𝔽25)—32no—
88C2×D3×C8—92no—
89C2×D3×C8—181no—
90#872—92no—
91C3×C4⋊C8—92no—
92Dic3⋊C8—32no—
93Dic3⋊C8—32no—
94Dic3⋊C8—92no—
95Dic3⋊C8—92no—
96Dic3⋊C8—182no—
97C4⋊(C3⋊C8)—92no—
98C22×C3⋊D3C4×D311yescharacteristic
99C22×C3×D3—32no—
100C22×C3×D3—61no—
101C2×C32×C4—22no—
102C2×C32×C4—21no—
103C2×C32×C4—41no—
104C2×C32×C4C22×D311yescharacteristic
105C2×C32×C4C22×D312yes—
106C2×C3⋊Dic3C4×D312yes—
107C2×C3⋊Dic3C4×D312yes—
108C2×C3⋊Dic3C4×D311yescharacteristic
109C4×C3⋊D3Dic612yes—
110C4×C3⋊D3#6712yes—
111C2×C3×Dic3—32no—
112C2×C3×Dic3—32no—
113C2×C3×Dic3—32no—
114C2×C3×Dic3—32no—
115C2×C3×Dic3—32no—
116C2×C3×Dic3—62no—
117C2×C3×Dic3—61no—
118C2×C3×Dic3—62no—
119C3×C4×D3—34no—
120C3×C4×D3—34no—
121C3×C4×D3—62no—
122C3×C4×D3—62no—
123C32×C8—61no—
124C32×C8—62no—
125C3⋊(C3⋊C8)—61no—
126C3⋊(C3⋊C8)—62no—
127C3⋊(C3⋊C8)—62no—
128C3⋊(C3⋊C8)—122no—
129C3⋊(C3⋊C8)—122no—
130C3×C3⋊C8—22no—
131C3×C3⋊C8—22no—
132C3×C3⋊C8—22no—
133C3×C3⋊C8—22no—
134C3×C3⋊C8—42no—
135#424—271noSylow
136C2×C33C2×C4⋊C411yescharacteristic
137C2×C33#13111yescharacteristic
138C2×C33#13111yescharacteristic
139C3×C3⋊D3—22no—
140C22×C3×C4—91no—
141C2×C4×D3—32no—
142C2×C4×D3—61no—
143C3×M16—182no—
144C2×C3×C8—32no—
145C2×C3×C8—32no—
146C2×C3×C8—32no—
147C2×C3×C8—61no—
148C2×C3×C8—62no—
149C3×C42—92no—
150#217—92no—
151(C2×D3)⋊C4—92no—
152(C2×D3)⋊C4—92no—
153C4⋊Dic3—91no—
154C4⋊Dic3—91no—
155Dic3⋊C4—92no—
156Dic3⋊C4—92no—
157C4×Dic3—32no—
158C4×Dic3—32no—
159C4×Dic3—62no—
160C2×C3⋊C8—92no—
161C2×C3⋊C8—91no—
162C2×C3⋊C8—181no—
163C2×C3⋊C8—182no—
164C2×C3⋊C8—182no—
165C2×C3⋊C8—361no—
166C2×C3⋊C8D3212yes—
167C2×C3⋊C8D3211yescharacteristic
168ΓL(1,𝔽25)—64no—
169D3×C8—184no—
170D3×C8—184no—
171C22×C32—22no—
172C22×C32—21no—
173C22×C32—41no—
174C22×C32C2×C4×D312yes—
175C22×C32C2×C4×D311yescharacteristic
176C2×C3⋊D3—21no—
177C2×C3⋊D3—21no—
178C2×C3⋊D3Dic3⋊C412yes—
179C2×C3×D3—34no—
180C2×C3×D3—62no—
181C2×C3×D3—62no—
182C2×C3×D3—62no—
183C2×C3×D3—62no—
184C2×C3×D3—62no—
185C32×C4—22no—
186C32×C4—21no—
187C32×C4—22no—
188C32×C4—21no—
189C32×C4—41no—
190C32×C4—41no—
191C32×C4C2×Dic611yescharacteristic
192C32×C4D3×D412yes—
193C32×C4D3×Q812yes—
194C32×C4C2×#6711yescharacteristic
195C3⋊Dic3—22no—
196C3⋊Dic3—22no—
197C3⋊Dic3Dic3⋊C412yes—
198C3×Dic3—34no—
199C3×Dic3—62no—
200C3×Dic3—64no—
201C3×Dic3—62no—
202C3×Dic3—62no—
203C3×Dic3—62no—
204C3×Dic3—62no—
205C3×Dic3—122no—
206C2×M16—91no—
207C22×C8—271no—
208#117—271no—
209C4⋊C8—92no—
210C4⋊C8—272no—
211C33#42411yescharacteristic, complemented, Sylow
212C23×C3—91no—
213C22×D3—32no—
214C22×D3—61no—
215C2×C3×C4—22no—
216C2×C3×C4—21no—
217C2×C3×C4—41no—
218C2×C3×C4—91no—
219C2×C3×C4—92no—
220C2×C3×C4—92no—
221C2×C3×C4—92no—
222C2×C3×C4—92no—
223C2×C3×C4C2×D3212yes—
224C2×C3×C4C2×D3211yescharacteristic
225C2×Dic3—32no—
226C2×Dic3—32no—
227C2×Dic3—32no—
228C2×Dic3—32no—
229C2×Dic3—32no—
230C2×Dic3—62no—
231C2×Dic3—61no—
232C2×Dic3—62no—
233C4×D3—34no—
234C4×D3—34no—
235C4×D3—62no—
236C4×D3—62no—
237C3×C8—62no—
238C3×C8—62no—
239C3×C8—62no—
240C3×C8—62no—
241C3×C8—122no—
242C3⋊C8—22no—
243C3⋊C8—21no—
244C3⋊C8—182no—
245C3⋊C8—182no—
246C3⋊C8—181no—
247C3⋊C8—362no—
248C3⋊C8—362no—
249C2×C32—21no—
250C2×C32—22no—
251C2×C32—22no—
252C2×C32—21no—
253C2×C32—22no—
254C2×C32—21no—
255C2×C32—41no—
256C2×C32—41no—
257C2×C32—41no—
258C2×C32D3×C4⋊C412yes—
259C2×C32#90112yes—
260C2×C32#90711yescharacteristic
261C2×C32#90712yes—
262C2×C32#90711yescharacteristic
263C2×C32C2×Dic3⋊C411yescharacteristic
264C3⋊D3—22no—
265C3×D3—64no—
266C3×D3—64no—
267C22×C4—91no—
268M16—182no—
269C2×C8—31no—
270C2×C8—32no—
271C2×C8—271no—
272C2×C8—542no—
273C4⋊C4—271no—
274C4⋊C4—271no—
275C22⋊C4—272no—
276C42—92no—
277C22×C3—22no—
278C22×C3—21no—
279C22×C3—41no—
280C22×C3—92no—
281C22×C3—181no—
282C22×C3—181no—
283C22×C3C4×D3211yescharacteristic
284C22×C3C4×D3212yes—
285C2×D3—34no—
286C2×D3—62no—
287C2×D3—62no—
288C2×D3—62no—
289C2×D3—62no—
290C2×D3—62no—
291C3×C4—21no—
292C3×C4—22no—
293C3×C4—21no—
294C3×C4—22no—
295C3×C4—41no—
296C3×C4—41no—
297C3×C4—92no—
298C3×C4—182no—
299C3×C4—182no—
300C3×C4D3×Dic612yes—
301C3×C4#379711yescharacteristic
302C3×C4D3×#6712yes—
303C3×C4#381111yescharacteristic
304Dic3—34no—
305Dic3—62no—
306Dic3—64no—
307Dic3—62no—
308Dic3—62no—
309Dic3—62no—
310Dic3—62no—
311Dic3—122no—
312C32—22no—
313C32—21no—
314C32—41no—
315C32#483912yescomplemented
316C32#511911yescharacteristic, complemented
317C23—91no—
318C2×C4—91no—
319C2×C4—92no—
320C2×C4—92no—
321C2×C4—92no—
322C2×C4—92no—
323C2×C4D3311yescharacteristic
324C8—61no—
325C8—62no—
326C8—541no—
327C2×C3—22no—
328C2×C3—21no—
329C2×C3—22no—
330C2×C3—21no—
331C2×C3—21no—
332C2×C3—22no—
333C2×C3—41no—
334C2×C3—41no—
335C2×C3—41no—
336C2×C3—182no—
337C2×C3#6380311yescharacteristic
338C2×C3#6380412yes—
339C2×C3#6380512yes—
340C2×C3#6380511yescharacteristic
341C2×C3D3×Dic3⋊C412yes—
342C2×C3#6388311yescharacteristic
343D3—64no—
344D3—64no—
345C22—92no—
346C22—181no—
347C22—181no—
348C22#8894911yescharacteristic
349C4—92no—
350C4—182no—
351C4—182no—
352C4#8895211yescharacteristic
353C4#8895711yescharacteristic
354C3—21no—
355C3—22no—
356C3—41no—
357C3#1059069512yescomplemented
358C3#1059070111yescharacteristic, complemented
359C2—182no—
360C2#1172282611yescharacteristic
361C2#1172282711yescharacteristic
362C2#1172291011yescharacteristic
subgroupG/H
center, Frattini, hypercenterC2×C4D33
commutator, socleC22×C33C22×C4
Fitting, cosocleC2×C33×C4C23
prime radicalC22×C3×C3⋊D3C2×C4
hypocenterC33#424
LCSG ▷ C22×C33 ▷ C33
UCSC1 ◁ C2×C4

Complements

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

NH ≅ G/Ntype
4. #11722051C2semidirect
5. C3×#63572C2semidirect
28. #63572D3semidirect
211. C33#424semidirect
315. C32#4839semidirect
316. C32#5119semidirect
357. C3#10590695semidirect
358. C3#10590701semidirect

Maximal subgroups

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

subgroupindex
#117236732
C2×C3⋊D3×C3⋊C82
C2×C3×#37122
#117220512
C3×#635722
#105907013
#105906953