Finite Group Databasebeta

← #13167383

Subgroups — #13167383

Order 1728.

684 conjugacy classes of subgroups; only the 103 normal subgroups are shown.

normal subgroups only

This group has 684 conjugacy classes of subgroups — too many to draw. Only its 103 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×#89040C211yesmaximal, characteristic
2C2×C3×C4×#157C211yesmaximal, characteristic
3#11723689C211yesmaximal, characteristic
4#11722791C214yesmaximal
5#10593342—31nomaximal
6C22×C3×#157C2212yes—
7C2×C3×C4×C3⋊D3C2211yescharacteristic
8C2×C3×#634—21no—
9C2×#6296—21no—
10C2×#6294C2212yes—
11C2×#6294C2212yes—
12#89040—22no—
13#89040C414yescomplemented
14C3×C4×#157—22no—
15C3×C4⋊(C3⋊Dic3)—21no—
16#88815—21no—
17C2×#3797—31no—
18C2×D3×Dic6—61no—
19C2×C4×#157D311yescharacteristic
20#63900—31no—
21D3×C4⋊Dic3—61no—
22#63737—34no—
23C22×C3×C3⋊D3C2311yescharacteristic
24C2×C3×#157—22no—
25C2×C3×#157—22no—
26C2×C3×#157Q814yescomplemented
27C2×C33×C4D411yescharacteristic
28C2×C3×C3⋊Dic3—22no—
29C2×C3×C3⋊Dic3—22no—
30C2×C3×C3⋊Dic3D411yescharacteristic
31C3×C4×C3⋊D3C2×C412yes—
32C3×C4×C3⋊D3D412yes—
33C3×#634—24no—
34#6296—24no—
35#6294—22no—
36#6294—24no—
37#6294C2×C414yes—
38#4929—91nomaximal
39C22×#157C2×D312yes—
40C2×C4×C3⋊D3C2×D311yescharacteristic
41C2×#634—21no—
42C2×C3×C4×D3—61no—
43C2×C3×C4×D3—61no—
44C2×C3×Dic6—61no—
45C2×C3×Dic6—61no—
46C2×C3×Dic6—121no—
47C2×#627—62no—
48C2×#627—61no—
49C2×#624—32no—
50C2×#624—32no—
51C2×D3×Dic3—62no—
52C2×D3×Dic3—62no—
53#3797—34no—
54#3797—62no—
55D3×Dic6—64no—
56D3×Dic6—64no—
57C4×#157—22no—
58C4⋊(C3⋊Dic3)—21no—
59C3×C4⋊Dic3—61no—
60C3×C4⋊Dic3—61no—
61C3×C4⋊Dic3—121no—
62#3724—61no—
63Dic3⋊Dic3—62no—
64C22×C33C2×D411yescharacteristic
65C2×C3×C3⋊D3C22×C411yescharacteristic
66C2×C3×C3⋊D3C2×D411yescharacteristic
67C2×C3×C3⋊D3C2×Q812yes—
68C2×C3×C3⋊D3#4112yes—
69C3×#157—24no—
70C33×C4C22⋊C412yes—
71C3×C3⋊Dic3—24no—
72C3×C3⋊Dic3—42no—
73C3×C3⋊Dic3C22⋊C412yes—
74C2×D3×Q8—181no—
75C22×Dic6—91no—
76C2×C3×C42—91no—
77C2×C4⋊Dic3—91no—
78D3×C4⋊C4—181no—
79C3⋊#95—94no—
80C22×C3⋊D3C22×D311yescharacteristic
81C22×C3×D3—61no—
82C22×C3×D3—61no—
83C2×#157—22no—
84C2×#157—22no—
85C2×#157Dic614yescomplemented
86C2×C32×C4—21no—
87C2×C32×C4—21no—
88C2×C32×C4—41no—
89C2×C32×C4—41no—
90C2×C32×C4D1211yescharacteristic
91C2×C3⋊Dic3—22no—
92C2×C3⋊Dic3—22no—
93C2×C3⋊Dic3D1211yescharacteristic
94C4×C3⋊D3C4×D312yes—
95C4×C3⋊D3#6712yes—
96#634—24no—
97C2×C3×Dic3—61no—
98C2×C3×Dic3—61no—
99C2×C3×Dic3—62no—
100C2×C3×Dic3—62no—
101C2×C3×Dic3—62no—
102C2×C3×Dic3—62no—
103C2×C3×Dic3—122no—
104C2×C3×Dic3—122no—
105C3×C4×D3—62no—
106C3×C4×D3—62no—
107C3×C4×D3—62no—
108C3×C4×D3—62no—
109C3×Dic6—64no—
110C3×Dic6—64no—
111C3×Dic6—124no—
112#627—68no—
113#627—64no—
114#624—34no—
115#624—62no—
116#624—64no—
117D3×Dic3—64no—
118D3×Dic3—64no—
119D3×Dic3—124no—
120#391—271noSylow
121C2×C33C2×C22⋊C411yescharacteristic
122C2×C33#12411yescharacteristic
123C2×C33C4⋊Q811yescharacteristic
124C3×C3⋊D3C4×Q812yes—
125C3×C3⋊D3C22⋊Q812yes—
126C2×C3×Q8—181no—
127C22×C3×C4—91no—
128C22×C3×C4—92no—
129C22×Dic3—92no—
130C22×Dic3—92no—
131D3×Q8—184no—
132D3×Q8—184no—
133C2×C4×D3—61no—
134C2×C4×D3—61no—
135C2×C4×D3—182no—
136C2×C4×D3—182no—
137C2×Dic6—61no—
138C2×Dic6—61no—
139C2×Dic6—94no—
140C2×Dic6—121no—
141C2×Dic6—182no—
142C2×Dic6—181no—
143C2×Dic6—182no—
144C3×C4⋊C4—181no—
145C3×C42—182no—
146C4⋊Dic3—61no—
147C4⋊Dic3—61no—
148C4⋊Dic3—121no—
149C4⋊Dic3—181no—
150Dic3⋊C4—182no—
151C22×C32—21no—
152C22×C32—21no—
153C22×C32—41no—
154C22×C32—41no—
155C22×C32C2×D1211yescharacteristic
156C2×C3⋊D3C2×Dic612yes—
157C2×C3⋊D3C2×C4×D311yescharacteristic
158C2×C3⋊D3#23512yes—
159C2×C3⋊D3C2×#6711yescharacteristic
160C2×C3×D3—61no—
161C2×C3×D3—61no—
162C2×C3×D3—61no—
163C2×C3×D3—61no—
164C2×C3×D3—62no—
165C2×C3×D3—62no—
166C2×C3×D3—62no—
167C2×C3×D3—62no—
168#157—24no—
169C32×C4—22no—
170C32×C4—22no—
171C32×C4—42no—
172C32×C4—42no—
173C32×C4(C2×D3)⋊C412yes—
174C3⋊Dic3—24no—
175C3⋊Dic3—42no—
176C3⋊Dic3(C2×D3)⋊C412yes—
177C3×Dic3—62no—
178C3×Dic3—62no—
179C3×Dic3—64no—
180C3×Dic3—64no—
181C3×Dic3—124no—
182C3×Dic3—122no—
183C3×Dic3—122no—
184C3×Dic3—242no—
185C22×Q8—271no—
186C2×C4⋊C4—271no—
187C2×C42—91no—
188#95—274no—
189C33#39111yescharacteristic, complemented, Sylow
190C23×C3—91no—
191C22×D3—61no—
192C22×D3—61no—
193C3×Q8—184no—
194C2×C3×C4—21no—
195C2×C3×C4—21no—
196C2×C3×C4—41no—
197C2×C3×C4—41no—
198C2×C3×C4—91no—
199C2×C3×C4—92no—
200C2×C3×C4—92no—
201C2×C3×C4—94no—
202C2×C3×C4—182no—
203C2×C3×C4—182no—
204C2×C3×C4—182no—
205C2×C3×C4—182no—
206C2×C3×C4GO+(4,𝔽2)11yescharacteristic
207C2×Dic3—61no—
208C2×Dic3—61no—
209C2×Dic3—62no—
210C2×Dic3—62no—
211C2×Dic3—62no—
212C2×Dic3—62no—
213C2×Dic3—94no—
214C2×Dic3—122no—
215C2×Dic3—122no—
216C2×Dic3—182no—
217C2×Dic3—184no—
218C4×D3—62no—
219C4×D3—62no—
220C4×D3—62no—
221C4×D3—62no—
222C4×D3—184no—
223C4×D3—184no—
224C4×D3—364no—
225Dic6—64no—
226Dic6—64no—
227Dic6—124no—
228Dic6—184no—
229Dic6—188no—
230C2×C32—21no—
231C2×C32—21no—
232C2×C32—21no—
233C2×C32—21no—
234C2×C32—21no—
235C2×C32—21no—
236C2×C32—41no—
237C2×C32—41no—
238C2×C32—41no—
239C2×C32—41no—
240C2×C32—41no—
241C2×C32—41no—
242C2×C32#86911yescharacteristic
243C2×C32#87511yescharacteristic
244C2×C32C2×(C2×D3)⋊C411yescharacteristic
245C3⋊D3C4×Dic612yes—
246C3⋊D3#92412yes—
247C3×D3—62no—
248C3×D3—62no—
249C3×D3—62no—
250C3×D3—62no—
251C2×Q8—181no—
252C2×Q8—274no—
253C2×Q8—541no—
254C2×Q8—542no—
255C22×C4—91no—
256C22×C4—92no—
257C22×C4—272no—
258C22×C4—272no—
259C4⋊C4—181no—
260C4⋊C4—541no—
261C42—182no—
262C22×C3—21no—
263C22×C3—21no—
264C22×C3—41no—
265C22×C3—41no—
266C22×C3—92no—
267C22×C3—92no—
268C22×C3—91no—
269C22×C3—91no—
270C22×C3C2×GO+(4,𝔽2)11yescharacteristic
271C2×D3—61no—
272C2×D3—61no—
273C2×D3—61no—
274C2×D3—61no—
275C2×D3—62no—
276C2×D3—62no—
277C2×D3—62no—
278C2×D3—62no—
279C3×C4—22no—
280C3×C4—22no—
281C3×C4—42no—
282C3×C4—42no—
283C3×C4—92no—
284C3×C4—184no—
285C3×C4—184no—
286C3×C4—362no—
287C3×C4D32⋊C412yes—
288Dic3—62no—
289Dic3—62no—
290Dic3—64no—
291Dic3—64no—
292Dic3—124no—
293Dic3—122no—
294Dic3—122no—
295Dic3—242no—
296C32—21no—
297C32—21no—
298C32—41no—
299C32—41no—
300C32#492911yescharacteristic, complemented
301C23—91no—
302Q8—184no—
303Q8—544no—
304C2×C4—91no—
305C2×C4—92no—
306C2×C4—92no—
307C2×C4—94no—
308C2×C4—182no—
309C2×C4—182no—
310C2×C4—182no—
311C2×C4—182no—
312C2×C4—274no—
313C2×C4—542no—
314C2×C4—544no—
315C2×C4#157⋊D311yescharacteristic
316C2×C3—21no—
317C2×C3—21no—
318C2×C3—21no—
319C2×C3—21no—
320C2×C3—21no—
321C2×C3—21no—
322C2×C3—41no—
323C2×C3—41no—
324C2×C3—41no—
325C2×C3—41no—
326C2×C3—41no—
327C2×C3—41no—
328C2×C3—92no—
329C2×C3—92no—
330C2×C3#6422111yescharacteristic
331C2×C3#6422211yescharacteristic
332C2×C3C2×D32⋊C411yescharacteristic
333D3—62no—
334D3—62no—
335D3—62no—
336D3—62no—
337C22—92no—
338C22—92no—
339C22—91no—
340C22—91no—
341C22C2×#157⋊D311yescharacteristic
342C4—92no—
343C4—184no—
344C4—184no—
345C4—362no—
346C4#8893912yes—
347C3—21no—
348C3—21no—
349C3—41no—
350C3—41no—
351C3#1059334211yescharacteristic, complemented
352C2—92no—
353C2—92no—
354C2#1172433411yescharacteristic
355C2#1172433511yescharacteristic
356C2C2×#8893911yescharacteristic
subgroupG/H
center, FrattiniC22C2×#157⋊D3
commutatorC2×C3×C3⋊D3C22×C4
FittingC2×C33×C4D4
socleC22×C33C2×D4
prime radical, cosocleC22×C3×C3⋊D3C23
hypercenterC2×C4#157⋊D3
hypocenterC33#391
LCSG ▷ C2×C3×C3⋊D3 ▷ C33
UCSC1 ◁ C22 ◁ C2×C4

Complements

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

NH ≅ G/Ntype
13. #89040C4semidirect
26. C2×C3×#157Q8semidirect
85. C2×#157Dic6semidirect
189. C33#391semidirect
300. C32#4929semidirect
351. C3#10593342semidirect

Maximal subgroups

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

subgroupindex
C2×#890402
C2×C3×C4×#1572
#117236892
#117227912
#105933423
#49299