Finite Group Databasebeta

← #13178728

Subgroups — #13178728

Order 1728.

1034 conjugacy classes of subgroups; only the 167 normal subgroups are shown.

normal subgroups only

This group has 1034 conjugacy classes of subgroups — too many to draw. Only its 167 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
1C3×#64156C211yesmaximal, characteristic
2C3×#64153C212yesmaximal, complemented
3#11723688C214yesmaximal, complemented
4#11723686C212yesmaximal
5#11723679C212yesmaximal, complemented
6#11723676C214yesmaximal, complemented
7#10595393—32nomaximal
8#10595352—31nomaximal
9C2×C33×Q8C2211yescharacteristic
10C2×C3×C4⋊(C3⋊D3)C2211yescharacteristic
11C2×C3×#634C2212yes—
12C2×#6294C2212yes—
13C3×#3752C2214yes—
14C3×#3750C2214yes—
15C3×#3750C2214yes—
16C3×C4×C3⋊Dic3C2212yes—
17C3×C4×C3⋊Dic3C2211yescharacteristic
18#88815C2212yes—
19#88815C2214yes—
20#88815C2212yes—
21#88814C2212yes—
22#88814C2214yes—
23#64156D311yescharacteristic
24#64153D312yescomplemented
25C3×#947—32no—
26C3×#947—61no—
27C3×#944—32no—
28C3×#944—32no—
29C3×#944—62no—
30#63897—32no—
31#63896—34no—
32#63888—34no—
33#63887—34no—
34#63884—34no—
35#63870—32no—
36#63868—34no—
37#63867—34no—
38#63862—34no—
39#63859—34no—
40C22×C3×C3⋊D3C2312yes—
41C33×Q8—41no—
42C2×C33×C4C2311yescharacteristic
43C2×C33×C4C2312yes—
44C2×C3×C3⋊Dic3C2314yes—
45C2×C3×C3⋊Dic3C2312yes—
46C2×C3×C3⋊Dic3C2314yes—
47C3×C4⋊(C3⋊D3)—41no—
48C3×#634—42no—
49#6294—42no—
50#5735—92no—
51#5672—91no—
52C2×C32×Q8—22no—
53C2×C32×Q8—21no—
54C2×C32×Q8—41no—
55C2×C32×Q8C2×D311yescharacteristic
56C2×C32×Q8C2×D312yes—
57C2×C4⋊(C3⋊D3)C2×D311yescharacteristic
58C2×#634C2×D312yes—
59C2×C3×D12—32no—
60C2×C3×D12—61no—
61C2×C3×Dic6—32no—
62C2×C3×Dic6—32no—
63C2×C3×Dic6—62no—
64C2×#624—32no—
65C2×D3×Dic3—34no—
66#3752C2×D314yes—
67#3750C2×D314yes—
68#3750C2×D314yes—
69C4×C3⋊Dic3C2×D312yes—
70C4×C3⋊Dic3C2×D311yescharacteristic
71C3×(C2×D3)⋊C4—34no—
72C3×(C2×D3)⋊C4—34no—
73C3×(C2×D3)⋊C4—64no—
74C3×Dic3⋊C4—34no—
75C3×Dic3⋊C4—34no—
76C3×Dic3⋊C4—34no—
77C3×Dic3⋊C4—34no—
78C3×Dic3⋊C4—64no—
79C3×Dic3⋊C4—64no—
80C3×C4×Dic3—32no—
81C3×C4×Dic3—32no—
82C3×C4×Dic3—32no—
83C3×C4×Dic3—62no—
84C3×C4×Dic3—61no—
85#3724—34no—
86#3724—32no—
87#3724—32no—
88Dic3⋊Dic3—34no—
89Dic3⋊Dic3—34no—
90Dic3⋊Dic3—34no—
91Dic3⋊Dic3—34no—
92#3722—32no—
93#3722—34no—
94(C2×D3)⋊Dic3—34no—
95(C2×D3)⋊Dic3—34no—
96(C2×D3)⋊Dic3—34no—
97C22×C33C2411yescharacteristic
98C2×C3×C3⋊D3—22no—
99C2×C3×C3⋊D3—24no—
100C33×C4—22no—
101C33×C4—21no—
102C3×C3⋊Dic3—24no—
103C3×C3⋊Dic3—24no—
104C3×C3⋊Dic3—22no—
105C3×C4⋊Q8—92no—
106C3×#124—91no—
107#947—32no—
108#947—61no—
109#946—94no—
110#944—32no—
111#944—32no—
112#944—62no—
113#935—92no—
114#898—94no—
115#896—94no—
116#894—94no—
117#889—94no—
118#879—94no—
119#878—94no—
120#876—94no—
121#870—92no—
122C22×C3⋊D3C22×D312yes—
123C22×C3×D3—34no—
124C22×C3×D3—62no—
125C32×Q8—42no—
126C32×Q8—41no—
127C32×Q8—44no—
128C32×Q8—42no—
129C32×Q8—44no—
130C2×C32×C4—22no—
131C2×C32×C4—21no—
132C2×C32×C4—22no—
133C2×C32×C4—22no—
134C2×C32×C4—22no—
135C2×C32×C4—42no—
136C2×C32×C4—41no—
137C2×C32×C4C22×D312yes—
138C2×C32×C4C22×D312yes—
139C2×C32×C4C22×D312yes—
140C2×C32×C4C22×D311yescharacteristic
141C2×C32×C4C22×D312yes—
142C2×C3⋊Dic3C22×D314yes—
143C2×C3⋊Dic3C22×D314yes—
144C2×C3⋊Dic3C22×D312yes—
145C4⋊(C3⋊D3)—41no—
146#634—42no—
147C2×C3×Dic3—34no—
148C2×C3×Dic3—34no—
149C2×C3×Dic3—34no—
150C2×C3×Dic3—34no—
151C2×C3×Dic3—34no—
152C2×C3×Dic3—64no—
153C2×C3×Dic3—62no—
154C2×C3×Dic3—64no—
155C3×D12—122no—
156C3×D12—122no—
157C3×Dic6—122no—
158C3×Dic6—122no—
159C3×Dic6—124no—
160#624—122no—
161D3×Dic3—124no—
162#563—271noSylow
163C2×C33H(5,𝔽2)11yescharacteristic
164C2×C332−1+412yes—
165C3×C3⋊D3—42no—
166C2×C3×Q8—22no—
167C2×C3×Q8—21no—
168C2×C3×Q8—41no—
169C2×C3×Q8—92no—
170C2×C3×Q8D3211yescharacteristic
171C2×C3×Q8D3212yes—
172C2×C3×D4—91no—
173C22×Dic3—92no—
174C2×D12—32no—
175C2×D12—61no—
176C2×C4×D3—94no—
177C2×Dic6—32no—
178C2×Dic6—32no—
179C2×Dic6—62no—
180C3×C4⋊C4—94no—
181C3×C4⋊C4—94no—
182C3×C22⋊C4—94no—
183C3×C42—91no—
184C3×C42—92no—
185#217—94no—
186#217—92no—
187(C2×D3)⋊C4—34no—
188(C2×D3)⋊C4—34no—
189(C2×D3)⋊C4—64no—
190(C2×D3)⋊C4—94no—
191(C2×D3)⋊C4—94no—
192(C2×D3)⋊C4—94no—
193C4⋊Dic3—92no—
194C4⋊Dic3—92no—
195C4⋊Dic3—94no—
196Dic3⋊C4—34no—
197Dic3⋊C4—34no—
198Dic3⋊C4—34no—
199Dic3⋊C4—34no—
200Dic3⋊C4—64no—
201Dic3⋊C4—64no—
202Dic3⋊C4—94no—
203Dic3⋊C4—94no—
204Dic3⋊C4—94no—
205Dic3⋊C4—94no—
206C4×Dic3—32no—
207C4×Dic3—32no—
208C4×Dic3—32no—
209C4×Dic3—62no—
210C4×Dic3—61no—
211C22×C32—22no—
212C22×C32—21no—
213C22×C32—41no—
214C22×C32C23×D312yes—
215C22×C32C23×D311yescharacteristic
216C2×C3⋊D3—22no—
217C2×C3⋊D3—24no—
218C2×C3×D3—64no—
219C2×C3×D3—68no—
220C2×C3×D3—122no—
221C2×C3×D3—124no—
222C32×C4—22no—
223C32×C4—21no—
224C32×C4—24no—
225C32×C4—22no—
226C32×C4—22no—
227C32×C4—22no—
228C32×C4—22no—
229C32×C4—42no—
230C32×C4—42no—
231C32×C4—42no—
232C32×C4—44no—
233C32×C4—42no—
234C3⋊Dic3—24no—
235C3⋊Dic3—24no—
236C3⋊Dic3—22no—
237C3×Dic3—68no—
238C3×Dic3—64no—
239C3×Dic3—64no—
240C3×Dic3—64no—
241C3×Dic3—64no—
242C3×Dic3—64no—
243C3×Dic3—124no—
244C3×Dic3—122no—
245C4⋊Q8—92no—
246#126—274no—
247#125—272no—
248#124—91no—
249#123—272no—
250C22⋊Q8—274no—
251C33#56311yescharacteristic, complemented, Sylow
252C23×C3—92no—
253C22×D3—34no—
254C22×D3—62no—
255C3×Q8—41no—
256C3×Q8—42no—
257C3×Q8—44no—
258C3×Q8—42no—
259C3×Q8—44no—
260C3×Q8—362no—
261C3×D4—361no—
262C2×C3×C4—22no—
263C2×C3×C4—21no—
264C2×C3×C4—22no—
265C2×C3×C4—22no—
266C2×C3×C4—22no—
267C2×C3×C4—42no—
268C2×C3×C4—41no—
269C2×C3×C4—92no—
270C2×C3×C4—94no—
271C2×C3×C4—94no—
272C2×C3×C4C2×D3212yes—
273C2×C3×C4C2×D3212yes—
274C2×C3×C4C2×D3212yes—
275C2×C3×C4C2×D3211yescharacteristic
276C2×C3×C4C2×D3212yes—
277C2×Dic3—34no—
278C2×Dic3—34no—
279C2×Dic3—34no—
280C2×Dic3—34no—
281C2×Dic3—34no—
282C2×Dic3—64no—
283C2×Dic3—62no—
284C2×Dic3—64no—
285C2×Dic3—362no—
286D12—122no—
287D12—122no—
288C4×D3—364no—
289Dic6—122no—
290Dic6—122no—
291Dic6—124no—
292C2×C32—24no—
293C2×C32—22no—
294C2×C32—22no—
295C2×C32—21no—
296C2×C32—41no—
297C2×C32—42no—
298C2×C32#100411yescharacteristic
299C2×C32#100412yes—
300C2×C32#100714yes—
301C2×C32#101012yes—
302C3⋊D3—42no—
303C3×D3—124no—
304C3×D3—242no—
305C2×Q8—92no—
306C2×Q8#111311yescharacteristic
307C2×D4—91no—
308C22×C4—272no—
309C4⋊C4—94no—
310C4⋊C4—94no—
311C4⋊C4—272no—
312C4⋊C4—274no—
313C4⋊C4—272no—
314C22⋊C4—94no—
315C22⋊C4—274no—
316C22⋊C4—272no—
317C42—91no—
318C42—92no—
319C22×C3—22no—
320C22×C3—21no—
321C22×C3—41no—
322C22×C3—184no—
323C22×C3—182no—
324C22×C3C22×D3212yes—
325C22×C3C22×D3211yescharacteristic
326C2×D3—68no—
327C2×D3—64no—
328C2×D3—122no—
329C2×D3—124no—
330C3×C4—24no—
331C3×C4—22no—
332C3×C4—22no—
333C3×C4—22no—
334C3×C4—22no—
335C3×C4—21no—
336C3×C4—22no—
337C3×C4—42no—
338C3×C4—42no—
339C3×C4—42no—
340C3×C4—42no—
341C3×C4—44no—
342C3×C4—184no—
343C3×C4—182no—
344C3×C4—184no—
345Dic3—68no—
346Dic3—64no—
347Dic3—64no—
348Dic3—64no—
349Dic3—64no—
350Dic3—64no—
351Dic3—124no—
352Dic3—122no—
353C32—21no—
354C32—22no—
355C32—41no—
356C32#567211yescharacteristic, complemented
357C32#573512yescomplemented
358C23—92no—
359Q8—41no—
360Q8—362no—
361D4—361no—
362C2×C4—92no—
363C2×C4—94no—
364C2×C4—94no—
365C2×C4—1082no—
366C2×C4C2×#111312yes—
367C2×C4C2×#111311yescharacteristic
368C2×C3—24no—
369C2×C3—22no—
370C2×C3—22no—
371C2×C3—21no—
372C2×C3—42no—
373C2×C3—41no—
374C2×C3—362no—
375C2×C3#6431514yes—
376C2×C3#6431612yes—
377C2×C3#6433011yescharacteristic
378C2×C3#6433012yes—
379D3—124no—
380D3—242no—
381C22—184no—
382C22—182no—
383C22C22×#111311yescharacteristic
384C4—21no—
385C4—22no—
386C4—184no—
387C4—182no—
388C4—184no—
389C3—21no—
390C3—22no—
391C3—41no—
392C3#1059535211yescharacteristic, complemented
393C3#1059539312yescomplemented
394C2—362no—
395C2#1172462412yes—
396C2#1172463311yescharacteristic
subgroupG/H
center, FrattiniC22C22×#1113
commutator, socle, cosocleC22×C33C24
FittingC2×C33×Q8C22
prime radicalC2×C3×C4⋊(C3⋊D3)C22
hypercenterC2×Q8#1113
hypocenterC33#563
LCSG ▷ C22×C33 ▷ C33
UCSC1 ◁ C22 ◁ C2×Q8

Complements

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

NH ≅ G/Ntype
2. C3×#64153C2semidirect
3. #11723688C2semidirect
5. #11723679C2semidirect
6. #11723676C2semidirect
24. #64153D3semidirect
251. C33#563semidirect
356. C32#5672semidirect
357. C32#5735semidirect
392. C3#10595352semidirect
393. C3#10595393semidirect

Maximal subgroups

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

subgroupindex
C3×#641562
C3×#641532
#117236882
#117236862
#117236792
#117236762
#105953933
#105953523