Finite Group Databasebeta

← #13156002

Subgroups — #13156002

Order 1728.

684 conjugacy classes of subgroups; only the 110 normal subgroups are shown.

normal subgroups only

This group has 684 conjugacy classes of subgroups — too many to draw. Only its 110 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#11723678C211yesmaximal, characteristic, complemented
2D12×C3⋊Dic3C211yesmaximal, characteristic
3C2×C3×#3716C212yesmaximal
4#11722040C212yesmaximal, complemented
5C3×#63572C211yesmaximal, characteristic, complemented
6#10591497—31nomaximal
7#10590752—32nomaximal
8C2×C3×#634C2212yes—
9C2×C32×D12C2211yescharacteristic
10C2×D3×C3⋊Dic3—21no—
11C4⋊(C3⋊(C3⋊Dic3))C2211yescharacteristic
12C3×C4×C3⋊Dic3C2211yescharacteristic
13C2×C32×C3⋊C8C2212yes—
14#88815—21no—
15#88815—21no—
16(C2×D3)⋊(C3⋊Dic3)—21no—
17C3×#3716—24no—
18D4×C3⋊Dic3—31no—
19C2×C3×#214—32no—
20C2×C3×#204—32no—
21Dic3×D12—32no—
22Dic3×D12—61no—
23#63873—31no—
24#63866—32no—
25C2×#3716D312yes—
26#63643—32no—
27C3×Dic3⋊C8—32no—
28#63572D311yescharacteristic, complemented
29#63566—32no—
30#63566—32no—
31#63566—62no—
32C22×C32×D3—21no—
33C2×C33×C4C2311yescharacteristic
34C2×C3⋊(C3⋊Dic3)—21no—
35C2×C3×C3⋊Dic3—22no—
36C2×C3×C3⋊Dic3D411yescharacteristic
37C2×C3×C3⋊Dic3D411yescharacteristic
38C3×#634—22no—
39C3×#634—22no—
40C32×D12—21no—
41C32×D12D412yes—
42D3×C3⋊Dic3—42no—
43C32×C3⋊C8—22no—
44#5162—92no—
45#4841—91no—
46C2×C32×D4—31no—
47C22×C3⋊Dic3—61no—
48C2×#634C2×D312yes—
49C2×C3×D12—21no—
50C2×C3×D12C2×D312yes—
51C2×C3×Dic6—32no—
52C2×C3×Dic6—32no—
53C2×C3×Dic6—62no—
54C2×D3×Dic3—62no—
55C2×D3×Dic3—62no—
56C2×C32×C8—32no—
57#3757—61no—
58C4⋊(C3⋊Dic3)—31no—
59C4⋊(C3⋊Dic3)—32no—
60C4⋊(C3⋊Dic3)—61no—
61C4⋊(C3⋊Dic3)—62no—
62C4⋊(C3⋊Dic3)—121no—
63C4×C3⋊Dic3C2×D311yescharacteristic
64C3×#214—64no—
65C3×C4×Dic3—32no—
66C3×C4×Dic3—61no—
67C2×C3×C3⋊C8—22no—
68C2×C3×C3⋊C8C2×D312yes—
69C2×C3×C3⋊C8C2×D312yes—
70C3×#204—64no—
71#3724—61no—
72#3724—61no—
73Dic3⋊Dic3—62no—
74Dic3⋊Dic3—62no—
75(C2×D3)⋊Dic3—62no—
76(C2×D3)⋊Dic3—62no—
77#3716—24no—
78C22×C33C2×D411yescharacteristic
79C2×C32×D3—22no—
80C2×C32×D3—41no—
81C2×C32×D3—41no—
82C33×C4C2×D411yescharacteristic
83C33×C4#4111yescharacteristic
84C3⋊(C3⋊Dic3)—41no—
85C3×C3⋊Dic3—21no—
86C3×C3⋊Dic3—42no—
87C3×C3⋊Dic3SD1612yescomplemented
88C2×C3×SD16—92no—
89#944—92no—
90D4×Dic3—92no—
91D4×Dic3—181no—
92C2×#214—32no—
93C2×#204—32no—
94C4×D12—91no—
95#869—91no—
96C3×C4⋊C8—91no—
97#835—92no—
98#816—92no—
99#816—92no—
100#816—182no—
101Dic3⋊C8—32no—
102C23×C32—61no—
103C22×C3×D3—22no—
104C22×C3×D3—22no—
105C32×D4—32no—
106C32×D4—61no—
107C2×C32×C4—22no—
108C2×C32×C4—21no—
109C2×C32×C4—41no—
110C2×C32×C4C22×D311yescharacteristic
111C2×C32×C4C22×D312yes—
112C2×C3⋊Dic3—22no—
113C2×C3⋊Dic3—61no—
114C2×C3⋊Dic3—62no—
115C2×C3⋊Dic3—62no—
116C2×C3⋊Dic3—122no—
117C2×C3⋊Dic3—122no—
118C2×C3⋊Dic3—122no—
119C2×C3⋊Dic3D1211yescharacteristic
120C2×C3⋊Dic3D1211yescharacteristic
121#634—22no—
122#634—22no—
123C2×C3×Dic3—32no—
124C2×C3×Dic3—32no—
125C2×C3×Dic3—61no—
126C2×C3×Dic3—61no—
127C2×C3×Dic3—62no—
128C2×C3×Dic3—62no—
129C2×C3×Dic3—122no—
130C3×D12—22no—
131C3×D12—22no—
132C3×D12—22no—
133C3×D12#6714yes—
134C3×Dic6—64no—
135C3×Dic6—62no—
136C3×Dic6—62no—
137C3×Dic6—62no—
138C3×Dic6—62no—
139C3×Dic6—122no—
140D3×Dic3—124no—
141D3×Dic3—124no—
142C32×C8—62no—
143C3×C3⋊C8—22no—
144C3×C3⋊C8—22no—
145C3×C3⋊C8—42no—
146#461—271noSylow
147C2×C33#12111yescharacteristic
148C2×C33C2×SD1611yescharacteristic
149C2×C33#13711yescharacteristic
150C32×D3—42no—
151C2×C3×Q8—92no—
152C2×C3×D4—32no—
153C2×C3×D4—61no—
154C22×Dic3—182no—
155C22×Dic3—182no—
156C2×D12D3211yescharacteristic
157C2×C4×D3—181no—
158C2×Dic6—32no—
159C2×Dic6—32no—
160C2×Dic6—62no—
161C3×SD16—184no—
162C2×C3×C8—32no—
163C2×C3×C8—32no—
164C2×C3×C8—62no—
165C3×C42—91no—
166#217—182no—
167#217—182no—
168#214—64no—
169(C2×D3)⋊C4—181no—
170C4⋊Dic3—91no—
171C4⋊Dic3—92no—
172C4⋊Dic3—181no—
173C4⋊Dic3—182no—
174C4⋊Dic3—181no—
175C4⋊Dic3—181no—
176C4⋊Dic3—361no—
177Dic3⋊C4—182no—
178Dic3⋊C4—182no—
179C4×Dic3—32no—
180C4×Dic3—61no—
181C2×C3⋊C8D3212yes—
182#204—64no—
183C22×C32—22no—
184C22×C32—21no—
185C22×C32—41no—
186C22×C32—62no—
187C22×C32—121no—
188C22×C32—121no—
189C22×C32C2×D1211yescharacteristic
190C22×C32D3×D412yes—
191C2×C3×D3—24no—
192C2×C3×D3—24no—
193C2×C3×D3—42no—
194C2×C3×D3—42no—
195C2×C3×D3—42no—
196C2×C3×D3—42no—
197C32×C4—22no—
198C32×C4—21no—
199C32×C4—21no—
200C32×C4—22no—
201C32×C4—41no—
202C32×C4—41no—
203C32×C4D3×D411yescharacteristic
204C32×C4#23712yes—
205C32×C4#23911yescharacteristic
206C32×C4C2×#6712yes—
207C3⋊Dic3—21no—
208C3⋊Dic3—42no—
209C3⋊Dic3—121no—
210C3⋊Dic3—122no—
211C3⋊Dic3—122no—
212C3⋊Dic3—242no—
213C3⋊Dic3—242no—
214C3⋊Dic3#20412yescomplemented
215C3×Dic3—34no—
216C3×Dic3—62no—
217C3×Dic3—62no—
218C3×Dic3—121no—
219C3×Dic3—124no—
220C3×Dic3—122no—
221C3×Dic3—122no—
222C2×SD16—92no—
223C4⋊Q8—271no—
224C4×D4—271no—
225C4⋊C8—91no—
226#103—272no—
227C33#46111yescharacteristic, complemented, Sylow
228C23×C3—62no—
229C23×C3—62no—
230C22×D3—21no—
231C3×Q8—182no—
232C3×Q8—182no—
233C3×D4—34no—
234C3×D4—62no—
235C3×D4—62no—
236C3×D4—62no—
237C2×C3×C4—21no—
238C2×C3×C4—22no—
239C2×C3×C4—41no—
240C2×C3×C4—91no—
241C2×C3×C4—91no—
242C2×C3×C4—182no—
243C2×C3×C4C2×D3212yes—
244C2×C3×C4C2×D3211yescharacteristic
245C2×Dic3—32no—
246C2×Dic3—32no—
247C2×Dic3—61no—
248C2×Dic3—61no—
249C2×Dic3—62no—
250C2×Dic3—62no—
251C2×Dic3—122no—
252C2×Dic3—182no—
253C2×Dic3—182no—
254C2×Dic3—181no—
255C2×Dic3—362no—
256C2×Dic3—362no—
257C2×Dic3—364no—
258C2×Dic3—364no—
259D12—21no—
260D12#62512yes—
261C4×D3—362no—
262Dic6—64no—
263Dic6—62no—
264Dic6—62no—
265Dic6—62no—
266Dic6—62no—
267Dic6—122no—
268C3×C8—62no—
269C3×C8—62no—
270C3×C8—122no—
271C3⋊C8—22no—
272C2×C32—22no—
273C2×C32—22no—
274C2×C32—22no—
275C2×C32—21no—
276C2×C32—21no—
277C2×C32—21no—
278C2×C32—41no—
279C2×C32—41no—
280C2×C32—41no—
281C2×C32—122no—
282C2×C32#89511yescharacteristic
283C2×C32C2×#20411yescharacteristic
284C2×C32#90911yescharacteristic
285C2×C32D3×SD1612yes—
286C2×C32#91512yes—
287C2×C32#93912yes—
288C3×D3—44no—
289C3×D3—44no—
290C2×Q8—92no—
291C2×D4—31no—
292C22×C4—541no—
293SD16—184no—
294C2×C8—32no—
295C4⋊C4—271no—
296C4⋊C4—541no—
297C4⋊C4—541no—
298C22⋊C4—541no—
299C42—91no—
300C22×C3—21no—
301C22×C3—22no—
302C22×C3—41no—
303C22×C3—64no—
304C22×C3—64no—
305C22×C3—122no—
306C22×C3—122no—
307C22×C3—122no—
308C22×C3—122no—
309C22×C3D3×D1212yes—
310C22×C3#381111yescharacteristic
311C2×D3—22no—
312C2×D3—41no—
313C2×D3—41no—
314C3×C4—22no—
315C3×C4—21no—
316C3×C4—22no—
317C3×C4—21no—
318C3×C4—41no—
319C3×C4—41no—
320C3×C4—92no—
321C3×C4—181no—
322C3×C4—362no—
323C3×C4#379612yes—
324C3×C4#380511yescharacteristic
325C3×C4C2×#62511yescharacteristic
326C3×C4D3×#6712yes—
327Dic3—34no—
328Dic3—62no—
329Dic3—62no—
330Dic3—121no—
331Dic3—124no—
332Dic3—122no—
333Dic3—122no—
334Dic3—362no—
335Dic3—362no—
336Dic3—361no—
337Dic3—722no—
338Dic3—722no—
339C32—21no—
340C32—22no—
341C32—41no—
342C32#484111yescharacteristic, complemented
343C32#516212yescomplemented
344C23—61no—
345Q8—182no—
346Q8—182no—
347D4—32no—
348D4—61no—
349C2×C4—91no—
350C2×C4—91no—
351C2×C4—182no—
352C2×C4—541no—
353C2×C4—1082no—
354C2×C4D3311yescharacteristic
355C8—62no—
356C2×C3—21no—
357C2×C3—22no—
358C2×C3—21no—
359C2×C3—22no—
360C2×C3—22no—
361C2×C3—21no—
362C2×C3—41no—
363C2×C3—41no—
364C2×C3—41no—
365C2×C3—124no—
366C2×C3—124no—
367C2×C3D3×#20412yes—
368C2×C3#6380012yes—
369C2×C3#6388612yes—
370C2×C3#6393011yescharacteristic
371C2×C3#6393111yescharacteristic
372C2×C3#6398011yescharacteristic
373D3—42no—
374C22—62no—
375C22—121no—
376C22—121no—
377C22#8895511yescharacteristic
378C4—92no—
379C4—181no—
380C4—362no—
381C4—1081no—
382C4D3×#62511yescharacteristic
383C4#8896411yescharacteristic
384C3—21no—
385C3—22no—
386C3—41no—
387C3#1059075212yescomplemented
388C3#1059149711yescharacteristic, complemented
389C2—122no—
390C2#1172284211yescharacteristic
391C2#1172289511yescharacteristic
392C2#1172295511yescharacteristic
subgroupG/H
centerC22#88955
commutator, Fitting, cosocleC2×C33×C4C23
Frattini, hypercenterC2×C4D33
socleC22×C33C2×D4
prime radicalC2×C32×D12C22
hypocenterC33#461
LCSG ▷ C2×C33×C4 ▷ C2×C33 ▷ C33
UCSC1 ◁ C22 ◁ C2×C4

Complements

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

NH ≅ G/Ntype
1. #11723678C2semidirect
4. #11722040C2semidirect
5. C3×#63572C2semidirect
28. #63572D3semidirect
87. C3×C3⋊Dic3SD16semidirect
214. C3⋊Dic3#204semidirect
227. C33#461semidirect
342. C32#4841semidirect
343. C32#5162semidirect
387. C3#10590752semidirect
388. C3#10591497semidirect

Maximal subgroups

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

subgroupindex
#117236782
D12×C3⋊Dic32
C2×C3×#37162
#117220402
C3×#635722
#105914973
#105907523