Finite Group Databasebeta

← #13159935

Subgroups — #13159935

Order 1728.

944 conjugacy classes of subgroups; only the 156 normal subgroups are shown.

normal subgroups only

This group has 944 conjugacy classes of subgroups — too many to draw. Only its 156 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×C3×#3744C211yesmaximal, characteristic, complemented
2Dic3×C4⋊(C3⋊D3)C211yesmaximal, characteristic, complemented
3#11723441C211yesmaximal, characteristic, complemented
4C2×#88791C211yesmaximal, characteristic
5#11722191C211yesmaximal, characteristic, complemented
6C32×Dic3⋊C8C211yesmaximal, characteristic, complemented
7#11722040C211yesmaximal, characteristic, complemented
8#10592543—31nomaximal
9#10590754—34nomaximal
10C2×C4⋊(C3⋊(C3⋊D3))C2211yescharacteristic
11C2×C3×C4⋊(C3⋊D3)C2211yescharacteristic
12C2×C3×#634C2211yescharacteristic
13C2×Dic3×C3⋊D3—21no—
14C2×C33×C8C2211yescharacteristic
15C4⋊(C3⋊(C3⋊Dic3))C2211yescharacteristic
16C3×#3744—22no—
17C32×C4×Dic3C2211yescharacteristic
18C2×C32×C3⋊C8C2211yescharacteristic
19#88804—21no—
20#88804—21no—
21#88803—21no—
22#88791—22no—
23C2×#3745—31no—
24C2×#3744D311yescharacteristic, complemented
25#64084—31no—
26C4×C4⋊(C3⋊D3)—31no—
27C2×C3×#204—34no—
28Dic3×D12—34no—
29#63864—34no—
30C2×#3714—34no—
31C32×C4⋊C8—31no—
32#63648—31no—
33#63648—34no—
34#63648—64no—
35#63643—31no—
36C3×Dic3⋊C8D314yescomplemented
37#63566—34no—
38C22×C3⋊(C3⋊D3)—21no—
39C22×C3×C3⋊D3—21no—
40C2×C33×C4C2311yescharacteristic
41C2×C3⋊(C3⋊Dic3)—21no—
42C4⋊(C3⋊(C3⋊D3))—21no—
43C4⋊(C3⋊(C3⋊D3))—21no—
44C2×C3×C3⋊Dic3—21no—
45C3×C4⋊(C3⋊D3)—21no—
46C3×C4⋊(C3⋊D3)D412yes—
47C3×#634—21no—
48C3×#634—21no—
49C2×C32×Dic3D411yescharacteristic
50C2×C32×Dic3D411yescharacteristic
51Dic3×C3⋊D3—42no—
52C33×C8—21no—
53C32×C3⋊C8—21no—
54#5165—91no—
55#4844—94no—
56C2×C4⋊(C3⋊D3)—31no—
57C2×C4⋊(C3⋊D3)—34no—
58C2×C4⋊(C3⋊D3)—64no—
59C2×C4⋊(C3⋊D3)C2×D311yescharacteristic
60C2×C4×C3⋊D3—61no—
61C2×#634C2×D311yescharacteristic
62C2×C3×D12—34no—
63C2×C3×Dic6—34no—
64C2×D3×Dic3—64no—
65C2×C32×C8—24no—
66C2×C32×C8—31no—
67C2×C32×C8C2×D314yes—
68C2×C32×C8C2×D311yescharacteristic
69C32×C42—31no—
70#3752—61no—
71#3752—61no—
72#3752—61no—
73C4⋊(C3⋊Dic3)—31no—
74C4⋊(C3⋊Dic3)—34no—
75C4⋊(C3⋊Dic3)—64no—
76#3745—62no—
77#3744—22no—
78C3×C4×Dic3C2×D314yes—
79C2×C3×C3⋊C8C2×D314yes—
80C3×#204—68no—
81#3722—64no—
82#3722—64no—
83(C2×D3)⋊Dic3—64no—
84#3714—68no—
85C22×C33C2×D411yescharacteristic
86C2×C3⋊(C3⋊D3)—41no—
87C2×C3⋊(C3⋊D3)—41no—
88C2×C3⋊(C3⋊D3)—41no—
89C2×C3×C3⋊D3—22no—
90C2×C3×C3⋊D3—41no—
91C2×C3×C3⋊D3—41no—
92C33×C4C2×D411yescharacteristic
93C33×C4#4111yescharacteristic
94C3⋊(C3⋊Dic3)—41no—
95C3×C3⋊Dic3—41no—
96C32×Dic3—21no—
97C32×Dic3—21no—
98C2×C3×SD16—91no—
99#947—91no—
100D4×Dic3—91no—
101C2×#213—91no—
102C2×D24—94no—
103C2×#204—34no—
104#875—94no—
105C4×D12—94no—
106C3×C4⋊C8—34no—
107#835—91no—
108#821—94no—
109#821—91no—
110#821—184no—
111#816—94no—
112Dic3⋊C8C3⋊D311yescharacteristic, complemented
113C22×C3⋊D3—21no—
114C22×C3⋊D3—61no—
115C22×C3⋊D3—64no—
116C22×C3⋊D3—124no—
117C22×C3×D3—64no—
118C2×C32×C4—24no—
119C2×C32×C4—31no—
120C2×C32×C4—31no—
121C2×C32×C4C22×D314yes—
122C2×C32×C4C22×D311yescharacteristic
123C2×C3⋊Dic3—21no—
124C2×C3⋊Dic3—61no—
125C2×C3⋊Dic3—64no—
126C2×C3⋊Dic3—124no—
127C4⋊(C3⋊D3)—21no—
128C4⋊(C3⋊D3)—68no—
129C4⋊(C3⋊D3)—61no—
130C4⋊(C3⋊D3)—64no—
131C4⋊(C3⋊D3)—61no—
132C4⋊(C3⋊D3)—64no—
133C4⋊(C3⋊D3)—124no—
134C4⋊(C3⋊D3)#6712yes—
135C4×C3⋊D3—122no—
136#634—21no—
137#634—21no—
138C2×C3×Dic3—64no—
139C2×C3×Dic3D1214yes—
140C2×C3×Dic3D1214yes—
141C3×D12—38no—
142C3×D12—64no—
143C3×Dic6—64no—
144C3×Dic6—64no—
145D3×Dic3—128no—
146C32×C8—21no—
147C32×C8—24no—
148C32×C8—44no—
149C32×C8—61no—
150C3×C3⋊C8—24no—
151#463—271noSylow
152C2×C33#12111yescharacteristic
153C2×C33#13511yescharacteristic
154C2×C33AGL(1,ℤ/8)11yescharacteristic
155C3⋊(C3⋊D3)—81no—
156C3×C3⋊D3—42no—
157C2×C3×Q8—91no—
158C2×C3×D4—91no—
159C22×Dic3—181no—
160C2×D12—34no—
161C2×D12—94no—
162C2×D12—91no—
163C2×D12—184no—
164C2×C4×D3—184no—
165C2×Dic6—34no—
166C3×SD16—182no—
167C2×C3×C8—24no—
168C2×C3×C8—34no—
169C2×C3×C8D3214yes—
170C2×C3×C8C2×C3⋊D311yescharacteristic
171C3×C42—34no—
172#217—181no—
173#213—182no—
174(C2×D3)⋊C4—184no—
175(C2×D3)⋊C4—184no—
176(C2×D3)⋊C4—184no—
177(C2×D3)⋊C4—181no—
178(C2×D3)⋊C4—181no—
179C4⋊Dic3—94no—
180C4⋊Dic3—91no—
181C4⋊Dic3—184no—
182C4×Dic3C2×C3⋊D311yescharacteristic
183C2×C3⋊C8C2×C3⋊D311yescharacteristic
184D24—188no—
185#204—68no—
186C22×C32—24no—
187C22×C32C2×D1214yes—
188C22×C32D3×D411yescharacteristic
189C2×C3⋊D3—22no—
190C2×C3⋊D3—41no—
191C2×C3⋊D3—41no—
192C2×C3⋊D3—128no—
193C2×C3⋊D3—124no—
194C2×C3⋊D3—121no—
195C2×C3⋊D3—121no—
196C2×C3⋊D3—121no—
197C2×C3⋊D3—124no—
198C2×C3⋊D3—124no—
199C2×C3⋊D3—244no—
200C2×C3⋊D3—244no—
201C2×C3×D3—68no—
202C2×C3×D3—124no—
203C2×C3×D3—124no—
204C32×C4—24no—
205C32×C4—24no—
206C32×C4—61no—
207C32×C4—61no—
208C32×C4D3×D414yes—
209C32×C4#23711yescharacteristic
210C32×C4#23914yes—
211C32×C4C2×#6711yescharacteristic
212C3⋊Dic3—41no—
213C3⋊Dic3—121no—
214C3⋊Dic3—124no—
215C3⋊Dic3—244no—
216C3×Dic3—24no—
217C3×Dic3—24no—
218C3×Dic3—124no—
219C2×SD16—91no—
220C2×D8—271no—
221#124—271no—
222C4×D4—271no—
223C4⋊C8—31no—
224#103—271no—
225#102—271no—
226C33#46311yescharacteristic, complemented, Sylow
227C23×C3—181no—
228C22×D3—64no—
229C22×D3—184no—
230C22×D3—181no—
231C22×D3—364no—
232C3×Q8—181no—
233C3×Q8—181no—
234C3×D4—92no—
235C3×D4—181no—
236C2×C3×C4—24no—
237C2×C3×C4—34no—
238C2×C3×C4—34no—
239C2×C3×C4—181no—
240C2×C3×C4C2×D3214yes—
241C2×C3×C4C22×C3⋊D311yescharacteristic
242C2×Dic3—64no—
243C2×Dic3—184no—
244C2×Dic3—181no—
245C2×Dic3—364no—
246C2×Dic3—362no—
247C2×Dic3C4⋊(C3⋊D3)11yescharacteristic
248C2×Dic3C4⋊(C3⋊D3)11yescharacteristic
249D12—38no—
250D12—64no—
251D12—188no—
252D12—184no—
253D12—184no—
254D12—181no—
255D12—181no—
256D12—364no—
257C4×D3—368no—
258Dic6—64no—
259Dic6—64no—
260C3×C8—24no—
261C3×C8—21no—
262C3×C8—44no—
263C3×C8—64no—
264C3⋊C8—21no—
265C2×C32—24no—
266C2×C32—24no—
267C2×C32—24no—
268C2×C32#89514yes—
269C2×C32#90414yes—
270C2×C32#90814yes—
271C2×C32#91411yescharacteristic
272C2×C32#91611yescharacteristic
273C2×C32#93911yescharacteristic
274C3⋊D3—42no—
275C3⋊D3—248no—
276C3⋊D3—241no—
277C3⋊D3—244no—
278C3×D3—128no—
279C2×Q8—91no—
280C2×D4—91no—
281C2×D4—271no—
282C22×C4—541no—
283SD16—182no—
284D8—542no—
285C2×C8—31no—
286C2×C8D3×C3⋊D311yescharacteristic
287C4⋊C4—271no—
288C22⋊C4—541no—
289C22⋊C4—541no—
290C22⋊C4—541no—
291C42—31no—
292C22×C3—24no—
293C22×C3—182no—
294C22×C3—361no—
295C22×C3—361no—
296C22×C3D3×D1214yes—
297C22×C3C2×C4⋊(C3⋊D3)11yescharacteristic
298C2×D3—68no—
299C2×D3—124no—
300C2×D3—124no—
301C2×D3—368no—
302C2×D3—364no—
303C2×D3—364no—
304C2×D3—361no—
305C2×D3—361no—
306C2×D3—361no—
307C2×D3—364no—
308C2×D3—724no—
309C2×D3—724no—
310C3×C4—24no—
311C3×C4—24no—
312C3×C4—64no—
313C3×C4—64no—
314C3×C4—361no—
315C3×C4#379614yes—
316C3×C4D3×#6714yes—
317C3×C4D4×C3⋊D311yescharacteristic
318C3×C4#383211yescharacteristic
319Dic3—21no—
320Dic3—21no—
321Dic3—124no—
322Dic3—364no—
323Dic3—361no—
324Dic3—724no—
325C32—24no—
326C32#484414yescomplemented
327C32#516511yescharacteristic, complemented
328C23—181no—
329C23—541no—
330Q8—181no—
331Q8—181no—
332D4—92no—
333D4—181no—
334D4—541no—
335D4—541no—
336C2×C4—31no—
337C2×C4—31no—
338C2×C4—181no—
339C2×C4—541no—
340C2×C4—1082no—
341C2×C4C2×D3×C3⋊D311yescharacteristic
342C8—21no—
343C8—61no—
344C2×C3—24no—
345C2×C3—24no—
346C2×C3—24no—
347C2×C3—362no—
348C2×C3#6379414yes—
349C2×C3#6380614yes—
350C2×C3#6388614yes—
351C2×C3#6410411yescharacteristic
352C2×C3#6411311yescharacteristic
353C2×C3#6411711yescharacteristic
354D3—128no—
355D3—728no—
356D3—721no—
357D3—724no—
358C22—182no—
359C22—361no—
360C22—361no—
361C22—1081no—
362C22—1081no—
363C22—1081no—
364C22D3×C4⋊(C3⋊D3)11yescharacteristic
365C4—61no—
366C4—61no—
367C4—361no—
368C4—1081no—
369C4#8901711yescharacteristic
370C4C3⋊D3×#6711yescharacteristic
371C3—24no—
372C3#1059075414yescomplemented
373C3#1059254311yescharacteristic, complemented
374C2—362no—
375C2—2161no—
376C2#1172331511yescharacteristic
377C2#1172333511yescharacteristic
378C2#1172348211yescharacteristic
subgroupG/H
centerC22D3×C4⋊(C3⋊D3)
commutator, cosocleC2×C33×C4C23
FrattiniC2×C4C2×D3×C3⋊D3
FittingC2×C33×C8C22
socleC22×C33C2×D4
prime radicalC2×#88791C2
hypercenterC2×C8D3×C3⋊D3
hypocenterC33#463
LCSG ▷ C2×C33×C4 ▷ C2×C33 ▷ C33
UCSC1 ◁ C22 ◁ C2×C4 ◁ C2×C8

Complements

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

NH ≅ G/Ntype
1. C2×C3×#3744C2semidirect
2. Dic3×C4⋊(C3⋊D3)C2semidirect
3. #11723441C2semidirect
5. #11722191C2semidirect
6. C32×Dic3⋊C8C2semidirect
7. #11722040C2semidirect
24. C2×#3744D3semidirect
36. C3×Dic3⋊C8D3semidirect
112. Dic3⋊C8C3⋊D3semidirect
226. C33#463semidirect
326. C32#4844semidirect
327. C32#5165semidirect
372. C3#10590754semidirect
373. C3#10592543semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#37442
Dic3×C4⋊(C3⋊D3)2
#117234412
C2×#887912
#117221912
C32×Dic3⋊C82
#117220402
#105925433
#105907543