Finite Group Databasebeta

← #13159968

Subgroups — #13159968

Order 1728.

920 conjugacy classes of subgroups; only the 148 normal subgroups are shown.

normal subgroups only

This group has 920 conjugacy classes of subgroups — too many to draw. Only its 148 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×#3743C211yesmaximal, characteristic, complemented
2#11723526C211yesmaximal, characteristic
3#11723435C211yesmaximal, characteristic, complemented
4C2×#88795C211yesmaximal, characteristic, complemented
5#11722191C211yesmaximal, characteristic, complemented
6C32×#816C211yesmaximal, characteristic, complemented
7#11722057C211yesmaximal, characteristic, complemented
8#10592511—31nomaximal
9#10590850—34nomaximal
10C2×C4⋊(C3⋊(C3⋊D3))C2211yescharacteristic
11C2×C3×C4×C3⋊D3C2211yescharacteristic
12C2×C32×Dic6C2211yescharacteristic
13C2×#6292—21no—
14C2×C33×C8C2211yescharacteristic
15C4⋊(C3⋊(C3⋊Dic3))C2211yescharacteristic
16C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
17C3×#3743—22no—
18C32×C4⋊Dic3C2211yescharacteristic
19#88806—21no—
20#88804—21no—
21#88803—21no—
22#88795—22no—
23#64155—31no—
24C2×#3755—31no—
25C2×#3743D311yescharacteristic, complemented
26#64104—31no—
27C2×C3×ΓL(1,𝔽25)—34no—
28#63909—34no—
29#63856—34no—
30C2×#3717—34no—
31C32×#103—31no—
32#63648—31no—
33#63648—34no—
34#63648—64no—
35#63634—31no—
36C3×#816D314yescomplemented
37#63575—34no—
38C22×C3⋊(C3⋊D3)—21no—
39C22×C3×C3⋊D3D411yescharacteristic
40C2×C33×C4C2311yescharacteristic
41C2×C3⋊(C3⋊Dic3)—21no—
42C4⋊(C3⋊(C3⋊D3))—21no—
43C4⋊(C3⋊(C3⋊D3))—21no—
44C2×C3×C3⋊Dic3D411yescharacteristic
45C3×C4×C3⋊D3—21no—
46C3×C4×C3⋊D3—21no—
47C2×C32×Dic3—21no—
48C2×C32×Dic3—21no—
49C32×Dic6—21no—
50C32×Dic6—21no—
51#6292—42no—
52C33×C8—21no—
53C3×C3⋊(C3⋊C8)D412yescomplemented
54#5134—91no—
55#4812—94no—
56C2×C32×Q8—31no—
57C2×C4⋊(C3⋊D3)—31no—
58C2×C4⋊(C3⋊D3)—34no—
59C2×C4⋊(C3⋊D3)—64no—
60C2×C4⋊(C3⋊D3)—61no—
61C2×C4×C3⋊D3C2×D311yescharacteristic
62C2×C3×C4×D3—34no—
63C2×C3×Dic6C2×D314yes—
64C2×#626—64no—
65C2×C32×C8—24no—
66C2×C32×C8C2×D314yes—
67C2×C32×C8C2×D311yescharacteristic
68C32×C4⋊C4—31no—
69#3755—62no—
70#3752—61no—
71#3752—61no—
72C4⋊(C3⋊Dic3)—31no—
73C4⋊(C3⋊Dic3)—34no—
74C4⋊(C3⋊Dic3)—64no—
75#3750—61no—
76C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
77#3743—22no—
78C3×C4⋊Dic3C2×D314yes—
79C2×C3×C3⋊C8—34no—
80C3×ΓL(1,𝔽25)—68no—
81#3724—64no—
82#3722—64no—
83(C2×D3)⋊Dic3—64no—
84#3717—68no—
85C22×C33C2×D411yescharacteristic
86C2×C3⋊(C3⋊D3)—41no—
87C2×C3⋊(C3⋊D3)—41no—
88C2×C3⋊(C3⋊D3)—41no—
89C2×C3×C3⋊D3—21no—
90C2×C3×C3⋊D3—21no—
91C2×C3×C3⋊D3—21no—
92C33×C4C2×D411yescharacteristic
93C33×C4#4111yescharacteristic
94C3⋊(C3⋊Dic3)—41no—
95C3×C3⋊Dic3—21no—
96C32×Dic3—41no—
97C32×Dic3—41no—
98C2×C3×M16—91no—
99#946—94no—
100C2×#215—94no—
101#930—91no—
102#924—91no—
103C2×#204—91no—
104C2×ΓL(1,𝔽25)—34no—
105#895—94no—
106C3×#103—34no—
107#821—94no—
108#821—91no—
109#821—184no—
110#818—91no—
111#816C3⋊D311yescharacteristic, complemented
112#807—94no—
113C22×C3⋊D3—61no—
114C22×C3⋊D3—64no—
115C22×C3⋊D3—124no—
116C22×C3⋊D3D1211yescharacteristic
117C22×C3×D3—34no—
118C32×Q8—61no—
119C32×Q8—61no—
120C2×C32×C4—24no—
121C2×C32×C4—61no—
122C2×C32×C4—61no—
123C2×C32×C4C22×D314yes—
124C2×C32×C4C22×D311yescharacteristic
125C2×C3⋊Dic3—61no—
126C2×C3⋊Dic3—64no—
127C2×C3⋊Dic3—124no—
128C2×C3⋊Dic3D1211yescharacteristic
129C4⋊(C3⋊D3)—68no—
130C4⋊(C3⋊D3)—64no—
131C4⋊(C3⋊D3)—61no—
132C4⋊(C3⋊D3)—61no—
133C4⋊(C3⋊D3)—64no—
134C4⋊(C3⋊D3)—124no—
135C4⋊(C3⋊D3)—122no—
136C4×C3⋊D3—21no—
137C4×C3⋊D3—21no—
138C2×C3×Dic3—24no—
139C2×C3×Dic3—24no—
140C2×C3×Dic3—34no—
141C3×C4×D3—64no—
142C3×C4×D3—64no—
143C3×Dic6—24no—
144C3×Dic6—24no—
145#626—128no—
146C32×C8—21no—
147C32×C8—24no—
148C32×C8—44no—
149C3⋊(C3⋊C8)#6712yescomplemented
150C3×C3⋊C8—38no—
151#468—271noSylow
152C2×C33#12111yescharacteristic
153C2×C33AGL(1,ℤ/8)11yescharacteristic
154C2×C33#13711yescharacteristic
155C3⋊(C3⋊D3)—81no—
156C3×C3⋊D3—41no—
157C2×C3×Q8—34no—
158C22×C3×C4—91no—
159C2×#67—181no—
160C2×D12—94no—
161C2×D12—91no—
162C2×D12—184no—
163C2×D12—184no—
164C2×C4×D3—34no—
165C2×Dic6C2×C3⋊D311yescharacteristic
166C3×M16—182no—
167C2×C3×C8—24no—
168C2×C3×C8—91no—
169C2×C3×C8D3214yes—
170C2×C3×C8C2×C3⋊D311yescharacteristic
171C3×C4⋊C4—34no—
172#217—181no—
173#215—188no—
174(C2×D3)⋊C4—181no—
175(C2×D3)⋊C4—184no—
176(C2×D3)⋊C4—184no—
177C4⋊Dic3—94no—
178C4⋊Dic3—91no—
179C4⋊Dic3—184no—
180C4⋊Dic3C2×C3⋊D311yescharacteristic
181Dic3⋊C4—181no—
182Dic3⋊C4—184no—
183C2×C3⋊C8—34no—
184#204—182no—
185ΓL(1,𝔽25)—68no—
186C22×C32—24no—
187C22×C32C2×D1211yescharacteristic
188C22×C32D3×D414yes—
189C2×C3⋊D3—21no—
190C2×C3⋊D3—21no—
191C2×C3⋊D3—21no—
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—64no—
202C2×C3×D3—64no—
203C2×C3×D3—64no—
204C32×C4—24no—
205C32×C4—24no—
206C32×C4—121no—
207C32×C4—121no—
208C32×C4#23514yes—
209C32×C4#23511yescharacteristic
210C32×C4D3×D414yes—
211C32×C4C2×#6711yescharacteristic
212C3⋊Dic3—21no—
213C3⋊Dic3—121no—
214C3⋊Dic3—124no—
215C3⋊Dic3—244no—
216C3×Dic3—44no—
217C3×Dic3—44no—
218C3×Dic3—64no—
219C2×SD16—271no—
220C2×M16—91no—
221C22⋊Q8—271no—
222#121—271no—
223#107—271no—
224#103—31no—
225#102—271no—
226C33#46811yescharacteristic, complemented, Sylow
227C23×C3—91no—
228C22×D3—34no—
229C22×D3—184no—
230C22×D3—181no—
231C22×D3—364no—
232C3×Q8—64no—
233C3×Q8—64no—
234C2×C3×C4—24no—
235C2×C3×C4—64no—
236C2×C3×C4—64no—
237C2×C3×C4—91no—
238C2×C3×C4—181no—
239C2×C3×C4—181no—
240C2×C3×C4C2×D3214yes—
241C2×C3×C4C22×C3⋊D311yescharacteristic
242#67—362no—
243C2×Dic3—21no—
244C2×Dic3—21no—
245C2×Dic3—34no—
246C2×Dic3—184no—
247C2×Dic3—181no—
248C2×Dic3—364no—
249D12—188no—
250D12—184no—
251D12—184no—
252D12—181no—
253D12—181no—
254D12—364no—
255D12—368no—
256C4×D3—64no—
257C4×D3—64no—
258Dic6—21no—
259Dic6—21no—
260C3×C8—24no—
261C3×C8—21no—
262C3×C8—44no—
263C3×C8—92no—
264C3⋊C8—38no—
265C2×C32—24no—
266C2×C32—24no—
267C2×C32—24no—
268C2×C32#88414yes—
269C2×C32#90811yescharacteristic
270C2×C32#90911yescharacteristic
271C2×C32#91414yes—
272C2×C32#91814yes—
273C2×C32#93011yescharacteristic
274C3⋊D3—41no—
275C3⋊D3—248no—
276C3⋊D3—241no—
277C3⋊D3—244no—
278C3×D3—124no—
279C2×Q8—31no—
280C2×D4—271no—
281C2×D4—541no—
282C22×C4—91no—
283SD16—542no—
284M16—182no—
285C2×C8—91no—
286C2×C8D3×C3⋊D311yescharacteristic
287C4⋊C4—31no—
288C4⋊C4—271no—
289C4⋊C4—541no—
290C22⋊C4—541no—
291C22⋊C4—541no—
292C22×C3—24no—
293C22×C3—181no—
294C22×C3—181no—
295C22×C3—181no—
296C22×C3D3×D1214yes—
297C22×C3D4×C3⋊D311yescharacteristic
298C2×D3—64no—
299C2×D3—64no—
300C2×D3—64no—
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—124no—
313C3×C4—124no—
314C3×C4—181no—
315C3×C4#379814yes—
316C3×C4D3×#6714yes—
317C3×C4#382811yescharacteristic
318C3×C4D4×C3⋊D311yescharacteristic
319Dic3—41no—
320Dic3—41no—
321Dic3—64no—
322Dic3—364no—
323Dic3—361no—
324Dic3—724no—
325C32—24no—
326C32#481214yescomplemented
327C32#513411yescharacteristic, complemented
328C23—91no—
329C23—541no—
330Q8—61no—
331Q8—61no—
332D4—541no—
333D4—541no—
334D4—1082no—
335C2×C4—61no—
336C2×C4—61no—
337C2×C4—91no—
338C2×C4—181no—
339C2×C4—181no—
340C2×C4—541no—
341C2×C4C2×D3×C3⋊D311yescharacteristic
342C8—21no—
343C8—92no—
344C2×C3—24no—
345C2×C3—24no—
346C2×C3—24no—
347C2×C3—361no—
348C2×C3#6379414yes—
349C2×C3#6380114yes—
350C2×C3#6390614yes—
351C2×C3#6409311yescharacteristic
352C2×C3#6412311yescharacteristic
353C2×C3#6412711yescharacteristic
354D3—124no—
355D3—728no—
356D3—724no—
357D3—721no—
358C22—181no—
359C22—181no—
360C22—181no—
361C22—1081no—
362C22—1081no—
363C22—1081no—
364C22C3⋊D3×D1211yescharacteristic
365C4—121no—
366C4—121no—
367C4—181no—
368C4—1081no—
369C4#8902011yescharacteristic
370C4C3⋊D3×#6711yescharacteristic
371C3—24no—
372C3#1059085014yescomplemented
373C3#1059251111yescharacteristic, complemented
374C2—361no—
375C2—2161no—
376C2#1172331911yescharacteristic
377C2#1172332911yescharacteristic
378C2#1172352011yescharacteristic
subgroupG/H
centerC22C3⋊D3×D12
commutator, cosocleC2×C33×C4C23
FrattiniC2×C4C2×D3×C3⋊D3
FittingC2×C33×C8C22
socleC22×C33C2×D4
prime radical#11723526C2
hypercenterC2×C8D3×C3⋊D3
hypocenterC33#468
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×#3743C2semidirect
3. #11723435C2semidirect
4. C2×#88795C2semidirect
5. #11722191C2semidirect
6. C32×#816C2semidirect
7. #11722057C2semidirect
25. C2×#3743D3semidirect
36. C3×#816D3semidirect
53. C3×C3⋊(C3⋊C8)D4semidirect
111. #816C3⋊D3semidirect
149. C3⋊(C3⋊C8)#67semidirect
226. C33#468semidirect
326. C32#4812semidirect
327. C32#5134semidirect
372. C3#10590850semidirect
373. C3#10592511semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#37432
#117235262
#117234352
C2×#887952
#117221912
C32×#8162
#117220572
#105925113
#105908503