Finite Group Databasebeta

← #13159348

Subgroups — #13159348

Order 1728.

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

normal subgroups only

This group has 920 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
1C32×#897C211yesmaximal, characteristic, complemented
2#11723516C211yesmaximal, characteristic
3C2×#88795C211yesmaximal, characteristic, complemented
4C2×#88786C211yesmaximal, characteristic, complemented
5#11722179C211yesmaximal, characteristic, complemented
6C3×#63634C211yesmaximal, characteristic, complemented
7#11722037C211yesmaximal, characteristic, complemented
8#10592748—31nomaximal
9#10590585—34nomaximal
10C2×C4⋊(C3⋊(C3⋊D3))C2211yescharacteristic
11C2×C32×C4×D3C2211yescharacteristic
12C2×C32×Dic6C2211yescharacteristic
13C2×#6291—21no—
14C33×C4⋊C4C2211yescharacteristic
15C2×C3⋊(C3⋊(C3⋊C8))C2211yescharacteristic
16C3×C4⋊(C3⋊Dic3)C2211yescharacteristic
17C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
18C32×(C2×D3)⋊C4—21no—
19C32×C4⋊Dic3—21no—
20#88804—21no—
21#88795—22no—
22#88786—22no—
23C32×C22⋊Q8—31no—
24C2×#3755—31no—
25#64139—31no—
26C2×#3748—31no—
27C3×#897D314yescomplemented
28#63909—34no—
29C2×#3717—34no—
30C2×#3711—34no—
31#63662—31no—
32#63636—31no—
33#63636—34no—
34#63636—64no—
35#63634D311yescharacteristic, complemented
36C3×#807—34no—
37#63565—34no—
38C22×C3⋊(C3⋊D3)—21no—
39C22×C32×D3D411yescharacteristic
40C2×C33×C4—21no—
41C2×C33×C4C2311yescharacteristic
42C4⋊(C3⋊(C3⋊D3))—21no—
43C4⋊(C3⋊(C3⋊D3))—21no—
44C2×C3×C3⋊Dic3—21no—
45C2×C32×Dic3—21no—
46C2×C32×Dic3D411yescharacteristic
47C32×C4×D3—21no—
48C32×C4×D3—21no—
49C32×Dic6—21no—
50C32×Dic6—21no—
51#6291—42no—
52C3⋊(C3⋊(C3⋊C8))—21no—
53C3×C3⋊(C3⋊C8)D412yescomplemented
54#5049—94no—
55#4886—91no—
56C2×C32×Q8—31no—
57C22×C32×C4—31no—
58C2×#638—61no—
59C2×C4⋊(C3⋊D3)—31no—
60C2×C4⋊(C3⋊D3)—34no—
61C2×C4⋊(C3⋊D3)—64no—
62C2×C3×C4×D3C2×D314yes—
63C2×C3×Dic6C2×D314yes—
64C2×#626—64no—
65C32×C4⋊C4—24no—
66C32×C4⋊C4—61no—
67C32×C4⋊C4C2×D314yes—
68C32×C4⋊C4C2×D311yescharacteristic
69C32×C22⋊C4—61no—
70#3755—62no—
71#3752—61no—
72C4⋊(C3⋊Dic3)C2×D311yescharacteristic
73#3748—62no—
74C2×C3⋊(C3⋊C8)—31no—
75C2×C3⋊(C3⋊C8)—34no—
76C2×C3⋊(C3⋊C8)—64no—
77C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
78C3×(C2×D3)⋊C4—24no—
79C3×C4⋊Dic3—24no—
80C3×C4⋊Dic3—34no—
81C2×C3×C3⋊C8—34no—
82#3722—64no—
83#3717—68no—
84#3711—68no—
85C22×C33C2×D411yescharacteristic
86C2×C3⋊(C3⋊D3)—41no—
87C2×C3⋊(C3⋊D3)—41no—
88C2×C3⋊(C3⋊D3)—41no—
89C2×C32×D3—21no—
90C2×C32×D3—21no—
91C2×C32×D3—21no—
92C33×C4—41no—
93C33×C4C2×D411yescharacteristic
94C33×C4#4111yescharacteristic
95C3×C3⋊Dic3—41no—
96C32×Dic3—21no—
97C32×Dic3—41no—
98C3×C22⋊Q8—34no—
99C2×#215—94no—
100#930—94no—
101C2×#208—94no—
102C2×#204—91no—
103C2×ΓL(1,𝔽25)—91no—
104#897C3⋊D311yescharacteristic, complemented
105#895—91no—
106C3×#107—91no—
107#835—94no—
108#810—91no—
109#809—94no—
110#809—91no—
111#809—184no—
112#807—34no—
113C23×C32—31no—
114C22×C3⋊D3—61no—
115C22×C3⋊D3—64no—
116C22×C3⋊D3—124no—
117C22×C3×D3#6714yes—
118C32×Q8—61no—
119C32×Q8—61no—
120C2×C32×C4—21no—
121C2×C32×C4—24no—
122C2×C32×C4—24no—
123C2×C32×C4—31no—
124C2×C32×C4—44no—
125C2×C32×C4—61no—
126C2×C32×C4—61no—
127C2×C32×C4—61no—
128C2×C32×C4C22×D314yes—
129C2×C32×C4C22×D311yescharacteristic
130#638—122no—
131C2×C3⋊Dic3—21no—
132C4⋊(C3⋊D3)—68no—
133C4⋊(C3⋊D3)—61no—
134C4⋊(C3⋊D3)—64no—
135C4⋊(C3⋊D3)—61no—
136C4⋊(C3⋊D3)—64no—
137C4⋊(C3⋊D3)—124no—
138C2×C3×Dic3—24no—
139C2×C3×Dic3—64no—
140C2×C3×Dic3#6714yes—
141C3×C4×D3—24no—
142C3×C4×D3—24no—
143C3×Dic6—24no—
144C3×Dic6—24no—
145#626—128no—
146C3⋊(C3⋊C8)—61no—
147C3⋊(C3⋊C8)—64no—
148C3⋊(C3⋊C8)—124no—
149C3⋊(C3⋊C8)D1212yescomplemented
150C3×C3⋊C8—38no—
151#468—271noSylow
152C2×C33#12111yescharacteristic
153C2×C33AGL(1,ℤ/8)11yescharacteristic
154C2×C33#13711yescharacteristic
155C3⋊(C3⋊D3)—81no—
156C32×D3—41no—
157C2×C3×Q8—34no—
158C22×C3×C4—34no—
159C2×#67—184no—
160C2×D12—94no—
161C2×D12—91no—
162C2×D12—184no—
163C2×D12—181no—
164C2×C4×D3C2×C3⋊D311yescharacteristic
165C2×Dic6C2×C3⋊D311yescharacteristic
166C2×C3×C8—91no—
167C3×C4⋊C4—24no—
168C3×C4⋊C4—64no—
169C3×C4⋊C4—91no—
170C3×C4⋊C4D3214yes—
171C3×C4⋊C4C2×C3⋊D311yescharacteristic
172C3×C22⋊C4—64no—
173#215—188no—
174(C2×D3)⋊C4—21no—
175(C2×D3)⋊C4—184no—
176(C2×D3)⋊C4—181no—
177C4⋊Dic3—21no—
178C4⋊Dic3—34no—
179#208—188no—
180C2×C3⋊C8—34no—
181C2×C3⋊C8—94no—
182C2×C3⋊C8—91no—
183C2×C3⋊C8—184no—
184#204—182no—
185ΓL(1,𝔽25)—182no—
186C22×C32—24no—
187C22×C32—61no—
188C22×C32—61no—
189C22×C32—61no—
190C22×C32D3×D411yescharacteristic
191C22×C32C2×#6714yes—
192C2×C3⋊D3—128no—
193C2×C3⋊D3—121no—
194C2×C3⋊D3—121no—
195C2×C3⋊D3—121no—
196C2×C3⋊D3—124no—
197C2×C3⋊D3—124no—
198C2×C3⋊D3—124no—
199C2×C3⋊D3—244no—
200C2×C3⋊D3—244no—
201C2×C3×D3—24no—
202C2×C3×D3—24no—
203C2×C3×D3—24no—
204C32×C4—24no—
205C32×C4—24no—
206C32×C4—41no—
207C32×C4—44no—
208C32×C4—61no—
209C32×C4—84no—
210C32×C4—121no—
211C32×C4C2×D1211yescharacteristic
212C32×C4D3×D414yes—
213C32×C4#23714yes—
214C32×C4#23911yescharacteristic
215C3⋊Dic3—41no—
216C3×Dic3—24no—
217C3×Dic3—44no—
218C3×Dic3—124no—
219C2×SD16—271no—
220C2×M16—271no—
221C22⋊Q8—31no—
222#121—271no—
223#107—91no—
224#103—271no—
225#102—271no—
226C33#46811yescharacteristic, complemented, Sylow
227C23×C3—34no—
228C22×D3—184no—
229C22×D3—181no—
230C22×D3—364no—
231C22×D3#63811yescharacteristic
232C3×Q8—64no—
233C3×Q8—64no—
234C2×C3×C4—24no—
235C2×C3×C4—21no—
236C2×C3×C4—24no—
237C2×C3×C4—34no—
238C2×C3×C4—44no—
239C2×C3×C4—64no—
240C2×C3×C4—64no—
241C2×C3×C4—64no—
242C2×C3×C4—181no—
243C2×C3×C4C2×D3214yes—
244C2×C3×C4C22×C3⋊D311yescharacteristic
245#67—368no—
246C2×Dic3—21no—
247C2×Dic3—64no—
248C2×Dic3#63811yescharacteristic
249D12—188no—
250D12—184no—
251D12—184no—
252D12—181no—
253D12—181no—
254D12—364no—
255D12—362no—
256C4×D3—21no—
257C4×D3—21no—
258Dic6—21no—
259Dic6—21no—
260C3×C8—92no—
261C3⋊C8—38no—
262C3⋊C8—184no—
263C3⋊C8—181no—
264C3⋊C8—364no—
265C2×C32—24no—
266C2×C32—24no—
267C2×C32—24no—
268C2×C32—121no—
269C2×C32#89511yescharacteristic
270C2×C32#91111yescharacteristic
271C2×C32#91811yescharacteristic
272C2×C32#93914yes—
273C2×C32#94214yes—
274C2×C32#94914yes—
275C3⋊D3—248no—
276C3⋊D3—241no—
277C3⋊D3—244no—
278C3×D3—44no—
279C2×Q8—31no—
280C2×D4—271no—
281C2×D4—541no—
282C22×C4—31no—
283SD16—542no—
284M16—542no—
285C2×C8—91no—
286C2×C8—271no—
287C4⋊C4—61no—
288C4⋊C4—91no—
289C4⋊C4D3×C3⋊D311yescharacteristic
290C22⋊C4—61no—
291C22⋊C4—541no—
292C22×C3—24no—
293C22×C3—64no—
294C22×C3—64no—
295C22×C3—64no—
296C22×C3D3×#6714yes—
297C22×C3C2×#63811yescharacteristic
298C2×D3—21no—
299C2×D3—21no—
300C2×D3—21no—
301C2×D3—368no—
302C2×D3—364no—
303C2×D3—361no—
304C2×D3—361no—
305C2×D3—361no—
306C2×D3—364no—
307C2×D3—364no—
308C2×D3—724no—
309C2×D3—724no—
310C3×C4—24no—
311C3×C4—24no—
312C3×C4—41no—
313C3×C4—44no—
314C3×C4—64no—
315C3×C4—84no—
316C3×C4—124no—
317C3×C4—361no—
318C3×C4#379614yes—
319C3×C4D3×D1214yes—
320C3×C4D4×C3⋊D311yescharacteristic
321C3×C4#383011yescharacteristic
322Dic3—21no—
323Dic3—41no—
324Dic3—124no—
325C32—24no—
326C32#488611yescharacteristic, complemented
327C32#504914yescomplemented
328C23—31no—
329C23—541no—
330Q8—61no—
331Q8—61no—
332D4—541no—
333D4—541no—
334D4—1082no—
335C2×C4—21no—
336C2×C4—31no—
337C2×C4—61no—
338C2×C4—61no—
339C2×C4—61no—
340C2×C4—181no—
341C2×C4C2×D3×C3⋊D311yescharacteristic
342C8—92no—
343C8—541no—
344C2×C3—24no—
345C2×C3—24no—
346C2×C3—24no—
347C2×C3—124no—
348C2×C3#6388614yes—
349C2×C3#6392514yes—
350C2×C3#6394814yes—
351C2×C3#6414811yescharacteristic
352C2×C3#6415111yescharacteristic
353C2×C3#6415811yescharacteristic
354D3—41no—
355D3—728no—
356D3—724no—
357D3—721no—
358C22—61no—
359C22—61no—
360C22—61no—
361C22—1081no—
362C22—1081no—
363C22—1081no—
364C22D3×#63811yescharacteristic
365C4—41no—
366C4—61no—
367C4—121no—
368C4—361no—
369C4#8901511yescharacteristic
370C4C3⋊D3×D1211yescharacteristic
371C3—24no—
372C3#1059058514yescomplemented
373C3#1059274811yescharacteristic, complemented
374C2—121no—
375C2—2161no—
376C2#1172347511yescharacteristic
377C2#1172354411yescharacteristic
378C2#1172358111yescharacteristic
subgroupG/H
centerC22D3×#638
commutator, cosocleC2×C33×C4C23
FrattiniC2×C4C2×D3×C3⋊D3
FittingC33×C4⋊C4C22
socleC22×C33C2×D4
prime radical#11723516C2
hypercenterC4⋊C4D3×C3⋊D3
hypocenterC33#468
LCSG ▷ C2×C33×C4 ▷ C2×C33 ▷ C33
UCSC1 ◁ C22 ◁ C2×C4 ◁ C4⋊C4

Complements

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

NH ≅ G/Ntype
1. C32×#897C2semidirect
3. C2×#88795C2semidirect
4. C2×#88786C2semidirect
5. #11722179C2semidirect
6. C3×#63634C2semidirect
7. #11722037C2semidirect
27. C3×#897D3semidirect
35. #63634D3semidirect
53. C3×C3⋊(C3⋊C8)D4semidirect
104. #897C3⋊D3semidirect
149. C3⋊(C3⋊C8)D12semidirect
226. C33#468semidirect
326. C32#4886semidirect
327. C32#5049semidirect
372. C3#10590585semidirect
373. C3#10592748semidirect

Maximal subgroups

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

subgroupindex
C32×#8972
#117235162
C2×#887952
C2×#887862
#117221792
C3×#636342
#117220372
#105927483
#105905853