Finite Group Databasebeta

← #13159034

Subgroups — #13159034

Order 1728.

976 conjugacy classes of subgroups; only the 176 normal subgroups are shown.

normal subgroups only

This group has 976 conjugacy classes of subgroups — too many to draw. Only its 176 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#11723916C211yesmaximal, characteristic
2C32×C4×Dic6C211yesmaximal, characteristic, complemented
3C2×#88799C211yesmaximal, characteristic, complemented
4C2×#88795C211yesmaximal, characteristic
5C3×C4⋊(C3⋊(C3⋊C8))C211yesmaximal, characteristic, complemented
6#11722041C211yesmaximal, characteristic, complemented
7#11722035C211yesmaximal, characteristic
8#10592727—31nomaximal
9#10590368—34nomaximal
10C2×C4⋊(C3⋊(C3⋊D3))C2211yescharacteristic
11C2×#6308C2211yescharacteristic
12C2×C32×Dic6C2211yescharacteristic
13C33×C42C2211yescharacteristic
14#88859—21no—
15#88859—21no—
16C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
17C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
18C32×C4⋊Dic3C2211yescharacteristic
19C32×Dic3⋊C4—21no—
20C32×C4×Dic3—21no—
21#88799—22no—
22#88795—22no—
23C32×C4×Q8—31no—
24C2×#3756—31no—
25C2×#3755—31no—
26#64084—31no—
27#64084—34no—
28#64084—64no—
29C3×C4×Dic6D314yescomplemented
30C2×#3719—34no—
31C2×#3717—34no—
32#63637—31no—
33#63636—31no—
34C4⋊(C3⋊(C3⋊C8))D311yescharacteristic, complemented
35C3×C4⋊(C3⋊C8)—34no—
36#63567—34no—
37#63564—34no—
38C22×C3⋊(C3⋊D3)—21no—
39C2×C33×C4D411yescharacteristic
40C2×C33×C4D411yescharacteristic
41C2×C33×C4C2311yescharacteristic
42C2×C3⋊(C3⋊Dic3)—21no—
43C4⋊(C3⋊(C3⋊D3))—21no—
44C4⋊(C3⋊(C3⋊D3))—21no—
45#6308—21no—
46#6308—21no—
47C2×C32×Dic3—21no—
48C2×C32×Dic3—21no—
49C32×Dic6—21no—
50C32×Dic6D412yes—
51C3×C3⋊(C3⋊C8)—21no—
52C3×C3⋊(C3⋊C8)—21no—
53#5028—94no—
54#4845—91no—
55C2×C32×Q8—31no—
56C2×C4⋊(C3⋊D3)—31no—
57C2×C4⋊(C3⋊D3)—34no—
58C2×C4⋊(C3⋊D3)—64no—
59C2×#634—31no—
60C2×#634—34no—
61C2×#634—64no—
62C2×C3×Dic6C2×D314yes—
63C32×C4⋊C4—31no—
64C32×C4⋊C4—61no—
65C32×C42—24no—
66C32×C42—61no—
67C32×C42C2×D311yescharacteristic
68C32×C42C2×D314yes—
69#3756—62no—
70#3755—62no—
71#3752—68no—
72#3752—68no—
73#3752—61no—
74#3752—61no—
75#3752—64no—
76#3752—64no—
77C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
78C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
79C3×C4⋊Dic3C2×D314yes—
80C3×Dic3⋊C4—24no—
81C3×C4×Dic3—24no—
82C2×C3×C3⋊C8—34no—
83C2×C3×C3⋊C8—34no—
84#3719—68no—
85#3717—68no—
86C22×C33C2×D411yescharacteristic
87C2×C3⋊(C3⋊D3)—41no—
88C2×C3⋊(C3⋊D3)—41no—
89C2×C3⋊(C3⋊D3)—41no—
90C33×C4—21no—
91C33×C4—21no—
92C33×C4C2×D411yescharacteristic
93C33×C4#4111yescharacteristic
94C3⋊(C3⋊Dic3)—41no—
95C32×Dic3—22no—
96C32×Dic3—41no—
97C3×C4×Q8—34no—
98C2×#216—94no—
99C2×#215—94no—
100C2×Dic12—91no—
101C2×#204—91no—
102#875—94no—
103#875—91no—
104#875—184no—
105C4×Dic6C3⋊D311yescharacteristic, complemented
106C3×C4⋊C8—91no—
107#821—91no—
108#816—91no—
109#810—94no—
110#809—94no—
111C4⋊(C3⋊C8)—34no—
112C22×C3⋊D3—68no—
113C22×C3⋊D3—61no—
114C22×C3⋊D3—64no—
115C32×Q8—32no—
116C32×Q8—61no—
117C2×C32×C4—24no—
118C2×C32×C4—24no—
119C2×C32×C4—24no—
120C2×C32×C4—61no—
121C2×C32×C4—61no—
122C2×C32×C4D1211yescharacteristic
123C2×C32×C4D1211yescharacteristic
124C2×C32×C4#6714yes—
125C2×C32×C4#6714yes—
126C2×C32×C4C22×D314yes—
127C2×C32×C4C22×D311yescharacteristic
128C2×C3⋊Dic3—68no—
129C2×C3⋊Dic3—61no—
130C2×C3⋊Dic3—64no—
131C4⋊(C3⋊D3)—61no—
132C4⋊(C3⋊D3)—61no—
133C4⋊(C3⋊D3)—64no—
134C4⋊(C3⋊D3)—64no—
135C4⋊(C3⋊D3)—124no—
136C4⋊(C3⋊D3)—124no—
137#634—61no—
138#634—61no—
139#634—64no—
140#634—64no—
141#634—124no—
142#634—124no—
143C2×C3×Dic3—24no—
144C2×C3×Dic3—24no—
145C3×Dic6—24no—
146C3×Dic6D1218yes—
147C3⋊(C3⋊C8)—21no—
148C3⋊(C3⋊C8)—21no—
149C3×C3⋊C8—64no—
150C3×C3⋊C8—64no—
151#464—271noSylow
152C2×C33#12111yescharacteristic
153C2×C33#13511yescharacteristic
154C2×C33#13711yescharacteristic
155C3⋊(C3⋊D3)—81no—
156C2×C3×Q8—34no—
157C2×D12—94no—
158C2×D12—91no—
159C2×D12—184no—
160C2×Dic6—94no—
161C2×Dic6—91no—
162C2×Dic6—184no—
163C2×Dic6C2×C3⋊D311yescharacteristic
164C2×C3×C8—91no—
165C2×C3×C8—91no—
166C3×C4⋊C4—34no—
167C3×C4⋊C4—64no—
168C3×C42—24no—
169C3×C42—64no—
170C3×C42D3214yes—
171C3×C42C2×C3⋊D311yescharacteristic
172#216—188no—
173#215—188no—
174(C2×D3)⋊C4—188no—
175(C2×D3)⋊C4—188no—
176(C2×D3)⋊C4—184no—
177(C2×D3)⋊C4—184no—
178(C2×D3)⋊C4—181no—
179(C2×D3)⋊C4—181no—
180C4⋊Dic3C2×C3⋊D311yescharacteristic
181Dic3⋊C4—21no—
182C4×Dic3—21no—
183C2×C3⋊C8—34no—
184C2×C3⋊C8—34no—
185Dic12—182no—
186#204—182no—
187C22×C32—24no—
188C22×C32C2×D1211yescharacteristic
189C22×C32C2×#6714yes—
190C2×C3⋊D3—128no—
191C2×C3⋊D3—128no—
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—
199C32×C4—21no—
200C32×C4—21no—
201C32×C4—24no—
202C32×C4—24no—
203C32×C4—24no—
204C32×C4—24no—
205C32×C4—44no—
206C32×C4—44no—
207C32×C4—62no—
208C32×C4—121no—
209C32×C4C2×D1214yes—
210C32×C4#23514yes—
211C32×C4D3×D411yescharacteristic
212C32×C4#23911yescharacteristic
213C3⋊Dic3—128no—
214C3⋊Dic3—121no—
215C3⋊Dic3—124no—
216C3×Dic3—28no—
217C3×Dic3—44no—
218C2×Q16—271no—
219C2×SD16—271no—
220#124—271no—
221C4×Q8—31no—
222C4⋊C8—91no—
223#103—271no—
224#102—271no—
225C33#46411yescharacteristic, complemented, Sylow
226C22×D3—188no—
227C22×D3—184no—
228C22×D3—181no—
229C3×Q8—38no—
230C3×Q8—64no—
231C2×C3×C4—24no—
232C2×C3×C4—24no—
233C2×C3×C4—24no—
234C2×C3×C4—64no—
235C2×C3×C4—64no—
236C2×C3×C4#62614yes—
237C2×C3×C4#62614yes—
238C2×C3×C4#63811yescharacteristic
239C2×C3×C4#63811yescharacteristic
240C2×C3×C4C2×D3214yes—
241C2×C3×C4C22×C3⋊D311yescharacteristic
242C2×Dic3—21no—
243C2×Dic3—21no—
244C2×Dic3—188no—
245C2×Dic3—184no—
246C2×Dic3—181no—
247D12—184no—
248D12—184no—
249D12—181no—
250D12—181no—
251D12—364no—
252D12—364no—
253Dic6—21no—
254Dic6—184no—
255Dic6—184no—
256Dic6—181no—
257Dic6—181no—
258Dic6—364no—
259Dic6—364no—
260Dic6C4⋊(C3⋊D3)12yes—
261C3×C8—181no—
262C3×C8—181no—
263C3⋊C8—64no—
264C3⋊C8—64no—
265C2×C32—24no—
266C2×C32—24no—
267C2×C32—24no—
268C2×C32#89511yescharacteristic
269C2×C32#90411yescharacteristic
270C2×C32#90911yescharacteristic
271C2×C32#93014yes—
272C2×C32#94214yes—
273C2×C32#95014yes—
274C3⋊D3—248no—
275C3⋊D3—244no—
276C3⋊D3—241no—
277C2×Q8—31no—
278C2×Q8—271no—
279C2×D4—271no—
280Q16—542no—
281SD16—542no—
282C2×C8—91no—
283C2×C8—91no—
284C4⋊C4—31no—
285C4⋊C4—61no—
286C22⋊C4—541no—
287C22⋊C4—541no—
288C42—61no—
289C42D3×C3⋊D311yescharacteristic
290C22×C3—24no—
291C22×C3C2×#62614yes—
292C22×C3C2×#63811yescharacteristic
293C2×D3—368no—
294C2×D3—368no—
295C2×D3—368no—
296C2×D3—364no—
297C2×D3—364no—
298C2×D3—361no—
299C2×D3—361no—
300C2×D3—361no—
301C2×D3—364no—
302C3×C4—24no—
303C3×C4—24no—
304C3×C4—21no—
305C3×C4—21no—
306C3×C4—24no—
307C3×C4—24no—
308C3×C4—44no—
309C3×C4—44no—
310C3×C4—68no—
311C3×C4—124no—
312C3×C4#379914yes—
313C3×C4D3×D1214yes—
314C3×C4C2×C4⋊(C3⋊D3)11yescharacteristic
315C3×C4#382811yescharacteristic
316Dic3—22no—
317Dic3—41no—
318Dic3—368no—
319Dic3—361no—
320Dic3—364no—
321C32—24no—
322C32#484511yescharacteristic, complemented
323C32#502814yescomplemented
324C23—541no—
325Q8—32no—
326Q8—61no—
327Q8—541no—
328Q8—541no—
329D4—541no—
330D4—541no—
331C2×C4—61no—
332C2×C4—61no—
333C2×C4—541no—
334C2×C4#629111yescharacteristic
335C2×C4#629111yescharacteristic
336C2×C4C2×D3×C3⋊D311yescharacteristic
337C8—181no—
338C8—181no—
339C2×C3—24no—
340C2×C3—24no—
341C2×C3—24no—
342C2×C3#6382914yes—
343C2×C3#6383314yes—
344C2×C3#6390914yes—
345C2×C3#6413911yescharacteristic
346C2×C3#6415111yescharacteristic
347C2×C3#6415911yescharacteristic
348D3—728no—
349D3—724no—
350D3—721no—
351C22—1081no—
352C22—1081no—
353C22—1081no—
354C22C2×#629111yescharacteristic
355C4—21no—
356C4—21no—
357C4—62no—
358C4—121no—
359C4—1081no—
360C4#8902211yescharacteristic
361C4D3×C4⋊(C3⋊D3)11yescharacteristic
362C3—24no—
363C3#1059036814yescomplemented
364C3#1059272711yescharacteristic, complemented
365C2—2161no—
366C2#1172337711yescharacteristic
367C2#1172338111yescharacteristic
368C2#1172351611yescharacteristic
subgroupG/H
centerC22C2×#6291
commutator, cosocleC2×C33×C4C23
FrattiniC2×C4C2×D3×C3⋊D3
FittingC33×C42C22
socleC22×C33C2×D4
prime radicalC2×C4⋊(C3⋊(C3⋊D3))C22
hypercenterC42D3×C3⋊D3
hypocenterC33#464
LCSG ▷ C2×C33×C4 ▷ C2×C33 ▷ C33
UCSC1 ◁ C22 ◁ C42

Complements

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

NH ≅ G/Ntype
2. C32×C4×Dic6C2semidirect
3. C2×#88799C2semidirect
5. C3×C4⋊(C3⋊(C3⋊C8))C2semidirect
6. #11722041C2semidirect
29. C3×C4×Dic6D3semidirect
34. C4⋊(C3⋊(C3⋊C8))D3semidirect
105. C4×Dic6C3⋊D3semidirect
225. C33#464semidirect
322. C32#4845semidirect
323. C32#5028semidirect
363. C3#10590368semidirect
364. C3#10592727semidirect

Maximal subgroups

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

subgroupindex
#117239162
C32×C4×Dic62
C2×#887992
C2×#887952
C3×C4⋊(C3⋊(C3⋊C8))2
#117220412
#117220352
#105927273
#105903683