Finite Group Databasebeta

← #13160082

Subgroups — #13160082

Order 1728.

1080 conjugacy classes of subgroups; only the 158 normal subgroups are shown.

normal subgroups only

This group has 1080 conjugacy classes of subgroups — too many to draw. Only its 158 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×#89021C211yesmaximal, characteristic
2C2×C3×#3744C211yesmaximal, characteristic, complemented
3#11723425C211yesmaximal, characteristic, complemented
4C2×#88800C211yesmaximal, characteristic, complemented
5#11722186C211yesmaximal, characteristic, complemented
6C32×(C2×D3)⋊C8C211yesmaximal, characteristic, complemented
7#11722034C211yesmaximal, characteristic, complemented
8#10592432—31nomaximal
9#10590960—34nomaximal
10C2×#6308C2211yescharacteristic
11C2×C3×C4⋊(C3⋊D3)C2211yescharacteristic
12C2×C3×#634C2211yescharacteristic
13C2×C32×C4×D3C2211yescharacteristic
14C2×#6290—21no—
15C2×Dic3×C3⋊D3—21no—
16#89021—22no—
17#89021—22no—
18C2×C33×C8C2211yescharacteristic
19C4⋊(C3⋊(C3⋊Dic3))C2211yescharacteristic
20C3×#3744—22no—
21C2×C32×C3⋊C8C2211yescharacteristic
22#88806—21no—
23(C2×D3)⋊(C3⋊Dic3)—21no—
24#88800—22no—
25C2×#3828—31no—
26C2×#3795—34no—
27#64133—31no—
28C2×#3746—31no—
29C2×#3744D311yescharacteristic, complemented
30C2×C3×#204—34no—
31#63856—34no—
32C2×#3719—34no—
33C32×C22⋊C8—31no—
34#63648—31no—
35#63643—31no—
36#63643—34no—
37#63643—64no—
38C3×(C2×D3)⋊C8D314yescomplemented
39#63563—34no—
40C22×C3×C3⋊D3—21no—
41C22×C32×D3D411yescharacteristic
42C2×C33×C4C2311yescharacteristic
43C2×C3⋊(C3⋊Dic3)—21no—
44C2×C3⋊(C3⋊Dic3)—21no—
45#6308—21no—
46#6308D412yes—
47C2×C3×C3⋊Dic3—21no—
48C3×C4⋊(C3⋊D3)—21no—
49C3×C4⋊(C3⋊D3)D412yes—
50C3×#634—21no—
51C3×#634—21no—
52C2×C32×Dic3D411yescharacteristic
53C32×C4×D3—21no—
54C32×C4×D3—21no—
55#6290—24no—
56Dic3×C3⋊D3—24no—
57C33×C8—21no—
58C32×C3⋊C8—21no—
59#5173—91no—
60#4733—94no—
61C22×C32×C4—31no—
62C2×#638—61no—
63#3828—62no—
64#3828—62no—
65C2×C4⋊(C3⋊D3)C2×D311yescharacteristic
66C2×C4×C3⋊D3—61no—
67C2×#634—31no—
68C2×#634—34no—
69C2×#634—64no—
70C2×#634C2×D311yescharacteristic
71C2×C3×D12—34no—
72C2×C3×C4×D3C2×D314yes—
73C2×C3×Dic6—34no—
74C2×#625—64no—
75C2×D3×Dic3—64no—
76#3795—68no—
77#3795—68no—
78C2×C32×C8—24no—
79C2×C32×C8—31no—
80C2×C32×C8C2×D314yes—
81C2×C32×C8C2×D311yescharacteristic
82#3757—61no—
83C4⋊(C3⋊Dic3)—31no—
84C4⋊(C3⋊Dic3)—34no—
85C4⋊(C3⋊Dic3)—64no—
86#3750—61no—
87#3746—62no—
88#3744—22no—
89C2×C3×C3⋊C8C2×D314yes—
90C3×#204—68no—
91#3724—64no—
92(C2×D3)⋊Dic3—64no—
93#3719—68no—
94C22×C33C2×D411yescharacteristic
95C2×C3×C3⋊D3—22no—
96C2×C3×C3⋊D3—41no—
97C2×C3×C3⋊D3—41no—
98C2×C32×D3—21no—
99C2×C32×D3—21no—
100C2×C32×D3—21no—
101C33×C4C2×D411yescharacteristic
102C33×C4C2×D411yescharacteristic
103C3⋊(C3⋊Dic3)—22no—
104C3⋊(C3⋊Dic3)—41no—
105C3×C3⋊Dic3—41no—
106C32×Dic3—21no—
107C2×#237—91no—
108C2×#235—94no—
109C2×C3×SD16—91no—
110#946—91no—
111C2×#216—91no—
112#924—94no—
113C2×Dic12—94no—
114C2×#204—34no—
115C3×C22⋊C8—34no—
116#832—91no—
117#821—94no—
118(C2×D3)⋊C8C3⋊D311yescharacteristic, complemented
119#816—94no—
120#816—91no—
121#816—184no—
122C23×C32—31no—
123C22×C3⋊D3—21no—
124C22×C3×D3—64no—
125C22×C3×D3D1214yes—
126C2×C32×C4—24no—
127C2×C32×C4—31no—
128C2×C32×C4—61no—
129C2×C32×C4—61no—
130C2×C32×C4C22×D314yes—
131C2×C32×C4C22×D311yescharacteristic
132#638—64no—
133C2×C3⋊Dic3—21no—
134C2×C3⋊Dic3—61no—
135C2×C3⋊Dic3—61no—
136C2×C3⋊Dic3—64no—
137C2×C3⋊Dic3—64no—
138C2×C3⋊Dic3—124no—
139C2×C3⋊Dic3—124no—
140C4⋊(C3⋊D3)—21no—
141C4⋊(C3⋊D3)#6712yes—
142C4×C3⋊D3—64no—
143#634—21no—
144#634—21no—
145#634—32no—
146#634—38no—
147#634—68no—
148#634—61no—
149#634—64no—
150#634—124no—
151C2×C3×Dic3—64no—
152C2×C3×Dic3D1214yes—
153C3×D12—38no—
154C3×D12—64no—
155C3×C4×D3—24no—
156C3×C4×D3—24no—
157C3×Dic6—64no—
158C3×Dic6—64no—
159#625—616no—
160D3×Dic3—616no—
161C32×C8—21no—
162C32×C8—24no—
163C32×C8—44no—
164C32×C8—61no—
165C3×C3⋊C8—24no—
166#452—271noSylow
167C2×C33#12011yescharacteristic
168C2×C33#13511yescharacteristic
169C2×C33#13711yescharacteristic
170C3×C3⋊D3—42no—
171C32×D3—41no—
172C2×C3×Q8—91no—
173C2×C3×D4—91no—
174C22×C3×C4—34no—
175C2×#67—184no—
176C2×#67—181no—
177C22×Dic3—181no—
178#237—182no—
179#237—182no—
180#235—188no—
181#235—188no—
182C2×D12—34no—
183C2×C4×D3—184no—
184C2×C4×D3C2×C3⋊D311yescharacteristic
185C2×Dic6—34no—
186C2×Dic6—94no—
187C2×Dic6—91no—
188C2×Dic6—184no—
189C3×SD16—182no—
190C2×C3×C8—24no—
191C2×C3×C8—34no—
192C2×C3×C8D3214yes—
193C2×C3×C8C2×C3⋊D311yescharacteristic
194#217—184no—
195#216—182no—
196(C2×D3)⋊C4—181no—
197C4⋊Dic3—94no—
198C4⋊Dic3—91no—
199C4⋊Dic3—184no—
200Dic3⋊C4—184no—
201Dic3⋊C4—181no—
202C2×C3⋊C8C2×C3⋊D311yescharacteristic
203Dic12—188no—
204#204—68no—
205C22×C32—24no—
206C22×C32—61no—
207C22×C32—61no—
208C22×C32—61no—
209C22×C32C2×D1214yes—
210C22×C32D3×D411yescharacteristic
211C2×C3⋊D3—22no—
212C2×C3⋊D3—41no—
213C2×C3⋊D3—41no—
214C2×C3×D3—24no—
215C2×C3×D3—24no—
216C2×C3×D3—24no—
217C2×C3×D3—68no—
218C2×C3×D3—124no—
219C2×C3×D3—124no—
220C32×C4—24no—
221C32×C4—24no—
222C32×C4—61no—
223C32×C4D3×D414yes—
224C32×C4D3×D414yes—
225C32×C4D3×D411yescharacteristic
226C32×C4C2×#6711yescharacteristic
227C3⋊Dic3—41no—
228C3⋊Dic3—62no—
229C3⋊Dic3—68no—
230C3⋊Dic3—128no—
231C3⋊Dic3—121no—
232C3⋊Dic3—124no—
233C3⋊Dic3—244no—
234C3×Dic3—24no—
235C3×Dic3—124no—
236C2×#41—271no—
237C2×Q16—271no—
238C2×SD16—91no—
239C22⋊Q8—271no—
240#103—271no—
241#102—271no—
242C22⋊C8—31no—
243C33#45211yescharacteristic, complemented, Sylow
244C23×C3—34no—
245C23×C3—181no—
246C22×D3—64no—
247C22×D3C4⋊(C3⋊D3)11yescharacteristic
248C3×Q8—181no—
249C3×Q8—181no—
250C3×D4—92no—
251C3×D4—181no—
252C2×C3×C4—24no—
253C2×C3×C4—34no—
254C2×C3×C4—64no—
255C2×C3×C4—64no—
256C2×C3×C4—181no—
257C2×C3×C4C2×D3214yes—
258C2×C3×C4C22×C3⋊D311yescharacteristic
259#67—184no—
260#67—1816no—
261C2×Dic3—64no—
262C2×Dic3—184no—
263C2×Dic3—184no—
264C2×Dic3—181no—
265C2×Dic3—184no—
266C2×Dic3—181no—
267C2×Dic3—364no—
268C2×Dic3—364no—
269C2×Dic3C4⋊(C3⋊D3)11yescharacteristic
270D12—38no—
271D12—64no—
272C4×D3—21no—
273C4×D3—21no—
274C4×D3—1816no—
275Dic6—64no—
276Dic6—64no—
277Dic6—98no—
278Dic6—92no—
279Dic6—188no—
280Dic6—184no—
281Dic6—181no—
282Dic6—364no—
283C3×C8—24no—
284C3×C8—21no—
285C3×C8—44no—
286C3×C8—64no—
287C3⋊C8—21no—
288C2×C32—24no—
289C2×C32—24no—
290C2×C32—24no—
291C2×C32—121no—
292C2×C32#88214yes—
293C2×C32#90414yes—
294C2×C32#90914yes—
295C2×C32#91511yescharacteristic
296C2×C32#91611yescharacteristic
297C2×C32#93711yescharacteristic
298C3⋊D3—42no—
299C3×D3—44no—
300C3×D3—128no—
301#41—542no—
302#41—542no—
303C2×Q8—91no—
304C2×Q8—271no—
305C2×D4—91no—
306C2×D4—541no—
307C22×C4—31no—
308C22×C4—541no—
309Q16—542no—
310SD16—182no—
311C2×C8—31no—
312C2×C8D3×C3⋊D311yescharacteristic
313C4⋊C4—271no—
314C4⋊C4—541no—
315C22⋊C4—541no—
316C22×C3—24no—
317C22×C3—64no—
318C22×C3—64no—
319C22×C3—64no—
320C22×C3—182no—
321C22×C3—361no—
322C22×C3—361no—
323C22×C3D3×D1214yes—
324C22×C3C2×C4⋊(C3⋊D3)11yescharacteristic
325C2×D3—21no—
326C2×D3—21no—
327C2×D3—21no—
328C2×D3—68no—
329C2×D3—124no—
330C2×D3—124no—
331C3×C4—24no—
332C3×C4—24no—
333C3×C4—64no—
334C3×C4—361no—
335C3×C4#380214yes—
336C3×C4D3×#6714yes—
337C3×C4D4×C3⋊D311yescharacteristic
338C3×C4D4×C3⋊D311yescharacteristic
339Dic3—21no—
340Dic3—124no—
341Dic3—182no—
342Dic3—188no—
343Dic3—368no—
344Dic3—364no—
345Dic3—361no—
346Dic3—724no—
347C32—24no—
348C32#473314yescomplemented
349C32#517311yescharacteristic, complemented
350C23—31no—
351C23—181no—
352Q8—181no—
353Q8—181no—
354Q8—272no—
355Q8—541no—
356D4—92no—
357D4—181no—
358D4—544no—
359C2×C4—31no—
360C2×C4—61no—
361C2×C4—61no—
362C2×C4—181no—
363C2×C4—544no—
364C2×C4—541no—
365C2×C4—541no—
366C2×C4C2×D3×C3⋊D311yescharacteristic
367C8—21no—
368C8—61no—
369C2×C3—24no—
370C2×C3—24no—
371C2×C3—24no—
372C2×C3—124no—
373C2×C3—362no—
374C2×C3#6380014yes—
375C2×C3#6380614yes—
376C2×C3#6392214yes—
377C2×C3#6409111yescharacteristic
378C2×C3#6411311yescharacteristic
379C2×C3#6411811yescharacteristic
380D3—41no—
381D3—128no—
382C22—61no—
383C22—61no—
384C22—61no—
385C22—182no—
386C22—361no—
387C22—361no—
388C22D3×C4⋊(C3⋊D3)11yescharacteristic
389C4—61no—
390C4—361no—
391C4—542no—
392C4—1081no—
393C4#8902611yescharacteristic
394C4C3⋊D3×#6711yescharacteristic
395C3—24no—
396C3#1059096014yescomplemented
397C3#1059243211yescharacteristic, complemented
398C2—121no—
399C2—362no—
400C2#1172332611yescharacteristic
401C2#1172333511yescharacteristic
402C2#1172354111yescharacteristic
subgroupG/H
centerC22D3×C4⋊(C3⋊D3)
commutator, cosocleC2×C33×C4C23
FrattiniC2×C4C2×D3×C3⋊D3
FittingC2×C33×C8C22
socleC22×C33C2×D4
prime radicalC2×#89021C2
hypercenterC2×C8D3×C3⋊D3
hypocenterC33#452
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
2. C2×C3×#3744C2semidirect
3. #11723425C2semidirect
4. C2×#88800C2semidirect
5. #11722186C2semidirect
6. C32×(C2×D3)⋊C8C2semidirect
7. #11722034C2semidirect
29. C2×#3744D3semidirect
38. C3×(C2×D3)⋊C8D3semidirect
118. (C2×D3)⋊C8C3⋊D3semidirect
243. C33#452semidirect
348. C32#4733semidirect
349. C32#5173semidirect
396. C3#10590960semidirect
397. C3#10592432semidirect

Maximal subgroups

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

subgroupindex
C2×#890212
C2×C3×#37442
#117234252
C2×#888002
#117221862
C32×(C2×D3)⋊C82
#117220342
#105924323
#105909603