Finite Group Databasebeta

← #13159429

Subgroups — #13159429

Order 1728.

1116 conjugacy classes of subgroups; only the 162 normal subgroups are shown.

normal subgroups only

This group has 1116 conjugacy classes of subgroups — too many to draw. Only its 162 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⋊D3×Dic6C211yesmaximal, characteristic, complemented
2C3×#64104C211yesmaximal, characteristic
3C2×#88796C211yesmaximal, characteristic, complemented
4C2×#88793C211yesmaximal, characteristic, complemented
5#11722180C211yesmaximal, characteristic, complemented
6C32×#810C211yesmaximal, characteristic, complemented
7#11722023C211yesmaximal, characteristic, complemented
8#10592505—31nomaximal
9#10590603—34nomaximal
10C2×#6308C2211yescharacteristic
11C2×C3×C4⋊(C3⋊D3)—21no—
12C2×C3×C4⋊(C3⋊D3)C2211yescharacteristic
13C2×C3×C4×C3⋊D3C2211yescharacteristic
14C2×C32×Dic6C2211yescharacteristic
15C2×#6293—21no—
16C2×Dic3×C3⋊D3—21no—
17C3⋊D3×Dic6—22no—
18C3⋊D3×Dic6—22no—
19C33×C4⋊C4C2211yescharacteristic
20C2×C3⋊(C3⋊(C3⋊C8))C2211yescharacteristic
21C3×#3752—21no—
22C2×C32×C3⋊C8C2211yescharacteristic
23#88796—22no—
24#88793—22no—
25C2×Q8×C3⋊D3—31no—
26C2×D3×Dic6—34no—
27C2×#3755—31no—
28C2×#3744—31no—
29#64104D311yescharacteristic
30C3×#895—34no—
31C2×#3716—34no—
32C2×#3715—34no—
33C32×#103—31no—
34#63647—31no—
35#63637—31no—
36#63637—34no—
37#63637—64no—
38C3×#810D314yescomplemented
39(C2×D3)⋊(C3⋊C8)—34no—
40C22×C3×C3⋊D3—21no—
41C22×C3×C3⋊D3D411yescharacteristic
42C2×C33×C4—21no—
43C2×C33×C4C2311yescharacteristic
44C2×C3⋊(C3⋊Dic3)—21no—
45#6308—21no—
46#6308D412yescomplemented
47C2×C3×C3⋊Dic3D411yescharacteristic
48C3×C4⋊(C3⋊D3)—21no—
49C3×C4⋊(C3⋊D3)—21no—
50C3×C4⋊(C3⋊D3)—42no—
51C3×C4×C3⋊D3—21no—
52C3×C4×C3⋊D3—21no—
53C2×C32×Dic3—21no—
54C32×Dic6—21no—
55C32×Dic6D412yescomplemented
56#6293—24no—
57Dic3×C3⋊D3—24no—
58C3⋊(C3⋊(C3⋊C8))—21no—
59C32×C3⋊C8—21no—
60#5040—91no—
61#4806—94no—
62C2×C32×Q8—31no—
63Q8×C3⋊D3—62no—
64Q8×C3⋊D3—62no—
65C2×C4⋊(C3⋊D3)—21no—
66C2×C4⋊(C3⋊D3)C2×D311yescharacteristic
67C2×C4×C3⋊D3—61no—
68C2×C4×C3⋊D3C2×D311yescharacteristic
69C2×#634—31no—
70C2×#634—34no—
71C2×#634—61no—
72C2×#634—64no—
73C2×C3×D12—34no—
74C2×C3×D12—64no—
75C2×C3×C4×D3—34no—
76C2×C3×Dic6C2×D314yes—
77C2×#627—64no—
78C2×D3×Dic3—64no—
79D3×Dic6—68no—
80D3×Dic6—68no—
81C2×C32×C8—31no—
82C32×C4⋊C4—24no—
83C32×C4⋊C4C2×D314yes—
84C32×C4⋊C4C2×D311yescharacteristic
85#3755—62no—
86#3752—21no—
87C2×C3⋊(C3⋊C8)—31no—
88C2×C3⋊(C3⋊C8)—34no—
89C2×C3⋊(C3⋊C8)—64no—
90#3744—62no—
91C3×(C2×D3)⋊C4—64no—
92C2×C3×C3⋊C8C2×D314yes—
93#3716—68no—
94#3715—68no—
95C22×C33C2×D411yescharacteristic
96C2×C3×C3⋊D3—21no—
97C2×C3×C3⋊D3—21no—
98C2×C3×C3⋊D3—41no—
99C2×C3×C3⋊D3—41no—
100C2×C3×C3⋊D3—41no—
101C2×C3×C3⋊D3SD1612yescomplemented
102C33×C4—41no—
103C33×C4C2×D411yescharacteristic
104C33×C4C2×D411yescharacteristic
105C3⋊(C3⋊Dic3)—22no—
106C3×C3⋊Dic3—21no—
107C32×Dic3—22no—
108C2×D3×Q8—94no—
109C22×Dic6—91no—
110C3×#121—91no—
111C2×#215—94no—
112C2×#214—91no—
113C2×#214—91no—
114C2×#204—94no—
115#895—34no—
116C3×#103—34no—
117#830—91no—
118(C2×D3)⋊C8—94no—
119#810—94no—
120#810—91no—
121#810—184no—
122#810C3⋊D311yescharacteristic, complemented
123C22×C3⋊D3—21no—
124C22×C3⋊D3#6711yescharacteristic
125C22×C3×D3—34no—
126C22×C3×D3—64no—
127C32×Q8—32no—
128C32×Q8—61no—
129C2×C32×C4—21no—
130C2×C32×C4—24no—
131C2×C32×C4—24no—
132C2×C32×C4—44no—
133C2×C32×C4—61no—
134C2×C32×C4C22×D314yes—
135C2×C32×C4C22×D311yescharacteristic
136C2×C3⋊Dic3—61no—
137C2×C3⋊Dic3—64no—
138C2×C3⋊Dic3—124no—
139C2×C3⋊Dic3#6711yescharacteristic
140C4⋊(C3⋊D3)—21no—
141C4⋊(C3⋊D3)—21no—
142C4⋊(C3⋊D3)—42no—
143C4×C3⋊D3—21no—
144C4×C3⋊D3—21no—
145C4×C3⋊D3—64no—
146#634—32no—
147#634—38no—
148#634—68no—
149#634—64no—
150#634—61no—
151#634—64no—
152#634—124no—
153C2×C3×Dic3—24no—
154C2×C3×Dic3—34no—
155C3×D12—64no—
156C3×D12—64no—
157C3×D12—128no—
158C3×C4×D3—64no—
159C3×C4×D3—64no—
160C3×Dic6—24no—
161C3×Dic6D1218yescomplemented
162#627—616no—
163D3×Dic3—616no—
164C32×C8—61no—
165C3⋊(C3⋊C8)—61no—
166C3⋊(C3⋊C8)—64no—
167C3⋊(C3⋊C8)—124no—
168C3×C3⋊C8—24no—
169#448—271noSylow
170C2×C33#12011yescharacteristic
171C2×C33C2×SD1611yescharacteristic
172C2×C33#13711yescharacteristic
173C3×C3⋊D3—22no—
174C3×C3⋊D3—81no—
175C2×C3×Q8—34no—
176C2×C3×D4—91no—
177C2×C3×D4—181no—
178C22×C3×C4—91no—
179C22×Dic3—181no—
180D3×Q8—188no—
181D3×Q8—188no—
182C2×D12—34no—
183C2×D12—64no—
184C2×C4×D3—34no—
185C2×C4×D3—184no—
186C2×Dic6—94no—
187C2×Dic6—91no—
188C2×Dic6—184no—
189C2×Dic6—182no—
190C2×Dic6—181no—
191C2×Dic6—184no—
192C2×Dic6—182no—
193C2×Dic6C2×C3⋊D311yescharacteristic
194C2×C3×C8—34no—
195C3×C4⋊C4—24no—
196C3×C4⋊C4D3214yes—
197C3×C4⋊C4C2×C3⋊D311yescharacteristic
198C3×C22⋊C4—181no—
199#215—188no—
200#214—182no—
201#214—182no—
202(C2×D3)⋊C4—64no—
203C2×C3⋊C8—94no—
204C2×C3⋊C8—91no—
205C2×C3⋊C8—184no—
206C2×C3⋊C8C2×C3⋊D311yescharacteristic
207#204—188no—
208C22×C32—24no—
209C22×C32D3×D414yes—
210C22×C32C2×#6711yescharacteristic
211C2×C3⋊D3—21no—
212C2×C3⋊D3—21no—
213C2×C3⋊D3—41no—
214C2×C3⋊D3—41no—
215C2×C3⋊D3—41no—
216C2×C3⋊D3#21412yescomplemented
217C2×C3×D3—38no—
218C2×C3×D3—64no—
219C2×C3×D3—64no—
220C2×C3×D3—124no—
221C2×C3×D3—124no—
222C2×C3×D3—124no—
223C32×C4—24no—
224C32×C4—24no—
225C32×C4—41no—
226C32×C4—44no—
227C32×C4—62no—
228C32×C4—84no—
229C32×C4C2×D1214yes—
230C32×C4D3×D414yes—
231C32×C4D3×D411yescharacteristic
232C32×C4D3×D411yescharacteristic
233C3⋊Dic3—21no—
234C3⋊Dic3—62no—
235C3⋊Dic3—68no—
236C3⋊Dic3—128no—
237C3×Dic3—28no—
238C3×Dic3—64no—
239C22×Q8—271no—
240C2×SD16—271no—
241C2×SD16—271no—
242#121—91no—
243#103—31no—
244#103—271no—
245C22⋊C8—271no—
246C33#44811yescharacteristic, complemented, Sylow
247C23×C3—91no—
248C23×C3—181no—
249C22×D3—34no—
250C22×D3—64no—
251C3×Q8—38no—
252C3×Q8—64no—
253C3×D4—181no—
254C3×D4—181no—
255C3×D4—362no—
256C2×C3×C4—24no—
257C2×C3×C4—21no—
258C2×C3×C4—24no—
259C2×C3×C4—44no—
260C2×C3×C4—64no—
261C2×C3×C4—91no—
262C2×C3×C4—181no—
263C2×C3×C4—181no—
264C2×C3×C4C2×D3214yes—
265C2×C3×C4C22×C3⋊D311yescharacteristic
266C2×Dic3—21no—
267C2×Dic3—34no—
268C2×Dic3—184no—
269C2×Dic3—184no—
270C2×Dic3—181no—
271C2×Dic3—364no—
272D12—64no—
273D12—64no—
274D12—128no—
275C4×D3—64no—
276C4×D3—64no—
277C4×D3—1816no—
278Dic6—21no—
279Dic6—98no—
280Dic6—92no—
281Dic6—188no—
282Dic6—184no—
283Dic6—1816no—
284Dic6—184no—
285Dic6—181no—
286Dic6—364no—
287Dic6C4⋊(C3⋊D3)12yescomplemented
288C3×C8—64no—
289C3⋊C8—21no—
290C3⋊C8—184no—
291C3⋊C8—181no—
292C3⋊C8—364no—
293C2×C32—24no—
294C2×C32—24no—
295C2×C32—24no—
296C2×C32#88214yes—
297C2×C32D3×SD1614yes—
298C2×C32#91814yes—
299C2×C32C2×#21411yescharacteristic
300C2×C32#93711yescharacteristic
301C2×C32#95111yescharacteristic
302C3⋊D3—22no—
303C3⋊D3—81no—
304C3×D3—68no—
305C3×D3—244no—
306C2×Q8—31no—
307C2×Q8—271no—
308C2×Q8—541no—
309C2×Q8—542no—
310C2×Q8—542no—
311C2×D4—91no—
312C2×D4—181no—
313C22×C4—91no—
314C22×C4—541no—
315SD16—542no—
316SD16—542no—
317C2×C8—31no—
318C2×C8—271no—
319C4⋊C4D3×C3⋊D311yescharacteristic
320C22⋊C4—181no—
321C22×C3—24no—
322C22×C3—92no—
323C22×C3—181no—
324C22×C3—181no—
325C22×C3—361no—
326C22×C3—361no—
327C22×C3—361no—
328C22×C3D3×#6714yes—
329C22×C3D4×C3⋊D311yescharacteristic
330C2×D3—38no—
331C2×D3—64no—
332C2×D3—64no—
333C2×D3—124no—
334C2×D3—124no—
335C2×D3—124no—
336C3×C4—24no—
337C3×C4—24no—
338C3×C4—41no—
339C3×C4—44no—
340C3×C4—68no—
341C3×C4—84no—
342C3×C4—181no—
343C3×C4D3×D1214yes—
344C3×C4#380214yes—
345C3×C4C2×C4⋊(C3⋊D3)11yescharacteristic
346C3×C4D4×C3⋊D311yescharacteristic
347Dic3—22no—
348Dic3—64no—
349Dic3—182no—
350Dic3—188no—
351Dic3—368no—
352C32—24no—
353C32#480614yescomplemented
354C32#504011yescharacteristic, complemented
355C23—91no—
356C23—181no—
357Q8—32no—
358Q8—61no—
359Q8—272no—
360Q8—544no—
361Q8—541no—
362D4—181no—
363D4—181no—
364D4—362no—
365C2×C4—21no—
366C2×C4—61no—
367C2×C4—91no—
368C2×C4—181no—
369C2×C4—181no—
370C2×C4—544no—
371C2×C4—541no—
372C2×C4C2×D3×C3⋊D311yescharacteristic
373C8—61no—
374C8—541no—
375C2×C3—24no—
376C2×C3—24no—
377C2×C3—24no—
378C2×C3—182no—
379C2×C3—721no—
380C2×C3#6392214yes—
381C2×C3D3×#21414yes—
382C2×C3#6394314yes—
383C2×C3#6409111yescharacteristic
384C2×C3SD16×C3⋊D311yescharacteristic
385C2×C3#6412711yescharacteristic
386D3—68no—
387D3—244no—
388C22—92no—
389C22—181no—
390C22—181no—
391C22—361no—
392C22—361no—
393C22—361no—
394C22C3⋊D3×#6711yescharacteristic
395C4—41no—
396C4—62no—
397C4—181no—
398C4—542no—
399C4D3×C4⋊(C3⋊D3)11yescharacteristic
400C4#8902611yescharacteristic
401C3—24no—
402C3#1059060314yescomplemented
403C3#1059250511yescharacteristic, complemented
404C2—182no—
405C2—721no—
406C2#1172354111yescharacteristic
407C2C3⋊D3×#21411yescharacteristic
408C2#1172357611yescharacteristic
subgroupG/H
centerC22C3⋊D3×#67
commutator, cosocleC2×C33×C4C23
FrattiniC2×C4C2×D3×C3⋊D3
FittingC33×C4⋊C4C22
socleC22×C33C2×D4
prime radicalC3×#64104C2
hypercenterC4⋊C4D3×C3⋊D3
hypocenterC33#448
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. C2×C3⋊D3×Dic6C2semidirect
3. C2×#88796C2semidirect
4. C2×#88793C2semidirect
5. #11722180C2semidirect
6. C32×#810C2semidirect
7. #11722023C2semidirect
38. C3×#810D3semidirect
46. #6308D4semidirect
55. C32×Dic6D4semidirect
101. C2×C3×C3⋊D3SD16semidirect
122. #810C3⋊D3semidirect
161. C3×Dic6D12semidirect
216. C2×C3⋊D3#214semidirect
246. C33#448semidirect
287. Dic6C4⋊(C3⋊D3)semidirect
353. C32#4806semidirect
354. C32#5040semidirect
402. C3#10590603semidirect
403. C3#10592505semidirect

Maximal subgroups

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

subgroupindex
C2×C3⋊D3×Dic62
C3×#641042
C2×#887962
C2×#887932
#117221802
C32×#8102
#117220232
#105925053
#105906033