Finite Group Databasebeta

← #13159430

Subgroups — #13159430

Order 1728.

1056 conjugacy classes of subgroups; only the 162 normal subgroups are shown.

normal subgroups only

This group has 1056 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
2C3×#64106C211yesmaximal, characteristic
3C2×#88800C211yesmaximal, characteristic, complemented
4C2×#88798C211yesmaximal, characteristic, complemented
5#11722180C211yesmaximal, characteristic, complemented
6C32×#810C211yesmaximal, characteristic, complemented
7#11722023C211yesmaximal, characteristic
8#10592508—31nomaximal
9#10590604—34nomaximal
10C2×#6308C2211yescharacteristic
11C2×C3×C4×C3⋊D3C2211yescharacteristic
12C2×C3×#634C2211yescharacteristic
13C2×C32×Dic6C2211yescharacteristic
14C2×#6293—21no—
15C2×Dic3×C3⋊D3—21no—
16C3⋊D3×Dic6—22no—
17C3⋊D3×Dic6—22no—
18C33×C4⋊C4C2211yescharacteristic
19C2×C3⋊(C3⋊(C3⋊C8))C2211yescharacteristic
20C3×#3752—21no—
21C3×C4⋊(C3⋊Dic3)—21no—
22C2×C32×C3⋊C8C2211yescharacteristic
23#88800—22no—
24#88798—22no—
25C2×Q8×C3⋊D3—31no—
26C2×D3×Dic6—34no—
27C2×#3756—31no—
28C2×#3746—31no—
29#64106D311yescharacteristic
30C3×#897—34no—
31C2×#3719—34no—
32C2×#3718—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⋊D3D411yescharacteristic
41C2×C33×C4—21no—
42C2×C33×C4C2311yescharacteristic
43C2×C3⋊(C3⋊Dic3)—21no—
44#6308—21no—
45#6308D412yes—
46C2×C3×C3⋊Dic3—21no—
47C2×C3×C3⋊Dic3D411yescharacteristic
48C3×C4×C3⋊D3—21no—
49C3×C4×C3⋊D3—21no—
50C3×#634—21no—
51C3×#634—21no—
52C2×C32×Dic3—21no—
53C32×Dic6—21no—
54C32×Dic6D412yes—
55#6293—24no—
56Dic3×C3⋊D3—24no—
57C3⋊(C3⋊(C3⋊C8))—21no—
58C32×C3⋊C8—21no—
59#5050—91no—
60#4809—94no—
61C2×C32×Q8—31no—
62Q8×C3⋊D3—62no—
63Q8×C3⋊D3—62no—
64C2×C4×C3⋊D3—61no—
65C2×C4×C3⋊D3C2×D311yescharacteristic
66C2×#634—31no—
67C2×#634—34no—
68C2×#634—61no—
69C2×#634—64no—
70C2×#634C2×D311yescharacteristic
71C2×C3×C4×D3—34no—
72C2×C3×Dic6—34no—
73C2×C3×Dic6C2×D314yes—
74C2×#627—64no—
75C2×D3×Dic3—64no—
76D3×Dic6—68no—
77D3×Dic6—68no—
78C2×C32×C8—31no—
79C32×C4⋊C4—24no—
80C32×C4⋊C4C2×D314yes—
81C32×C4⋊C4C2×D311yescharacteristic
82#3756—62no—
83#3752—21no—
84C4⋊(C3⋊Dic3)—21no—
85C2×C3⋊(C3⋊C8)—31no—
86C2×C3⋊(C3⋊C8)—34no—
87C2×C3⋊(C3⋊C8)—64no—
88#3746—62no—
89C3×(C2×D3)⋊C4—64no—
90C3×C4⋊Dic3—64no—
91C2×C3×C3⋊C8C2×D314yes—
92#3719—68no—
93#3718—68no—
94C22×C33C2×D411yescharacteristic
95C2×C3×C3⋊D3—21no—
96C2×C3×C3⋊D3—21no—
97C2×C3×C3⋊D3Q1612yescomplemented
98C33×C4—41no—
99C33×C4C2×D411yescharacteristic
100C33×C4C2×D411yescharacteristic
101C3⋊(C3⋊Dic3)—22no—
102C3×C3⋊Dic3—21no—
103C3×C3⋊Dic3—41no—
104C32×Dic3—22no—
105C2×D3×Q8—94no—
106C22×Dic6—91no—
107C3×C22⋊Q8—91no—
108C2×#216—94no—
109C2×#216—91no—
110C2×#216—91no—
111C2×Dic12—94no—
112#897—34no—
113C3×#103—34no—
114#830—91no—
115(C2×D3)⋊C8—94no—
116#810—94no—
117#810—91no—
118#810—184no—
119#810C3⋊D311yescharacteristic, complemented
120C22×C3⋊D3#6711yescharacteristic
121C22×C3×D3—34no—
122C32×Q8—32no—
123C32×Q8—61no—
124C2×C32×C4—21no—
125C2×C32×C4—24no—
126C2×C32×C4—24no—
127C2×C32×C4—44no—
128C2×C32×C4—61no—
129C2×C32×C4C22×D314yes—
130C2×C32×C4C22×D311yescharacteristic
131C2×C3⋊Dic3—21no—
132C2×C3⋊Dic3—61no—
133C2×C3⋊Dic3—64no—
134C2×C3⋊Dic3—124no—
135C2×C3⋊Dic3#6711yescharacteristic
136C4×C3⋊D3—21no—
137C4×C3⋊D3—21no—
138C4×C3⋊D3—64no—
139#634—21no—
140#634—21no—
141#634—32no—
142#634—38no—
143#634—68no—
144#634—64no—
145#634—61no—
146#634—64no—
147#634—124no—
148C2×C3×Dic3—24no—
149C2×C3×Dic3—34no—
150C2×C3×Dic3—64no—
151C3×C4×D3—64no—
152C3×C4×D3—64no—
153C3×Dic6—24no—
154C3×Dic6—64no—
155C3×Dic6—64no—
156C3×Dic6D1218yes—
157#627—616no—
158D3×Dic3—616no—
159C32×C8—61no—
160C3⋊(C3⋊C8)—61no—
161C3⋊(C3⋊C8)—64no—
162C3⋊(C3⋊C8)—124no—
163C3×C3⋊C8—24no—
164#451—271noSylow
165C2×C33#12011yescharacteristic
166C2×C33C2×Q1611yescharacteristic
167C2×C33#13711yescharacteristic
168C3×C3⋊D3—22no—
169C2×C3×Q8—34no—
170C2×C3×Q8—91no—
171C22×C3×C4—91no—
172C22×Dic3—181no—
173D3×Q8—188no—
174D3×Q8—188no—
175C2×C4×D3—34no—
176C2×C4×D3—184no—
177C2×Dic6—34no—
178C2×Dic6—94no—
179C2×Dic6—91no—
180C2×Dic6—184no—
181C2×Dic6—182no—
182C2×Dic6—181no—
183C2×Dic6—184no—
184C2×Dic6—182no—
185C2×Dic6C2×C3⋊D311yescharacteristic
186C2×C3×C8—34no—
187C3×C4⋊C4—24no—
188C3×C4⋊C4—181no—
189C3×C4⋊C4D3214yes—
190C3×C4⋊C4C2×C3⋊D311yescharacteristic
191C3×C22⋊C4—181no—
192#216—182no—
193#216—188no—
194#216—182no—
195(C2×D3)⋊C4—64no—
196C4⋊Dic3—64no—
197C2×C3⋊C8—94no—
198C2×C3⋊C8—91no—
199C2×C3⋊C8—184no—
200C2×C3⋊C8C2×C3⋊D311yescharacteristic
201Dic12—188no—
202C22×C32—24no—
203C22×C32D3×D414yes—
204C22×C32C2×#6711yescharacteristic
205C2×C3⋊D3—21no—
206C2×C3⋊D3—21no—
207C2×C3⋊D3#21612yescomplemented
208C2×C3×D3—38no—
209C2×C3×D3—64no—
210C2×C3×D3—64no—
211C32×C4—24no—
212C32×C4—24no—
213C32×C4—41no—
214C32×C4—44no—
215C32×C4—62no—
216C32×C4—84no—
217C32×C4C2×D1214yes—
218C32×C4D3×D414yes—
219C32×C4D3×D411yescharacteristic
220C32×C4D3×D411yescharacteristic
221C3⋊Dic3—21no—
222C3⋊Dic3—41no—
223C3⋊Dic3—62no—
224C3⋊Dic3—68no—
225C3⋊Dic3—128no—
226C3×Dic3—28no—
227C3×Dic3—64no—
228C3×Dic3—124no—
229C22×Q8—271no—
230C2×Q16—271no—
231C2×Q16—271no—
232C22⋊Q8—91no—
233#103—31no—
234#103—271no—
235C22⋊C8—271no—
236C33#45111yescharacteristic, complemented, Sylow
237C23×C3—91no—
238C22×D3—34no—
239C3×Q8—38no—
240C3×Q8—64no—
241C3×Q8—181no—
242C3×Q8—181no—
243C2×C3×C4—24no—
244C2×C3×C4—21no—
245C2×C3×C4—24no—
246C2×C3×C4—44no—
247C2×C3×C4—64no—
248C2×C3×C4—91no—
249C2×C3×C4—181no—
250C2×C3×C4—181no—
251C2×C3×C4—181no—
252C2×C3×C4C2×D3214yes—
253C2×C3×C4C22×C3⋊D311yescharacteristic
254C2×Dic3—21no—
255C2×Dic3—34no—
256C2×Dic3—64no—
257C2×Dic3—184no—
258C2×Dic3—184no—
259C2×Dic3—181no—
260C2×Dic3—364no—
261C4×D3—64no—
262C4×D3—64no—
263C4×D3—1816no—
264Dic6—21no—
265Dic6—64no—
266Dic6—64no—
267Dic6—98no—
268Dic6—92no—
269Dic6—188no—
270Dic6—184no—
271Dic6—1816no—
272Dic6—184no—
273Dic6—181no—
274Dic6—364no—
275Dic6C4⋊(C3⋊D3)12yes—
276C3×C8—64no—
277C3⋊C8—21no—
278C3⋊C8—184no—
279C3⋊C8—181no—
280C3⋊C8—364no—
281C2×C32—24no—
282C2×C32—24no—
283C2×C32—24no—
284C2×C32#88214yes—
285C2×C32#91514yes—
286C2×C32D3×Q1614yes—
287C2×C32#93711yescharacteristic
288C2×C32C2×#21611yescharacteristic
289C2×C32#95111yescharacteristic
290C3⋊D3—22no—
291C3×D3—68no—
292C2×Q8—31no—
293C2×Q8—91no—
294C2×Q8—271no—
295C2×Q8—541no—
296C2×Q8—542no—
297C2×Q8—542no—
298C22×C4—91no—
299C22×C4—541no—
300Q16—542no—
301Q16—542no—
302C2×C8—31no—
303C2×C8—271no—
304C4⋊C4—181no—
305C4⋊C4D3×C3⋊D311yescharacteristic
306C22⋊C4—181no—
307C22×C3—24no—
308C22×C3—92no—
309C22×C3—181no—
310C22×C3—181no—
311C22×C3D3×#6714yes—
312C22×C3D4×C3⋊D311yescharacteristic
313C2×D3—38no—
314C2×D3—64no—
315C2×D3—64no—
316C3×C4—24no—
317C3×C4—24no—
318C3×C4—41no—
319C3×C4—44no—
320C3×C4—68no—
321C3×C4—84no—
322C3×C4—181no—
323C3×C4—361no—
324C3×C4D3×D1214yes—
325C3×C4#380214yes—
326C3×C4C2×C4⋊(C3⋊D3)11yescharacteristic
327C3×C4D4×C3⋊D311yescharacteristic
328Dic3—22no—
329Dic3—64no—
330Dic3—124no—
331Dic3—182no—
332Dic3—188no—
333Dic3—368no—
334C32—24no—
335C32#480914yescomplemented
336C32#505011yescharacteristic, complemented
337C23—91no—
338Q8—32no—
339Q8—61no—
340Q8—181no—
341Q8—181no—
342Q8—272no—
343Q8—544no—
344Q8—541no—
345C2×C4—21no—
346C2×C4—61no—
347C2×C4—91no—
348C2×C4—181no—
349C2×C4—181no—
350C2×C4—181no—
351C2×C4—544no—
352C2×C4—541no—
353C2×C4C2×D3×C3⋊D311yescharacteristic
354C8—61no—
355C8—541no—
356C2×C3—24no—
357C2×C3—24no—
358C2×C3—24no—
359C2×C3—182no—
360C2×C3#6392214yes—
361C2×C3#6392914yes—
362C2×C3D3×#21614yes—
363C2×C3#6409111yescharacteristic
364C2×C3#6412411yescharacteristic
365C2×C3Q16×C3⋊D311yescharacteristic
366D3—68no—
367C22—92no—
368C22—181no—
369C22—181no—
370C22C3⋊D3×#6711yescharacteristic
371C4—41no—
372C4—62no—
373C4—181no—
374C4—361no—
375C4—542no—
376C4D3×C4⋊(C3⋊D3)11yescharacteristic
377C4#8902611yescharacteristic
378C3—24no—
379C3#1059060414yescomplemented
380C3#1059250811yescharacteristic, complemented
381C2—182no—
382C2#1172354111yescharacteristic
383C2#1172355211yescharacteristic
384C2C3⋊D3×#21611yescharacteristic
subgroupG/H
centerC22C3⋊D3×#67
commutator, cosocleC2×C33×C4C23
FrattiniC2×C4C2×D3×C3⋊D3
FittingC33×C4⋊C4C22
socleC22×C33C2×D4
prime radicalC22×C3×C3⋊D3D4
hypercenterC4⋊C4D3×C3⋊D3
hypocenterC33#451
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
3. C2×#88800C2semidirect
4. C2×#88798C2semidirect
5. #11722180C2semidirect
6. C32×#810C2semidirect
38. C3×#810D3semidirect
97. C2×C3×C3⋊D3Q16semidirect
119. #810C3⋊D3semidirect
207. C2×C3⋊D3#216semidirect
236. C33#451semidirect
335. C32#4809semidirect
336. C32#5050semidirect
379. C3#10590604semidirect
380. C3#10592508semidirect

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×#641062
C2×#888002
C2×#887982
#117221802
C32×#8102
#117220232
#105925083
#105906043