Finite Group Databasebeta

← #13159450

Subgroups — #13159450

Order 1728.

1112 conjugacy classes of subgroups; only the 158 normal subgroups are shown.

normal subgroups only

This group has 1112 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×#89018C211yesmaximal, characteristic
2C3×#64106C211yesmaximal, characteristic, complemented
3C2×#88798C211yesmaximal, characteristic, complemented
4C2×#88797C211yesmaximal, characteristic, complemented
5#11722179C211yesmaximal, characteristic, complemented
6C32×#810C211yesmaximal, characteristic, complemented
7#11722023C211yesmaximal, characteristic, complemented
8#10592507—31nomaximal
9#10590616—34nomaximal
10C2×C4⋊(C3⋊(C3⋊D3))C2211yescharacteristic
11C2×C3×C4×C3⋊D3C2211yescharacteristic
12C2×C3×#634C2211yescharacteristic
13C2×C32×Dic6C2211yescharacteristic
14C2×#6292—21no—
15C2×#6289—21no—
16#89018—22no—
17#89018—22no—
18C33×C4⋊C4C2211yescharacteristic
19C2×C3⋊(C3⋊(C3⋊C8))C2211yescharacteristic
20C3×#3752—21no—
21C3×C4⋊(C3⋊Dic3)—21no—
22C2×C32×C3⋊C8C2211yescharacteristic
23#88798—22no—
24#88797—22no—
25C2×#3832—31no—
26C2×#3799—34no—
27C2×#3756—31no—
28C2×#3744—31no—
29#64106D311yescharacteristic, complemented
30C3×#897—34no—
31C2×#3718—34no—
32C2×#3717—34no—
33C32×#103—31no—
34#63647—31no—
35#63636—31no—
36#63636—34no—
37#63636—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
44C4⋊(C3⋊(C3⋊D3))—21no—
45C4⋊(C3⋊(C3⋊D3))D412yes—
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#6292—24no—
56#6289—24no—
57C3⋊(C3⋊(C3⋊C8))—21no—
58C32×C3⋊C8—21no—
59#5047—91no—
60#4808—94no—
61C2×C32×Q8—31no—
62#3832—62no—
63#3832—62no—
64C2×C4⋊(C3⋊D3)—31no—
65C2×C4⋊(C3⋊D3)—34no—
66C2×C4⋊(C3⋊D3)—61no—
67C2×C4⋊(C3⋊D3)—64no—
68C2×C4×C3⋊D3—61no—
69C2×C4×C3⋊D3C2×D311yescharacteristic
70C2×#634C2×D311yescharacteristic
71C2×C3×C4×D3—34no—
72C2×C3×Dic6—34no—
73C2×C3×Dic6C2×D314yes—
74C2×#626—64no—
75C2×#624—64no—
76#3799—68no—
77#3799—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#3744—62no—
89C3×(C2×D3)⋊C4—64no—
90C3×C4⋊Dic3—64no—
91C2×C3×C3⋊C8C2×D314yes—
92#3718—68no—
93#3717—68no—
94C22×C33C2×D411yescharacteristic
95C2×C3⋊(C3⋊D3)—22no—
96C2×C3⋊(C3⋊D3)—41no—
97C2×C3⋊(C3⋊D3)—41no—
98C2×C3×C3⋊D3—21no—
99C2×C3×C3⋊D3—21no—
100C2×C3×C3⋊D3—21no—
101C33×C4—41no—
102C33×C4C2×D411yescharacteristic
103C33×C4C2×D411yescharacteristic
104C3×C3⋊Dic3—21no—
105C3×C3⋊Dic3—41no—
106C32×Dic3—22no—
107C2×#239—94no—
108C2×#235—91no—
109C3×C22⋊Q8—91no—
110C2×#216—94no—
111C2×#216—91no—
112C2×#215—91no—
113C2×#204—94no—
114#897—34no—
115C3×#103—34no—
116#830—91no—
117(C2×D3)⋊C8—94no—
118#810C3⋊D311yescharacteristic, complemented
119#809—94no—
120#809—91no—
121#809—184no—
122C22×C3⋊D3—61no—
123C22×C3⋊D3—64no—
124C22×C3⋊D3—124no—
125C22×C3⋊D3#6711yescharacteristic
126C22×C3×D3—34no—
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—21no—
137C2×C3⋊Dic3#6711yescharacteristic
138C4⋊(C3⋊D3)—32no—
139C4⋊(C3⋊D3)—38no—
140C4⋊(C3⋊D3)—68no—
141C4⋊(C3⋊D3)—64no—
142C4⋊(C3⋊D3)—61no—
143C4⋊(C3⋊D3)—64no—
144C4⋊(C3⋊D3)—124no—
145C4×C3⋊D3—21no—
146C4×C3⋊D3—21no—
147C4×C3⋊D3—64no—
148#634—21no—
149#634—21no—
150C2×C3×Dic3—24no—
151C2×C3×Dic3—34no—
152C2×C3×Dic3—64no—
153C3×C4×D3—64no—
154C3×C4×D3—64no—
155C3×Dic6—24no—
156C3×Dic6—64no—
157C3×Dic6—64no—
158C3×Dic6D1218yes—
159#626—616no—
160#624—616no—
161C32×C8—61no—
162C3⋊(C3⋊C8)—61no—
163C3⋊(C3⋊C8)—64no—
164C3⋊(C3⋊C8)—124no—
165C3×C3⋊C8—24no—
166#452—271noSylow
167C2×C33#12011yescharacteristic
168C2×C33#13511yescharacteristic
169C2×C33#13711yescharacteristic
170C3⋊(C3⋊D3)—42no—
171C3×C3⋊D3—41no—
172C2×C3×Q8—34no—
173C2×C3×Q8—91no—
174C22×C3×C4—91no—
175C2×#67—181no—
176#239—188no—
177#239—188no—
178#235—182no—
179#235—182no—
180C2×D12—94no—
181C2×D12—91no—
182C2×D12—184no—
183C2×D12—184no—
184C2×C4×D3—34no—
185C2×C4×D3—181no—
186C2×C4×D3—184no—
187C2×Dic6—34no—
188C2×Dic6C2×C3⋊D311yescharacteristic
189C2×C3×C8—34no—
190C3×C4⋊C4—24no—
191C3×C4⋊C4—181no—
192C3×C4⋊C4D3214yes—
193C3×C4⋊C4C2×C3⋊D311yescharacteristic
194C3×C22⋊C4—181no—
195#216—188no—
196#216—182no—
197#215—182no—
198(C2×D3)⋊C4—64no—
199C4⋊Dic3—64no—
200C2×C3⋊C8—94no—
201C2×C3⋊C8—91no—
202C2×C3⋊C8—184no—
203C2×C3⋊C8C2×C3⋊D311yescharacteristic
204#204—188no—
205C22×C32—24no—
206C22×C32D3×D414yes—
207C22×C32C2×#6711yescharacteristic
208C2×C3⋊D3—21no—
209C2×C3⋊D3—21no—
210C2×C3⋊D3—21no—
211C2×C3⋊D3—62no—
212C2×C3⋊D3—68no—
213C2×C3⋊D3—128no—
214C2×C3⋊D3—121no—
215C2×C3⋊D3—121no—
216C2×C3⋊D3—124no—
217C2×C3⋊D3—124no—
218C2×C3⋊D3—244no—
219C2×C3⋊D3—244no—
220C2×C3×D3—64no—
221C2×C3×D3—64no—
222C2×C3×D3—64no—
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—41no—
235C3×Dic3—28no—
236C3×Dic3—64no—
237C3×Dic3—124no—
238C2×#41—271no—
239C2×Q16—271no—
240C2×SD16—271no—
241C22⋊Q8—91no—
242#103—31no—
243#102—271no—
244C22⋊C8—271no—
245C33#45211yescharacteristic, complemented, Sylow
246C23×C3—91no—
247C22×D3—34no—
248C22×D3—184no—
249C22×D3—181no—
250C22×D3—364no—
251C3×Q8—38no—
252C3×Q8—64no—
253C3×Q8—181no—
254C3×Q8—181no—
255C2×C3×C4—24no—
256C2×C3×C4—21no—
257C2×C3×C4—24no—
258C2×C3×C4—44no—
259C2×C3×C4—64no—
260C2×C3×C4—91no—
261C2×C3×C4—181no—
262C2×C3×C4—181no—
263C2×C3×C4—181no—
264C2×C3×C4C2×D3214yes—
265C2×C3×C4C22×C3⋊D311yescharacteristic
266#67—184no—
267C2×Dic3—21no—
268C2×Dic3—34no—
269C2×Dic3—64no—
270D12—98no—
271D12—92no—
272D12—188no—
273D12—1816no—
274D12—184no—
275D12—181no—
276D12—364no—
277C4×D3—64no—
278C4×D3—64no—
279C4×D3—184no—
280C4×D3—1816no—
281Dic6—21no—
282Dic6—64no—
283Dic6—64no—
284Dic6C4⋊(C3⋊D3)12yes—
285C3×C8—64no—
286C3⋊C8—21no—
287C3⋊C8—184no—
288C3⋊C8—181no—
289C3⋊C8—364no—
290C2×C32—24no—
291C2×C32—24no—
292C2×C32—24no—
293C2×C32#88214yes—
294C2×C32#91614yes—
295C2×C32#91814yes—
296C2×C32#93711yescharacteristic
297C2×C32#94211yescharacteristic
298C2×C32#95011yescharacteristic
299C3⋊D3—41no—
300C3⋊D3—122no—
301C3⋊D3—128no—
302C3⋊D3—248no—
303C3×D3—124no—
304#41—542no—
305#41—542no—
306C2×Q8—31no—
307C2×Q8—91no—
308C2×D4—271no—
309C2×D4—541no—
310C22×C4—91no—
311C22×C4—541no—
312Q16—542no—
313SD16—542no—
314C2×C8—31no—
315C2×C8—271no—
316C4⋊C4—181no—
317C4⋊C4D3×C3⋊D311yescharacteristic
318C22⋊C4—181no—
319C22×C3—24no—
320C22×C3—181no—
321C22×C3—181no—
322C22×C3—181no—
323C22×C3D3×#6714yes—
324C22×C3D4×C3⋊D311yescharacteristic
325C2×D3—64no—
326C2×D3—64no—
327C2×D3—64no—
328C2×D3—182no—
329C2×D3—188no—
330C2×D3—368no—
331C2×D3—361no—
332C2×D3—361no—
333C2×D3—364no—
334C2×D3—364no—
335C2×D3—724no—
336C2×D3—724no—
337C3×C4—24no—
338C3×C4—24no—
339C3×C4—41no—
340C3×C4—44no—
341C3×C4—68no—
342C3×C4—84no—
343C3×C4—181no—
344C3×C4—361no—
345C3×C4D3×D1214yes—
346C3×C4#380214yes—
347C3×C4C2×C4⋊(C3⋊D3)11yescharacteristic
348C3×C4D4×C3⋊D311yescharacteristic
349Dic3—22no—
350Dic3—64no—
351Dic3—124no—
352C32—24no—
353C32#480814yescomplemented
354C32#504711yescharacteristic, complemented
355C23—91no—
356C23—541no—
357Q8—32no—
358Q8—61no—
359Q8—181no—
360Q8—181no—
361D4—272no—
362D4—544no—
363D4—541no—
364C2×C4—21no—
365C2×C4—61no—
366C2×C4—91no—
367C2×C4—181no—
368C2×C4—181no—
369C2×C4—181no—
370C2×C4—544no—
371C2×C4C2×D3×C3⋊D311yescharacteristic
372C8—61no—
373C8—541no—
374C2×C3—24no—
375C2×C3—24no—
376C2×C3—24no—
377C2×C3—361no—
378C2×C3#6392214yes—
379C2×C3#6393514yes—
380C2×C3#6394814yes—
381C2×C3#6409111yescharacteristic
382C2×C3#6412511yescharacteristic
383C2×C3#6412711yescharacteristic
384D3—124no—
385D3—368no—
386D3—362no—
387D3—728no—
388C22—181no—
389C22—181no—
390C22—181no—
391C22—542no—
392C22—1081no—
393C22—1081no—
394C22C3⋊D3×#6711yescharacteristic
395C4—41no—
396C4—62no—
397C4—181no—
398C4—361no—
399C4D3×C4⋊(C3⋊D3)11yescharacteristic
400C4#8902611yescharacteristic
401C3—24no—
402C3#1059061614yescomplemented
403C3#1059250711yescharacteristic, complemented
404C2—361no—
405C2—1082no—
406C2#1172354111yescharacteristic
407C2#1172356111yescharacteristic
408C2#1172358611yescharacteristic
subgroupG/H
centerC22C3⋊D3×#67
commutator, cosocleC2×C33×C4C23
FrattiniC2×C4C2×D3×C3⋊D3
FittingC33×C4⋊C4C22
socleC22×C33C2×D4
prime radicalC2×#89018C2
hypercenterC4⋊C4D3×C3⋊D3
hypocenterC33#452
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
2. C3×#64106C2semidirect
3. C2×#88798C2semidirect
4. C2×#88797C2semidirect
5. #11722179C2semidirect
6. C32×#810C2semidirect
7. #11722023C2semidirect
29. #64106D3semidirect
38. C3×#810D3semidirect
118. #810C3⋊D3semidirect
245. C33#452semidirect
353. C32#4808semidirect
354. C32#5047semidirect
402. C3#10590616semidirect
403. C3#10592507semidirect

Maximal subgroups

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

subgroupindex
C2×#890182
C3×#641062
C2×#887982
C2×#887972
#117221792
C32×#8102
#117220232
#105925073
#105906163