Finite Group Databasebeta

← #13161304

Subgroups — #13161304

Order 1728.

1084 conjugacy classes of subgroups; only the 158 normal subgroups are shown.

normal subgroups only

This group has 1084 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
1#11723988C211yesmaximal, characteristic, complemented
2C2×C3×#3755C211yesmaximal, characteristic, complemented
3C2×C32×#216C211yesmaximal, characteristic, complemented
4#11723436C211yesmaximal, characteristic, complemented
5C2×#88795C211yesmaximal, characteristic, complemented
6C2×#88791C211yesmaximal, characteristic
7#11722018C211yesmaximal, characteristic, complemented
8#10592890—31nomaximal
9#10591610—34nomaximal
10C2×C33×Q8C2211yescharacteristic
11C2×C4⋊(C3⋊(C3⋊D3))C2211yescharacteristic
12C2×C3×C4⋊(C3⋊D3)C2211yescharacteristic
13C2×C32×Dic6C2211yescharacteristic
14#88859—22no—
15C4×C3⋊(C3⋊Dic3)C2211yescharacteristic
16C3×#3755—22no—
17C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
18C32×#216—22no—
19C2×C32×C3⋊C8C2211yescharacteristic
20#88803—22no—
21#88795—22no—
22#88791—22no—
23C2×C32×Q16—31no—
24#64156—31no—
25#64156—31no—
26#64156—34no—
27#64156—64no—
28C2×#3755—31no—
29C2×#3755D311yescharacteristic, complemented
30C2×#3745—31no—
31C2×C3×#216D314yescomplemented
32C2×C3×#215—34no—
33#63863—34no—
34C2×#3717—34no—
35C2×#3714—34no—
36C8×C3⋊Dic3—31no—
37#63553—34no—
38C22×C3⋊(C3⋊D3)—21no—
39C22×C3×C3⋊D3—21no—
40C33×Q8—21no—
41C33×Q8—21no—
42C2×C33×C4—21no—
43C2×C33×C4C2311yescharacteristic
44C2×C3⋊(C3⋊Dic3)D412yes—
45C4⋊(C3⋊(C3⋊D3))—21no—
46C4⋊(C3⋊(C3⋊D3))—21no—
47C3×C4⋊(C3⋊D3)—21no—
48C3×C4⋊(C3⋊D3)—21no—
49C2×C32×Dic3—21no—
50C32×Dic6—21no—
51C32×Dic6—21no—
52C3×C3⋊(C3⋊C8)D412yes—
53C32×C3⋊C8D412yes—
54#5191—94no—
55#5168—91no—
56C2×C32×Q8—24no—
57C2×C32×Q8—31no—
58C2×C32×Q8C2×D314yes—
59C2×C32×Q8C2×D311yescharacteristic
60C2×C4⋊(C3⋊D3)—31no—
61C2×C4⋊(C3⋊D3)—34no—
62C2×C4⋊(C3⋊D3)—64no—
63C2×C4⋊(C3⋊D3)C2×D311yescharacteristic
64C2×C3×D12—34no—
65C2×C3×Dic6C2×D314yes—
66C32×Q16—62no—
67C2×C32×C8—31no—
68#3755—22no—
69#3755—62no—
70#3752—62no—
71#3752—68no—
72#3752—62no—
73#3752—616no—
74C4×C3⋊Dic3—31no—
75C4×C3⋊Dic3—34no—
76C4×C3⋊Dic3—64no—
77C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
78#3745—62no—
79C3×#216—28no—
80C3×#215—68no—
81C2×C3×C3⋊C8—34no—
82C2×C3×C3⋊C8C2×D314yes—
83(C2×D3)⋊Dic3—68no—
84#3717—68no—
85#3714—68no—
86C22×C33C2×D411yescharacteristic
87C2×C3⋊(C3⋊D3)—41no—
88C2×C3⋊(C3⋊D3)—42no—
89C2×C3×C3⋊D3—41no—
90C2×C3×C3⋊D3—42no—
91C33×C4—41no—
92C33×C4C2×D411yescharacteristic
93C33×C4C2×D411yescharacteristic
94C3⋊(C3⋊Dic3)—22no—
95C32×Dic3—41no—
96C2×C3×Q16—34no—
97C2×C3×SD16—91no—
98#947—94no—
99#947—94no—
100#947—91no—
101#947—184no—
102C2×#216C3⋊D311yescharacteristic, complemented
103C2×#215—34no—
104C2×#215—94no—
105#936—91no—
106C2×#213—91no—
107C2×D24—94no—
108C2×#204—91no—
109C8×Dic3—94no—
110C8×Dic3—91no—
111C22×C3⋊D3—21no—
112C22×C3⋊D3—68no—
113C22×C3⋊D3—64no—
114C22×C3⋊D3—61no—
115C22×C3×D3—64no—
116C32×Q8—21no—
117C32×Q8—21no—
118C32×Q8—24no—
119C32×Q8—24no—
120C32×Q8—44no—
121C32×Q8—44no—
122C32×Q8—61no—
123C32×Q8—61no—
124C2×C32×C4—28no—
125C2×C32×C4—24no—
126C2×C32×C4—21no—
127C2×C32×C4—24no—
128C2×C32×C4—61no—
129C2×C32×C4C22×D314yes—
130C2×C32×C4C22×D311yescharacteristic
131C2×C3⋊Dic3—32no—
132C2×C3⋊Dic3—38no—
133C2×C3⋊Dic3—68no—
134C4⋊(C3⋊D3)—21no—
135C4⋊(C3⋊D3)—21no—
136C4⋊(C3⋊D3)—61no—
137C4⋊(C3⋊D3)—61no—
138C4⋊(C3⋊D3)—64no—
139C4⋊(C3⋊D3)—64no—
140C4⋊(C3⋊D3)—124no—
141C4⋊(C3⋊D3)—124no—
142C2×C3×Dic3—24no—
143C3×D12—64no—
144C3×D12—64no—
145C3×Dic6—24no—
146C3×Dic6—24no—
147C32×C8—32no—
148C3⋊(C3⋊C8)#6712yes—
149C3×C3⋊C8—38no—
150C3×C3⋊C8#6718yes—
151#495—271noSylow
152C2×C33#12711yescharacteristic
153C2×C33#13512yes—
154C3⋊(C3⋊D3)—81no—
155C3×C3⋊D3—81no—
156C2×C3×Q8—24no—
157C2×C3×Q8—34no—
158C2×C3×Q8D3214yes—
159C2×C3×Q8C2×C3⋊D311yescharacteristic
160C2×C3×D4—91no—
161C2×D12—34no—
162C2×D12—94no—
163C2×D12—91no—
164C2×D12—184no—
165C2×Dic6C2×C3⋊D311yescharacteristic
166C3×Q16—68no—
167C3×SD16—182no—
168C2×C3×C8—34no—
169C2×C3×C8—91no—
170#217—182no—
171#216—22no—
172#215—68no—
173#215—188no—
174#213—182no—
175(C2×D3)⋊C4—188no—
176(C2×D3)⋊C4—182no—
177(C2×D3)⋊C4—188no—
178(C2×D3)⋊C4—1816no—
179C4×Dic3—94no—
180C4×Dic3—91no—
181C4×Dic3—184no—
182C2×C3⋊C8—34no—
183C2×C3⋊C8C2×C3⋊D311yescharacteristic
184D24—188no—
185#204—182no—
186C22×C32—24no—
187C22×C32D3×D414yes—
188C22×C32D3×D411yescharacteristic
189C2×C3⋊D3—42no—
190C2×C3⋊D3—41no—
191C2×C3⋊D3—128no—
192C2×C3⋊D3—1216no—
193C2×C3⋊D3—122no—
194C2×C3⋊D3—128no—
195C2×C3⋊D3—124no—
196C2×C3⋊D3—121no—
197C2×C3×D3—128no—
198C2×C3×D3—124no—
199C32×C4—24no—
200C32×C4—24no—
201C32×C4—48no—
202C32×C4—44no—
203C32×C4—41no—
204C32×C4—121no—
205C32×C4D3×D414yes—
206C32×C4D3×D411yescharacteristic
207C32×C4C2×#6714yes—
208C32×C4C2×#6711yescharacteristic
209C3⋊Dic3—62no—
210C3⋊Dic3—68no—
211C3⋊Dic3—128no—
212C3×Dic3—44no—
213C2×Q16—31no—
214C2×SD16—91no—
215C2×SD16—271no—
216C2×D8—271no—
217#124—271no—
218#124—271no—
219C4×C8—271no—
220C33#49511yescharacteristic, complemented, Sylow
221C23×C3—181no—
222C22×D3—64no—
223C22×D3—188no—
224C22×D3—184no—
225C22×D3—181no—
226C3×Q8—24no—
227C3×Q8—24no—
228C3×Q8—21no—
229C3×Q8—21no—
230C3×Q8—44no—
231C3×Q8—44no—
232C3×Q8—64no—
233C3×Q8—64no—
234C3×D4—181no—
235C3×D4—181no—
236C2×C3×C4—28no—
237C2×C3×C4—24no—
238C2×C3×C4—21no—
239C2×C3×C4—24no—
240C2×C3×C4—64no—
241C2×C3×C4C2×D3214yes—
242C2×C3×C4C22×C3⋊D311yescharacteristic
243C2×Dic3—21no—
244C2×Dic3—98no—
245C2×Dic3—92no—
246C2×Dic3—188no—
247D12—64no—
248D12—64no—
249D12—184no—
250D12—184no—
251D12—181no—
252D12—181no—
253D12—364no—
254D12—364no—
255Dic6—21no—
256Dic6—21no—
257C3×C8—38no—
258C3×C8—92no—
259C3⋊C8—38no—
260C3⋊C8#63812yes—
261C2×C32—28no—
262C2×C32—24no—
263C2×C32#91612yes—
264C2×C32#91918yes—
265C2×C32#94014yes—
266C2×C32#94011yescharacteristic
267C3⋊D3—81no—
268C3⋊D3—248no—
269C3⋊D3—241no—
270C3⋊D3—244no—
271C3×D3—244no—
272C2×Q8—31no—
273C2×Q8D3×C3⋊D311yescharacteristic
274C2×D4—91no—
275C2×D4—271no—
276Q16—62no—
277SD16—182no—
278SD16—542no—
279D8—542no—
280C2×C8—31no—
281C2×C8—91no—
282C22⋊C4—542no—
283C22⋊C4—542no—
284C42—271no—
285C22×C3—24no—
286C22×C3—362no—
287C22×C3—361no—
288C22×C3#381114yes—
289C22×C3D4×C3⋊D311yescharacteristic
290C2×D3—128no—
291C2×D3—124no—
292C2×D3—368no—
293C2×D3—3616no—
294C2×D3—362no—
295C2×D3—361no—
296C2×D3—368no—
297C2×D3—364no—
298C3×C4—24no—
299C3×C4—24no—
300C3×C4—48no—
301C3×C4—41no—
302C3×C4—44no—
303C3×C4—124no—
304C3×C4D3×#6714yes—
305C3×C4D3×#6714yes—
306C3×C4D4×C3⋊D311yescharacteristic
307C3×C4C2×#63811yescharacteristic
308Dic3—41no—
309Dic3—182no—
310Dic3—188no—
311Dic3—368no—
312C32—24no—
313C32#516811yescharacteristic, complemented
314C32#519114yescomplemented
315C23—181no—
316C23—541no—
317Q8—21no—
318Q8—21no—
319Q8—61no—
320Q8—61no—
321D4—181no—
322D4—181no—
323D4—541no—
324D4—541no—
325C2×C4—21no—
326C2×C4—61no—
327C2×C4—272no—
328C2×C4C2×D3×C3⋊D311yescharacteristic
329C8—32no—
330C8—92no—
331C2×C3—28no—
332C2×C3—24no—
333C2×C3—721no—
334C2×C3#6395018yes—
335C2×C3#6397914yes—
336C2×C3#6412812yes—
337C2×C3#6414911yescharacteristic
338D3—244no—
339D3—728no—
340D3—724no—
341D3—721no—
342C22—362no—
343C22—361no—
344C22—1082no—
345C22—1081no—
346C22#8903911yescharacteristic
347C4—41no—
348C4—121no—
349C4—542no—
350C4D3×#63811yescharacteristic
351C4C3⋊D3×#6711yescharacteristic
352C3—24no—
353C3#1059161014yescomplemented
354C3#1059289011yescharacteristic, complemented
355C2—721no—
356C2—2161no—
357C2#1172358912yes—
358C2#1172363311yescharacteristic
subgroupG/H
centerC22#89039
commutator, cosocleC2×C33×C4C23
FrattiniC2×C4C2×D3×C3⋊D3
FittingC2×C33×Q8C22
socleC22×C33C2×D4
prime radicalC2×#88791C2
hypercenterC2×Q8D3×C3⋊D3
hypocenterC33#495
LCSG ▷ C2×C33×C4 ▷ C2×C33 ▷ C33
UCSC1 ◁ C22 ◁ C2×C4 ◁ C2×Q8

Complements

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

NH ≅ G/Ntype
1. #11723988C2semidirect
2. C2×C3×#3755C2semidirect
3. C2×C32×#216C2semidirect
4. #11723436C2semidirect
5. C2×#88795C2semidirect
7. #11722018C2semidirect
29. C2×#3755D3semidirect
31. C2×C3×#216D3semidirect
102. C2×#216C3⋊D3semidirect
220. C33#495semidirect
313. C32#5168semidirect
314. C32#5191semidirect
353. C3#10591610semidirect
354. C3#10592890semidirect

Maximal subgroups

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

subgroupindex
#117239882
C2×C3×#37552
C2×C32×#2162
#117234362
C2×#887952
C2×#887912
#117220182
#105928903
#105916103