Finite Group Databasebeta

← #13160058

Subgroups — #13160058

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
2C2×C32×Dic12C211yesmaximal, characteristic, complemented
3#11723435C211yesmaximal, characteristic
4C2×#88799C211yesmaximal, characteristic, complemented
5#11722186C211yesmaximal, characteristic, complemented
6C3×#63647C211yesmaximal, characteristic
7#11722038C211yesmaximal, characteristic, complemented
8#10592885—31nomaximal
9#10590949—34nomaximal
10C2×#6308C2211yescharacteristic
11C2×C3×C4×C3⋊D3C2211yescharacteristic
12C2×C32×Dic6C2211yescharacteristic
13C2×C32×Dic6C2211yescharacteristic
14C2×#6293—21no—
15C2×Dic3×C3⋊D3—21no—
16C3⋊D3×Dic6—22no—
17C3⋊D3×Dic6—22no—
18C2×C33×C8C2211yescharacteristic
19C4⋊(C3⋊(C3⋊Dic3))C2211yescharacteristic
20C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
21C32×Dic12—22no—
22#88806—21no—
23#88803—21no—
24#88799—22no—
25C2×Q8×C3⋊D3—31no—
26C2×D3×Dic6—34no—
27C2×C32×Q16—31no—
28#64155—31no—
29C2×#3756—31no—
30C2×C3×Dic12D314yescomplemented
31#63856—34no—
32C2×#3719—34no—
33#63662—31no—
34#63647D311yescharacteristic
35#63643—31no—
36#63643—34no—
37#63643—64no—
38C3×(C2×D3)⋊C8—34no—
39#63566—34no—
40C22×C3×C3⋊D3D411yescharacteristic
41C2×C33×C4C2311yescharacteristic
42C2×C3⋊(C3⋊Dic3)—21no—
43C2×C3⋊(C3⋊Dic3)—21no—
44#6308—21no—
45#6308D412yes—
46C2×C3×C3⋊Dic3D411yescharacteristic
47C3×C4×C3⋊D3—21no—
48C3×C4×C3⋊D3—21no—
49C2×C32×Dic3—21no—
50C2×C32×Dic3—21no—
51C32×Dic6—21no—
52C32×Dic6—21no—
53C32×Dic6—21no—
54C32×Dic6D412yes—
55#6293—24no—
56Dic3×C3⋊D3—24no—
57C33×C8—21no—
58C3×C3⋊(C3⋊C8)—21no—
59#5186—94no—
60#4739—91no—
61C2×C32×Q8—31no—
62C2×C32×Q8—31no—
63Q8×C3⋊D3—62no—
64Q8×C3⋊D3—62no—
65C2×C4×C3⋊D3—61no—
66C2×C4×C3⋊D3C2×D311yescharacteristic
67C2×#634—31no—
68C2×#634—34no—
69C2×#634—61no—
70C2×#634—64no—
71C2×C3×C4×D3—34no—
72C2×C3×Dic6C2×D314yes—
73C2×C3×Dic6C2×D314yes—
74C2×#627—64no—
75C2×D3×Dic3—64no—
76D3×Dic6—68no—
77D3×Dic6—68no—
78C32×Q16—62no—
79C2×C32×C8—24no—
80C2×C32×C8C2×D314yes—
81C2×C32×C8C2×D311yescharacteristic
82#3756—62no—
83#3752—61no—
84C4⋊(C3⋊Dic3)—31no—
85C4⋊(C3⋊Dic3)—34no—
86C4⋊(C3⋊Dic3)—64no—
87#3750—61no—
88C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
89C2×C3×C3⋊C8—34no—
90C3×Dic12—28no—
91#3724—64no—
92(C2×D3)⋊Dic3—64no—
93#3719—68no—
94C22×C33C2×D411yescharacteristic
95C2×C3×C3⋊D3—21no—
96C2×C3×C3⋊D3—21no—
97C2×C3×C3⋊D3Q1612yescomplemented
98C33×C4C2×D411yescharacteristic
99C33×C4C2×D411yescharacteristic
100C3⋊(C3⋊Dic3)—22no—
101C3⋊(C3⋊Dic3)—41no—
102C3×C3⋊Dic3—21no—
103C32×Dic3—22no—
104C32×Dic3—41no—
105C2×D3×Q8—94no—
106C22×Dic6—91no—
107C2×C3×Q16—34no—
108#946—94no—
109C2×#216—94no—
110#924—91no—
111C2×Dic12—91no—
112C2×Dic12C3⋊D311yescharacteristic, complemented
113C3×C22⋊C8—91no—
114#835—94no—
115(C2×D3)⋊C8—34no—
116#816—91no—
117#816—94no—
118#816—91no—
119#816—184no—
120C22×C3⋊D3D1211yescharacteristic
121C22×C3×D3—34no—
122C32×Q8—32no—
123C32×Q8—61no—
124C32×Q8—61no—
125C32×Q8—61no—
126C2×C32×C4—24no—
127C2×C32×C4—61no—
128C2×C32×C4—61no—
129C2×C32×C4C22×D314yes—
130C2×C32×C4C22×D311yescharacteristic
131C2×C3⋊Dic3—61no—
132C2×C3⋊Dic3—61no—
133C2×C3⋊Dic3—64no—
134C2×C3⋊Dic3—64no—
135C2×C3⋊Dic3—124no—
136C2×C3⋊Dic3—124no—
137C2×C3⋊Dic3D1211yescharacteristic
138C4×C3⋊D3—21no—
139C4×C3⋊D3—21no—
140C4×C3⋊D3—64no—
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—24no—
150C2×C3×Dic3—34no—
151C3×C4×D3—64no—
152C3×C4×D3—64no—
153C3×Dic6—24no—
154C3×Dic6—24no—
155C3×Dic6—24no—
156C3×Dic6#6718yes—
157#627—616no—
158D3×Dic3—616no—
159C32×C8—21no—
160C32×C8—24no—
161C32×C8—44no—
162C3⋊(C3⋊C8)—21no—
163C3×C3⋊C8—64no—
164#451—271noSylow
165C2×C33#12011yescharacteristic
166C2×C33C2×Q1611yescharacteristic
167C2×C33#13711yescharacteristic
168C3×C3⋊D3—22no—
169C2×C3×Q8—34no—
170C2×C3×Q8—34no—
171C22×C3×C4—91no—
172C22×Dic3—181no—
173D3×Q8—188no—
174D3×Q8—188no—
175C2×C4×D3—34no—
176C2×C4×D3—184no—
177C2×Dic6—94no—
178C2×Dic6—91no—
179C2×Dic6—184no—
180C2×Dic6—182no—
181C2×Dic6—181no—
182C2×Dic6—184no—
183C2×Dic6—182no—
184C2×Dic6C2×C3⋊D311yescharacteristic
185C2×Dic6C2×C3⋊D311yescharacteristic
186C3×Q16—68no—
187C2×C3×C8—24no—
188C2×C3×C8—91no—
189C2×C3×C8D3214yes—
190C2×C3×C8C2×C3⋊D311yescharacteristic
191#217—181no—
192#216—188no—
193(C2×D3)⋊C4—184no—
194C4⋊Dic3—94no—
195C4⋊Dic3—91no—
196C4⋊Dic3—184no—
197Dic3⋊C4—181no—
198Dic3⋊C4—184no—
199C2×C3⋊C8—34no—
200Dic12—22no—
201Dic12—182no—
202C22×C32—24no—
203C22×C32C2×D1211yescharacteristic
204C22×C32D3×D414yes—
205C2×C3⋊D3—21no—
206C2×C3⋊D3—21no—
207C2×C3⋊D3Dic1212yescomplemented
208C2×C3×D3—38no—
209C2×C3×D3—64no—
210C2×C3×D3—64no—
211C32×C4—24no—
212C32×C4—24no—
213C32×C4—62no—
214C32×C4—121no—
215C32×C4D3×D414yes—
216C32×C4D3×D411yescharacteristic
217C32×C4D3×D411yescharacteristic
218C32×C4C2×#6714yes—
219C3⋊Dic3—21no—
220C3⋊Dic3—62no—
221C3⋊Dic3—68no—
222C3⋊Dic3—128no—
223C3⋊Dic3—121no—
224C3⋊Dic3—124no—
225C3⋊Dic3—244no—
226C3×Dic3—28no—
227C3×Dic3—44no—
228C3×Dic3—64no—
229C22×Q8—271no—
230C2×Q16—31no—
231C2×Q16—271no—
232C22⋊Q8—271no—
233#103—271no—
234#103—271no—
235C22⋊C8—91no—
236C33#45111yescharacteristic, complemented, Sylow
237C23×C3—91no—
238C22×D3—34no—
239C3×Q8—38no—
240C3×Q8—64no—
241C3×Q8—64no—
242C3×Q8—64no—
243C2×C3×C4—24no—
244C2×C3×C4—64no—
245C2×C3×C4—64no—
246C2×C3×C4—91no—
247C2×C3×C4—181no—
248C2×C3×C4—181no—
249C2×C3×C4C2×D3214yes—
250C2×C3×C4C22×C3⋊D311yescharacteristic
251C2×Dic3—21no—
252C2×Dic3—21no—
253C2×Dic3—34no—
254C2×Dic3—184no—
255C2×Dic3—184no—
256C2×Dic3—181no—
257C2×Dic3—184no—
258C2×Dic3—181no—
259C2×Dic3—364no—
260C2×Dic3—364no—
261C4×D3—64no—
262C4×D3—64no—
263C4×D3—1816no—
264Dic6—21no—
265Dic6—21no—
266Dic6—21no—
267Dic6—98no—
268Dic6—92no—
269Dic6—188no—
270Dic6—184no—
271Dic6—1816no—
272Dic6—184no—
273Dic6—181no—
274Dic6—364no—
275Dic6#63812yes—
276C3×C8—24no—
277C3×C8—21no—
278C3×C8—44no—
279C3×C8—181no—
280C3⋊C8—64no—
281C2×C32—24no—
282C2×C32—24no—
283C2×C32—24no—
284C2×C32#88211yescharacteristic
285C2×C32C2×Dic1211yescharacteristic
286C2×C32#90911yescharacteristic
287C2×C32D3×Q1614yes—
288C2×C32#91814yes—
289C2×C32#93714yes—
290C3⋊D3—22no—
291C3×D3—68no—
292C2×Q8—31no—
293C2×Q8—31no—
294C2×Q8—271no—
295C2×Q8—541no—
296C2×Q8—542no—
297C2×Q8—542no—
298C22×C4—91no—
299C22×C4—541no—
300Q16—62no—
301Q16—542no—
302C2×C8—91no—
303C2×C8D3×C3⋊D311yescharacteristic
304C4⋊C4—271no—
305C4⋊C4—541no—
306C22⋊C4—541no—
307C22×C3—24no—
308C22×C3—92no—
309C22×C3—181no—
310C22×C3—181no—
311C22×C3D3×D1214yes—
312C22×C3D4×C3⋊D311yescharacteristic
313C2×D3—38no—
314C2×D3—64no—
315C2×D3—64no—
316C3×C4—24no—
317C3×C4—24no—
318C3×C4—68no—
319C3×C4—124no—
320C3×C4—181no—
321C3×C4#380214yes—
322C3×C4D3×#6714yes—
323C3×C4D4×C3⋊D311yescharacteristic
324C3×C4C2×#63811yescharacteristic
325Dic3—22no—
326Dic3—41no—
327Dic3—64no—
328Dic3—182no—
329Dic3—188no—
330Dic3—368no—
331Dic3—364no—
332Dic3—361no—
333Dic3—724no—
334C32—24no—
335C32#473911yescharacteristic, complemented
336C32#518614yescomplemented
337C23—91no—
338Q8—32no—
339Q8—61no—
340Q8—61no—
341Q8—61no—
342Q8—272no—
343Q8—544no—
344Q8—541no—
345C2×C4—61no—
346C2×C4—61no—
347C2×C4—91no—
348C2×C4—181no—
349C2×C4—181no—
350C2×C4—544no—
351C2×C4—541no—
352C2×C4—541no—
353C2×C4C2×D3×C3⋊D311yescharacteristic
354C8—21no—
355C8—181no—
356C2×C3—24no—
357C2×C3—24no—
358C2×C3—24no—
359C2×C3—182no—
360C2×C3D3×Dic1214yes—
361C2×C3#6380114yes—
362C2×C3#6392214yes—
363C2×C3Q16×C3⋊D311yescharacteristic
364C2×C3#6412711yescharacteristic
365C2×C3#6414611yescharacteristic
366D3—68no—
367C22—92no—
368C22—181no—
369C22—181no—
370C22C3⋊D3×D1211yescharacteristic
371C4—62no—
372C4—121no—
373C4—181no—
374C4—542no—
375C4—1081no—
376C4#8902611yescharacteristic
377C4D3×#63811yescharacteristic
378C3—24no—
379C3#1059094914yescomplemented
380C3#1059288511yescharacteristic, complemented
381C2—182no—
382C2C3⋊D3×Dic1211yescharacteristic
383C2#1172332911yescharacteristic
384C2#1172354011yescharacteristic
subgroupG/H
centerC22C3⋊D3×D12
commutator, cosocleC2×C33×C4C23
FrattiniC2×C4C2×D3×C3⋊D3
FittingC2×C33×C8C22
socleC22×C33C2×D4
prime radicalC22×C3×C3⋊D3D4
hypercenterC2×C8D3×C3⋊D3
hypocenterC33#451
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×C32×Dic12C2semidirect
4. C2×#88799C2semidirect
5. #11722186C2semidirect
7. #11722038C2semidirect
30. C2×C3×Dic12D3semidirect
97. C2×C3×C3⋊D3Q16semidirect
112. C2×Dic12C3⋊D3semidirect
207. C2×C3⋊D3Dic12semidirect
236. C33#451semidirect
335. C32#4739semidirect
336. C32#5186semidirect
379. C3#10590949semidirect
380. C3#10592885semidirect

Maximal subgroups

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

subgroupindex
C2×C3⋊D3×Dic62
C2×C32×Dic122
#117234352
C2×#887992
#117221862
C3×#636472
#117220382
#105928853
#105909493