Finite Group Databasebeta

← #13159933

Subgroups — #13159933

Order 1728.

896 conjugacy classes of subgroups; only the 146 normal subgroups are shown.

normal subgroups only

This group has 896 conjugacy classes of subgroups — too many to draw. Only its 146 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×#3744C211yesmaximal, characteristic, complemented
2#11723501C211yesmaximal, characteristic, complemented
3#11723434C211yesmaximal, characteristic, complemented
4C2×#88789C211yesmaximal, characteristic
5#11722184C211yesmaximal, characteristic, complemented
6C32×#821C211yesmaximal, characteristic, complemented
7#11722039C211yesmaximal, characteristic, complemented
8#10592486—31nomaximal
9#10590750—34nomaximal
10C2×C3×C4⋊(C3⋊D3)C2211yescharacteristic
11C2×C3×#634C2211yescharacteristic
12C2×C32×D12C2211yescharacteristic
13C2×Dic3×C3⋊D3—21no—
14C2×C33×C8C2211yescharacteristic
15C4×C3⋊(C3⋊Dic3)C2211yescharacteristic
16C2×C3⋊(C3⋊(C3⋊C8))C2211yescharacteristic
17C3×#3744—22no—
18C32×C4⋊Dic3C2211yescharacteristic
19#88803—21no—
20(C2×D3)⋊(C3⋊Dic3)—21no—
21(C2×D3)⋊(C3⋊Dic3)—21no—
22#88789—22no—
23#64145—31no—
24C2×#3753—31no—
25C2×#3744D311yescharacteristic, complemented
26#64102—31no—
27C2×C3×#204—34no—
28#63898—34no—
29#63863—34no—
30C2×#3713—34no—
31C32×#102—31no—
32#63641—31no—
33#63641—34no—
34#63641—64no—
35#63637—31no—
36C3×#821D314yescomplemented
37#63565—34no—
38C22×C3×C3⋊D3—21no—
39C22×C32×D3—21no—
40C2×C33×C4C2311yescharacteristic
41C2×C3⋊(C3⋊Dic3)D411yescharacteristic
42C2×C3⋊(C3⋊Dic3)D411yescharacteristic
43C2×C3×C3⋊Dic3—21no—
44C3×C4⋊(C3⋊D3)—21no—
45C3×C4⋊(C3⋊D3)D412yes—
46C3×#634—21no—
47C3×#634—21no—
48C2×C32×Dic3—21no—
49C32×D12—21no—
50C32×D12—21no—
51Dic3×C3⋊D3—42no—
52C33×C8—21no—
53C3⋊(C3⋊(C3⋊C8))—21no—
54#5165—91no—
55#4787—94no—
56C2×C32×D4—31no—
57C2×C4⋊(C3⋊D3)C2×D311yescharacteristic
58C2×C4×C3⋊D3—61no—
59C2×#634C2×D311yescharacteristic
60C2×C3×D12—34no—
61C2×C3×D12C2×D314yes—
62C2×C3×Dic6—34no—
63C2×D3×Dic3—64no—
64C2×C32×C8—24no—
65C2×C32×C8C2×D314yes—
66C2×C32×C8C2×D311yescharacteristic
67C32×C4⋊C4—31no—
68#3757—61no—
69#3757—61no—
70#3753—62no—
71#3752—61no—
72C4×C3⋊Dic3—31no—
73C4×C3⋊Dic3—34no—
74C4×C3⋊Dic3—64no—
75C2×C3⋊(C3⋊C8)—31no—
76C2×C3⋊(C3⋊C8)—34no—
77C2×C3⋊(C3⋊C8)—64no—
78#3744—22no—
79C3×C4⋊Dic3C2×D314yes—
80C3×#204—68no—
81(C2×D3)⋊Dic3—64no—
82(C2×D3)⋊Dic3—64no—
83(C2×D3)⋊Dic3—64no—
84#3713—68no—
85C22×C33C2×D411yescharacteristic
86C2×C3×C3⋊D3—22no—
87C2×C3×C3⋊D3—41no—
88C2×C3×C3⋊D3—41no—
89C2×C32×D3—41no—
90C2×C32×D3—41no—
91C2×C32×D3—41no—
92C33×C4C2×D411yescharacteristic
93C33×C4#4111yescharacteristic
94C3⋊(C3⋊Dic3)—21no—
95C3⋊(C3⋊Dic3)—21no—
96C3×C3⋊Dic3—41no—
97C32×Dic3—41no—
98C2×C3×SD16—91no—
99#947—91no—
100#936—94no—
101D4×Dic3—91no—
102C2×#213—94no—
103C2×#213—91no—
104C2×#204—34no—
105#893—94no—
106C3×#102—34no—
107#835—91no—
108#821C3⋊D311yescharacteristic, complemented
109Dic3⋊C8—94no—
110Dic3⋊C8—91no—
111Dic3⋊C8—184no—
112#810—94no—
113C23×C32—61no—
114C22×C3⋊D3—21no—
115C22×C3×D3—24no—
116C22×C3×D3—64no—
117C32×D4—61no—
118C32×D4—61no—
119C2×C32×C4—24no—
120C2×C32×C4—61no—
121C2×C32×C4C22×D314yes—
122C2×C32×C4C22×D311yescharacteristic
123C2×C3⋊Dic3—21no—
124C2×C3⋊Dic3—31no—
125C2×C3⋊Dic3—31no—
126C2×C3⋊Dic3—34no—
127C2×C3⋊Dic3—34no—
128C2×C3⋊Dic3—64no—
129C2×C3⋊Dic3—64no—
130C4⋊(C3⋊D3)—21no—
131C4⋊(C3⋊D3)#6712yes—
132C4×C3⋊D3—122no—
133#634—21no—
134#634—21no—
135C2×C3×Dic3—24no—
136C2×C3×Dic3—64no—
137C3×D12—24no—
138C3×D12—24no—
139C3×D12—38no—
140C3×D12—64no—
141C3×Dic6—64no—
142C3×Dic6—64no—
143D3×Dic3—128no—
144C32×C8—21no—
145C32×C8—24no—
146C32×C8—44no—
147C3⋊(C3⋊C8)—61no—
148C3⋊(C3⋊C8)—64no—
149C3⋊(C3⋊C8)—124no—
150#463—271noSylow
151C2×C33#12111yescharacteristic
152C2×C33#13511yescharacteristic
153C2×C33AGL(1,ℤ/8)11yescharacteristic
154C3×C3⋊D3—42no—
155C32×D3—81no—
156C2×C3×Q8—91no—
157C2×C3×D4—34no—
158C2×C3×D4—91no—
159C22×Dic3—181no—
160C2×D12—34no—
161C2×D12C2×C3⋊D311yescharacteristic
162C2×C4×D3—184no—
163C2×Dic6—34no—
164C3×SD16—182no—
165C2×C3×C8—24no—
166C2×C3×C8D3214yes—
167C2×C3×C8C2×C3⋊D311yescharacteristic
168C3×C4⋊C4—34no—
169#217—181no—
170#217—184no—
171#217—184no—
172#213—182no—
173#213—188no—
174(C2×D3)⋊C4—184no—
175(C2×D3)⋊C4—181no—
176(C2×D3)⋊C4—181no—
177C4⋊Dic3C2×C3⋊D311yescharacteristic
178C4×Dic3—94no—
179C4×Dic3—91no—
180C4×Dic3—184no—
181C2×C3⋊C8—94no—
182C2×C3⋊C8—91no—
183C2×C3⋊C8—184no—
184#204—68no—
185C22×C32—24no—
186C22×C32—121no—
187C22×C32—121no—
188C22×C32—121no—
189C22×C32D3×D414yes—
190C22×C32D3×D411yescharacteristic
191C2×C3⋊D3—22no—
192C2×C3⋊D3—41no—
193C2×C3⋊D3—41no—
194C2×C3×D3—44no—
195C2×C3×D3—44no—
196C2×C3×D3—44no—
197C2×C3×D3—68no—
198C2×C3×D3—124no—
199C2×C3×D3—124no—
200C32×C4—24no—
201C32×C4—24no—
202C32×C4—121no—
203C32×C4#23514yes—
204C32×C4D3×D414yes—
205C32×C4#23711yescharacteristic
206C32×C4C2×#6711yescharacteristic
207C3⋊Dic3—41no—
208C3⋊Dic3—68no—
209C3⋊Dic3—61no—
210C3⋊Dic3—64no—
211C3⋊Dic3—61no—
212C3⋊Dic3—64no—
213C3⋊Dic3—124no—
214C3×Dic3—44no—
215C3×Dic3—124no—
216C2×SD16—91no—
217C2×D8—271no—
218#124—271no—
219C4×D4—271no—
220C4⋊C8—271no—
221#103—271no—
222#102—31no—
223C33#46311yescharacteristic, complemented, Sylow
224C23×C3—64no—
225C23×C3—181no—
226C22×D3—21no—
227C22×D3—64no—
228C3×Q8—181no—
229C3×Q8—181no—
230C3×D4—64no—
231C3×D4—64no—
232C3×D4—92no—
233C3×D4—181no—
234C2×C3×C4—24no—
235C2×C3×C4—64no—
236C2×C3×C4—181no—
237C2×C3×C4C2×D3214yes—
238C2×C3×C4C22×C3⋊D311yescharacteristic
239C2×Dic3—21no—
240C2×Dic3—64no—
241C2×Dic3—94no—
242C2×Dic3—94no—
243C2×Dic3—91no—
244C2×Dic3—91no—
245C2×Dic3—184no—
246C2×Dic3—184no—
247C2×Dic3—362no—
248D12—21no—
249D12—21no—
250D12—38no—
251D12—64no—
252C4×D3—368no—
253Dic6—64no—
254Dic6—64no—
255C3×C8—24no—
256C3×C8—21no—
257C3×C8—44no—
258C3⋊C8—184no—
259C3⋊C8—181no—
260C3⋊C8—364no—
261C2×C32—24no—
262C2×C32—24no—
263C2×C32—24no—
264C2×C32—241no—
265C2×C32#88414yes—
266C2×C32#91114yes—
267C2×C32#91411yescharacteristic
268C2×C32#91614yes—
269C2×C32#91611yescharacteristic
270C2×C32#93911yescharacteristic
271C3⋊D3—42no—
272C3×D3—84no—
273C3×D3—128no—
274C2×Q8—91no—
275C2×D4—31no—
276C2×D4—91no—
277C22×C4—541no—
278SD16—182no—
279D8—542no—
280C2×C8—271no—
281C2×C8D3×C3⋊D311yescharacteristic
282C4⋊C4—31no—
283C22⋊C4—541no—
284C22⋊C4—541no—
285C22⋊C4—541no—
286C42—271no—
287C22×C3—24no—
288C22×C3—124no—
289C22×C3—124no—
290C22×C3—124no—
291C22×C3—182no—
292C22×C3—361no—
293C22×C3—361no—
294C22×C3#380214yes—
295C22×C3D4×C3⋊D311yescharacteristic
296C2×D3—41no—
297C2×D3—41no—
298C2×D3—41no—
299C2×D3—68no—
300C2×D3—124no—
301C2×D3—124no—
302C3×C4—24no—
303C3×C4—24no—
304C3×C4—124no—
305C3×C4—361no—
306C3×C4#379514yes—
307C3×C4D3×#6714yes—
308C3×C4#382811yescharacteristic
309C3×C4D4×C3⋊D311yescharacteristic
310Dic3—41no—
311Dic3—124no—
312Dic3—188no—
313Dic3—184no—
314Dic3—184no—
315Dic3—181no—
316Dic3—181no—
317Dic3—364no—
318C32—24no—
319C32#478714yescomplemented
320C32#516511yescharacteristic, complemented
321C23—61no—
322C23—181no—
323Q8—181no—
324Q8—181no—
325D4—61no—
326D4—61no—
327D4—92no—
328D4—181no—
329C2×C4—61no—
330C2×C4—181no—
331C2×C4—271no—
332C2×C4—271no—
333C2×C4—1082no—
334C2×C4C2×D3×C3⋊D311yescharacteristic
335C8—21no—
336C8—541no—
337C2×C3—24no—
338C2×C3—24no—
339C2×C3—24no—
340C2×C3—244no—
341C2×C3—362no—
342C2×C3#6379814yes—
343C2×C3#6380914yes—
344C2×C3#6388514yes—
345C2×C3#6409311yescharacteristic
346C2×C3#6412011yescharacteristic
347C2×C3#6412511yescharacteristic
348D3—81no—
349D3—128no—
350C22—121no—
351C22—121no—
352C22—121no—
353C22—182no—
354C22—361no—
355C22—361no—
356C22#8902611yescharacteristic
357C4—121no—
358C4—361no—
359C4—541no—
360C4—541no—
361C4#8902111yescharacteristic
362C4C3⋊D3×#6711yescharacteristic
363C3—24no—
364C3#1059075014yescomplemented
365C3#1059248611yescharacteristic, complemented
366C2—241no—
367C2—362no—
368C2#1172332411yescharacteristic
369C2#1172334211yescharacteristic
370C2#1172351911yescharacteristic
subgroupG/H
centerC22#89026
commutator, cosocleC2×C33×C4C23
FrattiniC2×C4C2×D3×C3⋊D3
FittingC2×C33×C8C22
socleC22×C33C2×D4
prime radicalC2×#88789C2
hypercenterC2×C8D3×C3⋊D3
hypocenterC33#463
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
1. C2×C3×#3744C2semidirect
2. #11723501C2semidirect
3. #11723434C2semidirect
5. #11722184C2semidirect
6. C32×#821C2semidirect
7. #11722039C2semidirect
25. C2×#3744D3semidirect
36. C3×#821D3semidirect
108. #821C3⋊D3semidirect
223. C33#463semidirect
319. C32#4787semidirect
320. C32#5165semidirect
364. C3#10590750semidirect
365. C3#10592486semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#37442
#117235012
#117234342
C2×#887892
#117221842
C32×#8212
#117220392
#105924863
#105907503