Finite Group Databasebeta

← #13161310

Subgroups — #13161310

Order 1728.

920 conjugacy classes of subgroups; only the 148 normal subgroups are shown.

normal subgroups only

This group has 920 conjugacy classes of subgroups — too many to draw. Only its 148 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#11723985C211yesmaximal, characteristic
2C2×C3×#3756C211yesmaximal, characteristic
3#11723410C211yesmaximal, characteristic
4C2×#88800C211yesmaximal, characteristic
5C32×#835C211yesmaximal, characteristic
6#11722050C211yesmaximal, characteristic
7#11722041C211yesmaximal, characteristic
8#10592494—31nomaximal
9#10591580—34nomaximal
10C2×C33×Q8C2211yescharacteristic
11C2×#6308C2211yescharacteristic
12C2×C3×#634C2211yescharacteristic
13#88857—21no—
14#88857—21no—
15C4×C3⋊(C3⋊Dic3)C2211yescharacteristic
16C3×#3756—22no—
17C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
18C32×C4⋊Dic3C2211yescharacteristic
19C2×C32×C3⋊C8C2211yescharacteristic
20#88807—21no—
21Dic3×C3⋊Dic3—21no—
22#88800—22no—
23#64153—31no—
24#64153—34no—
25#64153—64no—
26C2×#3756D311yescharacteristic
27C2×#3746—31no—
28#64096—31no—
29C2×C3×#216—34no—
30#63843—34no—
31C2×#3719—34no—
32C32×#103—31no—
33#63641—31no—
34#63637—31no—
35C3×#835D314yes—
36#63572—34no—
37#63567—34no—
38C33×Q8—21no—
39C33×Q8—21no—
40C2×C33×C4—21no—
41C2×C33×C4C2311yescharacteristic
42C2×C3⋊(C3⋊Dic3)—21no—
43C2×C3⋊(C3⋊Dic3)D411yescharacteristic
44C2×C3⋊(C3⋊Dic3)D411yescharacteristic
45#6308—21no—
46#6308—21no—
47C2×C3×C3⋊Dic3—21no—
48C3×#634—21no—
49C3×#634D412yes—
50C2×C32×Dic3—21no—
51C3×C3⋊(C3⋊C8)—21no—
52C32×C3⋊C8—21no—
53#5182—91no—
54#4795—94no—
55C2×C32×Q8—24no—
56C2×C32×Q8C2×D314yes—
57C2×C32×Q8C2×D311yescharacteristic
58C2×#634—31no—
59C2×#634—34no—
60C2×#634—64no—
61C2×#634C2×D311yescharacteristic
62C2×C3×Dic6—34no—
63C2×C32×C8—31no—
64C32×C4⋊C4—31no—
65#3756—22no—
66#3750—61no—
67#3750—61no—
68#3750—61no—
69#3750—64no—
70#3750—64no—
71#3750—68no—
72#3750—68no—
73C4×C3⋊Dic3—31no—
74C4×C3⋊Dic3—34no—
75C4×C3⋊Dic3—61no—
76C4×C3⋊Dic3—64no—
77C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
78#3746—62no—
79C3×#216—68no—
80C3×C4⋊Dic3C2×D314yes—
81C2×C3×C3⋊C8—34no—
82C2×C3×C3⋊C8C2×D314yes—
83Dic3⋊Dic3—64no—
84Dic32—64no—
85#3719—68no—
86C22×C33C2×D411yescharacteristic
87C33×C4—41no—
88C33×C4C2×D411yescharacteristic
89C33×C4#4111yescharacteristic
90C3⋊(C3⋊Dic3)—21no—
91C3⋊(C3⋊Dic3)—41no—
92C3⋊(C3⋊Dic3)Q1612yescomplemented
93C3×C3⋊Dic3—22no—
94C32×Dic3—41no—
95C2×C3×Q16—91no—
96Q8×Dic3—91no—
97#944—94no—
98#944—91no—
99#944—184no—
100C2×#216—34no—
101C2×#216—91no—
102C2×Dic12—94no—
103#887—94no—
104C3×#103—34no—
105#835C3⋊D311yescharacteristic
106#816—91no—
107Dic3⋊C8—94no—
108Dic3⋊C8—91no—
109#810—94no—
110C32×Q8—21no—
111C32×Q8—21no—
112C32×Q8—24no—
113C32×Q8—24no—
114C32×Q8—44no—
115C32×Q8—44no—
116C2×C32×C4—28no—
117C2×C32×C4—24no—
118C2×C32×C4—21no—
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—
130C2×C3⋊Dic3—68no—
131C2×C3⋊Dic3—64no—
132C2×C3⋊Dic3—61no—
133#634—21no—
134#634—61no—
135#634—61no—
136#634—64no—
137#634—64no—
138#634—124no—
139#634—124no—
140#634#6712yes—
141C2×C3×Dic3—24no—
142C2×C3×Dic3—64no—
143C3×Dic6—38no—
144C3×Dic6—64no—
145C32×C8—61no—
146C3⋊(C3⋊C8)—21no—
147C3×C3⋊C8—24no—
148C3×C3⋊C8—64no—
149#462—271noSylow
150C2×C33#12111yescharacteristic
151C2×C33C2×Q1611yescharacteristic
152C2×C33#13711yescharacteristic
153C2×C3×Q8—24no—
154C2×C3×Q8—91no—
155C2×C3×Q8D3214yes—
156C2×C3×Q8C2×C3⋊D311yescharacteristic
157C2×Dic6—34no—
158C2×Dic6—94no—
159C2×Dic6—91no—
160C2×Dic6—184no—
161C3×Q16—182no—
162C2×C3×C8—34no—
163C2×C3×C8—91no—
164C3×C4⋊C4—34no—
165#216—68no—
166#216—182no—
167C4⋊Dic3—181no—
168C4⋊Dic3C2×C3⋊D311yescharacteristic
169Dic3⋊C4—184no—
170Dic3⋊C4—184no—
171Dic3⋊C4—181no—
172Dic3⋊C4—184no—
173Dic3⋊C4—181no—
174Dic3⋊C4—188no—
175Dic3⋊C4—188no—
176C4×Dic3—94no—
177C4×Dic3—91no—
178C4×Dic3—184no—
179C4×Dic3—181no—
180C4×Dic3—184no—
181C2×C3⋊C8—34no—
182C2×C3⋊C8C2×C3⋊D311yescharacteristic
183Dic12—188no—
184C22×C32—24no—
185C22×C32D3×D414yes—
186C22×C32D3×D411yescharacteristic
187C32×C4—24no—
188C32×C4—24no—
189C32×C4—48no—
190C32×C4—44no—
191C32×C4—41no—
192C32×C4—121no—
193C32×C4#23514yes—
194C32×C4D3×D414yes—
195C32×C4#23711yescharacteristic
196C32×C4C2×#6711yescharacteristic
197C3⋊Dic3—22no—
198C3⋊Dic3—32no—
199C3⋊Dic3—38no—
200C3⋊Dic3—68no—
201C3⋊Dic3—68no—
202C3⋊Dic3—64no—
203C3⋊Dic3—61no—
204C3⋊Dic3—128no—
205C3⋊Dic3—124no—
206C3⋊Dic3—121no—
207C3×Dic3—44no—
208C3×Dic3—68no—
209C2×Q16—91no—
210C2×Q16—271no—
211C4⋊Q8—271no—
212C4×Q8—271no—
213C4⋊C8—271no—
214#103—31no—
215#103—271no—
216C33#46211yescharacteristic, complemented, Sylow
217C3×Q8—24no—
218C3×Q8—24no—
219C3×Q8—21no—
220C3×Q8—21no—
221C3×Q8—44no—
222C3×Q8—44no—
223C3×Q8—92no—
224C3×Q8—181no—
225C2×C3×C4—28no—
226C2×C3×C4—24no—
227C2×C3×C4—21no—
228C2×C3×C4—24no—
229C2×C3×C4—64no—
230C2×C3×C4—181no—
231C2×C3×C4C2×D3214yes—
232C2×C3×C4C22×C3⋊D311yescharacteristic
233C2×Dic3—21no—
234C2×Dic3—64no—
235C2×Dic3—94no—
236C2×Dic3—91no—
237C2×Dic3—94no—
238C2×Dic3—91no—
239C2×Dic3—184no—
240C2×Dic3—184no—
241C2×Dic3—188no—
242C2×Dic3—184no—
243C2×Dic3—181no—
244Dic6—38no—
245Dic6—64no—
246Dic6—184no—
247Dic6—184no—
248Dic6—181no—
249Dic6—181no—
250Dic6—364no—
251Dic6—364no—
252C3×C8—64no—
253C3×C8—181no—
254C3⋊C8—21no—
255C3⋊C8—64no—
256C2×C32—24no—
257C2×C32—24no—
258C2×C32—24no—
259C2×C32#88414yes—
260C2×C32#91514yes—
261C2×C32D3×Q1614yes—
262C2×C32D3×Q1611yescharacteristic
263C2×C32#91811yescharacteristic
264C2×C32#93911yescharacteristic
265C2×Q8—91no—
266C2×Q8—271no—
267C2×Q8D3×C3⋊D311yescharacteristic
268Q16—182no—
269Q16—542no—
270C2×C8—31no—
271C2×C8—91no—
272C4⋊C4—31no—
273C4⋊C4—541no—
274C4⋊C4—541no—
275C4⋊C4—541no—
276C42—271no—
277C42—541no—
278C22×C3—24no—
279C22×C3#381114yes—
280C22×C3D4×C3⋊D311yescharacteristic
281C3×C4—24no—
282C3×C4—24no—
283C3×C4—48no—
284C3×C4—41no—
285C3×C4—44no—
286C3×C4—124no—
287C3×C4—182no—
288C3×C4#380414yes—
289C3×C4D3×#6714yes—
290C3×C4#382811yescharacteristic
291C3×C4D4×C3⋊D311yescharacteristic
292Dic3—41no—
293Dic3—68no—
294Dic3—92no—
295Dic3—98no—
296Dic3—188no—
297Dic3—188no—
298Dic3—181no—
299Dic3—184no—
300Dic3—368no—
301Dic3—361no—
302Dic3—364no—
303C32—24no—
304C32#479514yescomplemented
305C32#518211yescharacteristic, complemented
306Q8—21no—
307Q8—21no—
308Q8—92no—
309Q8—181no—
310Q8—541no—
311Q8—541no—
312C2×C4—21no—
313C2×C4—61no—
314C2×C4—181no—
315C2×C4—271no—
316C2×C4—271no—
317C2×C4—541no—
318C2×C4C2×D3×C3⋊D311yescharacteristic
319C8—61no—
320C8—181no—
321C2×C3—24no—
322C2×C3—24no—
323C2×C3—24no—
324C2×C3#6394414yes—
325C2×C3#6395114yes—
326C2×C3#6397114yes—
327C2×C3#6409311yescharacteristic
328C2×C3#6412411yescharacteristic
329C2×C3Q16×C3⋊D311yescharacteristic
330C22#8903911yescharacteristic
331C4—41no—
332C4—121no—
333C4—182no—
334C4—272no—
335C4—541no—
336C4—1081no—
337C4#8903111yescharacteristic
338C4C3⋊D3×#6711yescharacteristic
339C3—24no—
340C3#1059158014yescomplemented
341C3#1059249411yescharacteristic, complemented
342C2#1172357711yescharacteristic
343C2#1172358811yescharacteristic
344C2#1172362411yescharacteristic
subgroupG/H
centerC22#89039
commutator, cosocleC2×C33×C4C23
FrattiniC2×C4C2×D3×C3⋊D3
FittingC2×C33×Q8C22
socle, prime radicalC22×C33C2×D4
hypercenterC2×Q8D3×C3⋊D3
hypocenterC33#462
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
92. C3⋊(C3⋊Dic3)Q16semidirect
216. C33#462semidirect
304. C32#4795semidirect
305. C32#5182semidirect
340. C3#10591580semidirect
341. C3#10592494semidirect

Maximal subgroups

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

subgroupindex
#117239852
C2×C3×#37562
#117234102
C2×#888002
C32×#8352
#117220502
#117220412
#105924943
#105915803