Finite Group Databasebeta

← #13156596

Subgroups — #13156596

Order 1728.

698 conjugacy classes of subgroups; only the 112 normal subgroups are shown.

normal subgroups only

This group has 698 conjugacy classes of subgroups — too many to draw. Only its 112 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×#88812C211yesmaximal, characteristic
2#11723451C212yesmaximal
3C2×C3×#3719C212yesmaximal
4C2×C3×#3718C211yesmaximal, characteristic
5#11722018C211yesmaximal, characteristic
6#10591628—32nomaximal
7#10590905—31nomaximal
8C2×C3×#634C2212yes—
9C2×C32×Dic6C2212yes—
10C4×C3⋊(C3⋊Dic3)C2211yescharacteristic
11C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
12C2×C32×C3⋊C8C2211yescharacteristic
13#88812—22no—
14#88806—24no—
15C3×#3719—24no—
16C3×#3718—22no—
17#64153—32no—
18C2×C3×#216—32no—
19C2×C3×#216—32no—
20C2×C3×Dic12—32no—
21#63873—32no—
22#63873—32no—
23#63873—62no—
24C2×#3719—32no—
25C2×#3719D312yes—
26C2×#3718—31no—
27C2×#3718D311yescharacteristic
28C8×C3⋊Dic3—31no—
29#63553—32no—
30#63553—61no—
31C2×C33×C4C2311yescharacteristic
32C2×C3⋊(C3⋊Dic3)D412yes—
33C2×C3×C3⋊Dic3—22no—
34C3×#634—22no—
35C3×#634—22no—
36C2×C32×Dic3—22no—
37C32×Dic6—22no—
38C32×Dic6—22no—
39C3×C3⋊(C3⋊C8)D412yes—
40C32×C3⋊C8D412yes—
41#5183—92no—
42#5183—91no—
43C2×C32×Q8—32no—
44C2×#634C2×D312yes—
45C2×C3×Dic6—22no—
46C2×C3×Dic6—32no—
47C2×C3×Dic6—32no—
48C2×C3×Dic6—62no—
49C2×C3×Dic6C2×D312yes—
50C2×C3×Dic6C2×D312yes—
51C2×C32×C8—31no—
52#3750—64no—
53C4×C3⋊Dic3—32no—
54C4×C3⋊Dic3—31no—
55C4×C3⋊Dic3—61no—
56C4×C3⋊Dic3—62no—
57C4×C3⋊Dic3—121no—
58C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
59C3×#216—64no—
60C3×#216—64no—
61C2×C3×C3⋊C8—21no—
62C2×C3×C3⋊C8—32no—
63C2×C3×C3⋊C8—61no—
64C2×C3×C3⋊C8C2×D312yes—
65C3×Dic12—64no—
66#3724—64no—
67#3724—64no—
68#3724—68no—
69#3719—24no—
70#3719—64no—
71#3718—22no—
72#3718—62no—
73C22×C33C2×D411yescharacteristic
74C33×C4C2×D411yescharacteristic
75C33×C4C2×D411yescharacteristic
76C3⋊(C3⋊Dic3)Q1614yescomplemented
77C3×C3⋊Dic3—42no—
78C32×Dic3—42no—
79C2×C3×Q16—92no—
80C2×C3×Q16—91no—
81#944—92no—
82#944—92no—
83#944—92no—
84#944—182no—
85C2×#216—32no—
86C2×#216—32no—
87C2×#216—92no—
88C2×Dic12—32no—
89C2×Dic12—91no—
90C8×Dic3—92no—
91C8×Dic3—91no—
92C8×Dic3—181no—
93C32×Q8—62no—
94C32×Q8—62no—
95C2×C32×C4—21no—
96C2×C32×C4—22no—
97C2×C32×C4—41no—
98C2×C32×C4—62no—
99C2×C32×C4C22×D312yes—
100C2×C32×C4C22×D311yescharacteristic
101C2×C3⋊Dic3—22no—
102C2×C3⋊Dic3—32no—
103C2×C3⋊Dic3—34no—
104C2×C3⋊Dic3—64no—
105C2×C3⋊Dic3—62no—
106C2×C3⋊Dic3—122no—
107#634—22no—
108#634—22no—
109C2×C3×Dic3—24no—
110C2×C3×Dic3—22no—
111C2×C3×Dic3—22no—
112C2×C3×Dic3—64no—
113C2×C3×Dic3—62no—
114C2×C3×Dic3—62no—
115C3×Dic6—22no—
116C3×Dic6—22no—
117C3×Dic6—22no—
118C3×Dic6—22no—
119C3×Dic6—42no—
120C3×Dic6—42no—
121C3×Dic6—62no—
122C3×Dic6—62no—
123C3×Dic6—62no—
124C3×Dic6—62no—
125C3×Dic6—122no—
126C3×Dic6—122no—
127C32×C8—32no—
128C3⋊(C3⋊C8)#6712yes—
129C3×C3⋊C8—22no—
130C3×C3⋊C8—34no—
131C3×C3⋊C8—62no—
132C3×C3⋊C8#6714yes—
133#494—271noSylow
134C2×C33#12711yescharacteristic
135C2×C33C2×Q1612yes—
136C2×C3×Q8—32no—
137C2×C3×Q8—32no—
138C2×C3×Q8—62no—
139C2×C3×Q8—92no—
140C2×Dic6—32no—
141C2×Dic6—32no—
142C2×Dic6—62no—
143C2×Dic6D3212yes—
144C3×Q16—184no—
145C3×Q16—182no—
146C2×C3×C8—32no—
147C2×C3×C8—61no—
148C2×C3×C8—91no—
149#216—64no—
150#216—64no—
151#216—184no—
152Dic3⋊C4—184no—
153Dic3⋊C4—184no—
154Dic3⋊C4—184no—
155Dic3⋊C4—188no—
156C4×Dic3—91no—
157C4×Dic3—92no—
158C4×Dic3—181no—
159C4×Dic3—182no—
160C4×Dic3—361no—
161C2×C3⋊C8—32no—
162C2×C3⋊C8—61no—
163C2×C3⋊C8D3211yescharacteristic
164Dic12—64no—
165Dic12—182no—
166C22×C32—21no—
167C22×C32—22no—
168C22×C32—41no—
169C22×C32D3×D412yes—
170C22×C32D3×D411yescharacteristic
171C32×C4—21no—
172C32×C4—21no—
173C32×C4—22no—
174C32×C4—22no—
175C32×C4—41no—
176C32×C4—41no—
177C32×C4—122no—
178C32×C4D3×D412yes—
179C32×C4D3×D411yescharacteristic
180C32×C4C2×#6712yes—
181C32×C4C2×#6711yescharacteristic
182C3⋊Dic3—38no—
183C3⋊Dic3—34no—
184C3⋊Dic3—42no—
185C3⋊Dic3—64no—
186C3⋊Dic3—68no—
187C3⋊Dic3—124no—
188C3×Dic3—44no—
189C3×Dic3—42no—
190C3×Dic3—42no—
191C3×Dic3—124no—
192C3×Dic3—122no—
193C3×Dic3—122no—
194C2×Q16—92no—
195C2×Q16—91no—
196C2×Q16—271no—
197C4⋊Q8—272no—
198C4×C8—271no—
199C33#49411yescharacteristic, complemented, Sylow
200C3×Q8—62no—
201C3×Q8—62no—
202C3×Q8—62no—
203C3×Q8—62no—
204C3×Q8—122no—
205C3×Q8—122no—
206C3×Q8—182no—
207C3×Q8—182no—
208C2×C3×C4—21no—
209C2×C3×C4—22no—
210C2×C3×C4—41no—
211C2×C3×C4—64no—
212C2×C3×C4—62no—
213C2×C3×C4—62no—
214C2×C3×C4—182no—
215C2×C3×C4C2×D3211yescharacteristic
216C2×C3×C4C2×D3212yes—
217C2×Dic3—22no—
218C2×Dic3—64no—
219C2×Dic3—62no—
220C2×Dic3—62no—
221C2×Dic3—92no—
222C2×Dic3—94no—
223C2×Dic3—182no—
224C2×Dic3—184no—
225C2×Dic3—362no—
226Dic6—22no—
227Dic6—22no—
228Dic6—62no—
229Dic6—62no—
230Dic6—62no—
231Dic6—62no—
232Dic6—122no—
233Dic6—122no—
234C3×C8—34no—
235C3×C8—62no—
236C3×C8—92no—
237C3⋊C8—34no—
238C3⋊C8—62no—
239C3⋊C8#62512yes—
240C2×C32—22no—
241C2×C32—21no—
242C2×C32—24no—
243C2×C32—22no—
244C2×C32—41no—
245C2×C32—42no—
246C2×C32D3×Q1614yes—
247C2×C32D3×Q1612yes—
248C2×C32#94012yes—
249C2×C32#94011yescharacteristic
250C2×Q8—32no—
251C2×Q8—92no—
252Q16—184no—
253Q16—182no—
254Q16—542no—
255C2×C8—31no—
256C2×C8—91no—
257C4⋊C4—544no—
258C42—271no—
259C22×C3—22no—
260C22×C3—21no—
261C22×C3—41no—
262C22×C3#380211yescharacteristic
263C22×C3#381112yes—
264C3×C4—21no—
265C3×C4—22no—
266C3×C4—21no—
267C3×C4—22no—
268C3×C4—41no—
269C3×C4—41no—
270C3×C4—124no—
271C3×C4—122no—
272C3×C4—122no—
273C3×C4—362no—
274C3×C4#380211yescharacteristic
275C3×C4C2×#62511yescharacteristic
276C3×C4D3×#6712yes—
277C3×C4D3×#6712yes—
278Dic3—42no—
279Dic3—94no—
280Dic3—98no—
281Dic3—124no—
282Dic3—122no—
283Dic3—122no—
284Dic3—184no—
285Dic3—188no—
286Dic3—364no—
287C32—22no—
288C32—21no—
289C32—41no—
290C32#518311yescharacteristic, complemented
291C32#518312yescomplemented
292Q8—62no—
293Q8—62no—
294Q8—182no—
295Q8—182no—
296C2×C4—62no—
297C2×C4—182no—
298C2×C4—272no—
299C2×C4D3311yescharacteristic
300C8—32no—
301C8—92no—
302C2×C3—22no—
303C2×C3—21no—
304C2×C3—24no—
305C2×C3—22no—
306C2×C3—41no—
307C2×C3—42no—
308C2×C3#6380212yes—
309C2×C3#6391411yescharacteristic
310C2×C3#6394414yes—
311C2×C3#6397912yes—
312C22#8895411yescharacteristic
313C4—122no—
314C4—274no—
315C4—362no—
316C4D3×#62511yescharacteristic
317C4#8895311yescharacteristic
318C3—22no—
319C3—21no—
320C3—41no—
321C3#1059090511yescharacteristic, complemented
322C3#1059162812yescomplemented
323C2#1172284812yes—
324C2#1172302911yescharacteristic
subgroupG/H
centerC22#88954
commutator, Fitting, cosocleC2×C33×C4C23
Frattini, hypercenterC2×C4D33
socle, prime radicalC22×C33C2×D4
hypocenterC33#494
LCSG ▷ C2×C33×C4 ▷ C2×C33 ▷ C33
UCSC1 ◁ C22 ◁ C2×C4

Complements

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

NH ≅ G/Ntype
76. C3⋊(C3⋊Dic3)Q16semidirect
199. C33#494semidirect
290. C32#5183semidirect
291. C32#5183semidirect
321. C3#10590905semidirect
322. C3#10591628semidirect

Maximal subgroups

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

subgroupindex
C2×#888122
#117234512
C2×C3×#37192
C2×C3×#37182
#117220182
#105916283
#105909053