Finite Group Databasebeta

← #13156590

Subgroups — #13156590

Order 1728.

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

normal subgroups only

This group has 938 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×#88810C211yesmaximal, characteristic, complemented
2#11723529C212yesmaximal, complemented
3C2×C3×#3714C212yesmaximal, complemented
4C2×C3×#3713C211yesmaximal, characteristic, complemented
5#11722018C211yesmaximal, characteristic, complemented
6#10591484—32nomaximal
7#10590886—31nomaximal
8C2×C3×C4⋊(C3⋊D3)C2212yes—
9C2×C32×D12C2212yes—
10C2×#6290—24no—
11C4×C3⋊(C3⋊Dic3)C2211yescharacteristic
12C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
13C2×C32×C3⋊C8C2211yescharacteristic
14#88810—22no—
15C3×#3714—24no—
16C3×#3713—22no—
17#64149—32no—
18C2×C3×#213—32no—
19C2×C3×#213—32no—
20C2×C3×D24—32no—
21#63914—32no—
22#63914—32no—
23#63914—62no—
24C2×#3714—32no—
25C2×#3714D312yescomplemented
26C2×#3713—31no—
27C2×#3713D311yescharacteristic, complemented
28C8×C3⋊Dic3—31no—
29#63553—32no—
30#63553—61no—
31C22×C3×C3⋊D3—22no—
32C22×C32×D3—22no—
33C2×C33×C4C2311yescharacteristic
34C2×C3⋊(C3⋊Dic3)D412yes—
35C3×C4⋊(C3⋊D3)—22no—
36C3×C4⋊(C3⋊D3)—22no—
37C32×D12—22no—
38C32×D12—22no—
39#6290—48no—
40C3×C3⋊(C3⋊C8)D412yescomplemented
41C32×C3⋊C8D412yescomplemented
42#5151—92no—
43#5151—91no—
44C2×C32×D4—32no—
45C2×#638—64no—
46C2×C4⋊(C3⋊D3)C2×D312yes—
47C2×C3×D12—22no—
48C2×C3×D12—32no—
49C2×C3×D12—32no—
50C2×C3×D12—62no—
51C2×C3×D12C2×D312yes—
52C2×C3×D12C2×D312yes—
53C2×#625—64no—
54C2×#625—64no—
55C2×#625—68no—
56C2×C32×C8—31no—
57C4×C3⋊Dic3—32no—
58C4×C3⋊Dic3—31no—
59C4×C3⋊Dic3—61no—
60C4×C3⋊Dic3—62no—
61C4×C3⋊Dic3—121no—
62C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
63C3×#213—64no—
64C3×#213—64no—
65C2×C3×C3⋊C8—21no—
66C2×C3×C3⋊C8—32no—
67C2×C3×C3⋊C8—61no—
68C2×C3×C3⋊C8C2×D312yes—
69C3×D24—64no—
70#3714—24no—
71#3714—64no—
72#3713—22no—
73#3713—62no—
74C22×C33C2×D411yescharacteristic
75C2×C3×C3⋊D3—42no—
76C2×C3×C3⋊D3—44no—
77C2×C32×D3—42no—
78C2×C32×D3—44no—
79C33×C4C2×D411yescharacteristic
80C33×C4C2×D411yescharacteristic
81C3⋊(C3⋊Dic3)D814yescomplemented
82C2×C3×D8—92no—
83C2×C3×D8—91no—
84#940—92no—
85#940—92no—
86#940—92no—
87#940—182no—
88C2×#213—32no—
89C2×#213—32no—
90C2×#213—92no—
91C2×D24—32no—
92C2×D24—91no—
93C8×Dic3—92no—
94C8×Dic3—91no—
95C8×Dic3—181no—
96C23×C32—62no—
97C22×C3⋊D3—22no—
98C22×C3×D3—24no—
99C22×C3×D3—22no—
100C22×C3×D3—22no—
101C22×C3×D3—64no—
102C22×C3×D3—62no—
103C22×C3×D3—62no—
104C32×D4—62no—
105C32×D4—62no—
106C2×C32×C4—21no—
107C2×C32×C4—22no—
108C2×C32×C4—41no—
109C2×C32×C4C22×D312yes—
110C2×C32×C4C22×D311yescharacteristic
111#638—128no—
112C2×C3⋊Dic3—32no—
113C2×C3⋊Dic3—34no—
114C2×C3⋊Dic3—62no—
115C2×C3⋊Dic3—64no—
116C2×C3⋊Dic3—122no—
117C4⋊(C3⋊D3)—22no—
118C4⋊(C3⋊D3)—22no—
119C3×D12—22no—
120C3×D12—22no—
121C3×D12—22no—
122C3×D12—22no—
123C3×D12—42no—
124C3×D12—42no—
125C3×D12—62no—
126C3×D12—62no—
127C3×D12—62no—
128C3×D12—62no—
129C3×D12—122no—
130C3×D12—122no—
131#625—128no—
132#625—128no—
133#625—1216no—
134C32×C8—32no—
135C3⋊(C3⋊C8)#6712yescomplemented
136C3×C3⋊C8—22no—
137C3×C3⋊C8—34no—
138C3×C3⋊C8—62no—
139C3×C3⋊C8#6714yescomplemented
140#493—271noSylow
141C2×C33#12711yescharacteristic
142C2×C33C2×D812yes—
143C3×C3⋊D3—82no—
144C32×D3—82no—
145C2×C3×D4—32no—
146C2×C3×D4—32no—
147C2×C3×D4—62no—
148C2×C3×D4—92no—
149C2×#67—184no—
150C2×#67—184no—
151C2×#67—184no—
152C2×#67—188no—
153C2×D12—32no—
154C2×D12—32no—
155C2×D12—62no—
156C2×D12D3212yes—
157C3×D8—184no—
158C3×D8—182no—
159C2×C3×C8—32no—
160C2×C3×C8—61no—
161C2×C3×C8—91no—
162#213—64no—
163#213—64no—
164#213—184no—
165C4×Dic3—91no—
166C4×Dic3—92no—
167C4×Dic3—181no—
168C4×Dic3—182no—
169C4×Dic3—361no—
170C2×C3⋊C8—32no—
171C2×C3⋊C8—61no—
172C2×C3⋊C8D3211yescharacteristic
173D24—64no—
174D24—182no—
175C22×C32—21no—
176C22×C32—22no—
177C22×C32—41no—
178C22×C32—124no—
179C22×C32—122no—
180C22×C32D3×D412yes—
181C22×C32D3×D411yescharacteristic
182C2×C3⋊D3—44no—
183C2×C3⋊D3—42no—
184C2×C3×D3—48no—
185C2×C3×D3—44no—
186C2×C3×D3—44no—
187C2×C3×D3—44no—
188C2×C3×D3—42no—
189C2×C3×D3—42no—
190C2×C3×D3—128no—
191C2×C3×D3—124no—
192C2×C3×D3—124no—
193C2×C3×D3—124no—
194C2×C3×D3—122no—
195C2×C3×D3—122no—
196C32×C4—21no—
197C32×C4—21no—
198C32×C4—22no—
199C32×C4—22no—
200C32×C4—41no—
201C32×C4—41no—
202C32×C4D3×D412yes—
203C32×C4D3×D411yescharacteristic
204C32×C4C2×#6712yes—
205C32×C4C2×#6711yescharacteristic
206C3⋊Dic3—34no—
207C3⋊Dic3—38no—
208C3⋊Dic3—68no—
209C3⋊Dic3—64no—
210C3⋊Dic3—124no—
211C2×D8—92no—
212C2×D8—91no—
213C2×D8—271no—
214#127—272no—
215C4×C8—271no—
216C33#49311yescharacteristic, complemented, Sylow
217C23×C3—64no—
218C23×C3—62no—
219C23×C3—62no—
220C23×C3—182no—
221C22×D3—22no—
222C22×D3—64no—
223C22×D3—62no—
224C22×D3—62no—
225C3×D4—62no—
226C3×D4—62no—
227C3×D4—62no—
228C3×D4—62no—
229C3×D4—122no—
230C3×D4—122no—
231C3×D4—182no—
232C3×D4—182no—
233C2×C3×C4—21no—
234C2×C3×C4—22no—
235C2×C3×C4—41no—
236C2×C3×C4C2×D3211yescharacteristic
237C2×C3×C4C2×D3212yes—
238#67—368no—
239#67—368no—
240#67—368no—
241#67—3616no—
242C2×Dic3—92no—
243C2×Dic3—94no—
244C2×Dic3—182no—
245C2×Dic3—184no—
246C2×Dic3—362no—
247D12—22no—
248D12—22no—
249D12—62no—
250D12—62no—
251D12—62no—
252D12—62no—
253D12—122no—
254D12—122no—
255C3×C8—34no—
256C3×C8—62no—
257C3×C8—92no—
258C3⋊C8—34no—
259C3⋊C8—62no—
260C3⋊C8#62512yescomplemented
261C2×C32—22no—
262C2×C32—21no—
263C2×C32—24no—
264C2×C32—22no—
265C2×C32—41no—
266C2×C32—42no—
267C2×C32—242no—
268C2×C32D3×D814yes—
269C2×C32D3×D812yes—
270C2×C32#94012yes—
271C2×C32#94011yescharacteristic
272C3⋊D3—82no—
273C3×D3—84no—
274C3×D3—82no—
275C3×D3—82no—
276C3×D3—244no—
277C3×D3—242no—
278C3×D3—242no—
279C2×D4—32no—
280C2×D4—92no—
281C2×D4—544no—
282D8—184no—
283D8—182no—
284D8—542no—
285C2×C8—31no—
286C2×C8—91no—
287C42—271no—
288C22×C3—22no—
289C22×C3—21no—
290C22×C3—41no—
291C22×C3—128no—
292C22×C3—124no—
293C22×C3—124no—
294C22×C3—122no—
295C22×C3—124no—
296C22×C3—122no—
297C22×C3—364no—
298C22×C3—362no—
299C22×C3#380211yescharacteristic
300C22×C3#381112yes—
301C2×D3—44no—
302C2×D3—42no—
303C2×D3—124no—
304C2×D3—128no—
305C2×D3—124no—
306C2×D3—122no—
307C2×D3—124no—
308C2×D3—122no—
309C3×C4—21no—
310C3×C4—22no—
311C3×C4—21no—
312C3×C4—22no—
313C3×C4—41no—
314C3×C4—41no—
315C3×C4#380211yescharacteristic
316C3×C4C2×#62511yescharacteristic
317C3×C4D3×#6712yes—
318C3×C4D3×#6712yes—
319Dic3—94no—
320Dic3—98no—
321Dic3—188no—
322Dic3—184no—
323Dic3—364no—
324C32—21no—
325C32—22no—
326C32—41no—
327C32#515112yescomplemented
328C32#515111yescharacteristic, complemented
329C23—62no—
330C23—182no—
331D4—62no—
332D4—62no—
333D4—182no—
334D4—182no—
335D4—1088no—
336C2×C4—272no—
337C2×C4D3311yescharacteristic
338C8—32no—
339C8—92no—
340C2×C3—22no—
341C2×C3—21no—
342C2×C3—24no—
343C2×C3—22no—
344C2×C3—41no—
345C2×C3—42no—
346C2×C3—244no—
347C2×C3—242no—
348C2×C3—242no—
349C2×C3—722no—
350C2×C3#6379712yes—
351C2×C3#6391411yescharacteristic
352C2×C3#6392614yes—
353C2×C3#6397912yes—
354D3—82no—
355D3—244no—
356D3—242no—
357D3—242no—
358C22—122no—
359C22—124no—
360C22—362no—
361C22—364no—
362C22#8895411yescharacteristic
363C4—274no—
364C4D3×#62511yescharacteristic
365C4#8895311yescharacteristic
366C3—22no—
367C3—21no—
368C3—41no—
369C3#1059088611yescharacteristic, complemented
370C3#1059148412yescomplemented
371C2—242no—
372C2—722no—
373C2#1172283212yes—
374C2#1172302911yescharacteristic
subgroupG/H
centerC22#88954
commutator, Fitting, cosocleC2×C33×C4C23
Frattini, hypercenterC2×C4D33
socleC22×C33C2×D4
hypocenterC33#493
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
1. C2×#88810C2semidirect
2. #11723529C2semidirect
3. C2×C3×#3714C2semidirect
4. C2×C3×#3713C2semidirect
5. #11722018C2semidirect
25. C2×#3714D3semidirect
27. C2×#3713D3semidirect
40. C3×C3⋊(C3⋊C8)D4semidirect
41. C32×C3⋊C8D4semidirect
81. C3⋊(C3⋊Dic3)D8semidirect
135. C3⋊(C3⋊C8)#67semidirect
139. C3×C3⋊C8#67semidirect
216. C33#493semidirect
260. C3⋊C8#625semidirect
327. C32#5151semidirect
328. C32#5151semidirect
369. C3#10590886semidirect
370. C3#10591484semidirect

Maximal subgroups

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

subgroupindex
C2×#888102
#117235292
C2×C3×#37142
C2×C3×#37132
#117220182
#105914843
#105908863