Finite Group Databasebeta

← #13156607

Subgroups — #13156607

Order 1728.

746 conjugacy classes of subgroups; only the 108 normal subgroups are shown.

normal subgroups only

This group has 746 conjugacy classes of subgroups — too many to draw. Only its 108 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
2#11723436C212yesmaximal, complemented
3C2×C3×#3718C211yesmaximal, characteristic, complemented
4C2×C3×#3717C212yesmaximal, complemented
5#11722018C211yesmaximal, characteristic, complemented
6#10591610—32nomaximal
7#10590867—31nomaximal
8C2×C3×C4⋊(C3⋊D3)C2212yes—
9C2×C32×Dic6C2212yes—
10C4×C3⋊(C3⋊Dic3)C2211yescharacteristic
11C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
12C2×C32×C3⋊C8C2211yescharacteristic
13#88810—22no—
14#88803—24no—
15C3×#3718—22no—
16C3×#3717—24no—
17#64156—32no—
18C2×C3×#216—32no—
19C2×C3×#215—32no—
20C2×C3×#204—32no—
21#63863—32no—
22#63863—32no—
23#63863—62no—
24C2×#3718D311yescharacteristic, complemented
25C2×#3717D312yescomplemented
26C2×#3714—32no—
27C2×#3713—31no—
28C8×C3⋊Dic3—31no—
29#63553—32no—
30#63553—61no—
31C22×C3×C3⋊D3—22no—
32C2×C33×C4C2311yescharacteristic
33C2×C3⋊(C3⋊Dic3)D412yes—
34C3×C4⋊(C3⋊D3)—22no—
35C3×C4⋊(C3⋊D3)—22no—
36C2×C32×Dic3—22no—
37C32×Dic6—22no—
38C32×Dic6—22no—
39C3×C3⋊(C3⋊C8)D412yes—
40C32×C3⋊C8D412yes—
41#5191—91no—
42#5168—92no—
43C2×C32×Q8—32no—
44C2×C4⋊(C3⋊D3)C2×D312yes—
45C2×C3×D12—32no—
46C2×C3×D12—32no—
47C2×C3×D12—62no—
48C2×C3×Dic6—22no—
49C2×C3×Dic6C2×D312yes—
50C2×C3×Dic6C2×D312yes—
51C2×C32×C8—31no—
52#3752—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×#215—64no—
61C2×C3×C3⋊C8—21no—
62C2×C3×C3⋊C8—32no—
63C2×C3×C3⋊C8—61no—
64C2×C3×C3⋊C8C2×D312yes—
65C3×#204—64no—
66(C2×D3)⋊Dic3—64no—
67(C2×D3)⋊Dic3—64no—
68(C2×D3)⋊Dic3—68no—
69#3718—22no—
70#3717—24no—
71#3714—64no—
72#3713—62no—
73C22×C33C2×D411yescharacteristic
74C2×C3×C3⋊D3—42no—
75C2×C3×C3⋊D3—44no—
76C33×C4C2×D411yescharacteristic
77C33×C4C2×D411yescharacteristic
78C3⋊(C3⋊Dic3)—22no—
79C32×Dic3—42no—
80C2×C3×Q16—91no—
81C2×C3×SD16—92no—
82#947—92no—
83#947—92no—
84#947—182no—
85C2×#216—32no—
86C2×#215—32no—
87#936—92no—
88C2×#213—92no—
89C2×D24—91no—
90C2×#204—32no—
91C8×Dic3—92no—
92C8×Dic3—91no—
93C8×Dic3—181no—
94C22×C3⋊D3—22no—
95C22×C3×D3—64no—
96C22×C3×D3—62no—
97C22×C3×D3—62no—
98C32×Q8—62no—
99C32×Q8—62no—
100C2×C32×C4—21no—
101C2×C32×C4—22no—
102C2×C32×C4—41no—
103C2×C32×C4—62no—
104C2×C32×C4C22×D312yes—
105C2×C32×C4C22×D311yescharacteristic
106C2×C3⋊Dic3—32no—
107C2×C3⋊Dic3—34no—
108C2×C3⋊Dic3—62no—
109C2×C3⋊Dic3—64no—
110C2×C3⋊Dic3—122no—
111C4⋊(C3⋊D3)—22no—
112C4⋊(C3⋊D3)—22no—
113C2×C3×Dic3—24no—
114C2×C3×Dic3—22no—
115C2×C3×Dic3—22no—
116C3×D12—62no—
117C3×D12—62no—
118C3×D12—62no—
119C3×D12—62no—
120C3×D12—122no—
121C3×D12—122no—
122C3×Dic6—22no—
123C3×Dic6—22no—
124C3×Dic6—22no—
125C3×Dic6—22no—
126C3×Dic6—42no—
127C3×Dic6—42no—
128C32×C8—32no—
129C3⋊(C3⋊C8)#6712yes—
130C3×C3⋊C8—22no—
131C3×C3⋊C8—34no—
132C3×C3⋊C8—62no—
133C3×C3⋊C8#6714yes—
134#495—271noSylow
135C2×C33#12711yescharacteristic
136C2×C33#13512yes—
137C3×C3⋊D3—82no—
138C2×C3×Q8—32no—
139C2×C3×Q8—32no—
140C2×C3×Q8—62no—
141C2×C3×D4—92no—
142C2×D12—32no—
143C2×D12—32no—
144C2×D12—62no—
145C2×Dic6D3212yes—
146C3×Q16—182no—
147C3×SD16—184no—
148C2×C3×C8—32no—
149C2×C3×C8—61no—
150C2×C3×C8—91no—
151#217—184no—
152#216—64no—
153#215—64no—
154#213—184no—
155(C2×D3)⋊C4—184no—
156(C2×D3)⋊C4—184no—
157(C2×D3)⋊C4—188no—
158C4×Dic3—92no—
159C4×Dic3—91no—
160C4×Dic3—181no—
161C4×Dic3—182no—
162C4×Dic3—361no—
163C2×C3⋊C8—32no—
164C2×C3⋊C8—61no—
165C2×C3⋊C8D3211yescharacteristic
166D24—182no—
167#204—64no—
168C22×C32—21no—
169C22×C32—22no—
170C22×C32—41no—
171C22×C32D3×D412yes—
172C22×C32D3×D411yescharacteristic
173C2×C3⋊D3—44no—
174C2×C3⋊D3—42no—
175C2×C3×D3—124no—
176C2×C3×D3—128no—
177C2×C3×D3—124no—
178C2×C3×D3—124no—
179C2×C3×D3—122no—
180C2×C3×D3—122no—
181C32×C4—21no—
182C32×C4—21no—
183C32×C4—22no—
184C32×C4—22no—
185C32×C4—41no—
186C32×C4—41no—
187C32×C4—122no—
188C32×C4D3×D412yes—
189C32×C4D3×D411yescharacteristic
190C32×C4C2×#6712yes—
191C32×C4C2×#6711yescharacteristic
192C3⋊Dic3—62no—
193C3⋊Dic3—64no—
194C3⋊Dic3—64no—
195C3⋊Dic3—124no—
196C3⋊Dic3—124no—
197C3×Dic3—44no—
198C3×Dic3—42no—
199C3×Dic3—42no—
200C2×Q16—91no—
201C2×SD16—92no—
202C2×D8—271no—
203#124—272no—
204C4×C8—271no—
205C33#49511yescharacteristic, complemented, Sylow
206C23×C3—182no—
207C22×D3—64no—
208C22×D3—62no—
209C22×D3—62no—
210C3×Q8—62no—
211C3×Q8—62no—
212C3×Q8—62no—
213C3×Q8—62no—
214C3×Q8—122no—
215C3×Q8—122no—
216C3×D4—182no—
217C3×D4—182no—
218C2×C3×C4—21no—
219C2×C3×C4—22no—
220C2×C3×C4—41no—
221C2×C3×C4—64no—
222C2×C3×C4—62no—
223C2×C3×C4—62no—
224C2×C3×C4C2×D3211yescharacteristic
225C2×C3×C4C2×D3212yes—
226C2×Dic3—22no—
227C2×Dic3—92no—
228C2×Dic3—94no—
229C2×Dic3—182no—
230C2×Dic3—184no—
231C2×Dic3—362no—
232D12—62no—
233D12—62no—
234D12—62no—
235D12—62no—
236D12—122no—
237D12—122no—
238Dic6—22no—
239Dic6—22no—
240C3×C8—34no—
241C3×C8—62no—
242C3×C8—92no—
243C3⋊C8—34no—
244C3⋊C8—62no—
245C3⋊C8#62512yes—
246C2×C32—22no—
247C2×C32—21no—
248C2×C32—24no—
249C2×C32—22no—
250C2×C32—41no—
251C2×C32—42no—
252C2×C32#91614yes—
253C2×C32#91912yes—
254C2×C32#94012yes—
255C2×C32#94011yescharacteristic
256C3⋊D3—82no—
257C3×D3—244no—
258C3×D3—242no—
259C3×D3—242no—
260C2×Q8—32no—
261C2×D4—92no—
262Q16—182no—
263SD16—184no—
264D8—542no—
265C2×C8—31no—
266C2×C8—91no—
267C22⋊C4—544no—
268C42—271no—
269C22×C3—22no—
270C22×C3—21no—
271C22×C3—41no—
272C22×C3—364no—
273C22×C3—362no—
274C22×C3#380211yescharacteristic
275C22×C3#381112yes—
276C2×D3—124no—
277C2×D3—128no—
278C2×D3—124no—
279C2×D3—122no—
280C2×D3—124no—
281C2×D3—122no—
282C3×C4—21no—
283C3×C4—22no—
284C3×C4—21no—
285C3×C4—22no—
286C3×C4—41no—
287C3×C4—41no—
288C3×C4—124no—
289C3×C4—122no—
290C3×C4—122no—
291C3×C4#380211yescharacteristic
292C3×C4C2×#62511yescharacteristic
293C3×C4D3×#6712yes—
294C3×C4D3×#6712yes—
295Dic3—42no—
296Dic3—184no—
297Dic3—182no—
298Dic3—184no—
299Dic3—364no—
300Dic3—364no—
301C32—22no—
302C32—21no—
303C32—41no—
304C32#516812yescomplemented
305C32#519111yescharacteristic, complemented
306C23—182no—
307Q8—62no—
308Q8—62no—
309D4—182no—
310D4—182no—
311C2×C4—62no—
312C2×C4—272no—
313C2×C4D3311yescharacteristic
314C8—32no—
315C8—92no—
316C2×C3—22no—
317C2×C3—21no—
318C2×C3—22no—
319C2×C3—24no—
320C2×C3—41no—
321C2×C3—42no—
322C2×C3—722no—
323C2×C3#6380912yes—
324C2×C3#6391411yescharacteristic
325C2×C3#6395014yes—
326C2×C3#6397912yes—
327D3—244no—
328D3—242no—
329D3—242no—
330C22—362no—
331C22—364no—
332C22#8895411yescharacteristic
333C4—122no—
334C4—542no—
335C4D3×#62511yescharacteristic
336C4#8895311yescharacteristic
337C3—22no—
338C3—21no—
339C3—41no—
340C3#1059086711yescharacteristic, complemented
341C3#1059161012yescomplemented
342C2—722no—
343C2#1172286312yes—
344C2#1172302911yescharacteristic
subgroupG/H
centerC22#88954
commutator, Fitting, cosocleC2×C33×C4C23
Frattini, hypercenterC2×C4D33
socleC22×C33C2×D4
prime radicalC2×#88810C2
hypocenterC33#495
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
2. #11723436C2semidirect
3. C2×C3×#3718C2semidirect
4. C2×C3×#3717C2semidirect
5. #11722018C2semidirect
24. C2×#3718D3semidirect
25. C2×#3717D3semidirect
205. C33#495semidirect
304. C32#5168semidirect
305. C32#5191semidirect
340. C3#10590867semidirect
341. C3#10591610semidirect

Maximal subgroups

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

subgroupindex
C2×#888102
#117234362
C2×C3×#37182
C2×C3×#37172
#117220182
#105916103
#105908673