Finite Group Databasebeta

← #13156647

Subgroups — #13156647

Order 1728.

762 conjugacy classes of subgroups; only the 106 normal subgroups are shown.

normal subgroups only

This group has 762 conjugacy classes of subgroups — too many to draw. Only its 106 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#11723440C211yesmaximal, characteristic
2C2×#88795C212yesmaximal
3C3×#63873C211yesmaximal, characteristic, complemented
4C2×C3×#3718C212yesmaximal, complemented
5#11722071C211yesmaximal, characteristic, complemented
6#10591629—31nomaximal
7#10591608—32nomaximal
8C2×C4⋊(C3⋊(C3⋊D3))C2211yescharacteristic
9C2×C32×Dic6C2212yes—
10C2×C32×Dic6C2211yescharacteristic
11C3×C4×C3⋊Dic3C2211yescharacteristic
12C2×C3×C3⋊(C3⋊C8)C2212yes—
13#88804—22no—
14#88795—24no—
15C3×#3724—22no—
16C3×#3718—24no—
17#64156—31no—
18C2×#3755—32no—
19C3×#944—32no—
20C2×C3×#216—32no—
21C2×C3×#216—32no—
22#63873D311yescharacteristic, complemented
23#63864—32no—
24#63864—61no—
25C2×#3718D312yescomplemented
26C2×#3717—32no—
27C2×#3717—32no—
28C2×#3717—62no—
29#63555—31no—
30#63554—32no—
31C22×C3⋊(C3⋊D3)—21no—
32C2×C33×C4C2311yescharacteristic
33C4⋊(C3⋊(C3⋊D3))—22no—
34C2×C3×C3⋊Dic3D412yes—
35C2×C32×Dic3—21no—
36C2×C32×Dic3—22no—
37C32×Dic6—22no—
38C32×Dic6—22no—
39C32×Dic6—22no—
40C3×C3⋊(C3⋊C8)D414yes—
41#5190—92no—
42#5086—91no—
43C2×C32×Q8—31no—
44C2×C32×Q8—32no—
45C2×C4⋊(C3⋊D3)—31no—
46C2×C4⋊(C3⋊D3)—32no—
47C2×C4⋊(C3⋊D3)—62no—
48C2×C4⋊(C3⋊D3)—61no—
49C2×C4⋊(C3⋊D3)—121no—
50C2×C3×Dic6—22no—
51C2×C3×Dic6—21no—
52C2×C3×Dic6C2×D312yes—
53C2×C3×Dic6C2×D312yes—
54C2×C3×Dic6C2×D312yes—
55#3755—64no—
56#3752—62no—
57C4×C3⋊Dic3C2×D311yescharacteristic
58C2×C3⋊(C3⋊C8)C2×D312yes—
59C3×#216—64no—
60C3×#216—64no—
61C3×Dic3⋊C4—64no—
62C3×C4×Dic3—32no—
63C3×C4×Dic3—61no—
64C2×C3×C3⋊C8—32no—
65C2×C3×C3⋊C8—32no—
66C2×C3×C3⋊C8—62no—
67#3724—22no—
68#3722—64no—
69#3722—64no—
70#3718—24no—
71#3717—64no—
72#3717—64no—
73#3717—68no—
74C22×C33C2×D411yescharacteristic
75C2×C3⋊(C3⋊D3)—41no—
76C2×C3⋊(C3⋊D3)—42no—
77C33×C4C2×D412yes—
78C3×C3⋊Dic3—22no—
79C32×Dic3—42no—
80C32×Dic3—41no—
81C2×C3×Q16—92no—
82C3×C4⋊Q8—91no—
83#947—92no—
84#947—181no—
85#944—32no—
86C2×#216—32no—
87C2×#216—32no—
88C2×#215—92no—
89C2×#215—92no—
90C2×#215—182no—
91C2×#204—92no—
92#875—91no—
93#815—92no—
94#803—91no—
95C22×C3⋊D3—61no—
96C22×C3⋊D3—62no—
97C22×C3⋊D3—62no—
98C22×C3⋊D3—122no—
99C22×C3⋊D3—122no—
100C32×Q8—62no—
101C32×Q8—62no—
102C32×Q8—62no—
103C2×C32×C4—22no—
104C2×C32×C4—21no—
105C2×C32×C4—41no—
106C2×C32×C4—61no—
107C2×C32×C4—62no—
108C2×C32×C4C22×D311yescharacteristic
109C2×C32×C4C22×D312yes—
110C2×C3⋊Dic3#6712yes—
111C4⋊(C3⋊D3)—62no—
112C4⋊(C3⋊D3)—64no—
113C4⋊(C3⋊D3)—62no—
114C4⋊(C3⋊D3)—62no—
115C4⋊(C3⋊D3)—122no—
116C4⋊(C3⋊D3)—122no—
117C4⋊(C3⋊D3)—124no—
118C2×C3×Dic3—22no—
119C2×C3×Dic3—22no—
120C2×C3×Dic3—22no—
121C2×C3×Dic3—22no—
122C2×C3×Dic3—34no—
123C2×C3×Dic3—42no—
124C2×C3×Dic3—62no—
125C3×Dic6—24no—
126C3×Dic6—22no—
127C3×Dic6—22no—
128C3×Dic6—22no—
129C3×Dic6—22no—
130C3×Dic6—22no—
131C3×Dic6—22no—
132C3×Dic6—42no—
133C3×Dic6—42no—
134C3⋊(C3⋊C8)#6714yes—
135C3×C3⋊C8—34no—
136C3×C3⋊C8—34no—
137C3×C3⋊C8—64no—
138#497—271noSylow
139C2×C33#12711yescharacteristic
140C2×C33#13712yes—
141C3⋊(C3⋊D3)—81no—
142C2×C3×Q8—32no—
143C2×C3×Q8—32no—
144C2×C3×Q8—32no—
145C2×C3×Q8—61no—
146C2×C3×Q8—62no—
147C2×D12—91no—
148C2×D12—92no—
149C2×D12—182no—
150C2×D12—181no—
151C2×D12—361no—
152C2×Dic6D3211yescharacteristic
153C2×Dic6D3212yes—
154C3×Q16—184no—
155C2×C3×C8—92no—
156C3×C4⋊C4—182no—
157C3×C42—91no—
158#216—64no—
159#216—64no—
160#215—184no—
161#215—184no—
162#215—188no—
163(C2×D3)⋊C4—182no—
164(C2×D3)⋊C4—184no—
165(C2×D3)⋊C4—184no—
166Dic3⋊C4—64no—
167C4×Dic3—32no—
168C4×Dic3—61no—
169C2×C3⋊C8—32no—
170C2×C3⋊C8—32no—
171C2×C3⋊C8—62no—
172#204—184no—
173C22×C32—22no—
174C22×C32—21no—
175C22×C32—41no—
176C22×C32D3×D412yes—
177C22×C32C2×#6711yescharacteristic
178C2×C3⋊D3—124no—
179C2×C3⋊D3—122no—
180C2×C3⋊D3—124no—
181C2×C3⋊D3—122no—
182C2×C3⋊D3—121no—
183C2×C3⋊D3—122no—
184C2×C3⋊D3—124no—
185C2×C3⋊D3—124no—
186C2×C3⋊D3—244no—
187C2×C3⋊D3—244no—
188C32×C4—22no—
189C32×C4—22no—
190C32×C4—22no—
191C32×C4—42no—
192C32×C4—121no—
193C32×C4—122no—
194C32×C4D3×D412yes—
195C32×C4D3×D412yes—
196C32×C4C2×#6712yes—
197C3⋊Dic3—22no—
198C3×Dic3—44no—
199C3×Dic3—42no—
200C3×Dic3—42no—
201C3×Dic3—42no—
202C3×Dic3—42no—
203C3×Dic3—64no—
204C3×Dic3—122no—
205C2×Q16—92no—
206C2×SD16—272no—
207C4⋊Q8—91no—
208#124—271no—
209#97—271no—
210C33#49711yescharacteristic, complemented, Sylow
211C22×D3—182no—
212C22×D3—182no—
213C22×D3—181no—
214C22×D3—362no—
215C22×D3—362no—
216C3×Q8—64no—
217C3×Q8—62no—
218C3×Q8—62no—
219C3×Q8—62no—
220C3×Q8—62no—
221C3×Q8—62no—
222C3×Q8—62no—
223C3×Q8—122no—
224C3×Q8—122no—
225C2×C3×C4—22no—
226C2×C3×C4—21no—
227C2×C3×C4—41no—
228C2×C3×C4—62no—
229C2×C3×C4—62no—
230C2×C3×C4—62no—
231C2×C3×C4—62no—
232C2×C3×C4—92no—
233C2×C3×C4—122no—
234C2×C3×C4C2×D3211yescharacteristic
235C2×C3×C4C2×D3212yes—
236C2×Dic3—22no—
237C2×Dic3—21no—
238C2×Dic3—34no—
239C2×Dic3—62no—
240D12—184no—
241D12—182no—
242D12—182no—
243D12—182no—
244D12—362no—
245D12—362no—
246D12—364no—
247Dic6—22no—
248Dic6—22no—
249Dic6—22no—
250C3×C8—94no—
251C3⋊C8—34no—
252C3⋊C8—34no—
253C3⋊C8—64no—
254C2×C32—22no—
255C2×C32—24no—
256C2×C32—22no—
257C2×C32—21no—
258C2×C32—41no—
259C2×C32—42no—
260C2×C32#91814yes—
261C2×C32#94012yes—
262C2×C32#94011yescharacteristic
263C2×C32#94212yes—
264C3⋊D3—242no—
265C3⋊D3—244no—
266C3⋊D3—244no—
267C3⋊D3—242no—
268C3⋊D3—241no—
269C2×Q8—31no—
270C2×Q8—32no—
271C2×D4—271no—
272Q16—184no—
273SD16—544no—
274C2×C8—92no—
275C4⋊C4—182no—
276C22⋊C4—542no—
277C42—91no—
278C22×C3—21no—
279C22×C3—22no—
280C22×C3—41no—
281C22×C3D3×#6712yes—
282C22×C3#381111yescharacteristic
283C2×D3—364no—
284C2×D3—362no—
285C2×D3—364no—
286C2×D3—364no—
287C2×D3—362no—
288C2×D3—364no—
289C2×D3—362no—
290C2×D3—361no—
291C2×D3—724no—
292C2×D3—724no—
293C3×C4—22no—
294C3×C4—22no—
295C3×C4—22no—
296C3×C4—42no—
297C3×C4—122no—
298C3×C4—124no—
299C3×C4—122no—
300C3×C4—122no—
301C3×C4—122no—
302C3×C4—182no—
303C3×C4D3×#6712yes—
304C3×C4D3×#6712yes—
305C3×C4#381112yes—
306Dic3—42no—
307Dic3—41no—
308Dic3—64no—
309Dic3—122no—
310C32—22no—
311C32—21no—
312C32—41no—
313C32#508611yescharacteristic, complemented
314C32#519012yescomplemented
315C23—541no—
316Q8—62no—
317Q8—62no—
318Q8—62no—
319D4—542no—
320C2×C4—61no—
321C2×C4—62no—
322C2×C4—92no—
323C2×C4D3311yescharacteristic
324C8—94no—
325C2×C3—22no—
326C2×C3—21no—
327C2×C3—22no—
328C2×C3—24no—
329C2×C3—41no—
330C2×C3—42no—
331C2×C3#6394512yes—
332C2×C3#6394814yes—
333C2×C3#6397912yes—
334C2×C3#6397911yescharacteristic
335D3—722no—
336D3—724no—
337D3—724no—
338D3—722no—
339D3—721no—
340C22—1081no—
341C22—1082no—
342C22#8895211yescharacteristic
343C4—121no—
344C4—122no—
345C4—182no—
346C4#8895212yes—
347C3—22no—
348C3—21no—
349C3—41no—
350C3#1059160812yescomplemented
351C3#1059162911yescharacteristic, complemented
352C2—2161no—
353C2#1172289012yes—
354C2#1172303111yescharacteristic
subgroupG/H
centerC22#88952
commutator, Fitting, cosocleC2×C33×C4C23
Frattini, hypercenterC2×C4D33
socleC22×C33C2×D4
prime radicalC2×C4⋊(C3⋊(C3⋊D3))C22
hypocenterC33#497
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
3. C3×#63873C2semidirect
4. C2×C3×#3718C2semidirect
5. #11722071C2semidirect
22. #63873D3semidirect
25. C2×#3718D3semidirect
210. C33#497semidirect
313. C32#5086semidirect
314. C32#5190semidirect
350. C3#10591608semidirect
351. C3#10591629semidirect

Maximal subgroups

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

subgroupindex
#117234402
C2×#887952
C3×#638732
C2×C3×#37182
#117220712
#105916293
#105916083