Finite Group Databasebeta

← #13161971

Subgroups — #13161971

Order 1728.

866 conjugacy classes of subgroups; only the 115 normal subgroups are shown.

normal subgroups only

This group has 866 conjugacy classes of subgroups — too many to draw. Only its 115 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
1C3×#64149C211yesmaximal, characteristic
2C2×C3×#3754C212yesmaximal, complemented
3#11723678C211yesmaximal, characteristic, complemented
4C2×#88811C212yesmaximal, complemented
5#11722070C211yesmaximal, characteristic, complemented
6#10591492—31nomaximal
7#10591490—32nomaximal
8C2×C33×D4C2211yescharacteristic
9C2×C3×#638—22no—
10C2×C3×C4⋊(C3⋊D3)C2211yescharacteristic
11C2×C3×#634C2212yes—
12C3×#3754—24no—
13C3×C4×C3⋊Dic3C2211yescharacteristic
14C2×C3×C3⋊(C3⋊C8)C2212yes—
15#88815—22no—
16#88811—24no—
17#64149D311yescharacteristic
18C2×#3754D312yescomplemented
19C3×#940—32no—
20C3×#940—61no—
21C2×C3×#214—32no—
22C2×C3×#214—32no—
23C2×C3×#214—62no—
24#63873—31no—
25#63866—32no—
26C2×#3717—32no—
27C2×#3716—32no—
28C2×#3715—32no—
29#63553—31no—
30Dic3×C3⋊C8—32no—
31C23×C33—21no—
32C22×C3×C3⋊D3—21no—
33C33×D4—22no—
34C2×C33×C4C2311yescharacteristic
35C3×#638—44no—
36C2×C3×C3⋊Dic3—22no—
37C2×C3×C3⋊Dic3D412yes—
38C3×C4⋊(C3⋊D3)—22no—
39C3×#634—22no—
40C3×#634—22no—
41C3×C3⋊(C3⋊C8)D414yescomplemented
42#5175—92no—
43#5076—91no—
44C2×C32×D4—22no—
45C2×C32×D4—21no—
46C2×C32×D4—41no—
47C2×C32×D4C2×D311yescharacteristic
48C2×C32×D4C2×D312yes—
49C2×#638—22no—
50C2×C4⋊(C3⋊D3)C2×D311yescharacteristic
51C2×#634C2×D312yes—
52C2×C3×#67—64no—
53C2×C3×#67—64no—
54C2×C3×D12—32no—
55C2×C3×D12—61no—
56C2×C3×Dic6—32no—
57C2×C3×Dic6—32no—
58C2×C3×Dic6—62no—
59#3754—24no—
60C4×C3⋊Dic3C2×D311yescharacteristic
61C2×C3⋊(C3⋊C8)C2×D312yes—
62C3×#214—64no—
63C3×#214—64no—
64C3×#214—68no—
65C3×C4×Dic3—32no—
66C3×C4×Dic3—61no—
67C2×C3×C3⋊C8—32no—
68C2×C3×C3⋊C8—32no—
69C2×C3×C3⋊C8—62no—
70#3724—62no—
71Dic3⋊Dic3—64no—
72#3717—64no—
73#3716—64no—
74#3715—64no—
75C22×C33—41no—
76C22×C33—42no—
77C22×C33C2×D411yescharacteristic
78C2×C3×C3⋊D3—41no—
79C2×C3×C3⋊D3—42no—
80C33×C4C2×D412yes—
81C3×C3⋊Dic3—42no—
82C3×C3⋊Dic3SD1614yescomplemented
83C2×C3×SD16—92no—
84C3×#127—91no—
85#944—92no—
86C2×#215—92no—
87#940—32no—
88#940—61no—
89C2×#214—32no—
90C2×#214—32no—
91C2×#214—62no—
92C2×#214—92no—
93C2×#204—92no—
94#869—91no—
95C8×Dic3—92no—
96C4×C3⋊C8—91no—
97C23×C32—21no—
98C23×C32—22no—
99C23×C32—22no—
100C23×C32—42no—
101C23×C32—42no—
102C22×C3⋊D3—21no—
103C22×C3×D3—62no—
104C22×C3×D3—62no—
105C32×D4—22no—
106C32×D4—24no—
107C32×D4—22no—
108C32×D4—22no—
109C32×D4—42no—
110C32×D4—42no—
111C32×D4—44no—
112C2×C32×C4—22no—
113C2×C32×C4—21no—
114C2×C32×C4—41no—
115C2×C32×C4C22×D311yescharacteristic
116C2×C32×C4C22×D312yes—
117#638—44no—
118C2×C3⋊Dic3—22no—
119C2×C3⋊Dic3#6712yes—
120C4⋊(C3⋊D3)—22no—
121#634—22no—
122#634—22no—
123C3×#67—128no—
124C3×#67—128no—
125C2×C3×Dic3—34no—
126C2×C3×Dic3—62no—
127C2×C3×Dic3—62no—
128C2×C3×Dic3—62no—
129C2×C3×Dic3—122no—
130C3×D12—62no—
131C3×D12—62no—
132C3×D12—122no—
133C3×Dic6—64no—
134C3×Dic6—62no—
135C3×Dic6—62no—
136C3×Dic6—62no—
137C3×Dic6—62no—
138C3×Dic6—122no—
139C3⋊(C3⋊C8)#6714yescomplemented
140C3×C3⋊C8—34no—
141C3×C3⋊C8—34no—
142C3×C3⋊C8—64no—
143#492—271noSylow
144C2×C33—81no—
145C2×C33#12711yescharacteristic
146C2×C33C2×SD1612yes—
147C3×C3⋊D3—81no—
148C2×C3×Q8—92no—
149C2×C3×D4—22no—
150C2×C3×D4—21no—
151C2×C3×D4—41no—
152C2×C3×D4—91no—
153C2×C3×D4—182no—
154C2×C3×D4D3211yescharacteristic
155C2×C3×D4D3212yes—
156C2×#67—64no—
157C2×#67—64no—
158C2×D12—32no—
159C2×D12—61no—
160C2×Dic6—32no—
161C2×Dic6—32no—
162C2×Dic6—62no—
163C3×SD16—184no—
164C2×C3×C8—92no—
165C3×C42—91no—
166#215—184no—
167#214—64no—
168#214—64no—
169#214—68no—
170#214—184no—
171C4⋊Dic3—182no—
172Dic3⋊C4—184no—
173C4×Dic3—32no—
174C4×Dic3—61no—
175C2×C3⋊C8—32no—
176C2×C3⋊C8—32no—
177C2×C3⋊C8—62no—
178#204—184no—
179C22×C32—22no—
180C22×C32—21no—
181C22×C32—41no—
182C22×C32—44no—
183C22×C32—42no—
184C22×C32—44no—
185C22×C32—42no—
186C22×C32—42no—
187C22×C32—41no—
188C22×C32—44no—
189C22×C32—44no—
190C22×C32—84no—
191C22×C32—84no—
192C22×C32D3×D412yes—
193C22×C32C2×#6711yescharacteristic
194C2×C3⋊D3—42no—
195C2×C3⋊D3—41no—
196C2×C3×D3—122no—
197C2×C3×D3—124no—
198C2×C3×D3—124no—
199C2×C3×D3—122no—
200C32×C4—22no—
201C32×C4—22no—
202C32×C4—22no—
203C32×C4—42no—
204C32×C4D3×D412yes—
205C32×C4D3×D412yes—
206C32×C4C2×#6712yes—
207C3⋊Dic3—42no—
208C3⋊Dic3#21414yescomplemented
209C3×Dic3—38no—
210C3×Dic3—64no—
211C3×Dic3—124no—
212C3×Dic3—122no—
213C3×Dic3—122no—
214C2×SD16—92no—
215C2×SD16—272no—
216C4⋊Q8—271no—
217#127—91no—
218C4×C8—271no—
219C33#49211yescharacteristic, complemented, Sylow
220C23×C3—22no—
221C23×C3—22no—
222C23×C3—21no—
223C23×C3—42no—
224C23×C3—42no—
225C23×C3—181no—
226C22×D3—62no—
227C22×D3—62no—
228C3×Q8—182no—
229C3×Q8—182no—
230C3×D4—24no—
231C3×D4—22no—
232C3×D4—22no—
233C3×D4—22no—
234C3×D4—42no—
235C3×D4—42no—
236C3×D4—44no—
237C3×D4—182no—
238C3×D4—364no—
239C2×C3×C4—22no—
240C2×C3×C4—21no—
241C2×C3×C4—41no—
242C2×C3×C4—92no—
243C2×C3×C4—182no—
244C2×C3×C4C2×D3211yescharacteristic
245C2×C3×C4C2×D3212yes—
246#67—128no—
247#67—128no—
248C2×Dic3—34no—
249C2×Dic3—62no—
250C2×Dic3—62no—
251C2×Dic3—62no—
252C2×Dic3—122no—
253D12—62no—
254D12—62no—
255D12—122no—
256Dic6—64no—
257Dic6—62no—
258Dic6—62no—
259Dic6—62no—
260Dic6—62no—
261Dic6—122no—
262C3×C8—94no—
263C3⋊C8—34no—
264C3⋊C8—34no—
265C3⋊C8—64no—
266C2×C32—24no—
267C2×C32—22no—
268C2×C32—22no—
269C2×C32—21no—
270C2×C32—41no—
271C2×C32—42no—
272C2×C32—82no—
273C2×C32—84no—
274C2×C32—84no—
275C2×C32—82no—
276C2×C32—81no—
277C2×C32D3×SD1614yes—
278C2×C32C2×#21412yes—
279C2×C32#94012yes—
280C2×C32#94011yescharacteristic
281C3⋊D3—81no—
282C3×D3—242no—
283C3×D3—242no—
284C2×Q8—92no—
285C2×D4—91no—
286C2×D4—182no—
287C2×D4#111311yescharacteristic
288SD16—184no—
289SD16—544no—
290C2×C8—92no—
291C4⋊C4—542no—
292C42—91no—
293C22×C3—21no—
294C22×C3—22no—
295C22×C3—44no—
296C22×C3—44no—
297C22×C3—42no—
298C22×C3—44no—
299C22×C3—44no—
300C22×C3—42no—
301C22×C3—42no—
302C22×C3—41no—
303C22×C3—41no—
304C22×C3—84no—
305C22×C3—84no—
306C22×C3—362no—
307C22×C3—361no—
308C22×C3D3×#6712yes—
309C22×C3#381111yescharacteristic
310C2×D3—122no—
311C2×D3—124no—
312C2×D3—124no—
313C2×D3—122no—
314C3×C4—22no—
315C3×C4—22no—
316C3×C4—22no—
317C3×C4—42no—
318C3×C4—94no—
319C3×C4—362no—
320C3×C4D3×#6712yes—
321C3×C4D3×#6712yes—
322C3×C4#381112yes—
323Dic3—38no—
324Dic3—64no—
325Dic3—124no—
326Dic3—122no—
327Dic3—122no—
328C32—22no—
329C32—21no—
330C32—41no—
331C32#507611yescharacteristic, complemented
332C32#517512yescomplemented
333C23—21no—
334C23—181no—
335Q8—182no—
336Q8—182no—
337D4—22no—
338D4—182no—
339D4—364no—
340C2×C4—92no—
341C2×C4—182no—
342C2×C4C2×#111311yescharacteristic
343C8—94no—
344C2×C3—22no—
345C2×C3—24no—
346C2×C3—22no—
347C2×C3—21no—
348C2×C3—42no—
349C2×C3—41no—
350C2×C3—82no—
351C2×C3—84no—
352C2×C3—84no—
353C2×C3—82no—
354C2×C3—81no—
355C2×C3—721no—
356C2×C3D3×#21414yes—
357C2×C3#6393012yes—
358C2×C3#6397912yes—
359C2×C3#6397911yescharacteristic
360D3—242no—
361D3—242no—
362C22—42no—
363C22—41no—
364C22—362no—
365C22—361no—
366C22#8904911yescharacteristic
367C4—94no—
368C4—362no—
369C4#8904912yes—
370C3—21no—
371C3—22no—
372C3—41no—
373C3#1059149012yescomplemented
374C3#1059149211yescharacteristic, complemented
375C2—81no—
376C2—721no—
377C2#1172370412yes—
378C2#1172372811yescharacteristic
subgroupG/H
centerC22#89049
commutator, cosocleC2×C33×C4C23
FrattiniC2×C4C2×#1113
FittingC2×C33×D4C22
socleC22×C33C2×D4
prime radicalC3×#64149C2
hypercenterC2×D4#1113
hypocenterC33#492
LCSG ▷ C2×C33×C4 ▷ C2×C33 ▷ C33
UCSC1 ◁ C22 ◁ C2×C4 ◁ C2×D4

Complements

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

NH ≅ G/Ntype
2. C2×C3×#3754C2semidirect
3. #11723678C2semidirect
4. C2×#88811C2semidirect
5. #11722070C2semidirect
18. C2×#3754D3semidirect
41. C3×C3⋊(C3⋊C8)D4semidirect
82. C3×C3⋊Dic3SD16semidirect
139. C3⋊(C3⋊C8)#67semidirect
208. C3⋊Dic3#214semidirect
219. C33#492semidirect
331. C32#5076semidirect
332. C32#5175semidirect
373. C3#10591490semidirect
374. C3#10591492semidirect

Maximal subgroups

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

subgroupindex
C3×#641492
C2×C3×#37542
#117236782
C2×#888112
#117220702
#105914923
#105914903