Finite Group Databasebeta

← #13156629

Subgroups — #13156629

Order 1728.

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

normal subgroups only

This group has 826 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#11723429C211yesmaximal, characteristic, complemented
2C2×#88794C212yesmaximal, complemented
3C3×#63914C211yesmaximal, characteristic, complemented
4C2×C3×#3713C212yesmaximal, complemented
5#11722071C211yesmaximal, characteristic, complemented
6#10591485—31nomaximal
7#10591469—32nomaximal
8C2×#6308C2211yescharacteristic
9C2×C32×D12C2212yes—
10C2×C32×D12C2211yescharacteristic
11C2×C3×#625—22no—
12C3×C4×C3⋊Dic3C2211yescharacteristic
13C2×C3×C3⋊(C3⋊C8)C2212yes—
14(C2×D3)⋊(C3⋊Dic3)—22no—
15#88794—24no—
16C3×#3713—24no—
17#64145—31no—
18C2×#3754—32no—
19C3×#940—32no—
20C2×C3×#213—32no—
21C2×C3×#213—32no—
22#63914D311yescharacteristic, complemented
23#63860—32no—
24#63860—61no—
25C2×#3716—32no—
26C2×#3716—32no—
27C2×#3716—62no—
28C2×#3713D312yescomplemented
29#63555—31no—
30#63554—32no—
31C22×C32×D3—21no—
32C22×C32×D3—22no—
33C2×C33×C4C2311yescharacteristic
34C2×C3⋊(C3⋊Dic3)—21no—
35#6308—22no—
36C2×C3×C3⋊Dic3D412yes—
37C32×D12—22no—
38C32×D12—22no—
39C32×D12—22no—
40C3×#625—44no—
41C3×C3⋊(C3⋊C8)D414yescomplemented
42#5154—92no—
43#5074—91no—
44C2×C32×D4—31no—
45C2×C32×D4—32no—
46C2×#634—31no—
47C2×#634—32no—
48C2×#634—62no—
49C2×#634—61no—
50C2×#634—121no—
51C2×C3×#67—64no—
52C2×C3×D12—22no—
53C2×C3×D12—21no—
54C2×C3×D12C2×D312yes—
55C2×C3×D12C2×D312yes—
56C2×C3×D12C2×D312yes—
57C2×#625—22no—
58#3757—62no—
59#3754—64no—
60C4×C3⋊Dic3C2×D311yescharacteristic
61C2×C3⋊(C3⋊C8)C2×D312yes—
62C3×#213—64no—
63C3×#213—64no—
64C3×C4×Dic3—32no—
65C3×C4×Dic3—61no—
66C2×C3×C3⋊C8—32no—
67C2×C3×C3⋊C8—32no—
68C2×C3×C3⋊C8—62no—
69(C2×D3)⋊Dic3—64no—
70(C2×D3)⋊Dic3—64no—
71#3716—64no—
72#3716—64no—
73#3716—68no—
74#3713—24no—
75C22×C33C2×D411yescharacteristic
76C2×C32×D3—42no—
77C2×C32×D3—41no—
78C2×C32×D3—42no—
79C2×C32×D3—44no—
80C33×C4C2×D412yes—
81C3⋊(C3⋊Dic3)—41no—
82C3×C3⋊Dic3—22no—
83C2×C3×D8—92no—
84C3×#127—91no—
85#940—32no—
86#936—92no—
87#936—181no—
88C2×#214—92no—
89C2×#214—92no—
90C2×#214—182no—
91C2×#213—32no—
92C2×#213—32no—
93C2×#204—92no—
94#875—91no—
95#815—92no—
96#803—91no—
97C23×C32—61no—
98C23×C32—62no—
99C22×C3×D3—22no—
100C22×C3×D3—22no—
101C22×C3×D3—22no—
102C22×C3×D3—22no—
103C22×C3×D3—42no—
104C32×D4—62no—
105C32×D4—62no—
106C32×D4—62no—
107C2×C32×C4—22no—
108C2×C32×C4—21no—
109C2×C32×C4—41no—
110C2×C32×C4C22×D311yescharacteristic
111C2×C32×C4C22×D312yes—
112C2×C3⋊Dic3—61no—
113C2×C3⋊Dic3—62no—
114C2×C3⋊Dic3—62no—
115C2×C3⋊Dic3—122no—
116C2×C3⋊Dic3—122no—
117C2×C3⋊Dic3#6712yes—
118#634—62no—
119#634—64no—
120#634—62no—
121#634—62no—
122#634—122no—
123#634—122no—
124#634—124no—
125C3×#67—128no—
126C2×C3×Dic3—34no—
127C2×C3×Dic3—62no—
128C3×D12—24no—
129C3×D12—22no—
130C3×D12—22no—
131C3×D12—22no—
132C3×D12—22no—
133C3×D12—22no—
134C3×D12—22no—
135C3×D12—42no—
136C3×D12—42no—
137#625—44no—
138C3⋊(C3⋊C8)#6714yescomplemented
139C3×C3⋊C8—34no—
140C3×C3⋊C8—34no—
141C3×C3⋊C8—64no—
142#496—271noSylow
143C2×C33#12711yescharacteristic
144C2×C33AGL(1,ℤ/8)12yes—
145C32×D3—81no—
146C32×D3—82no—
147C2×C3×D4—32no—
148C2×C3×D4—32no—
149C2×C3×D4—32no—
150C2×C3×D4—62no—
151C2×C3×D4—61no—
152C2×C3×D4—182no—
153C2×#67—64no—
154C2×D12D3211yescharacteristic
155C2×D12D3212yes—
156C2×Dic6—91no—
157C2×Dic6—92no—
158C2×Dic6—182no—
159C2×Dic6—181no—
160C2×Dic6—361no—
161C3×D8—184no—
162C2×C3×C8—92no—
163C3×C42—91no—
164#217—184no—
165#217—184no—
166#214—184no—
167#214—184no—
168#214—188no—
169#213—64no—
170#213—64no—
171(C2×D3)⋊C4—182no—
172C4×Dic3—32no—
173C4×Dic3—61no—
174C2×C3⋊C8—32no—
175C2×C3⋊C8—32no—
176C2×C3⋊C8—62no—
177#204—184no—
178C22×C32—22no—
179C22×C32—21no—
180C22×C32—41no—
181C22×C32—122no—
182C22×C32—122no—
183C22×C32—121no—
184C22×C32—124no—
185C22×C32D3×D412yes—
186C22×C32C2×#6711yescharacteristic
187C2×C3×D3—44no—
188C2×C3×D3—42no—
189C2×C3×D3—44no—
190C2×C3×D3—44no—
191C2×C3×D3—44no—
192C2×C3×D3—44no—
193C2×C3×D3—42no—
194C2×C3×D3—42no—
195C2×C3×D3—42no—
196C2×C3×D3—84no—
197C32×C4—22no—
198C32×C4—22no—
199C32×C4—22no—
200C32×C4—42no—
201C32×C4D3×D412yes—
202C32×C4D3×D412yes—
203C32×C4C2×#6712yes—
204C3⋊Dic3—22no—
205C3⋊Dic3—124no—
206C3⋊Dic3—122no—
207C3⋊Dic3—124no—
208C3⋊Dic3—122no—
209C3⋊Dic3—121no—
210C3×Dic3—64no—
211C3×Dic3—122no—
212C2×SD16—272no—
213C2×D8—92no—
214#127—91no—
215#124—271no—
216#97—271no—
217C33#49611yescharacteristic, complemented, Sylow
218C23×C3—62no—
219C23×C3—62no—
220C23×C3—62no—
221C23×C3—62no—
222C23×C3—122no—
223C22×D3—22no—
224C22×D3—21no—
225C3×D4—64no—
226C3×D4—62no—
227C3×D4—62no—
228C3×D4—62no—
229C3×D4—62no—
230C3×D4—62no—
231C3×D4—62no—
232C3×D4—122no—
233C3×D4—122no—
234C3×D4—364no—
235C2×C3×C4—22no—
236C2×C3×C4—21no—
237C2×C3×C4—41no—
238C2×C3×C4—92no—
239C2×C3×C4C2×D3211yescharacteristic
240C2×C3×C4C2×D3212yes—
241#67—128no—
242C2×Dic3—34no—
243C2×Dic3—62no—
244C2×Dic3—182no—
245C2×Dic3—182no—
246C2×Dic3—181no—
247C2×Dic3—362no—
248C2×Dic3—362no—
249D12—22no—
250D12—22no—
251D12—22no—
252Dic6—184no—
253Dic6—182no—
254Dic6—182no—
255Dic6—182no—
256Dic6—362no—
257Dic6—362no—
258Dic6—364no—
259C3×C8—94no—
260C3⋊C8—34no—
261C3⋊C8—34no—
262C3⋊C8—64no—
263C2×C32—22no—
264C2×C32—24no—
265C2×C32—22no—
266C2×C32—21no—
267C2×C32—41no—
268C2×C32—42no—
269C2×C32—242no—
270C2×C32—241no—
271C2×C32#91114yes—
272C2×C32#93212yes—
273C2×C32#94012yes—
274C2×C32#94011yescharacteristic
275C3×D3—82no—
276C3×D3—84no—
277C3×D3—82no—
278C3×D3—82no—
279C3×D3—82no—
280C2×Q8—271no—
281C2×D4—31no—
282C2×D4—32no—
283C2×D4—182no—
284SD16—544no—
285D8—184no—
286C2×C8—92no—
287C22⋊C4—542no—
288C42—91no—
289C22×C3—21no—
290C22×C3—22no—
291C22×C3—41no—
292C22×C3—124no—
293C22×C3—122no—
294C22×C3—124no—
295C22×C3—124no—
296C22×C3—124no—
297C22×C3—124no—
298C22×C3—122no—
299C22×C3—122no—
300C22×C3—122no—
301C22×C3—244no—
302C22×C3D3×#6712yes—
303C22×C3#381111yescharacteristic
304C2×D3—42no—
305C2×D3—44no—
306C2×D3—42no—
307C2×D3—41no—
308C3×C4—22no—
309C3×C4—22no—
310C3×C4—22no—
311C3×C4—42no—
312C3×C4—182no—
313C3×C4D3×#6712yes—
314C3×C4D3×#6712yes—
315C3×C4#381112yes—
316Dic3—64no—
317Dic3—122no—
318Dic3—364no—
319Dic3—362no—
320Dic3—364no—
321Dic3—362no—
322Dic3—361no—
323C32—22no—
324C32—21no—
325C32—41no—
326C32#507411yescharacteristic, complemented
327C32#515412yescomplemented
328C23—61no—
329C23—62no—
330Q8—542no—
331D4—62no—
332D4—62no—
333D4—62no—
334D4—364no—
335C2×C4—92no—
336C2×C4—541no—
337C2×C4D3311yescharacteristic
338C8—94no—
339C2×C3—22no—
340C2×C3—21no—
341C2×C3—22no—
342C2×C3—24no—
343C2×C3—41no—
344C2×C3—42no—
345C2×C3—242no—
346C2×C3—244no—
347C2×C3—242no—
348C2×C3—242no—
349C2×C3—242no—
350C2×C3#6392712yes—
351C2×C3#6393214yes—
352C2×C3#6397912yes—
353C2×C3#6397911yescharacteristic
354D3—82no—
355D3—81no—
356C22—121no—
357C22—122no—
358C22—124no—
359C22—122no—
360C22#8895211yescharacteristic
361C4—182no—
362C4—1081no—
363C4#8895212yes—
364C3—22no—
365C3—21no—
366C3—41no—
367C3#1059146912yescomplemented
368C3#1059148511yescharacteristic, complemented
369C2—242no—
370C2—241no—
371C2#1172287212yes—
372C2#1172303111yescharacteristic
subgroupG/H
centerC22#88952
commutator, Fitting, cosocleC2×C33×C4C23
Frattini, hypercenterC2×C4D33
socleC22×C33C2×D4
hypocenterC33#496
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. #11723429C2semidirect
2. C2×#88794C2semidirect
3. C3×#63914C2semidirect
4. C2×C3×#3713C2semidirect
5. #11722071C2semidirect
22. #63914D3semidirect
28. C2×#3713D3semidirect
41. C3×C3⋊(C3⋊C8)D4semidirect
138. C3⋊(C3⋊C8)#67semidirect
217. C33#496semidirect
326. C32#5074semidirect
327. C32#5154semidirect
367. C3#10591469semidirect
368. C3#10591485semidirect

Maximal subgroups

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

subgroupindex
#117234292
C2×#887942
C3×#639142
C2×C3×#37132
#117220712
#105914853
#105914693