Finite Group Databasebeta

← #13156409

Subgroups — #13156409

Order 1728.

648 conjugacy classes of subgroups; only the 102 normal subgroups are shown.

normal subgroups only

This group has 648 conjugacy classes of subgroups — too many to draw. Only its 102 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#11723522C211yesmaximal, characteristic
2C3×#63900C211yesmaximal, characteristic, complemented
3#11722072C211yesmaximal, characteristic, complemented
4#11722054C212yesmaximal, complemented
5C3×#63562C212yesmaximal, complemented
6#10591419—31nomaximal
7#10590500—32nomaximal
8C2×C3×C4×C3⋊D3C2211yescharacteristic
9C2×C32×D12C2211yescharacteristic
10C2×#6290—21no—
11C2×C3×#624—21no—
12C4⋊(C3⋊(C3⋊Dic3))C2211yescharacteristic
13C2×C3×C3⋊(C3⋊C8)C2212yes—
14C32×C4⋊Dic3C2212yes—
15(C2×D3)⋊(C3⋊Dic3)—21no—
16C3×#3724—21no—
17#64147—31no—
18C3×D3×C4⋊C4—32no—
19#63908—32no—
20#63908—61no—
21#63900D311yescharacteristic, complemented
22#63635—32no—
23C3×#832—32no—
24C3×#809—32no—
25#63573—32no—
26#63573—32no—
27#63573—62no—
28#63562D312yescomplemented
29#63557—31no—
30(C2×D3)⋊(C3⋊C8)—32no—
31C22×C3×C3⋊D3D411yescharacteristic
32C22×C32×D3—21no—
33C2×C33×C4C2311yescharacteristic
34C2×C3⋊(C3⋊Dic3)—21no—
35C2×C3×C3⋊Dic3D411yescharacteristic
36C3×C4×C3⋊D3—22no—
37C2×C32×Dic3—22no—
38C32×D12—22no—
39#6290—42no—
40C3×#624—42no—
41C3×C3⋊(C3⋊C8)—22no—
42#4968—91no—
43#4777—92no—
44C2×C32×D4—31no—
45C2×#638—61no—
46C2×C4×C3⋊D3C2×D311yescharacteristic
47C2×C3×D12—21no—
48C2×C3×D12C2×D312yes—
49C2×C3×C4×D3—32no—
50C2×C3×C4×D3—62no—
51C2×C3×C4×D3—61no—
52C2×#625—62no—
53C2×#625—62no—
54C2×#624—21no—
55C32×C4⋊C4—32no—
56#3757—61no—
57C4⋊(C3⋊Dic3)—31no—
58C4⋊(C3⋊Dic3)—32no—
59C4⋊(C3⋊Dic3)—61no—
60C4⋊(C3⋊Dic3)—62no—
61C4⋊(C3⋊Dic3)—121no—
62C2×C3⋊(C3⋊C8)C2×D312yes—
63C3×C4⋊Dic3—22no—
64C3×C4⋊Dic3C2×D312yes—
65C3×C4⋊Dic3C2×D312yes—
66C3×Dic3⋊C4—62no—
67C2×C3×C3⋊C8—32no—
68C2×C3×C3⋊C8—32no—
69C2×C3×C3⋊C8—62no—
70#3724—21no—
71(C2×D3)⋊Dic3—62no—
72(C2×D3)⋊Dic3—62no—
73C22×C33C2×D411yescharacteristic
74C2×C3×C3⋊D3—21no—
75C2×C3×C3⋊D3—21no—
76C2×C3×C3⋊D3SD1612yes—
77C2×C32×D3—41no—
78C2×C32×D3—41no—
79C2×C32×D3—41no—
80C33×C4#4112yes—
81C3⋊(C3⋊Dic3)—41no—
82C3×C3⋊Dic3—21no—
83C32×Dic3—42no—
84C2×C3×C4⋊C4—91no—
85#938—92no—
86#938—181no—
87#930—91no—
88D3×C4⋊C4—32no—
89C3×#102—92no—
90#832—32no—
91#830—91no—
92(C2×D3)⋊C8—92no—
93#817—92no—
94#809—32no—
95#808—92no—
96#808—92no—
97#808—182no—
98C23×C32—61no—
99C22×C3⋊D3#6711yescharacteristic
100C22×C3×D3—22no—
101C22×C3×D3—22no—
102C22×C3×D3—32no—
103C22×C3×D3—61no—
104C32×D4—62no—
105C2×C32×C4—22no—
106C2×C32×C4—21no—
107C2×C32×C4—41no—
108C2×C32×C4—62no—
109C2×C32×C4C22×D311yescharacteristic
110C2×C32×C4C22×D312yes—
111#638—122no—
112C2×C3⋊Dic3—61no—
113C2×C3⋊Dic3—62no—
114C2×C3⋊Dic3—62no—
115C2×C3⋊Dic3—122no—
116C2×C3⋊Dic3—122no—
117C2×C3⋊Dic3#6711yescharacteristic
118C4×C3⋊D3—22no—
119C2×C3×Dic3—22no—
120C2×C3×Dic3—22no—
121C2×C3×Dic3—32no—
122C2×C3×Dic3—42no—
123C2×C3×Dic3—61no—
124C3×D12—22no—
125C3×D12—22no—
126C3×D12—42no—
127C3×C4×D3—62no—
128C3×C4×D3—62no—
129C3×C4×D3—122no—
130C3×C4×D3—124no—
131#625—124no—
132#625—124no—
133#624—42no—
134C3⋊(C3⋊C8)—22no—
135C3×C3⋊C8—64no—
136C3×C3⋊C8—62no—
137C3×C3⋊C8—62no—
138#481—271noSylow
139C2×C33#12311yescharacteristic
140C2×C33C2×SD1611yescharacteristic
141C2×C33AGL(1,ℤ/8)11yescharacteristic
142C3×C3⋊D3—22no—
143C32×D3—81no—
144C2×C3×D4—32no—
145C2×C3×D4—61no—
146C22×C3×C4—91no—
147C22×C3×C4—181no—
148C2×#67—181no—
149C2×#67—182no—
150C2×#67—182no—
151C2×D12D3211yescharacteristic
152C2×C4×D3—32no—
153C2×C4×D3—61no—
154C2×C4×D3—62no—
155C2×C3×C8—92no—
156C3×C4⋊C4—32no—
157C3×C4⋊C4—32no—
158C3×C4⋊C4—62no—
159C3×C4⋊C4—181no—
160#217—182no—
161#217—182no—
162(C2×D3)⋊C4—181no—
163C4⋊Dic3—91no—
164C4⋊Dic3—92no—
165C4⋊Dic3—181no—
166C4⋊Dic3—182no—
167C4⋊Dic3—361no—
168C4⋊Dic3D3212yes—
169Dic3⋊C4—62no—
170C2×C3⋊C8—32no—
171C2×C3⋊C8—32no—
172C2×C3⋊C8—62no—
173C22×C32—22no—
174C22×C32—21no—
175C22×C32—41no—
176C22×C32—121no—
177C22×C32—121no—
178C22×C32—121no—
179C22×C32D3×D412yes—
180C22×C32C2×#6711yescharacteristic
181C2×C3⋊D3—21no—
182C2×C3⋊D3—21no—
183C2×C3⋊D3#21512yes—
184C2×C3×D3—34no—
185C2×C3×D3—42no—
186C2×C3×D3—42no—
187C2×C3×D3—42no—
188C2×C3×D3—42no—
189C2×C3×D3—42no—
190C2×C3×D3—42no—
191C2×C3×D3—62no—
192C2×C3×D3—62no—
193C2×C3×D3—62no—
194C2×C3×D3—121no—
195C2×C3×D3—121no—
196C32×C4—22no—
197C32×C4—22no—
198C32×C4—22no—
199C32×C4—42no—
200C32×C4—122no—
201C32×C4#23512yes—
202C32×C4#23512yes—
203C32×C4#23712yes—
204C3⋊Dic3—21no—
205C3⋊Dic3—121no—
206C3⋊Dic3—122no—
207C3⋊Dic3—122no—
208C3⋊Dic3—242no—
209C3⋊Dic3—242no—
210C3×Dic3—42no—
211C3×Dic3—42no—
212C3×Dic3—62no—
213C3×Dic3—82no—
214C3×Dic3—121no—
215#121—271no—
216C2×C4⋊C4—91no—
217#106—272no—
218#102—92no—
219C22⋊C8—271no—
220C33#48111yescharacteristic, complemented, Sylow
221C23×C3—62no—
222C23×C3—62no—
223C23×C3—91no—
224C22×D3—21no—
225C22×D3—32no—
226C22×D3—61no—
227C3×D4—62no—
228C3×D4—62no—
229C3×D4—122no—
230C2×C3×C4—22no—
231C2×C3×C4—21no—
232C2×C3×C4—41no—
233C2×C3×C4—62no—
234C2×C3×C4—62no—
235C2×C3×C4—91no—
236C2×C3×C4—122no—
237C2×C3×C4—182no—
238C2×C3×C4—362no—
239C2×C3×C4C2×D3212yes—
240C2×C3×C4C2×D3211yescharacteristic
241#67—364no—
242#67—362no—
243#67—364no—
244C2×Dic3—22no—
245C2×Dic3—32no—
246C2×Dic3—61no—
247C2×Dic3—182no—
248C2×Dic3—182no—
249C2×Dic3—181no—
250C2×Dic3—362no—
251C2×Dic3—362no—
252D12—22no—
253C4×D3—62no—
254C4×D3—62no—
255C4×D3—122no—
256C4×D3—124no—
257C3×C8—182no—
258C3⋊C8—64no—
259C3⋊C8—62no—
260C3⋊C8—62no—
261C2×C32—22no—
262C2×C32—21no—
263C2×C32—21no—
264C2×C32—22no—
265C2×C32—22no—
266C2×C32—21no—
267C2×C32—41no—
268C2×C32—41no—
269C2×C32—41no—
270C2×C32—241no—
271C2×C32#88312yes—
272C2×C32#91112yes—
273C2×C32D3×SD1612yes—
274C2×C32#92911yescharacteristic
275C2×C32#93211yescharacteristic
276C2×C32C2×#21511yescharacteristic
277C3⋊D3—22no—
278C3×D3—64no—
279C3×D3—82no—
280C3×D3—82no—
281C3×D3—122no—
282C2×D4—31no—
283C2×D4—541no—
284C22×C4—91no—
285C22×C4—181no—
286C2×C8—92no—
287C4⋊C4—32no—
288C4⋊C4—181no—
289C4⋊C4—271no—
290C22⋊C4—541no—
291C22×C3—21no—
292C22×C3—22no—
293C22×C3—41no—
294C22×C3—92no—
295C22×C3—122no—
296C22×C3—122no—
297C22×C3—122no—
298C22×C3—122no—
299C22×C3—122no—
300C22×C3—122no—
301C22×C3—181no—
302C22×C3—181no—
303C22×C3D3×#6712yes—
304C22×C3#381111yescharacteristic
305C2×D3—34no—
306C2×D3—41no—
307C2×D3—41no—
308C2×D3—41no—
309C2×D3—62no—
310C2×D3—62no—
311C2×D3—62no—
312C2×D3—121no—
313C2×D3—121no—
314C3×C4—22no—
315C3×C4—22no—
316C3×C4—22no—
317C3×C4—42no—
318C3×C4—122no—
319C3×C4—122no—
320C3×C4—181no—
321C3×C4—242no—
322C3×C4#379512yes—
323C3×C4#379812yes—
324C3×C4#380412yes—
325Dic3—42no—
326Dic3—62no—
327Dic3—121no—
328Dic3—362no—
329Dic3—361no—
330Dic3—362no—
331Dic3—722no—
332Dic3—722no—
333C32—22no—
334C32—21no—
335C32—41no—
336C32#477712yescomplemented
337C32#496811yescharacteristic, complemented
338C23—61no—
339C23—91no—
340D4—62no—
341D4—1082no—
342C2×C4—62no—
343C2×C4—91no—
344C2×C4—182no—
345C2×C4—362no—
346C2×C4—541no—
347C2×C4D3311yescharacteristic
348C8—182no—
349C2×C3—21no—
350C2×C3—22no—
351C2×C3—22no—
352C2×C3—21no—
353C2×C3—21no—
354C2×C3—22no—
355C2×C3—41no—
356C2×C3—41no—
357C2×C3—41no—
358C2×C3—182no—
359C2×C3—242no—
360C2×C3—242no—
361C2×C3#6390512yes—
362C2×C3#6392711yescharacteristic
363C2×C3#6393011yescharacteristic
364C2×C3#6393212yes—
365C2×C3D3×#21512yes—
366C2×C3#6397511yescharacteristic
367D3—64no—
368D3—81no—
369D3—122no—
370C22—92no—
371C22—121no—
372C22—121no—
373C22—121no—
374C22—181no—
375C22—181no—
376C22#8895211yescharacteristic
377C4—122no—
378C4—181no—
379C4—1081no—
380C4#8896012yes—
381C3—21no—
382C3—22no—
383C3—41no—
384C3#1059050012yescomplemented
385C3#1059141911yescharacteristic, complemented
386C2—182no—
387C2—241no—
388C2#1172283811yescharacteristic
389C2#1172287211yescharacteristic
390C2#1172299211yescharacteristic
subgroupG/H
centerC22#88952
commutator, Fitting, cosocleC2×C33×C4C23
Frattini, hypercenterC2×C4D33
socleC22×C33C2×D4
prime radical#11723522C2
hypocenterC33#481
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. C3×#63900C2semidirect
3. #11722072C2semidirect
4. #11722054C2semidirect
5. C3×#63562C2semidirect
21. #63900D3semidirect
28. #63562D3semidirect
220. C33#481semidirect
336. C32#4777semidirect
337. C32#4968semidirect
384. C3#10590500semidirect
385. C3#10591419semidirect

Maximal subgroups

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

subgroupindex
#117235222
C3×#639002
#117220722
#117220542
C3×#635622
#105914193
#105905003