Finite Group Databasebeta

← #13156410

Subgroups — #13156410

Order 1728.

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

normal subgroups only

This group has 704 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#11723526C211yesmaximal, characteristic
2C3×#63900C211yesmaximal, characteristic, complemented
3#11722072C211yesmaximal, characteristic, complemented
4#11722035C212yesmaximal, complemented
5C3×#63574C212yesmaximal, complemented
6#10590445—31nomaximal
7#10590423—32nomaximal
8C2×C4⋊(C3⋊(C3⋊D3))C2211yescharacteristic
9C2×C3×C4×C3⋊D3C2211yescharacteristic
10C2×#6292—21no—
11C2×C3×#624—21no—
12C2×C3×C3⋊(C3⋊C8)C2212yes—
13C32×C4⋊Dic3C2211yescharacteristic
14C32×C4⋊Dic3C2212yes—
15#88804—21no—
16C3×#3724—21no—
17#64104—31no—
18C3×D3×C4⋊C4—32no—
19#63909—32no—
20#63909—61no—
21#63900D311yescharacteristic, complemented
22#63636—32no—
23C3×#808—32no—
24C3×#808—32no—
25#63574D312yescomplemented
26#63564—32no—
27#63564—32no—
28#63564—62no—
29#63557—31no—
30(C2×D3)⋊(C3⋊C8)—32no—
31C22×C3⋊(C3⋊D3)—21no—
32C22×C3×C3⋊D3D411yescharacteristic
33C2×C33×C4C2311yescharacteristic
34C4⋊(C3⋊(C3⋊D3))—22no—
35C2×C3×C3⋊Dic3D411yescharacteristic
36C3×C4×C3⋊D3—22no—
37C2×C32×Dic3—21no—
38C2×C32×Dic3—22no—
39#6292—42no—
40C3×#624—42no—
41C3×C3⋊(C3⋊C8)—22no—
42#4968—91no—
43#4863—92no—
44C2×C4⋊(C3⋊D3)—31no—
45C2×C4⋊(C3⋊D3)—32no—
46C2×C4⋊(C3⋊D3)—62no—
47C2×C4⋊(C3⋊D3)—61no—
48C2×C4⋊(C3⋊D3)—61no—
49C2×C4⋊(C3⋊D3)—121no—
50C2×C4×C3⋊D3C2×D311yescharacteristic
51C2×C3×C4×D3—32no—
52C2×C3×C4×D3—62no—
53C2×C3×C4×D3—61no—
54C2×#626—62no—
55C2×#626—62no—
56C2×#624—21no—
57C32×C4⋊C4—32no—
58C32×C4⋊C4—31no—
59#3752—61no—
60C2×C3⋊(C3⋊C8)C2×D312yes—
61C3×C4⋊Dic3—21no—
62C3×C4⋊Dic3—22no—
63C3×C4⋊Dic3C2×D312yes—
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#3722—62no—
72#3722—62no—
73C22×C33C2×D411yescharacteristic
74C2×C3⋊(C3⋊D3)—41no—
75C2×C3⋊(C3⋊D3)—41no—
76C2×C3⋊(C3⋊D3)—41no—
77C2×C3×C3⋊D3—21no—
78C2×C3×C3⋊D3—21no—
79C2×C3×C3⋊D3SD1612yes—
80C33×C4#4112yes—
81C3×C3⋊Dic3—21no—
82C32×Dic3—41no—
83C32×Dic3—42no—
84C2×C3×C4⋊C4—91no—
85#930—91no—
86#895—92no—
87#895—181no—
88D3×C4⋊C4—32no—
89C3×#106—92no—
90#830—91no—
91#821—92no—
92(C2×D3)⋊C8—92no—
93#809—92no—
94#809—92no—
95#809—182no—
96#808—32no—
97#808—32no—
98C22×C3⋊D3—61no—
99C22×C3⋊D3—62no—
100C22×C3⋊D3—62no—
101C22×C3⋊D3—122no—
102C22×C3⋊D3—122no—
103C22×C3⋊D3#6711yescharacteristic
104C22×C3×D3—32no—
105C22×C3×D3—61no—
106C2×C32×C4—22no—
107C2×C32×C4—21no—
108C2×C32×C4—41no—
109C2×C32×C4—61no—
110C2×C32×C4—62no—
111C2×C32×C4C22×D311yescharacteristic
112C2×C32×C4C22×D312yes—
113C2×C3⋊Dic3#6711yescharacteristic
114C4⋊(C3⋊D3)—62no—
115C4⋊(C3⋊D3)—64no—
116C4⋊(C3⋊D3)—62no—
117C4⋊(C3⋊D3)—62no—
118C4⋊(C3⋊D3)—122no—
119C4⋊(C3⋊D3)—122no—
120C4⋊(C3⋊D3)—124no—
121C4⋊(C3⋊D3)—122no—
122C4×C3⋊D3—22no—
123C2×C3×Dic3—22no—
124C2×C3×Dic3—22no—
125C2×C3×Dic3—22no—
126C2×C3×Dic3—22no—
127C2×C3×Dic3—32no—
128C2×C3×Dic3—42no—
129C2×C3×Dic3—61no—
130C3×C4×D3—62no—
131C3×C4×D3—62no—
132C3×C4×D3—122no—
133C3×C4×D3—124no—
134#626—124no—
135#626—124no—
136#624—42no—
137C3⋊(C3⋊C8)—22no—
138C3×C3⋊C8—62no—
139C3×C3⋊C8—62no—
140C3×C3⋊C8—122no—
141#481—271noSylow
142C2×C33#12311yescharacteristic
143C2×C33C2×SD1611yescharacteristic
144C2×C33AGL(1,ℤ/8)11yescharacteristic
145C3⋊(C3⋊D3)—81no—
146C3×C3⋊D3—22no—
147C22×C3×C4—91no—
148C22×C3×C4—181no—
149C2×#67—181no—
150C2×D12—91no—
151C2×D12—92no—
152C2×D12—182no—
153C2×D12—181no—
154C2×D12—182no—
155C2×D12—182no—
156C2×D12—361no—
157C2×C4×D3—32no—
158C2×C4×D3—61no—
159C2×C4×D3—62no—
160C2×C3×C8—92no—
161C3×C4⋊C4—32no—
162C3×C4⋊C4—32no—
163C3×C4⋊C4—32no—
164C3×C4⋊C4—61no—
165C3×C4⋊C4—62no—
166C3×C4⋊C4—181no—
167(C2×D3)⋊C4—181no—
168(C2×D3)⋊C4—182no—
169(C2×D3)⋊C4—182no—
170C4⋊Dic3D3211yescharacteristic
171C4⋊Dic3D3212yes—
172Dic3⋊C4—62no—
173C2×C3⋊C8—32no—
174C2×C3⋊C8—32no—
175C2×C3⋊C8—62no—
176C22×C32—22no—
177C22×C32—21no—
178C22×C32—41no—
179C22×C32D3×D412yes—
180C22×C32C2×#6711yescharacteristic
181C2×C3⋊D3—21no—
182C2×C3⋊D3—21no—
183C2×C3⋊D3—124no—
184C2×C3⋊D3—122no—
185C2×C3⋊D3—124no—
186C2×C3⋊D3—122no—
187C2×C3⋊D3—121no—
188C2×C3⋊D3—121no—
189C2×C3⋊D3—121no—
190C2×C3⋊D3—122no—
191C2×C3⋊D3—122no—
192C2×C3⋊D3—122no—
193C2×C3⋊D3—122no—
194C2×C3⋊D3—242no—
195C2×C3⋊D3—242no—
196C2×C3⋊D3—242no—
197C2×C3⋊D3—242no—
198C2×C3⋊D3#21512yes—
199C2×C3×D3—34no—
200C2×C3×D3—62no—
201C2×C3×D3—62no—
202C2×C3×D3—62no—
203C2×C3×D3—121no—
204C2×C3×D3—121no—
205C32×C4—22no—
206C32×C4—22no—
207C32×C4—22no—
208C32×C4—42no—
209C32×C4—121no—
210C32×C4—122no—
211C32×C4#23512yes—
212C32×C4#23512yes—
213C32×C4#23912yes—
214C3⋊Dic3—21no—
215C3×Dic3—42no—
216C3×Dic3—42no—
217C3×Dic3—42no—
218C3×Dic3—42no—
219C3×Dic3—62no—
220C3×Dic3—82no—
221C3×Dic3—121no—
222#121—271no—
223C2×C4⋊C4—91no—
224#106—92no—
225#102—272no—
226C22⋊C8—271no—
227C33#48111yescharacteristic, complemented, Sylow
228C23×C3—91no—
229C22×D3—32no—
230C22×D3—61no—
231C22×D3—182no—
232C22×D3—182no—
233C22×D3—181no—
234C22×D3—362no—
235C22×D3—362no—
236C2×C3×C4—22no—
237C2×C3×C4—21no—
238C2×C3×C4—41no—
239C2×C3×C4—62no—
240C2×C3×C4—62no—
241C2×C3×C4—62no—
242C2×C3×C4—62no—
243C2×C3×C4—91no—
244C2×C3×C4—122no—
245C2×C3×C4—182no—
246C2×C3×C4—362no—
247C2×C3×C4C2×D3212yes—
248C2×C3×C4C2×D3211yescharacteristic
249#67—362no—
250C2×Dic3—22no—
251C2×Dic3—21no—
252C2×Dic3—32no—
253C2×Dic3—61no—
254D12—184no—
255D12—182no—
256D12—182no—
257D12—182no—
258D12—362no—
259D12—362no—
260D12—364no—
261D12—364no—
262D12—364no—
263C4×D3—62no—
264C4×D3—62no—
265C4×D3—122no—
266C4×D3—124no—
267C3×C8—182no—
268C3⋊C8—62no—
269C3⋊C8—62no—
270C3⋊C8—122no—
271C2×C32—22no—
272C2×C32—21no—
273C2×C32—21no—
274C2×C32—22no—
275C2×C32—22no—
276C2×C32—21no—
277C2×C32—41no—
278C2×C32—41no—
279C2×C32—41no—
280C2×C32#89412yes—
281C2×C32D3×SD1612yes—
282C2×C32#91412yes—
283C2×C32#92911yescharacteristic
284C2×C32#93211yescharacteristic
285C2×C32C2×#21511yescharacteristic
286C3⋊D3—22no—
287C3⋊D3—242no—
288C3⋊D3—244no—
289C3⋊D3—244no—
290C3⋊D3—241no—
291C3⋊D3—242no—
292C3×D3—64no—
293C3×D3—122no—
294C2×D4—271no—
295C2×D4—541no—
296C22×C4—91no—
297C22×C4—181no—
298C2×C8—92no—
299C4⋊C4—32no—
300C4⋊C4—31no—
301C4⋊C4—181no—
302C22⋊C4—541no—
303C22×C3—21no—
304C22×C3—22no—
305C22×C3—41no—
306C22×C3—92no—
307C22×C3—181no—
308C22×C3—181no—
309C22×C3D3×#6712yes—
310C22×C3#381111yescharacteristic
311C2×D3—34no—
312C2×D3—62no—
313C2×D3—62no—
314C2×D3—62no—
315C2×D3—121no—
316C2×D3—121no—
317C2×D3—364no—
318C2×D3—362no—
319C2×D3—362no—
320C2×D3—362no—
321C2×D3—364no—
322C2×D3—362no—
323C2×D3—361no—
324C2×D3—361no—
325C2×D3—362no—
326C2×D3—362no—
327C2×D3—361no—
328C2×D3—722no—
329C2×D3—722no—
330C2×D3—722no—
331C2×D3—722no—
332C3×C4—22no—
333C3×C4—22no—
334C3×C4—22no—
335C3×C4—42no—
336C3×C4—122no—
337C3×C4—122no—
338C3×C4—122no—
339C3×C4—122no—
340C3×C4—181no—
341C3×C4—242no—
342C3×C4#379812yes—
343C3×C4#379912yes—
344C3×C4#379912yes—
345Dic3—41no—
346Dic3—42no—
347Dic3—62no—
348Dic3—121no—
349C32—22no—
350C32—21no—
351C32—41no—
352C32#486312yescomplemented
353C32#496811yescharacteristic, complemented
354C23—91no—
355C23—541no—
356D4—542no—
357D4—1082no—
358C2×C4—61no—
359C2×C4—62no—
360C2×C4—91no—
361C2×C4—182no—
362C2×C4—362no—
363C2×C4D3311yescharacteristic
364C8—182no—
365C2×C3—21no—
366C2×C3—22no—
367C2×C3—22no—
368C2×C3—21no—
369C2×C3—21no—
370C2×C3—22no—
371C2×C3—41no—
372C2×C3—41no—
373C2×C3—41no—
374C2×C3—182no—
375C2×C3#6385012yes—
376C2×C3#6389711yescharacteristic
377C2×C3#6393011yescharacteristic
378C2×C3#6393412yes—
379C2×C3D3×#21512yes—
380C2×C3#6394111yescharacteristic
381D3—64no—
382D3—122no—
383D3—722no—
384D3—724no—
385D3—724no—
386D3—722no—
387D3—721no—
388C22—92no—
389C22—181no—
390C22—181no—
391C22—1081no—
392C22—1081no—
393C22—1081no—
394C22#8895211yescharacteristic
395C4—121no—
396C4—122no—
397C4—181no—
398C4#8896512yes—
399C3—21no—
400C3—22no—
401C3—41no—
402C3#1059042312yescomplemented
403C3#1059044511yescharacteristic, complemented
404C2—182no—
405C2—2161no—
406C2#1172283811yescharacteristic
407C2#1172287311yescharacteristic
408C2#1172299311yescharacteristic
subgroupG/H
centerC22#88952
commutator, Fitting, cosocleC2×C33×C4C23
Frattini, hypercenterC2×C4D33
socleC22×C33C2×D4
prime radical#11723526C2
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. #11722035C2semidirect
5. C3×#63574C2semidirect
21. #63900D3semidirect
25. #63574D3semidirect
227. C33#481semidirect
352. C32#4863semidirect
353. C32#4968semidirect
402. C3#10590423semidirect
403. C3#10590445semidirect

Maximal subgroups

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

subgroupindex
#117235262
C3×#639002
#117220722
#117220352
C3×#635742
#105904453
#105904233