Finite Group Databasebeta

← #13161501

Subgroups — #13161501

Order 1728.

710 conjugacy classes of subgroups.

The subgroup lattice has 466 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#64081C211yesmaximal, characteristic
2C2×#88809C211yesmaximal, characteristic
3C2×#88808C211yesmaximal, characteristic
4C3×#63630C211yesmaximal, characteristic, complemented
5C3×C4×C3⋊(C3⋊C8)C211yesmaximal, characteristic, complemented
6#11722071C211yesmaximal, characteristic, complemented
7#11722070C211yesmaximal, characteristic, complemented
8#10590321—31nomaximal
9#10590319—31nomaximal
10#10590318—31nomaximal
11C2×C3×C4×C3⋊D3C2211yescharacteristic
12C33×C42C2211yescharacteristic
13C3×#3752C412yes—
14C3×#3750C412yes—
15C3×C4×C3⋊Dic3C2211yescharacteristic
16C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
17C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
18C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
19C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
20#88809C414yescomplemented
21#88808C414yescomplemented
22#64081D311yescharacteristic
23C3×#872—31no—
24C3×#872—31no—
25C3×#872—61no—
26C2×#3712—31no—
27C2×#3711—31no—
28C2×#3711—31no—
29C2×#3710—31no—
30C2×D3×C3⋊C8—31no—
31C2×D3×C3⋊C8—31no—
32#63630D311yescharacteristic, complemented
33C4×C3⋊(C3⋊C8)D311yescharacteristic, complemented
34C3×#803—31no—
35C3×#803—31no—
36C3×#803—61no—
37C3×C4×C3⋊C8—31no—
38C3×C4×C3⋊C8—31no—
39C3×C4×C3⋊C8—61no—
40#63555—31no—
41#63554—31no—
42#63554—31no—
43#63553—31no—
44Dic3×C3⋊C8—31no—
45Dic3×C3⋊C8—31no—
46C22×C3×C3⋊D3C2×C411yescharacteristic
47C2×C33×C4C2×C412yes—
48C2×C33×C4C2311yescharacteristic
49C2×C3×C3⋊Dic3C2×C412yes—
50C2×C3×C3⋊Dic3C2×C411yescharacteristic
51C3×C4×C3⋊D3C2×C412yes—
52C3×C4×C3⋊D3C2×C412yes—
53C3×C3⋊(C3⋊C8)C2×C412yes—
54C3×C3⋊(C3⋊C8)C2×C412yes—
55C3×C3⋊(C3⋊C8)C2×C412yes—
56C3×C3⋊(C3⋊C8)C2×C412yes—
57#4997—91no—
58#4708—91no—
59#4689—91no—
60C2×C4×C3⋊D3C2×D311yescharacteristic
61C2×C3×C4×D3—31no—
62C2×C3×C4×D3—31no—
63C2×C3×C4×D3—61no—
64C32×C42—21no—
65C32×C42—21no—
66C32×C42—21no—
67C32×C42—41no—
68C32×C42C2×D311yescharacteristic
69C32×C42C2×D311yescharacteristic
70C32×C42C2×D311yescharacteristic
71#3752Dic312yes—
72#3750Dic312yes—
73C4×C3⋊Dic3C2×D311yescharacteristic
74C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
75C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
76C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
77C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
78C3×(C2×D3)⋊C4—32no—
79C3×(C2×D3)⋊C4—32no—
80C3×(C2×D3)⋊C4—62no—
81C3×Dic3⋊C4—32no—
82C3×Dic3⋊C4—32no—
83C3×Dic3⋊C4—62no—
84C3×C4×Dic3—31no—
85C3×C4×Dic3—31no—
86C3×C4×Dic3—61no—
87C2×C3×C3⋊C8—31no—
88C2×C3×C3⋊C8—31no—
89C2×C3×C3⋊C8—31no—
90C2×C3×C3⋊C8—31no—
91C2×C3×C3⋊C8—31no—
92C2×C3×C3⋊C8—31no—
93C2×C3×C3⋊C8—31no—
94C2×C3×C3⋊C8—31no—
95C2×C3×C3⋊C8—61no—
96C2×C3×C3⋊C8—61no—
97C2×C3×C3⋊C8—61no—
98C2×C3×C3⋊C8—61no—
99#3712—34no—
100#3711—34no—
101#3711—34no—
102#3710—34no—
103D3×C3⋊C8—34no—
104D3×C3⋊C8—34no—
105C22×C33C22×C411yescharacteristic
106C2×C3×C3⋊D3—22no—
107C2×C3×C3⋊D3C4212yes—
108C33×C4—22no—
109C33×C4C22×C411yescharacteristic
110C33×C4C22×C411yescharacteristic
111C3×C3⋊Dic3—22no—
112C3×C3⋊Dic3C4212yes—
113C3×#117—91no—
114C2×#208—91no—
115C22×C3⋊C8—91no—
116C2×ΓL(1,𝔽25)—91no—
117C2×ΓL(1,𝔽25)—91no—
118C2×D3×C8—91no—
119C2×D3×C8—91no—
120#872—31no—
121#872—31no—
122#872—61no—
123C3×#97—91no—
124C3×C4×C8—91no—
125#815—91no—
126#815—91no—
127C8×Dic3—91no—
128C8×Dic3—91no—
129#803—31no—
130#803—31no—
131#803—61no—
132#803—91no—
133C4×C3⋊C8—31no—
134C4×C3⋊C8—31no—
135C4×C3⋊C8—61no—
136C4×C3⋊C8—91no—
137C22×C3⋊D3C2×Dic311yescharacteristic
138C22×C3×D3—31no—
139C22×C3×D3—31no—
140C22×C3×D3—61no—
141C2×C32×C4—22no—
142C2×C32×C4—22no—
143C2×C32×C4—22no—
144C2×C32×C4—21no—
145C2×C32×C4—21no—
146C2×C32×C4—21no—
147C2×C32×C4—42no—
148C2×C32×C4—41no—
149C2×C32×C4C4×D312yes—
150C2×C32×C4C4×D312yes—
151C2×C32×C4C2×Dic312yes—
152C2×C32×C4C22×D311yescharacteristic
153C2×C32×C4C22×D311yescharacteristic
154C2×C32×C4C22×D311yescharacteristic
155C2×C3⋊Dic3C2×Dic312yes—
156C2×C3⋊Dic3C2×Dic311yescharacteristic
157C4×C3⋊D3C4×D312yes—
158C4×C3⋊D3C4×D312yes—
159C2×C3×Dic3—31no—
160C2×C3×Dic3—31no—
161C2×C3×Dic3—32no—
162C2×C3×Dic3—32no—
163C2×C3×Dic3—61no—
164C2×C3×Dic3—62no—
165C3×C4×D3—32no—
166C3×C4×D3—32no—
167C3×C4×D3—32no—
168C3×C4×D3—32no—
169C3×C4×D3—62no—
170C3×C4×D3—62no—
171C3⋊(C3⋊C8)C4×D312yes—
172C3⋊(C3⋊C8)C4×D312yes—
173C3⋊(C3⋊C8)C4×D312yes—
174C3⋊(C3⋊C8)C4×D312yes—
175C3×C3⋊C8—32no—
176C3×C3⋊C8—32no—
177C3×C3⋊C8—32no—
178C3×C3⋊C8—32no—
179C3×C3⋊C8—32no—
180C3×C3⋊C8—32no—
181C3×C3⋊C8—32no—
182C3×C3⋊C8—32no—
183C3×C3⋊C8—62no—
184C3×C3⋊C8—62no—
185C3×C3⋊C8—62no—
186C3×C3⋊C8—62no—
187#405—271noSylow
188C2×C33C2×C4211yescharacteristic
189C2×C33#13112yes—
190C3×C3⋊D3—22no—
191C22×C3×C4—91no—
192C2×C4×D3—31no—
193C2×C4×D3—31no—
194C2×C4×D3—61no—
195C2×C3×C8—91no—
196C2×C3×C8—91no—
197C2×C3×C8—91no—
198C2×C3×C8—91no—
199C3×C4⋊C4—92no—
200C3×C22⋊C4—92no—
201C3×C42—21no—
202C3×C42—21no—
203C3×C42—21no—
204C3×C42—41no—
205C3×C42—91no—
206C3×C42D3211yescharacteristic
207C3×C42D3211yescharacteristic
208C3×C42D3211yescharacteristic
209(C2×D3)⋊C4—32no—
210(C2×D3)⋊C4—32no—
211(C2×D3)⋊C4—62no—
212Dic3⋊C4—32no—
213Dic3⋊C4—32no—
214Dic3⋊C4—62no—
215C4×Dic3—31no—
216C4×Dic3—31no—
217C4×Dic3—61no—
218#208—94no—
219C2×C3⋊C8—31no—
220C2×C3⋊C8—31no—
221C2×C3⋊C8—31no—
222C2×C3⋊C8—31no—
223C2×C3⋊C8—31no—
224C2×C3⋊C8—31no—
225C2×C3⋊C8—31no—
226C2×C3⋊C8—31no—
227C2×C3⋊C8—61no—
228C2×C3⋊C8—61no—
229C2×C3⋊C8—61no—
230C2×C3⋊C8—61no—
231C2×C3⋊C8—94no—
232ΓL(1,𝔽25)—94no—
233ΓL(1,𝔽25)—94no—
234D3×C8—94no—
235D3×C8—94no—
236C22×C32—21no—
237C22×C32—21no—
238C22×C32—21no—
239C22×C32—41no—
240C22×C32C2×C4×D311yescharacteristic
241C22×C32C2×C4×D311yescharacteristic
242C22×C32C22×Dic311yescharacteristic
243C2×C3⋊D3—22no—
244C2×C3⋊D3C4×Dic312yes—
245C2×C3×D3—32no—
246C2×C3×D3—32no—
247C2×C3×D3—62no—
248C2×C3×D3—62no—
249C2×C3×D3—62no—
250C2×C3×D3—122no—
251C32×C4—22no—
252C32×C4—24no—
253C32×C4—22no—
254C32×C4—22no—
255C32×C4—21no—
256C32×C4—21no—
257C32×C4—21no—
258C32×C4—21no—
259C32×C4—21no—
260C32×C4—21no—
261C32×C4—41no—
262C32×C4—41no—
263C32×C4—42no—
264C32×C4—42no—
265C32×C4—44no—
266C32×C4C2×C4×D311yescharacteristic
267C32×C4C2×C4×D311yescharacteristic
268C32×C4C2×C4×D311yescharacteristic
269C32×C4C2×C4×D311yescharacteristic
270C32×C4C2×C4×D311yescharacteristic
271C32×C4C2×C4×D311yescharacteristic
272C3⋊Dic3—22no—
273C3⋊Dic3C4×Dic312yes—
274C3×Dic3—32no—
275C3×Dic3—32no—
276C3×Dic3—62no—
277C3×Dic3—62no—
278C3×Dic3—62no—
279C3×Dic3—122no—
280C2×M16—271no—
281C22×C8—271no—
282#117—91no—
283#97—91no—
284#97—271no—
285C4×C8—91no—
286C4×C8—271no—
287C33#40511yescharacteristic, complemented, Sylow
288C23×C3—91no—
289C22×D3—31no—
290C22×D3—31no—
291C22×D3—61no—
292C2×C3×C4—22no—
293C2×C3×C4—22no—
294C2×C3×C4—22no—
295C2×C3×C4—21no—
296C2×C3×C4—21no—
297C2×C3×C4—21no—
298C2×C3×C4—42no—
299C2×C3×C4—41no—
300C2×C3×C4—91no—
301C2×C3×C4—92no—
302C2×C3×C4—92no—
303C2×C3×C4—92no—
304C2×C3×C4D3×Dic312yes—
305C2×C3×C4D3×Dic312yes—
306C2×C3×C4#62412yes—
307C2×C3×C4C2×D3211yescharacteristic
308C2×C3×C4C2×D3211yescharacteristic
309C2×C3×C4C2×D3211yescharacteristic
310C2×Dic3—31no—
311C2×Dic3—31no—
312C2×Dic3—32no—
313C2×Dic3—32no—
314C2×Dic3—61no—
315C2×Dic3—62no—
316C4×D3—32no—
317C4×D3—32no—
318C4×D3—32no—
319C4×D3—32no—
320C4×D3—62no—
321C4×D3—62no—
322C3×C8—92no—
323C3×C8—92no—
324C3×C8—92no—
325C3×C8—92no—
326C3⋊C8—32no—
327C3⋊C8—32no—
328C3⋊C8—32no—
329C3⋊C8—32no—
330C3⋊C8—32no—
331C3⋊C8—32no—
332C3⋊C8—32no—
333C3⋊C8—32no—
334C3⋊C8—62no—
335C3⋊C8—62no—
336C3⋊C8—62no—
337C3⋊C8—62no—
338C2×C32—22no—
339C2×C32—22no—
340C2×C32—22no—
341C2×C32—21no—
342C2×C32—21no—
343C2×C32—21no—
344C2×C32—41no—
345C2×C32—42no—
346C2×C32C42×D311yescharacteristic
347C2×C32C42×D311yescharacteristic
348C2×C32#90112yes—
349C2×C32#90712yes—
350C2×C32C2×C4×Dic311yescharacteristic
351C2×C32#94812yes—
352C3⋊D3—22no—
353C3×D3—62no—
354C3×D3—62no—
355C3×D3—122no—
356C22×C4—91no—
357M16—274no—
358C2×C8—91no—
359C2×C8—91no—
360C2×C8—91no—
361C2×C8—91no—
362C2×C8—274no—
363C4⋊C4—92no—
364C22⋊C4—92no—
365C42—91no—
366C42#111311yescharacteristic
367C22×C3—21no—
368C22×C3—21no—
369C22×C3—21no—
370C22×C3—41no—
371C22×C3—92no—
372C22×C3—182no—
373C22×C3C2×D3×Dic311yescharacteristic
374C22×C3C2×D3×Dic311yescharacteristic
375C22×C3C2×#62411yescharacteristic
376C2×D3—32no—
377C2×D3—32no—
378C2×D3—62no—
379C2×D3—62no—
380C2×D3—62no—
381C2×D3—122no—
382C3×C4—24no—
383C3×C4—22no—
384C3×C4—22no—
385C3×C4—22no—
386C3×C4—21no—
387C3×C4—21no—
388C3×C4—21no—
389C3×C4—21no—
390C3×C4—21no—
391C3×C4—21no—
392C3×C4—42no—
393C3×C4—42no—
394C3×C4—44no—
395C3×C4—41no—
396C3×C4—41no—
397C3×C4—92no—
398C3×C4—182no—
399C3×C4C4×D3211yescharacteristic
400C3×C4C4×D3211yescharacteristic
401C3×C4C4×D3211yescharacteristic
402C3×C4C4×D3211yescharacteristic
403C3×C4C4×D3211yescharacteristic
404C3×C4C4×D3211yescharacteristic
405Dic3—32no—
406Dic3—32no—
407Dic3—62no—
408Dic3—62no—
409Dic3—62no—
410Dic3—122no—
411C32—21no—
412C32—21no—
413C32—21no—
414C32—41no—
415C32#468911yescharacteristic, complemented
416C32#470811yescharacteristic, complemented
417C32#499711yescharacteristic, complemented
418C23—91no—
419C2×C4—91no—
420C2×C4—92no—
421C2×C4—92no—
422C2×C4—92no—
423C2×C4#629412yes—
424C2×C4C2×#111311yescharacteristic
425C8—92no—
426C8—92no—
427C8—92no—
428C8—92no—
429C2×C3—22no—
430C2×C3—22no—
431C2×C3—22no—
432C2×C3—21no—
433C2×C3—21no—
434C2×C3—21no—
435C2×C3—42no—
436C2×C3—41no—
437C2×C3—182no—
438C2×C3#6381412yes—
439C2×C3#6381512yes—
440C2×C3#6381712yes—
441C2×C3C4×D3×Dic311yescharacteristic
442C2×C3C4×D3×Dic311yescharacteristic
443C2×C3C4×#62411yescharacteristic
444D3—62no—
445D3—62no—
446D3—122no—
447C22—92no—
448C22—182no—
449C22C2×#629411yescharacteristic
450C4—22no—
451C4—92no—
452C4—182no—
453C4C4×#111311yescharacteristic
454C4C4×#111311yescharacteristic
455C3—21no—
456C3—21no—
457C3—21no—
458C3—41no—
459C3#1059031811yescharacteristic, complemented
460C3#1059031911yescharacteristic, complemented
461C3#1059032111yescharacteristic, complemented
462C2—182no—
463C2#1172365712yes—
464C2C4×#629411yescharacteristic
subgroupG/H
center, FrattiniC2×C4C2×#1113
commutatorC2×C33C2×C42
FittingC33×C42C22
socleC22×C33C22×C4
prime radicalC22×C3×C3⋊D3C2×C4
cosocleC2×C33×C4C23
hypercenterC42#1113
hypocenterC33#405
LCSG ▷ C2×C33 ▷ C33
UCSC1 ◁ C2×C4 ◁ C42

Complements

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

NH ≅ G/Ntype
4. C3×#63630C2semidirect
5. C3×C4×C3⋊(C3⋊C8)C2semidirect
6. #11722071C2semidirect
7. #11722070C2semidirect
20. #88809C4semidirect
21. #88808C4semidirect
32. #63630D3semidirect
33. C4×C3⋊(C3⋊C8)D3semidirect
287. C33#405semidirect
415. C32#4689semidirect
416. C32#4708semidirect
417. C32#4997semidirect
459. C3#10590318semidirect
460. C3#10590319semidirect
461. C3#10590321semidirect

Maximal subgroups

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

subgroupindex
C3×#640812
C2×#888092
C2×#888082
C3×#636302
C3×C4×C3⋊(C3⋊C8)2
#117220712
#117220702
#105903213
#105903193
#105903183