Finite Group Databasebeta

← #13155294

Subgroups — #13155294

Order 1728.

694 conjugacy classes of subgroups.

The subgroup lattice has 458 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#88808C211yesmaximal, characteristic
2C2×#88787C211yesmaximal, characteristic
3C3×#63874C211yesmaximal, characteristic
4#11722019C211yesmaximal, characteristic, complemented
5Dic3×C3⋊(C3⋊C8)C211yesmaximal, characteristic, complemented
6C3×#63554C211yesmaximal, characteristic, complemented
7C3×Dic3×C3⋊C8C211yesmaximal, characteristic, complemented
8#10590910—31nomaximal
9#10590319—31nomaximal
10#10590318—31nomaximal
11C2×C3×C4×C3⋊D3C2211yescharacteristic
12C2×C3⋊(C3⋊(C3⋊C8))C2211yescharacteristic
13C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
14C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
15C32×C4×Dic3C2211yescharacteristic
16C32×C4×Dic3C2211yescharacteristic
17C2×C32×C3⋊C8C2211yescharacteristic
18#88808C414yescomplemented
19#88787C414yescomplemented
20C3×#3724C412yes—
21C3×#3722C412yes—
22C2×#3743—31no—
23C3×#872—31no—
24C3×#872—31no—
25#63874D311yescharacteristic
26C2×#3711—31no—
27C2×#3711—31no—
28C2×#3711—61no—
29C2×#3710—31no—
30C2×D3×C3⋊C8—31no—
31C2×D3×C3⋊C8—31no—
32#63630—31no—
33C4×C3⋊(C3⋊C8)—31no—
34C3×#815—31no—
35C3×C8×Dic3—31no—
36C3×#803—31no—
37C3×C4×C3⋊C8—31no—
38#63554—31no—
39#63554—31no—
40#63554—61no—
41#63554D311yescharacteristic, complemented
42Dic3×C3⋊C8—31no—
43Dic3×C3⋊C8—31no—
44Dic3×C3⋊C8—61no—
45Dic3×C3⋊C8D311yescharacteristic, complemented
46C22×C3×C3⋊D3C2×C411yescharacteristic
47C2×C33×C4C2311yescharacteristic
48C2×C3×C3⋊Dic3C2×C411yescharacteristic
49C3×C4×C3⋊D3C2×C412yes—
50C3×C4×C3⋊D3C2×C412yes—
51C2×C32×Dic3C2×C412yes—
52C2×C32×Dic3C2×C412yes—
53C3⋊(C3⋊(C3⋊C8))C2×C412yes—
54C3×C3⋊(C3⋊C8)C2×C412yes—
55C3×C3⋊(C3⋊C8)C2×C412yes—
56C32×C3⋊C8C2×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—
64C2×C32×C8—31no—
65C32×C42—31no—
66C32×C42—31no—
67C2×C3⋊(C3⋊C8)—31no—
68C2×C3⋊(C3⋊C8)—31no—
69C2×C3⋊(C3⋊C8)—31no—
70C2×C3⋊(C3⋊C8)—61no—
71C2×C3⋊(C3⋊C8)—61no—
72C2×C3⋊(C3⋊C8)—61no—
73C2×C3⋊(C3⋊C8)—121no—
74C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
75C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
76#3743—34no—
77C3×(C2×D3)⋊C4—32no—
78C3×(C2×D3)⋊C4—32no—
79C3×Dic3⋊C4—32no—
80C3×Dic3⋊C4—32no—
81C3×C4×Dic3—21no—
82C3×C4×Dic3—21no—
83C3×C4×Dic3C2×D311yescharacteristic
84C3×C4×Dic3C2×D311yescharacteristic
85C3×C4×Dic3C2×D311yescharacteristic
86C3×C4×Dic3C2×D311yescharacteristic
87C2×C3×C3⋊C8—21no—
88C2×C3×C3⋊C8—31no—
89C2×C3×C3⋊C8—31no—
90C2×C3×C3⋊C8—31no—
91C2×C3×C3⋊C8—31no—
92C2×C3×C3⋊C8—61no—
93C2×C3×C3⋊C8—61no—
94C2×C3×C3⋊C8C2×D311yescharacteristic
95C2×C3×C3⋊C8C2×D311yescharacteristic
96#3724Dic312yes—
97#3722Dic312yes—
98#3711—34no—
99#3711—34no—
100#3711—64no—
101#3710—34no—
102D3×C3⋊C8—34no—
103D3×C3⋊C8—34no—
104C22×C33C22×C411yescharacteristic
105C2×C3×C3⋊D3—22no—
106C2×C3×C3⋊D3C4212yes—
107C33×C4C22×C411yescharacteristic
108C33×C4C22×C411yescharacteristic
109C3×C3⋊Dic3C4212yes—
110C32×Dic3—22no—
111C32×Dic3—22no—
112C3×#117—91no—
113C2×#208—91no—
114C22×C3⋊C8—91no—
115C2×ΓL(1,𝔽25)—91no—
116C2×ΓL(1,𝔽25)—91no—
117C2×ΓL(1,𝔽25)—181no—
118C2×D3×C8—91no—
119C2×D3×C8—91no—
120#872—31no—
121#872—31no—
122C3×#97—91no—
123C3×C4×C8—91no—
124#815—31no—
125#815—91no—
126C8×Dic3—31no—
127C8×Dic3—91no—
128#803—31no—
129#803—91no—
130#803—91no—
131#803—181no—
132C4×C3⋊C8—31no—
133C4×C3⋊C8—91no—
134C4×C3⋊C8—91no—
135C4×C3⋊C8—181no—
136C22×C3⋊D3C2×Dic311yescharacteristic
137C22×C3×D3—31no—
138C22×C3×D3—31no—
139C22×C3×D3—61no—
140C2×C32×C4—21no—
141C2×C32×C4—21no—
142C2×C32×C4—21no—
143C2×C32×C4—32no—
144C2×C32×C4—32no—
145C2×C32×C4—41no—
146C2×C32×C4C22×D311yescharacteristic
147C2×C32×C4C22×D311yescharacteristic
148C2×C32×C4C22×D311yescharacteristic
149C2×C3⋊Dic3C2×Dic311yescharacteristic
150C4×C3⋊D3C4×D312yes—
151C4×C3⋊D3C4×D312yes—
152C2×C3×Dic3—22no—
153C2×C3×Dic3—22no—
154C2×C3×Dic3—31no—
155C2×C3×Dic3—31no—
156C2×C3×Dic3—61no—
157C2×C3×Dic3C4×D312yes—
158C2×C3×Dic3C4×D312yes—
159C2×C3×Dic3C2×Dic312yes—
160C2×C3×Dic3C2×Dic312yes—
161C3×C4×D3—32no—
162C3×C4×D3—32no—
163C3×C4×D3—32no—
164C3×C4×D3—32no—
165C3×C4×D3—62no—
166C3×C4×D3—62no—
167C32×C8—32no—
168C3⋊(C3⋊C8)—32no—
169C3⋊(C3⋊C8)—32no—
170C3⋊(C3⋊C8)—32no—
171C3⋊(C3⋊C8)—62no—
172C3⋊(C3⋊C8)—62no—
173C3⋊(C3⋊C8)—62no—
174C3⋊(C3⋊C8)—122no—
175C3⋊(C3⋊C8)C4×D312yes—
176C3⋊(C3⋊C8)C4×D312yes—
177C3×C3⋊C8—22no—
178C3×C3⋊C8—32no—
179C3×C3⋊C8—32no—
180C3×C3⋊C8—32no—
181C3×C3⋊C8—32no—
182C3×C3⋊C8—62no—
183C3×C3⋊C8—62no—
184C3×C3⋊C8C4×D312yes—
185C3×C3⋊C8C4×D312yes—
186#405—271noSylow
187C2×C33C2×C4211yescharacteristic
188C2×C33#13112yes—
189C3×C3⋊D3—22no—
190C22×C3×C4—91no—
191C2×C4×D3—31no—
192C2×C4×D3—31no—
193C2×C4×D3—61no—
194C2×C3×C8—31no—
195C2×C3×C8—31no—
196C2×C3×C8—61no—
197C2×C3×C8—91no—
198C2×C3×C8—91no—
199C3×C4⋊C4—92no—
200C3×C22⋊C4—92no—
201C3×C42—31no—
202C3×C42—31no—
203C3×C42—31no—
204C3×C42—31no—
205C3×C42—61no—
206C3×C42—61no—
207(C2×D3)⋊C4—32no—
208(C2×D3)⋊C4—32no—
209Dic3⋊C4—32no—
210Dic3⋊C4—32no—
211C4×Dic3D3211yescharacteristic
212C4×Dic3D3211yescharacteristic
213#208—94no—
214C2×C3⋊C8—31no—
215C2×C3⋊C8—31no—
216C2×C3⋊C8—31no—
217C2×C3⋊C8—31no—
218C2×C3⋊C8—61no—
219C2×C3⋊C8—61no—
220C2×C3⋊C8—94no—
221C2×C3⋊C8—91no—
222C2×C3⋊C8—91no—
223C2×C3⋊C8—91no—
224C2×C3⋊C8—181no—
225C2×C3⋊C8—181no—
226C2×C3⋊C8—181no—
227C2×C3⋊C8—361no—
228C2×C3⋊C8D3211yescharacteristic
229ΓL(1,𝔽25)—94no—
230ΓL(1,𝔽25)—94no—
231ΓL(1,𝔽25)—184no—
232D3×C8—94no—
233D3×C8—94no—
234C22×C32—21no—
235C22×C32—21no—
236C22×C32—21no—
237C22×C32—41no—
238C22×C32C2×C4×D311yescharacteristic
239C22×C32C2×C4×D311yescharacteristic
240C22×C32C22×Dic311yescharacteristic
241C2×C3⋊D3—22no—
242C2×C3⋊D3C4×Dic312yes—
243C2×C3×D3—32no—
244C2×C3×D3—32no—
245C2×C3×D3—62no—
246C2×C3×D3—62no—
247C2×C3×D3—62no—
248C2×C3×D3—122no—
249C32×C4—21no—
250C32×C4—21no—
251C32×C4—21no—
252C32×C4—21no—
253C32×C4—21no—
254C32×C4—21no—
255C32×C4—41no—
256C32×C4—41no—
257C32×C4—62no—
258C32×C4—62no—
259C32×C4C2×C4×D311yescharacteristic
260C32×C4C2×C4×D311yescharacteristic
261C32×C4C2×C4×D311yescharacteristic
262C32×C4C2×C4×D311yescharacteristic
263C32×C4C2×C4×D311yescharacteristic
264C32×C4C2×C4×D311yescharacteristic
265C3⋊Dic3C4×Dic312yes—
266C3×Dic3—24no—
267C3×Dic3—22no—
268C3×Dic3—22no—
269C3×Dic3—22no—
270C3×Dic3—22no—
271C3×Dic3—32no—
272C3×Dic3—32no—
273C3×Dic3—42no—
274C3×Dic3—62no—
275C2×M16—271no—
276C22×C8—271no—
277#117—91no—
278#97—91no—
279#97—271no—
280C4×C8—91no—
281C4×C8—271no—
282C33#40511yescharacteristic, complemented, Sylow
283C23×C3—91no—
284C22×D3—31no—
285C22×D3—31no—
286C22×D3—61no—
287C2×C3×C4—21no—
288C2×C3×C4—21no—
289C2×C3×C4—21no—
290C2×C3×C4—32no—
291C2×C3×C4—32no—
292C2×C3×C4—32no—
293C2×C3×C4—32no—
294C2×C3×C4—41no—
295C2×C3×C4—62no—
296C2×C3×C4—62no—
297C2×C3×C4—91no—
298C2×C3×C4—92no—
299C2×C3×C4—92no—
300C2×C3×C4C2×D3211yescharacteristic
301C2×C3×C4C2×D3211yescharacteristic
302C2×C3×C4C2×D3211yescharacteristic
303C2×Dic3—31no—
304C2×Dic3—31no—
305C2×Dic3—61no—
306C2×Dic3D3×Dic312yes—
307C2×Dic3D3×Dic312yes—
308C4×D3—32no—
309C4×D3—32no—
310C4×D3—32no—
311C4×D3—32no—
312C4×D3—62no—
313C4×D3—62no—
314C3×C8—32no—
315C3×C8—32no—
316C3×C8—62no—
317C3×C8—92no—
318C3×C8—92no—
319C3⋊C8—32no—
320C3⋊C8—32no—
321C3⋊C8—32no—
322C3⋊C8—32no—
323C3⋊C8—62no—
324C3⋊C8—62no—
325C3⋊C8—92no—
326C3⋊C8—92no—
327C3⋊C8—92no—
328C3⋊C8—182no—
329C3⋊C8—182no—
330C3⋊C8—182no—
331C3⋊C8—362no—
332C3⋊C8#62412yes—
333C2×C32—22no—
334C2×C32—22no—
335C2×C32—22no—
336C2×C32—21no—
337C2×C32—21no—
338C2×C32—21no—
339C2×C32—41no—
340C2×C32—42no—
341C2×C32C42×D311yescharacteristic
342C2×C32C42×D311yescharacteristic
343C2×C32#90112yes—
344C2×C32#90712yes—
345C2×C32C2×C4×Dic311yescharacteristic
346C2×C32#94812yes—
347C3⋊D3—22no—
348C3×D3—62no—
349C3×D3—62no—
350C3×D3—122no—
351C22×C4—91no—
352M16—274no—
353C2×C8—31no—
354C2×C8—91no—
355C2×C8—91no—
356C2×C8—274no—
357C2×C8—271no—
358C4⋊C4—92no—
359C22⋊C4—92no—
360C42—31no—
361C42—31no—
362C22×C3—21no—
363C22×C3—21no—
364C22×C3—21no—
365C22×C3—41no—
366C22×C3—92no—
367C22×C3—182no—
368C22×C3C4×D3211yescharacteristic
369C22×C3C2×D3×Dic311yescharacteristic
370C22×C3C2×D3×Dic311yescharacteristic
371C2×D3—32no—
372C2×D3—32no—
373C2×D3—62no—
374C2×D3—62no—
375C2×D3—62no—
376C2×D3—122no—
377C3×C4—21no—
378C3×C4—21no—
379C3×C4—21no—
380C3×C4—21no—
381C3×C4—21no—
382C3×C4—21no—
383C3×C4—41no—
384C3×C4—41no—
385C3×C4—64no—
386C3×C4—62no—
387C3×C4—62no—
388C3×C4—62no—
389C3×C4—62no—
390C3×C4—92no—
391C3×C4—122no—
392C3×C4C4×D3211yescharacteristic
393C3×C4C4×D3211yescharacteristic
394C3×C4C4×D3211yescharacteristic
395C3×C4C4×D3211yescharacteristic
396C3×C4C4×D3211yescharacteristic
397C3×C4C2×#62411yescharacteristic
398Dic3—22no—
399Dic3—22no—
400Dic3—32no—
401Dic3—32no—
402Dic3—62no—
403C32—21no—
404C32—21no—
405C32—21no—
406C32—41no—
407C32#468911yescharacteristic, complemented
408C32#470811yescharacteristic, complemented
409C32#499711yescharacteristic, complemented
410C23—91no—
411C2×C4—32no—
412C2×C4—32no—
413C2×C4—91no—
414C2×C4—92no—
415C2×C4—92no—
416C2×C4D3311yescharacteristic
417C8—32no—
418C8—92no—
419C8—92no—
420C8—272no—
421C2×C3—22no—
422C2×C3—22no—
423C2×C3—22no—
424C2×C3—21no—
425C2×C3—21no—
426C2×C3—21no—
427C2×C3—42no—
428C2×C3—41no—
429C2×C3—182no—
430C2×C3#6380412yes—
431C2×C3#6381412yes—
432C2×C3#6381512yes—
433C2×C3C4×D3×Dic311yescharacteristic
434C2×C3C4×D3×Dic311yescharacteristic
435C2×C3C4×#62411yescharacteristic
436D3—62no—
437D3—62no—
438D3—122no—
439C22—92no—
440C22—182no—
441C22D32×Dic311yescharacteristic
442C4—62no—
443C4—62no—
444C4—92no—
445C4D3×#62411yescharacteristic
446C4#8894911yescharacteristic
447C3—21no—
448C3—21no—
449C3—21no—
450C3—41no—
451C3#1059031811yescharacteristic, complemented
452C3#1059031911yescharacteristic, complemented
453C3#1059091011yescharacteristic, complemented
454C2—182no—
455C2#1172282212yes—
456C2Dic3×#62411yescharacteristic
subgroupG/H
center, Frattini, hypercenterC2×C4D33
commutatorC2×C33C2×C42
Fitting, cosocleC2×C33×C4C23
socleC22×C33C22×C4
prime radicalC22×C3×C3⋊D3C2×C4
hypocenterC33#405
LCSG ▷ C2×C33 ▷ C33
UCSC1 ◁ C2×C4

Complements

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

NH ≅ G/Ntype
4. #11722019C2semidirect
5. Dic3×C3⋊(C3⋊C8)C2semidirect
6. C3×#63554C2semidirect
7. C3×Dic3×C3⋊C8C2semidirect
18. #88808C4semidirect
19. #88787C4semidirect
41. #63554D3semidirect
45. Dic3×C3⋊C8D3semidirect
282. C33#405semidirect
407. C32#4689semidirect
408. C32#4708semidirect
409. C32#4997semidirect
451. C3#10590318semidirect
452. C3#10590319semidirect
453. C3#10590910semidirect

Maximal subgroups

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

subgroupindex
C2×#888082
C2×#887872
C3×#638742
#117220192
Dic3×C3⋊(C3⋊C8)2
C3×#635542
C3×Dic3×C3⋊C82
#105909103
#105903193
#105903183