Finite Group Databasebeta

← #13155296

Subgroups — #13155296

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
1C2×#88809C211yesmaximal, characteristic
2#11723437C211yesmaximal, characteristic
3C2×C3×#3710C211yesmaximal, characteristic
4#11722021C211yesmaximal, characteristic, complemented
5#11722018C211yesmaximal, characteristic, complemented
6C3×#63554C211yesmaximal, characteristic, complemented
7C3×Dic3×C3⋊C8C211yesmaximal, characteristic, complemented
8#10590693—31nomaximal
9#10590692—31nomaximal
10#10590321—31nomaximal
11C2×C3×C4×C3⋊D3C2211yescharacteristic
12C4×C3⋊(C3⋊Dic3)C2211yescharacteristic
13C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
14C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
15C32×C4×Dic3C2211yescharacteristic
16C2×C32×C3⋊C8C2211yescharacteristic
17C2×C32×C3⋊C8C2211yescharacteristic
18#88809C414yescomplemented
19#88807C412yes—
20#88803C412yes—
21C3×#3710C414yescomplemented
22#64081—31no—
23C2×C3×D3×C8—31no—
24C2×C3×D3×C8—31no—
25#63855—31no—
26#63855—31no—
27#63855—61no—
28C2×#3712—31no—
29C2×#3711—31no—
30C2×#3711—31no—
31C2×#3710D311yescharacteristic
32#63642—31no—
33C8×C3⋊Dic3—31no—
34C3×#815—31no—
35C3×C8×Dic3—31no—
36C3×#803—31no—
37C3×C4×C3⋊C8—31no—
38#63555—31no—
39#63555—31no—
40#63555—61no—
41#63554D311yescharacteristic, complemented
42#63553—31no—
43#63553—31no—
44#63553—61no—
45Dic3×C3⋊C8D311yescharacteristic, complemented
46C22×C3×C3⋊D3C2×C411yescharacteristic
47C2×C33×C4C2311yescharacteristic
48C2×C3⋊(C3⋊Dic3)C2×C412yes—
49C2×C3×C3⋊Dic3C2×C411yescharacteristic
50C3×C4×C3⋊D3C2×C412yes—
51C3×C4×C3⋊D3C2×C412yes—
52C2×C32×Dic3C2×C412yes—
53C3×C3⋊(C3⋊C8)C2×C412yes—
54C3×C3⋊(C3⋊C8)C2×C412yes—
55C32×C3⋊C8C2×C412yes—
56C32×C3⋊C8C2×C412yes—
57#5101—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—
65C2×C32×C8—31no—
66C32×C42—31no—
67#3752—32no—
68#3750—32no—
69C4×C3⋊Dic3—31no—
70C4×C3⋊Dic3—31no—
71C4×C3⋊Dic3—31no—
72C4×C3⋊Dic3—61no—
73C4×C3⋊Dic3—61no—
74C4×C3⋊Dic3—61no—
75C4×C3⋊Dic3—121no—
76C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
77C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
78C3×C4×Dic3—21no—
79C3×C4×Dic3C2×D311yescharacteristic
80C3×C4×Dic3C2×D311yescharacteristic
81C2×C3×C3⋊C8—21no—
82C2×C3×C3⋊C8—21no—
83C2×C3×C3⋊C8—31no—
84C2×C3×C3⋊C8—31no—
85C2×C3×C3⋊C8—31no—
86C2×C3×C3⋊C8—31no—
87C2×C3×C3⋊C8—61no—
88C2×C3×C3⋊C8—61no—
89C2×C3×C3⋊C8C2×D311yescharacteristic
90C2×C3×C3⋊C8C2×D311yescharacteristic
91C2×C3×C3⋊C8C2×D311yescharacteristic
92C2×C3×C3⋊C8C2×D311yescharacteristic
93C3×D3×C8—34no—
94C3×D3×C8—34no—
95Dic3⋊Dic3—32no—
96Dic3⋊Dic3—32no—
97Dic3⋊Dic3—62no—
98(C2×D3)⋊Dic3—32no—
99(C2×D3)⋊Dic3—32no—
100(C2×D3)⋊Dic3—62no—
101#3712—34no—
102#3711—34no—
103#3711—34no—
104#3710Dic314yescomplemented
105C22×C33C22×C411yescharacteristic
106C2×C3×C3⋊D3—22no—
107C2×C3×C3⋊D3C4212yes—
108C33×C4C22×C411yescharacteristic
109C33×C4C22×C411yescharacteristic
110C3⋊(C3⋊Dic3)—22no—
111C3×C3⋊Dic3C4212yes—
112C32×Dic3—22no—
113C22×C3×C8—91no—
114#926—91no—
115C2×#208—91no—
116C2×ΓL(1,𝔽25)—91no—
117C2×ΓL(1,𝔽25)—91no—
118C2×D3×C8—31no—
119C2×D3×C8—31no—
120#872—91no—
121#872—91no—
122#872—181no—
123C3×#97—91no—
124C3×C4×C8—91no—
125#815—31no—
126#815—91no—
127#815—91no—
128#815—91no—
129#815—181no—
130C8×Dic3—31no—
131C8×Dic3—91no—
132C8×Dic3—91no—
133C8×Dic3—91no—
134C8×Dic3—181no—
135#803—31no—
136C4×C3⋊C8—31no—
137C22×C3⋊D3C4×D311yescharacteristic
138C22×C3×D3—31no—
139C22×C3×D3—31no—
140C22×C3×D3—61no—
141C2×C32×C4—21no—
142C2×C32×C4—21no—
143C2×C32×C4—21no—
144C2×C32×C4—32no—
145C2×C32×C4—41no—
146C2×C32×C4C22×D311yescharacteristic
147C2×C32×C4C22×D311yescharacteristic
148C2×C32×C4C22×D311yescharacteristic
149C2×C3⋊Dic3—32no—
150C2×C3⋊Dic3—32no—
151C2×C3⋊Dic3—32no—
152C2×C3⋊Dic3—62no—
153C2×C3⋊Dic3—62no—
154C2×C3⋊Dic3—62no—
155C2×C3⋊Dic3—122no—
156C2×C3⋊Dic3C4×D311yescharacteristic
157C4×C3⋊D3C4×D312yes—
158C4×C3⋊D3C2×Dic312yes—
159C2×C3×Dic3—22no—
160C2×C3×Dic3—31no—
161C2×C3×Dic3—31no—
162C2×C3×Dic3—61no—
163C2×C3×Dic3C4×D312yes—
164C2×C3×Dic3C4×D312yes—
165C3×C4×D3—32no—
166C3×C4×D3—32no—
167C3×C4×D3—32no—
168C3×C4×D3—32no—
169C3×C4×D3—62no—
170C3×C4×D3—62no—
171C32×C8—32no—
172C32×C8—32no—
173C3⋊(C3⋊C8)C4×D312yes—
174C3⋊(C3⋊C8)C4×D312yes—
175C3×C3⋊C8—22no—
176C3×C3⋊C8—22no—
177C3×C3⋊C8—32no—
178C3×C3⋊C8—32no—
179C3×C3⋊C8—32no—
180C3×C3⋊C8—32no—
181C3×C3⋊C8—62no—
182C3×C3⋊C8—62no—
183C3×C3⋊C8C4×D312yes—
184C3×C3⋊C8C4×D312yes—
185C3×C3⋊C8C2×Dic312yes—
186C3×C3⋊C8C2×Dic312yes—
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—31no—
196C2×C3×C8—31no—
197C2×C3×C8—31no—
198C2×C3×C8—31no—
199C2×C3×C8—61no—
200C2×C3×C8—61no—
201C2×C3×C8—94no—
202C2×C3×C8—91no—
203C2×C3×C8—91no—
204C3×C42—31no—
205C3×C42—31no—
206C3×C42—61no—
207#217—92no—
208(C2×D3)⋊C4—92no—
209(C2×D3)⋊C4—92no—
210(C2×D3)⋊C4—182no—
211C4⋊Dic3—92no—
212Dic3⋊C4—92no—
213Dic3⋊C4—92no—
214Dic3⋊C4—182no—
215C4×Dic3—91no—
216C4×Dic3—91no—
217C4×Dic3—91no—
218C4×Dic3—181no—
219C4×Dic3—181no—
220C4×Dic3—181no—
221C4×Dic3—361no—
222C4×Dic3D3211yescharacteristic
223#208—94no—
224C2×C3⋊C8—31no—
225C2×C3⋊C8—31no—
226C2×C3⋊C8—31no—
227C2×C3⋊C8—31no—
228C2×C3⋊C8—61no—
229C2×C3⋊C8—61no—
230C2×C3⋊C8D3211yescharacteristic
231C2×C3⋊C8D3211yescharacteristic
232ΓL(1,𝔽25)—94no—
233ΓL(1,𝔽25)—94no—
234D3×C8—34no—
235D3×C8—34no—
236C22×C32—21no—
237C22×C32—21no—
238C22×C32—21no—
239C22×C32—41no—
240C22×C32C2×C4×D311yescharacteristic
241C22×C32C2×C4×D311yescharacteristic
242C22×C32C2×C4×D311yescharacteristic
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—21no—
252C32×C4—21no—
253C32×C4—21no—
254C32×C4—21no—
255C32×C4—21no—
256C32×C4—21no—
257C32×C4—41no—
258C32×C4—41no—
259C32×C4—62no—
260C32×C4C2×C4×D311yescharacteristic
261C32×C4C2×C4×D311yescharacteristic
262C32×C4C2×C4×D311yescharacteristic
263C32×C4C2×C4×D311yescharacteristic
264C32×C4C2×C4×D311yescharacteristic
265C32×C4C22×Dic311yescharacteristic
266C3⋊Dic3—62no—
267C3⋊Dic3—64no—
268C3⋊Dic3—62no—
269C3⋊Dic3—62no—
270C3⋊Dic3—122no—
271C3⋊Dic3—122no—
272C3⋊Dic3—124no—
273C3⋊Dic3C4×Dic312yes—
274C3×Dic3—22no—
275C3×Dic3—22no—
276C3×Dic3—32no—
277C3×Dic3—32no—
278C3×Dic3—42no—
279C3×Dic3—62no—
280C2×M16—271no—
281C22×C8—91no—
282#117—271no—
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—21no—
293C2×C3×C4—21no—
294C2×C3×C4—21no—
295C2×C3×C4—32no—
296C2×C3×C4—32no—
297C2×C3×C4—41no—
298C2×C3×C4—62no—
299C2×C3×C4—91no—
300C2×C3×C4—92no—
301C2×C3×C4—92no—
302C2×C3×C4C2×D3211yescharacteristic
303C2×C3×C4C2×D3211yescharacteristic
304C2×C3×C4C2×D3211yescharacteristic
305C2×Dic3—31no—
306C2×Dic3—31no—
307C2×Dic3—61no—
308C2×Dic3—92no—
309C2×Dic3—92no—
310C2×Dic3—92no—
311C2×Dic3—182no—
312C2×Dic3—182no—
313C2×Dic3—182no—
314C2×Dic3—362no—
315C2×Dic3#62412yes—
316C4×D3—32no—
317C4×D3—32no—
318C4×D3—32no—
319C4×D3—32no—
320C4×D3—62no—
321C4×D3—62no—
322C3×C8—32no—
323C3×C8—32no—
324C3×C8—32no—
325C3×C8—32no—
326C3×C8—62no—
327C3×C8—62no—
328C3×C8—92no—
329C3×C8—92no—
330C3⋊C8—32no—
331C3⋊C8—32no—
332C3⋊C8—32no—
333C3⋊C8—32no—
334C3⋊C8—62no—
335C3⋊C8—62no—
336C3⋊C8D3×Dic312yes—
337C3⋊C8D3×Dic312yes—
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#90112yes—
350C2×C32#90712yes—
351C2×C32C2×C4×Dic311yescharacteristic
352C3⋊D3—22no—
353C3×D3—62no—
354C3×D3—62no—
355C3×D3—122no—
356C22×C4—91no—
357M16—274no—
358C2×C8—31no—
359C2×C8—31no—
360C2×C8—94no—
361C2×C8—91no—
362C2×C8—91no—
363C4⋊C4—272no—
364C22⋊C4—272no—
365C42—31no—
366C42—271no—
367C22×C3—21no—
368C22×C3—21no—
369C22×C3—21no—
370C22×C3—41no—
371C22×C3—92no—
372C22×C3—182no—
373C22×C3C4×D3211yescharacteristic
374C22×C3C4×D3211yescharacteristic
375C22×C3C2×#62411yescharacteristic
376C2×D3—32no—
377C2×D3—32no—
378C2×D3—62no—
379C2×D3—62no—
380C2×D3—62no—
381C2×D3—122no—
382C3×C4—21no—
383C3×C4—21no—
384C3×C4—21no—
385C3×C4—21no—
386C3×C4—21no—
387C3×C4—21no—
388C3×C4—41no—
389C3×C4—41no—
390C3×C4—62no—
391C3×C4—62no—
392C3×C4—92no—
393C3×C4—122no—
394C3×C4C4×D3211yescharacteristic
395C3×C4C4×D3211yescharacteristic
396C3×C4C4×D3211yescharacteristic
397C3×C4C4×D3211yescharacteristic
398C3×C4C2×D3×Dic311yescharacteristic
399C3×C4C2×D3×Dic311yescharacteristic
400Dic3—22no—
401Dic3—32no—
402Dic3—32no—
403Dic3—62no—
404Dic3—184no—
405Dic3—182no—
406Dic3—182no—
407Dic3—182no—
408Dic3—362no—
409Dic3—362no—
410Dic3—364no—
411C32—21no—
412C32—21no—
413C32—21no—
414C32—41no—
415C32#468911yescharacteristic, complemented
416C32#470811yescharacteristic, complemented
417C32#510111yescharacteristic, complemented
418C23—91no—
419C2×C4—32no—
420C2×C4—91no—
421C2×C4—92no—
422C2×C4—92no—
423C2×C4—272no—
424C2×C4D3311yescharacteristic
425C8—32no—
426C8—32no—
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#6380312yes—
439C2×C3#6380412yes—
440C2×C3#6381712yes—
441C2×C3C4×D3×Dic311yescharacteristic
442C2×C3C4×D3×Dic311yescharacteristic
443C2×C3C4×#62411yescharacteristic
444D3—62no—
445D3—62no—
446D3—122no—
447C22—92no—
448C22—182no—
449C22D3×#62411yescharacteristic
450C4—62no—
451C4—92no—
452C4—542no—
453C4D32×Dic311yescharacteristic
454C4#8894911yescharacteristic
455C3—21no—
456C3—21no—
457C3—21no—
458C3—41no—
459C3#1059032111yescharacteristic, complemented
460C3#1059069211yescharacteristic, complemented
461C3#1059069311yescharacteristic, complemented
462C2—182no—
463C2#1172282412yes—
464C2Dic3×#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. #11722021C2semidirect
5. #11722018C2semidirect
6. C3×#63554C2semidirect
7. C3×Dic3×C3⋊C8C2semidirect
18. #88809C4semidirect
21. C3×#3710C4semidirect
41. #63554D3semidirect
45. Dic3×C3⋊C8D3semidirect
104. #3710Dic3semidirect
287. C33#405semidirect
415. C32#4689semidirect
416. C32#4708semidirect
417. C32#5101semidirect
459. C3#10590321semidirect
460. C3#10590692semidirect
461. C3#10590693semidirect

Maximal subgroups

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

subgroupindex
C2×#888092
#117234372
C2×C3×#37102
#117220212
#117220182
C3×#635542
C3×Dic3×C3⋊C82
#105906933
#105906923
#105903213