Finite Group Databasebeta

← #13168107

Subgroups — #13168107

Order 1728.

1610 conjugacy classes of subgroups; only the 212 normal subgroups are shown.

normal subgroups only

This group has 1610 conjugacy classes of subgroups — too many to draw. Only its 212 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
1C2×#89040C211yesmaximal, characteristic, complemented
2C2×C3×#3802C211yesmaximal, characteristic
3C4×Dic3×C3⋊D3C211yesmaximal, characteristic, complemented
4#11723434C212yesmaximal, complemented
5C3×#63860C212yesmaximal, complemented
6#11723006C214yesmaximal, complemented
7#11722902C214yesmaximal, complemented
8D3×#936—32nomaximal
9#10594009—31nomaximal
10C22×C3×D32C2212yes—
11C2×C3×C4×C3⋊D3C2211yescharacteristic
12C2×C3×#634C2212yescomplemented
13C2×C32×D12C2212yes—
14C2×#6296C2212yescomplemented
15C2×#6294C2212yes—
16#89040—24no—
17C2×Dic3×C3⋊D3C2212yes—
18C2×C3×#625C2212yes—
19C3×#3802—24no—
20C4×C3⋊(C3⋊Dic3)C2211yescharacteristic, complemented
21C32×C4×Dic3C2211yescharacteristic, complemented
22(C2×D3)⋊(C3⋊Dic3)C2218yescomplemented
23Dic3×C3⋊Dic3C2212yescomplemented
24C3×(C2×D3)⋊Dic3C2218yescomplemented
25C2×C3×D3×D4—32no—
26C2×#3802D311yescharacteristic
27C2×#3797—31no—
28C2×D3×Dic6—32no—
29#64145—32no—
30C42×C3⋊D3—31no—
31C3×#936—32no—
32C3×#875—32no—
33D3×#217—38no—
34#63959—38no—
35#63921—34no—
36#63915—34no—
37C4×D3×Dic3—32no—
38C4×D3×Dic3—61no—
39#63863—32no—
40#63863—32no—
41#63863—62no—
42#63860D312yescomplemented
43C22×C3×C3⋊D3C2311yescharacteristic
44C22×C32×D3C2314yes—
45C2×C3×D32—24no—
46C2×C3×D32—28no—
47C2×C33×C4C2311yescharacteristic
48C2×C3⋊(C3⋊Dic3)C2312yes—
49C2×C3×C3⋊Dic3C2314yes—
50C2×C3×C3⋊Dic3C2311yescharacteristic
51C3×C4×C3⋊D3D414yes—
52C3×#634—24no—
53C2×C32×Dic3C2312yes—
54C32×D12—24no—
55#6296—24no—
56#6294—24no—
57Dic3×C3⋊D3—24no—
58C3×#625—24no—
59C2×#936—91no—
60D3×#124—92no—
61C23×C3×D3—34no—
62C22×D32C2×D312yes—
63C2×C32×D4—32no—
64C2×C4×C3⋊D3—32no—
65C2×C4×C3⋊D3C2×D311yescharacteristic
66C2×#634C2×D312yescomplemented
67C2×C3×#67—34no—
68C3×D3×D4—68no—
69C2×C3×D12—22no—
70C2×C3×D12C2×D312yes—
71C2×C3×D12C2×D312yes—
72C2×C3×C4×D3—32no—
73C2×C3×C4×D3—61no—
74C2×C3×Dic6—32no—
75C2×C3×Dic6—32no—
76C2×C3×Dic6—62no—
77C2×#627—34no—
78C2×#627—32no—
79C2×#625C2×D312yes—
80C2×#624—32no—
81C2×D3×Dic3—34no—
82C2×D3×Dic3—34no—
83C2×D3×Dic3—62no—
84#3802—24no—
85#3797—64no—
86D3×Dic6—68no—
87C32×C42—31no—
88#3757—38no—
89C4×C3⋊Dic3—31no—
90C4×C3⋊Dic3—32no—
91C4×C3⋊Dic3—32no—
92C4×C3⋊Dic3—62no—
93C4×C3⋊Dic3—61no—
94C4×C3⋊Dic3—121no—
95C3×#217—38no—
96C3×(C2×D3)⋊C4—38no—
97C3×C4×Dic3—21no—
98C3×C4×Dic3C2×D312yescomplemented
99(C2×D3)⋊Dic3—38no—
100(C2×D3)⋊Dic3—38no—
101(C2×D3)⋊Dic3—68no—
102(C2×D3)⋊Dic3C2×D318yescomplemented
103Dic32—34no—
104Dic32—62no—
105C22×C33C2411yescharacteristic
106C2×C3×C3⋊D3C2×D412yes—
107C2×C3×C3⋊D3#4114yes—
108C2×C32×D3—24no—
109C2×C32×D3—28no—
110C3×D32—48no—
111C33×C4C2×D412yes—
112C3⋊(C3⋊Dic3)—22no—
113C3×C3⋊Dic3—24no—
114C3×C3⋊Dic3C2×D412yes—
115C32×Dic3—22no—
116C22×C3×D4—91no—
117C2×D3×Q8—92no—
118C2×D3×D4—32no—
119C22×Dic6—91no—
120C3×#124—92no—
121C2×#217—94no—
122#947—92no—
123#936—32no—
124#936—94no—
125#936—92no—
126#936—92no—
127#936—182no—
128C2×C4×Dic3—91no—
129#885—98no—
130D3×C22⋊C4—98no—
131#875—32no—
132C42×D3—92no—
133C42×D3—181no—
134C23×C32—34no—
135C22×C3⋊D3C22×D311yescharacteristic
136C22×C3×D3—24no—
137C22×C3×D3—32no—
138C22×C3×D3—61no—
139C22×C3×D3—616no—
140C22×C3×D3—68no—
141C22×C3×D3C22×D314yes—
142C22×C3×D3C22×D314yes—
143C2×D32—28no—
144C2×D32—24no—
145C32×D4—64no—
146C2×C32×C4—21no—
147C2×C32×C4—22no—
148C2×C32×C4—32no—
149C2×C32×C4—41no—
150C2×C32×C4C22×D312yes—
151C2×C32×C4C22×D311yescharacteristic
152C2×C3⋊Dic3—34no—
153C2×C3⋊Dic3—32no—
154C2×C3⋊Dic3—62no—
155C2×C3⋊Dic3—64no—
156C2×C3⋊Dic3—122no—
157C2×C3⋊Dic3C22×D314yes—
158C2×C3⋊Dic3C22×D311yescharacteristic
159C4×C3⋊D3—64no—
160C4×C3⋊D3#6714yes—
161#634—24no—
162C3×#67—68no—
163C2×C3×Dic3—22no—
164C2×C3×Dic3—32no—
165C2×C3×Dic3—34no—
166C2×C3×Dic3—34no—
167C2×C3×Dic3—61no—
168C2×C3×Dic3—64no—
169C2×C3×Dic3C22×D314yes—
170C3×D12—24no—
171C3×D12—24no—
172C3×D12—44no—
173C3×C4×D3—38no—
174C3×C4×D3—64no—
175C3×Dic6—64no—
176C3×Dic6—64no—
177C3×Dic6—124no—
178#627—68no—
179#627—64no—
180#625—24no—
181#624—64no—
182D3×Dic3—68no—
183D3×Dic3—68no—
184D3×Dic3—68no—
185C2×#124—271noSylow
186C2×C33C22×D411yescharacteristic
187C2×C33C2×#4112yes—
188C3×C3⋊D3#12414yescomplemented
189C32×D3—44no—
190C24×C3—92no—
191C23×D3—34no—
192C2×C3×Q8—92no—
193C2×C3×D4—32no—
194C2×C3×D4—32no—
195C2×C3×D4—62no—
196C2×C3×D4—92no—
197C2×C3×D4—184no—
198C22×C3×C4—91no—
199C2×#67—34no—
200C22×Dic3—92no—
201C22×Dic3—92no—
202D3×Q8—188no—
203D3×D4—68no—
204C2×D12D3212yes—
205C2×C4×D3—32no—
206C2×C4×D3—61no—
207C2×C4×D3—94no—
208C2×C4×D3—94no—
209C2×C4×D3—182no—
210C2×Dic6—32no—
211C2×Dic6—32no—
212C2×Dic6—62no—
213C2×Dic6—94no—
214C2×Dic6—92no—
215C2×Dic6—184no—
216C3×C22⋊C4—98no—
217C3×C42—32no—
218C3×C42—61no—
219#217—38no—
220#217—98no—
221#217—98no—
222#217—188no—
223(C2×D3)⋊C4—38no—
224(C2×D3)⋊C4—98no—
225C4×Dic3—94no—
226C4×Dic3—92no—
227C4×Dic3—91no—
228C4×Dic3—92no—
229C4×Dic3—182no—
230C4×Dic3—181no—
231C4×Dic3—182no—
232C4×Dic3—361no—
233C4×Dic3D3211yescharacteristic, complemented
234C22×C32—21no—
235C22×C32—22no—
236C22×C32—41no—
237C22×C32—64no—
238C22×C32—68no—
239C22×C32C23×D312yes—
240C22×C32C23×D311yescharacteristic
241C2×C3⋊D3#23714yes—
242C2×C3⋊D3C2×#6712yes—
243C2×C3×D3—24no—
244C2×C3×D3—28no—
245C2×C3×D3—28no—
246C2×C3×D3—24no—
247C2×C3×D3—34no—
248C2×C3×D3—38no—
249C2×C3×D3—44no—
250C2×C3×D3—48no—
251C2×C3×D3—62no—
252C2×C3×D3—64no—
253C2×C3×D3—1216no—
254D32—48no—
255C32×C4—22no—
256C32×C4—24no—
257C32×C4—42no—
258C32×C4—62no—
259C32×C4D3×D414yes—
260C32×C4C2×#6712yes—
261C3⋊Dic3—24no—
262C3⋊Dic3—62no—
263C3⋊Dic3—64no—
264C3⋊Dic3—64no—
265C3⋊Dic3—124no—
266C3⋊Dic3—124no—
267C3⋊Dic3C2×#6712yes—
268C3×Dic3—24no—
269C3×Dic3—24no—
270C3×Dic3—34no—
271C3×Dic3—62no—
272C3×Dic3—64no—
273C3×Dic3—64no—
274C3×Dic3—124no—
275C22×Q8—271no—
276C22×D4—91no—
277#124—92no—
278#124—274no—
279#124—272no—
280C2×C22⋊C4—274no—
281C2×C42—271no—
282C33C2×#12411yescharacteristic, complemented, Sylow
283C23×C3—34no—
284C23×C3—34no—
285C23×C3—64no—
286C23×C3—91no—
287C23×C3—184no—
288C23×C3—188no—
289C22×D3—32no—
290C22×D3—61no—
291C22×D3—68no—
292C22×D3—616no—
293C22×D3C2×D3214yes—
294C3×Q8—184no—
295C3×D4—64no—
296C3×D4—64no—
297C3×D4—124no—
298C3×D4—184no—
299C2×C3×C4—21no—
300C2×C3×C4—22no—
301C2×C3×C4—34no—
302C2×C3×C4—41no—
303C2×C3×C4—62no—
304C2×C3×C4—91no—
305C2×C3×C4—94no—
306C2×C3×C4—94no—
307C2×C3×C4C2×D3211yescharacteristic
308C2×C3×C4C2×D3212yes—
309#67—68no—
310C2×Dic3—32no—
311C2×Dic3—34no—
312C2×Dic3—34no—
313C2×Dic3—61no—
314C2×Dic3—64no—
315C2×Dic3—94no—
316C2×Dic3—92no—
317C2×Dic3—182no—
318C2×Dic3—184no—
319C2×Dic3—184no—
320C2×Dic3—184no—
321C2×Dic3—362no—
322C2×Dic3C2×D3212yes—
323D12—24no—
324C4×D3—38no—
325C4×D3—64no—
326C4×D3—188no—
327C4×D3—188no—
328C4×D3—188no—
329Dic6—64no—
330Dic6—64no—
331Dic6—124no—
332Dic6—188no—
333Dic6—184no—
334C2×C32—21no—
335C2×C32—22no—
336C2×C32—24no—
337C2×C32—22no—
338C2×C32—41no—
339C2×C32—42no—
340C2×C32—124no—
341C2×C32C2×D3×D412yes—
342C2×C32C2×#23712yes—
343C2×C32D3×#4114yes—
344C2×C32C22×#6711yescharacteristic
345C3⋊D3#93614yescomplemented
346C3×D3—38no—
347C3×D3—44no—
348C3×D3—44no—
349C3×D3—64no—
350C3×D3—84no—
351C24—92no—
352C2×Q8—92no—
353C2×Q8—272no—
354C2×Q8—544no—
355C2×D4—32no—
356C2×D4—92no—
357C2×D4—184no—
358C22×C4—91no—
359C22×C4—272no—
360C22×C4—272no—
361C22⋊C4—98no—
362C22⋊C4—278no—
363C42—31no—
364C42—272no—
365C42—271no—
366C22×C3—22no—
367C22×C3—21no—
368C22×C3—41no—
369C22×C3—64no—
370C22×C3—68no—
371C22×C3—68no—
372C22×C3—64no—
373C22×C3—92no—
374C22×C3—94no—
375C22×C3—124no—
376C22×C3—128no—
377C22×C3—368no—
378C22×C3C22×D3211yescharacteristic
379C22×C3C22×D3212yes—
380C2×D3—28no—
381C2×D3—24no—
382C2×D3—38no—
383C2×D3—34no—
384C2×D3—62no—
385C2×D3—64no—
386C2×D3—1216no—
387C3×C4—24no—
388C3×C4—22no—
389C3×C4—42no—
390C3×C4—64no—
391C3×C4—64no—
392C3×C4—92no—
393C3×C4—184no—
394C3×C4#380212yes—
395C3×C4D3×#6714yes—
396Dic3—22no—
397Dic3—34no—
398Dic3—62no—
399Dic3—64no—
400Dic3—64no—
401Dic3—124no—
402Dic3—184no—
403Dic3—182no—
404Dic3—184no—
405Dic3—364no—
406Dic3—364no—
407C32—22no—
408C32—21no—
409C32—41no—
410C32D3×#12412yescomplemented
411C32C2×#93611yescharacteristic, complemented
412C23—34no—
413C23—91no—
414C23—184no—
415C23—188no—
416Q8—184no—
417Q8—544no—
418D4—64no—
419D4—184no—
420C2×C4—32no—
421C2×C4—91no—
422C2×C4—94no—
423C2×C4—94no—
424C2×C4—272no—
425C2×C4—544no—
426C2×C4—544no—
427C2×C4D3311yescharacteristic
428C2×C3—22no—
429C2×C3—24no—
430C2×C3—21no—
431C2×C3—22no—
432C2×C3—42no—
433C2×C3—41no—
434C2×C3—94no—
435C2×C3—124no—
436C2×C3—124no—
437C2×C3—244no—
438C2×C3C2×#380211yescharacteristic
439C2×C3#6430612yes—
440C2×C3D3×#23714yes—
441C2×C3C2×D3×#6712yes—
442D3—38no—
443D3—44no—
444D3—64no—
445C22—68no—
446C22—64no—
447C22—94no—
448C22—92no—
449C22—368no—
450C22C2×D3311yescharacteristic
451C4—62no—
452C4—92no—
453C4—184no—
454C4—542no—
455C4#8895312yes—
456C3—21no—
457C3—22no—
458C3—41no—
459C3#1059400911yescharacteristic, complemented
460C3D3×#93612yescomplemented
461C2—94no—
462C2—124no—
463C2#1172437312yes—
464C2C2×#8895311yescharacteristic
subgroupG/H
center, FrattiniC22C2×D33
commutator, socle, cosocleC22×C33C24
FittingC2×C33×C4C23
prime radicalC2×C3×#3802C2
hypercenterC2×C4D33
hypocenterC33C2×#124
LCSG ▷ C22×C33 ▷ C33
UCSC1 ◁ C22 ◁ C2×C4

Complements

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

NH ≅ G/Ntype
1. C2×#89040C2semidirect
3. C4×Dic3×C3⋊D3C2semidirect
4. #11723434C2semidirect
5. C3×#63860C2semidirect
6. #11723006C2semidirect
7. #11722902C2semidirect
12. C2×C3×#634C22semidirect
14. C2×#6296C22semidirect
20. C4×C3⋊(C3⋊Dic3)C22semidirect
21. C32×C4×Dic3C22semidirect
22. (C2×D3)⋊(C3⋊Dic3)C22semidirect
23. Dic3×C3⋊Dic3C22semidirect
24. C3×(C2×D3)⋊Dic3C22semidirect
42. #63860D3semidirect
66. C2×#634C2×D3semidirect
98. C3×C4×Dic3C2×D3semidirect
102. (C2×D3)⋊Dic3C2×D3semidirect
188. C3×C3⋊D3#124semidirect
233. C4×Dic3D32semidirect
282. C33C2×#124semidirect
345. C3⋊D3#936semidirect
410. C32D3×#124semidirect
411. C32C2×#936semidirect
459. C3#10594009semidirect
460. C3D3×#936semidirect

Maximal subgroups

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

subgroupindex
C2×#890402
C2×C3×#38022
C4×Dic3×C3⋊D32
#117234342
C3×#638602
#117230062
#117229022
D3×#9363
#105940093