Finite Group Databasebeta

← #13169444

Subgroups — #13169444

Order 1728.

1096 conjugacy classes of subgroups; only the 184 normal subgroups are shown.

normal subgroups only

This group has 1096 conjugacy classes of subgroups — too many to draw. Only its 184 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
1C4×#6295C211yesmaximal, characteristic
2#11723404C212yesmaximal, complemented
3Dic6×C3⋊Dic3C211yesmaximal, characteristic, complemented
4C3×#63919C211yesmaximal, characteristic, complemented
5C3×#63871C212yesmaximal, complemented
6#11723027C214yesmaximal, complemented
7#11723007C212yesmaximal
8#11722941C212yesmaximal, complemented
9#10595413—31nomaximal
10#10594557—32nomaximal
11C2×C3×C4×C3⋊D3C2212yes—
12C2×C32×Dic6C2211yescharacteristic
13C2×#6295C2211yescharacteristic
14C2×C3×#627C2212yes—
15C4⋊(C3⋊(C3⋊Dic3))C2211yescharacteristic
16C3×C4×C3⋊Dic3C2211yescharacteristic
17C32×C4⋊Dic3C2212yes—
18#88815C2211yescharacteristic
19#88814C2212yes—
20Dic3⋊(C3⋊Dic3)C2212yes—
21#88803C2214yes—
22Dic3×C3⋊Dic3C2214yes—
23Dic3×C3⋊Dic3C2212yes—
24C3×#3724C2212yes—
25C3×Dic3⋊Dic3C2214yes—
26C3×#3722C2214yes—
27Q8×C3⋊Dic3—31no—
28#64101—32no—
29C3×#946—32no—
30C3×#897—32no—
31C3×#888—32no—
32#63919D311yescharacteristic, complemented
33#63915—32no—
34#63907—34no—
35C4×#626—32no—
36C4×#625—31no—
37#63871D312yescomplemented
38#63862—34no—
39#63861—34no—
40#63848—34no—
41#63843—34no—
42Dic3×Dic6—32no—
43Dic3×Dic6—61no—
44#63841—32no—
45#63841—32no—
46#63841—62no—
47#63838—32no—
48C22×C3×C3⋊D3C2312yes—
49C2×C33×C4C2311yescharacteristic
50C2×C3⋊(C3⋊Dic3)C2312yes—
51C2×C3×C3⋊Dic3C2311yescharacteristic
52C2×C3×C3⋊Dic3C2312yes—
53C2×C3×C3⋊Dic3C2311yescharacteristic
54C3×C4×C3⋊D3—24no—
55C2×C32×Dic3C2314yes—
56C2×C32×Dic3C2312yes—
57C32×Dic6—22no—
58#6295—22no—
59C3×#627—24no—
60#5732—91no—
61#5641—92no—
62C2×C32×Q8—31no—
63C2×C4×C3⋊D3C2×D312yes—
64C2×C3×C4×D3—32no—
65C2×C3×C4×D3—32no—
66C2×C3×C4×D3—62no—
67C2×C3×Dic6—21no—
68C2×C3×Dic6—34no—
69C2×C3×Dic6C2×D312yes—
70C2×#627C2×D312yes—
71C2×#626—32no—
72C2×#625—31no—
73C32×C4⋊C4—32no—
74#3752—34no—
75C4⋊(C3⋊Dic3)—31no—
76C4⋊(C3⋊Dic3)—32no—
77C4⋊(C3⋊Dic3)—32no—
78C4⋊(C3⋊Dic3)—62no—
79C4⋊(C3⋊Dic3)—61no—
80C4⋊(C3⋊Dic3)—121no—
81C4×C3⋊Dic3—32no—
82C4×C3⋊Dic3—34no—
83C4×C3⋊Dic3C2×D311yescharacteristic
84C3×(C2×D3)⋊C4—34no—
85C3×(C2×D3)⋊C4—34no—
86C3×C4⋊Dic3—22no—
87C3×C4⋊Dic3—34no—
88C3×C4⋊Dic3C2×D312yes—
89C3×C4⋊Dic3C2×D312yes—
90C3×Dic3⋊C4—34no—
91C3×Dic3⋊C4—34no—
92C3×C4×Dic3—32no—
93C3×C4×Dic3—61no—
94#3724—31no—
95#3724C2×D312yes—
96Dic3⋊Dic3—32no—
97Dic3⋊Dic3—34no—
98Dic3⋊Dic3—62no—
99Dic3⋊Dic3C2×D314yes—
100#3722—32no—
101#3722C2×D314yes—
102(C2×D3)⋊Dic3—32no—
103(C2×D3)⋊Dic3—32no—
104(C2×D3)⋊Dic3—34no—
105(C2×D3)⋊Dic3—34no—
106(C2×D3)⋊Dic3—64no—
107Dic32—34no—
108Dic32—34no—
109Dic32—34no—
110Dic32—64no—
111Dic32—62no—
112C22×C33C2411yescharacteristic
113C2×C3×C3⋊D3—22no—
114C2×C3×C3⋊D3—22no—
115C2×C3×C3⋊D3—22no—
116C33×C4#4112yes—
117C3⋊(C3⋊Dic3)—22no—
118C3×C3⋊Dic3—22no—
119C3×C3⋊Dic3—21no—
120C3×C3⋊Dic3#4112yes—
121C32×Dic3—22no—
122C32×Dic3—24no—
123C3×C4⋊Q8—91no—
124C3×C22⋊Q8—92no—
125#947—92no—
126#946—32no—
127Q8×Dic3—92no—
128Q8×Dic3—92no—
129Q8×Dic3—181no—
130C4×#67—92no—
131#926—92no—
132#898—94no—
133#898—94no—
134#897—32no—
135#892—92no—
136#892—92no—
137#892—182no—
138#888—32no—
139#887—94no—
140#885—94no—
141#879—94no—
142C4×D12—91no—
143C4×Dic6—91no—
144C22×C3⋊D3C22×D312yes—
145C22×C3×D3—32no—
146C22×C3×D3—32no—
147C22×C3×D3—62no—
148C32×Q8—62no—
149C2×C32×C4—21no—
150C2×C32×C4—22no—
151C2×C32×C4—32no—
152C2×C32×C4—34no—
153C2×C32×C4—41no—
154C2×C32×C4C22×D312yes—
155C2×C32×C4C22×D311yescharacteristic
156C2×C3⋊Dic3—32no—
157C2×C3⋊Dic3—34no—
158C2×C3⋊Dic3—64no—
159C2×C3⋊Dic3—62no—
160C2×C3⋊Dic3—122no—
161C2×C3⋊Dic3C22×D311yescharacteristic
162C2×C3⋊Dic3C22×D312yes—
163C2×C3⋊Dic3C22×D311yescharacteristic
164C4×C3⋊D3—24no—
165C2×C3×Dic3—22no—
166C2×C3×Dic3—24no—
167C2×C3×Dic3—32no—
168C2×C3×Dic3—32no—
169C2×C3×Dic3—32no—
170C2×C3×Dic3—32no—
171C2×C3×Dic3—61no—
172C2×C3×Dic3—62no—
173C2×C3×Dic3—61no—
174C2×C3×Dic3C22×D314yes—
175C2×C3×Dic3C22×D314yes—
176C2×C3×Dic3C22×D314yes—
177C3×C4×D3—68no—
178C3×C4×D3—64no—
179C3×C4×D3—64no—
180C3×Dic6—24no—
181C3×Dic6—24no—
182C3×Dic6—68no—
183#627—24no—
184#626—64no—
185#625—62no—
186#556—271noSylow
187C2×C33C2×#4111yescharacteristic
188C2×C33C2×#4111yescharacteristic
189C2×C332−1+411yescharacteristic
190C3×C3⋊D3—42no—
191C2×C3×Q8—32no—
192C2×C3×Q8—61no—
193C2×C3×Q8—92no—
194C22×C3×C4—92no—
195C2×#67—92no—
196C2×D12—91no—
197C2×C4×D3—32no—
198C2×C4×D3—32no—
199C2×C4×D3—62no—
200C2×Dic6—34no—
201C2×Dic6D3211yescharacteristic
202C3×C4⋊C4—32no—
203C3×C4⋊C4—32no—
204C3×C4⋊C4—62no—
205C3×C4⋊C4—94no—
206C3×C4⋊C4—92no—
207C3×C22⋊C4—94no—
208C3×C42—91no—
209#217—92no—
210#217—94no—
211(C2×D3)⋊C4—34no—
212(C2×D3)⋊C4—34no—
213(C2×D3)⋊C4—92no—
214(C2×D3)⋊C4—92no—
215(C2×D3)⋊C4—94no—
216(C2×D3)⋊C4—94no—
217(C2×D3)⋊C4—184no—
218C4⋊Dic3—34no—
219C4⋊Dic3—91no—
220C4⋊Dic3—94no—
221C4⋊Dic3—91no—
222C4⋊Dic3—92no—
223C4⋊Dic3—182no—
224C4⋊Dic3—182no—
225C4⋊Dic3—181no—
226C4⋊Dic3—361no—
227C4⋊Dic3D3212yes—
228Dic3⋊C4—34no—
229Dic3⋊C4—34no—
230Dic3⋊C4—92no—
231Dic3⋊C4—92no—
232C4×Dic3—32no—
233C4×Dic3—61no—
234C4×Dic3—94no—
235C4×Dic3—94no—
236C4×Dic3—94no—
237C4×Dic3—92no—
238C4×Dic3—94no—
239C4×Dic3—184no—
240C4×Dic3—182no—
241C22×C32—21no—
242C22×C32—22no—
243C22×C32—41no—
244C22×C32C23×D311yescharacteristic
245C22×C32C23×D312yes—
246C2×C3⋊D3—22no—
247C2×C3⋊D3—22no—
248C2×C3⋊D3—22no—
249C2×C3×D3—64no—
250C2×C3×D3—62no—
251C2×C3×D3—62no—
252C2×C3×D3—62no—
253C2×C3×D3—62no—
254C2×C3×D3—62no—
255C2×C3×D3—62no—
256C2×C3×D3—122no—
257C2×C3×D3—122no—
258C32×C4—24no—
259C32×C4—22no—
260C32×C4—42no—
261C32×C4—62no—
262C32×C4—64no—
263C32×C4#23714yes—
264C32×C4#23912yes—
265C3⋊Dic3—22no—
266C3⋊Dic3—21no—
267C3⋊Dic3—64no—
268C3⋊Dic3—62no—
269C3⋊Dic3—122no—
270C3⋊Dic3—124no—
271C3⋊Dic3—242no—
272C3⋊Dic3#23712yes—
273C3×Dic3—24no—
274C3×Dic3—24no—
275C3×Dic3—24no—
276C3×Dic3—24no—
277C3×Dic3—34no—
278C3×Dic3—44no—
279C3×Dic3—62no—
280C3×Dic3—64no—
281C3×Dic3—62no—
282C3×Dic3—62no—
283C3×Dic3—62no—
284C3×Dic3—121no—
285C4⋊Q8—91no—
286#126—274no—
287#124—272no—
288C22⋊Q8—92no—
289C4×Q8—271no—
290C4×Q8—272no—
291C4×D4—271no—
292#117—272no—
293C33#55611yescharacteristic, complemented, Sylow
294C23×C3—92no—
295C22×D3—32no—
296C22×D3—32no—
297C22×D3—62no—
298C3×Q8—64no—
299C3×Q8—64no—
300C3×Q8—184no—
301C2×C3×C4—21no—
302C2×C3×C4—22no—
303C2×C3×C4—34no—
304C2×C3×C4—34no—
305C2×C3×C4—34no—
306C2×C3×C4—41no—
307C2×C3×C4—62no—
308C2×C3×C4—64no—
309C2×C3×C4—91no—
310C2×C3×C4—92no—
311C2×C3×C4—91no—
312C2×C3×C4—184no—
313C2×C3×C4C2×D3212yes—
314C2×C3×C4C2×D3211yescharacteristic
315#67—184no—
316C2×Dic3—32no—
317C2×Dic3—32no—
318C2×Dic3—32no—
319C2×Dic3—32no—
320C2×Dic3—61no—
321C2×Dic3—62no—
322C2×Dic3—61no—
323C2×Dic3—92no—
324C2×Dic3—94no—
325C2×Dic3—182no—
326C2×Dic3—184no—
327C2×Dic3—362no—
328C2×Dic3C2×D3214yes—
329C2×Dic3C2×D3212yes—
330D12—182no—
331C4×D3—68no—
332C4×D3—64no—
333C4×D3—64no—
334Dic6—22no—
335Dic6—68no—
336C2×C32—22no—
337C2×C32—22no—
338C2×C32—21no—
339C2×C32—22no—
340C2×C32—21no—
341C2×C32—21no—
342C2×C32—41no—
343C2×C32—41no—
344C2×C32—41no—
345C2×C32C2×#23712yes—
346C2×C32C2×#23711yescharacteristic
347C2×C32C2×#23911yescharacteristic
348C2×C32#100712yes—
349C2×C32#100711yescharacteristic
350C2×C32D3×#4112yes—
351C3⋊D3—42no—
352C3×D3—124no—
353C3×D3—122no—
354C3×D3—122no—
355C2×Q8—31no—
356C2×Q8—92no—
357C2×D4—271no—
358C22×C4—92no—
359C4⋊C4—32no—
360C4⋊C4—94no—
361C4⋊C4—92no—
362C4⋊C4—272no—
363C4⋊C4—271no—
364C4⋊C4—271no—
365C22⋊C4—94no—
366C22⋊C4—274no—
367C22⋊C4—272no—
368C42—91no—
369C42—274no—
370C42—272no—
371C22×C3—21no—
372C22×C3—22no—
373C22×C3—41no—
374C22×C3—182no—
375C22×C3—182no—
376C22×C3—182no—
377C22×C3C22×D3212yes—
378C22×C3C22×D3211yescharacteristic
379C2×D3—64no—
380C2×D3—62no—
381C2×D3—62no—
382C2×D3—62no—
383C2×D3—62no—
384C2×D3—62no—
385C2×D3—62no—
386C2×D3—122no—
387C2×D3—122no—
388C3×C4—24no—
389C3×C4—22no—
390C3×C4—42no—
391C3×C4—64no—
392C3×C4—64no—
393C3×C4—64no—
394C3×C4—64no—
395C3×C4—92no—
396C3×C4—124no—
397C3×C4—182no—
398C3×C4—181no—
399C3×C4#379614yes—
400C3×C4#380512yes—
401Dic3—24no—
402Dic3—22no—
403Dic3—34no—
404Dic3—62no—
405Dic3—64no—
406Dic3—62no—
407Dic3—62no—
408Dic3—62no—
409Dic3—121no—
410Dic3—182no—
411Dic3—184no—
412Dic3—362no—
413Dic3—364no—
414Dic3—722no—
415C32—21no—
416C32—22no—
417C32—41no—
418C32#564112yescomplemented
419C32#573211yescharacteristic, complemented
420C23—92no—
421Q8—62no—
422Q8—184no—
423D4—542no—
424C2×C4—32no—
425C2×C4—34no—
426C2×C4—91no—
427C2×C4—91no—
428C2×C4—92no—
429C2×C4—184no—
430C2×C4—272no—
431C2×C4D3311yescharacteristic
432C2×C3—21no—
433C2×C3—22no—
434C2×C3—21no—
435C2×C3—22no—
436C2×C3—22no—
437C2×C3—21no—
438C2×C3—41no—
439C2×C3—41no—
440C2×C3—41no—
441C2×C3—362no—
442C2×C3C2×#379612yes—
443C2×C3D3×#23712yes—
444C2×C3#6431611yescharacteristic
445C2×C3#6431612yes—
446C2×C3#6431911yescharacteristic
447C2×C3C2×#380511yescharacteristic
448D3—124no—
449D3—122no—
450D3—122no—
451C22—182no—
452C22—182no—
453C22—182no—
454C22C2×D3311yescharacteristic
455C4—64no—
456C4—62no—
457C4—92no—
458C4—181no—
459C4—182no—
460C4—542no—
461C4#8896412yes—
462C3—22no—
463C3—21no—
464C3—41no—
465C3#1059455712yescomplemented
466C3#1059541311yescharacteristic, complemented
467C2—362no—
468C2#1172437411yescharacteristic
469C2#1172439911yescharacteristic
470C2C2×#8896411yescharacteristic
subgroupG/H
center, FrattiniC22C2×D33
commutator, socle, cosocleC22×C33C24
FittingC2×C33×C4C23
prime radicalC2×#6295C22
hypercenterC2×C4D33
hypocenterC33#556
LCSG ▷ C22×C33 ▷ C33
UCSC1 ◁ C22 ◁ C2×C4

Complements

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

NH ≅ G/Ntype
2. #11723404C2semidirect
3. Dic6×C3⋊Dic3C2semidirect
4. C3×#63919C2semidirect
5. C3×#63871C2semidirect
6. #11723027C2semidirect
8. #11722941C2semidirect
32. #63919D3semidirect
37. #63871D3semidirect
293. C33#556semidirect
418. C32#5641semidirect
419. C32#5732semidirect
465. C3#10594557semidirect
466. C3#10595413semidirect

Maximal subgroups

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

subgroupindex
C4×#62952
#117234042
Dic6×C3⋊Dic32
C3×#639192
C3×#638712
#117230272
#117230072
#117229412
#105954133
#105945573