Finite Group Databasebeta

← #13169439

Subgroups — #13169439

Order 1728.

1104 conjugacy classes of subgroups; only the 184 normal subgroups are shown.

normal subgroups only

This group has 1104 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#11723431C212yesmaximal, complemented
3#11723406C211yesmaximal, characteristic, complemented
4C3×#63919C211yesmaximal, characteristic, complemented
5C3×#63849C212yesmaximal, complemented
6#11723023C214yesmaximal, complemented
7#11723007C212yesmaximal
8Dic3×#627C212yesmaximal, complemented
9#10594581—31nomaximal
10#10594556—32nomaximal
11C2×#6308C2211yescharacteristic
12C2×C3×C4×C3⋊D3C2212yes—
13C2×#6295C2211yescharacteristic
14C2×C3×#627C2212yes—
15C3×C4×C3⋊Dic3C2211yescharacteristic
16C32×C4⋊Dic3C2211yescharacteristic
17C32×C4⋊Dic3C2212yes—
18#88815C2211yescharacteristic
19#88814C2212yes—
20#88806C2212yes—
21Dic3⋊(C3⋊Dic3)C2214yes—
22#88803C2214yes—
23Dic3×C3⋊Dic3C2212yes—
24C3×#3724C2212yes—
25C3×#3722C2214yes—
26C3×Dic32C2214yes—
27#64106—32no—
28#64096—31no—
29C3×#892—32no—
30C3×#892—32no—
31C3×#888—32no—
32#63919D311yescharacteristic, complemented
33C4×#627—32no—
34#63915—32no—
35#63907—34no—
36C4×#626—32no—
37C4×#625—31no—
38#63868—34no—
39#63861—34no—
40#63859—34no—
41#63857—32no—
42#63857—32no—
43#63857—62no—
44#63849D312yescomplemented
45#63844—32no—
46#63844—61no—
47Dic3×Dic6—34no—
48C22×C3×C3⋊D3C2312yes—
49C2×C33×C4C2311yescharacteristic
50C2×C3⋊(C3⋊Dic3)C2312yes—
51#6308—22no—
52C2×C3×C3⋊Dic3C2311yescharacteristic
53C2×C3×C3⋊Dic3C2312yes—
54C2×C3×C3⋊Dic3C2311yescharacteristic
55C3×C4×C3⋊D3—24no—
56C2×C32×Dic3C2314yes—
57C2×C32×Dic3C2312yes—
58#6295—22no—
59C3×#627—24no—
60#5732—91no—
61#5533—92no—
62C2×C4×C3⋊D3C2×D312yes—
63C2×#634—31no—
64C2×#634—32no—
65C2×#634—61no—
66C2×#634—62no—
67C2×#634—121no—
68C2×C3×C4×D3—32no—
69C2×C3×C4×D3—32no—
70C2×C3×C4×D3—62no—
71C2×C3×Dic6—34no—
72C2×#627C2×D312yes—
73C2×#626—32no—
74C2×#625—31no—
75C32×C4⋊C4—32no—
76C32×C4⋊C4—31no—
77#3752—34no—
78C4⋊(C3⋊Dic3)—34no—
79#3750—32no—
80C4×C3⋊Dic3—32no—
81C4×C3⋊Dic3C2×D311yescharacteristic
82C3×(C2×D3)⋊C4—34no—
83C3×(C2×D3)⋊C4—34no—
84C3×C4⋊Dic3—21no—
85C3×C4⋊Dic3—22no—
86C3×C4⋊Dic3C2×D312yes—
87C3×C4⋊Dic3C2×D312yes—
88C3×C4⋊Dic3C2×D312yes—
89C3×Dic3⋊C4—34no—
90C3×C4×Dic3—32no—
91C3×C4×Dic3—34no—
92C3×C4×Dic3—34no—
93C3×C4×Dic3—61no—
94#3724—31no—
95#3724—34no—
96#3724—62no—
97#3724C2×D312yes—
98Dic3⋊Dic3—32no—
99Dic3⋊Dic3—34no—
100Dic3⋊Dic3—34no—
101Dic3⋊Dic3—64no—
102#3722—32no—
103#3722C2×D314yes—
104(C2×D3)⋊Dic3—32no—
105(C2×D3)⋊Dic3—32no—
106(C2×D3)⋊Dic3—34no—
107(C2×D3)⋊Dic3—34no—
108(C2×D3)⋊Dic3—64no—
109Dic32—34no—
110Dic32—62no—
111Dic32C2×D314yes—
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—24no—
122C32×Dic3—22no—
123C3×C4⋊Q8—91no—
124C3×#117—92no—
125#947—92no—
126Q8×Dic3—92no—
127C4×#67—92no—
128#924—92no—
129#898—94no—
130#897—92no—
131#897—92no—
132#897—182no—
133#892—32no—
134#892—32no—
135#888—32no—
136#887—92no—
137#887—181no—
138#885—94no—
139#879—94no—
140#876—94no—
141C4×D12—91no—
142C4×Dic6—94no—
143C4×Dic6—91no—
144C22×C3⋊D3C22×D312yes—
145C22×C3×D3—32no—
146C22×C3×D3—32no—
147C22×C3×D3—62no—
148C2×C32×C4—21no—
149C2×C32×C4—22no—
150C2×C32×C4—32no—
151C2×C32×C4—34no—
152C2×C32×C4—41no—
153C2×C32×C4C22×D312yes—
154C2×C32×C4C22×D311yescharacteristic
155C2×C3⋊Dic3—34no—
156C2×C3⋊Dic3—32no—
157C2×C3⋊Dic3—62no—
158C2×C3⋊Dic3—64no—
159C2×C3⋊Dic3—122no—
160C2×C3⋊Dic3C22×D312yes—
161C2×C3⋊Dic3C22×D311yescharacteristic
162C2×C3⋊Dic3C22×D311yescharacteristic
163C4×C3⋊D3—24no—
164#634—62no—
165#634—64no—
166#634—64no—
167#634—124no—
168#634—124no—
169C2×C3×Dic3—22no—
170C2×C3×Dic3—24no—
171C2×C3×Dic3—32no—
172C2×C3×Dic3—32no—
173C2×C3×Dic3—32no—
174C2×C3×Dic3—32no—
175C2×C3×Dic3—61no—
176C2×C3×Dic3—62no—
177C2×C3×Dic3—61no—
178C2×C3×Dic3C22×D314yes—
179C2×C3×Dic3C22×D314yes—
180C2×C3×Dic3C22×D314yes—
181C3×C4×D3—64no—
182C3×C4×D3—64no—
183C3×C4×D3—124no—
184C3×Dic6—68no—
185#627—24no—
186#626—64no—
187#625—62no—
188#556—271noSylow
189C2×C33C2×#4111yescharacteristic
190C2×C33C2×#4111yescharacteristic
191C2×C332−1+411yescharacteristic
192C3×C3⋊D3—42no—
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×Dic6—91no—
202C2×Dic6—92no—
203C2×Dic6—181no—
204C2×Dic6—182no—
205C2×Dic6—361no—
206C3×C4⋊C4—32no—
207C3×C4⋊C4—32no—
208C3×C4⋊C4—32no—
209C3×C4⋊C4—61no—
210C3×C4⋊C4—62no—
211C3×C4⋊C4—92no—
212C3×C22⋊C4—94no—
213C3×C42—94no—
214C3×C42—91no—
215#217—92no—
216#217—94no—
217(C2×D3)⋊C4—34no—
218(C2×D3)⋊C4—34no—
219(C2×D3)⋊C4—92no—
220(C2×D3)⋊C4—92no—
221(C2×D3)⋊C4—94no—
222(C2×D3)⋊C4—94no—
223(C2×D3)⋊C4—184no—
224C4⋊Dic3—91no—
225C4⋊Dic3—94no—
226C4⋊Dic3—94no—
227C4⋊Dic3—184no—
228C4⋊Dic3D3211yescharacteristic
229C4⋊Dic3D3212yes—
230Dic3⋊C4—34no—
231Dic3⋊C4—92no—
232Dic3⋊C4—94no—
233Dic3⋊C4—94no—
234Dic3⋊C4—92no—
235Dic3⋊C4—182no—
236C4×Dic3—32no—
237C4×Dic3—34no—
238C4×Dic3—34no—
239C4×Dic3—61no—
240C4×Dic3—94no—
241C4×Dic3—92no—
242C4×Dic3—182no—
243C22×C32—21no—
244C22×C32—22no—
245C22×C32—41no—
246C22×C32C23×D312yes—
247C22×C32C23×D311yescharacteristic
248C2×C3⋊D3—22no—
249C2×C3⋊D3—22no—
250C2×C3⋊D3—22no—
251C2×C3×D3—64no—
252C2×C3×D3—62no—
253C2×C3×D3—62no—
254C2×C3×D3—62no—
255C2×C3×D3—62no—
256C2×C3×D3—62no—
257C2×C3×D3—62no—
258C2×C3×D3—122no—
259C2×C3×D3—122no—
260C32×C4—22no—
261C32×C4—24no—
262C32×C4—42no—
263C32×C4—62no—
264C32×C4—64no—
265C32×C4#23514yes—
266C32×C4#23912yes—
267C3⋊Dic3—22no—
268C3⋊Dic3—21no—
269C3⋊Dic3—62no—
270C3⋊Dic3—64no—
271C3⋊Dic3—64no—
272C3⋊Dic3—124no—
273C3⋊Dic3—124no—
274C3⋊Dic3#23712yes—
275C3×Dic3—24no—
276C3×Dic3—24no—
277C3×Dic3—24no—
278C3×Dic3—34no—
279C3×Dic3—42no—
280C3×Dic3—44no—
281C3×Dic3—62no—
282C3×Dic3—62no—
283C3×Dic3—62no—
284C3×Dic3—62no—
285C3×Dic3—122no—
286C3×Dic3—121no—
287C4⋊Q8—91no—
288#126—274no—
289#124—272no—
290C22⋊Q8—272no—
291C4×Q8—272no—
292C4×Q8—271no—
293C4×D4—271no—
294#117—92no—
295C33#55611yescharacteristic, complemented, Sylow
296C23×C3—92no—
297C22×D3—32no—
298C22×D3—32no—
299C22×D3—62no—
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×D3211yescharacteristic
314C2×C3×C4C2×D3212yes—
315#67—184no—
316C2×Dic3—32no—
317C2×Dic3—32no—
318C2×Dic3—32no—
319C2×Dic3—32no—
320C2×Dic3—61no—
321C2×Dic3—61no—
322C2×Dic3—62no—
323C2×Dic3—94no—
324C2×Dic3—92no—
325C2×Dic3—182no—
326C2×Dic3—184no—
327C2×Dic3—362no—
328C2×Dic3C2×D3212yes—
329C2×Dic3C2×D3214yes—
330D12—182no—
331C4×D3—64no—
332C4×D3—64no—
333C4×D3—124no—
334Dic6—68no—
335Dic6—184no—
336Dic6—182no—
337Dic6—184no—
338Dic6—364no—
339Dic6—364no—
340C2×C32—21no—
341C2×C32—22no—
342C2×C32—21no—
343C2×C32—22no—
344C2×C32—22no—
345C2×C32—21no—
346C2×C32—41no—
347C2×C32—41no—
348C2×C32—41no—
349C2×C32C2×#23512yes—
350C2×C32C2×#23711yescharacteristic
351C2×C32C2×#23911yescharacteristic
352C2×C32#100711yescharacteristic
353C2×C32D3×#4112yes—
354C2×C32#101012yes—
355C3⋊D3—42no—
356C3×D3—124no—
357C3×D3—122no—
358C3×D3—122no—
359C2×Q8—92no—
360C2×Q8—271no—
361C2×D4—271no—
362C22×C4—92no—
363C4⋊C4—31no—
364C4⋊C4—32no—
365C4⋊C4—92no—
366C4⋊C4—274no—
367C4⋊C4—272no—
368C4⋊C4—271no—
369C22⋊C4—94no—
370C22⋊C4—274no—
371C22⋊C4—272no—
372C42—94no—
373C42—91no—
374C42—272no—
375C22×C3—21no—
376C22×C3—22no—
377C22×C3—41no—
378C22×C3—182no—
379C22×C3—182no—
380C22×C3—182no—
381C22×C3C22×D3211yescharacteristic
382C22×C3C22×D3212yes—
383C2×D3—64no—
384C2×D3—62no—
385C2×D3—62no—
386C2×D3—62no—
387C2×D3—62no—
388C2×D3—62no—
389C2×D3—62no—
390C2×D3—122no—
391C2×D3—122no—
392C3×C4—22no—
393C3×C4—24no—
394C3×C4—42no—
395C3×C4—64no—
396C3×C4—64no—
397C3×C4—64no—
398C3×C4—92no—
399C3×C4—124no—
400C3×C4—122no—
401C3×C4—182no—
402C3×C4—181no—
403C3×C4#379812yes—
404C3×C4#379914yes—
405Dic3—22no—
406Dic3—24no—
407Dic3—34no—
408Dic3—62no—
409Dic3—62no—
410Dic3—62no—
411Dic3—62no—
412Dic3—122no—
413Dic3—121no—
414Dic3—184no—
415Dic3—182no—
416Dic3—184no—
417Dic3—364no—
418Dic3—364no—
419C32—21no—
420C32—22no—
421C32—41no—
422C32#553312yescomplemented
423C32#573211yescharacteristic, complemented
424C23—92no—
425Q8—184no—
426Q8—542no—
427D4—542no—
428C2×C4—32no—
429C2×C4—34no—
430C2×C4—91no—
431C2×C4—92no—
432C2×C4—91no—
433C2×C4—184no—
434C2×C4—272no—
435C2×C4D3311yescharacteristic
436C2×C3—21no—
437C2×C3—22no—
438C2×C3—21no—
439C2×C3—22no—
440C2×C3—22no—
441C2×C3—21no—
442C2×C3—41no—
443C2×C3—41no—
444C2×C3—41no—
445C2×C3—362no—
446C2×C3C2×#379811yescharacteristic
447C2×C3C2×#379912yes—
448C2×C3#6430911yescharacteristic
449C2×C3D3×#23712yes—
450C2×C3#6431512yes—
451C2×C3#6431911yescharacteristic
452D3—124no—
453D3—122no—
454D3—122no—
455C22—182no—
456C22—182no—
457C22—182no—
458C22C2×D3311yescharacteristic
459C4—64no—
460C4—62no—
461C4—92no—
462C4—181no—
463C4—182no—
464C4—542no—
465C4#8896512yes—
466C3—22no—
467C3—21no—
468C3—41no—
469C3#1059455612yescomplemented
470C3#1059458111yescharacteristic, complemented
471C2—362no—
472C2#1172437411yescharacteristic
473C2#1172439411yescharacteristic
474C2C2×#8896511yescharacteristic
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. #11723431C2semidirect
3. #11723406C2semidirect
4. C3×#63919C2semidirect
5. C3×#63849C2semidirect
6. #11723023C2semidirect
8. Dic3×#627C2semidirect
32. #63919D3semidirect
44. #63849D3semidirect
295. C33#556semidirect
422. C32#5533semidirect
423. C32#5732semidirect
469. C3#10594556semidirect
470. C3#10594581semidirect

Maximal subgroups

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

subgroupindex
C4×#62952
#117234312
#117234062
C3×#639192
C3×#638492
#117230232
#117230072
Dic3×#6272
#105945813
#105945563