Finite Group Databasebeta

← #13169193

Subgroups — #13169193

Order 1728.

1168 conjugacy classes of subgroups.

The subgroup lattice has 484 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C4×#6295C211yesmaximal, characteristic
2#11723431C212yesmaximal, complemented
3#11723406C211yesmaximal, characteristic, complemented
4C3×#63919C211yesmaximal, characteristic, complemented
5C3×#63900C212yesmaximal, complemented
6#11723018C212yesmaximal, complemented
7#11722970C214yesmaximal, complemented
8#11722931C212yesmaximal
9#10594537—32nomaximal
10#10594531—31nomaximal
11C2×#6308C2211yescharacteristic
12C2×C3×C4×C3⋊D3C2212yes—
13C2×#6295C2211yescharacteristic
14C2×C3×#627C2212yes—
15C2×C3×#624C2214yes—
16C3×C4×C3⋊Dic3C2211yescharacteristic
17C32×C4⋊Dic3C2211yescharacteristic
18C32×C4⋊Dic3C2212yes—
19#88815C2211yescharacteristic
20#88814C2212yes—
21#88806C2212yes—
22Dic3⋊(C3⋊Dic3)C2214yes—
23#88803C2214yes—
24Dic3×C3⋊Dic3C2212yes—
25C3×#3724C2212yes—
26C3×#3724C2214yes—
27#64106—32no—
28#64096—31no—
29C3×D3×C4⋊C4—32no—
30C3×D3×C4⋊C4—32no—
31C3×#888—32no—
32#63919D311yescharacteristic, complemented
33#63904—34no—
34C4×#626—32no—
35C4×#625—31no—
36C4×#625—32no—
37#63900D312yescomplemented
38#63888—34no—
39#63870—32no—
40#63857—32no—
41#63857—32no—
42#63857—62no—
43#63856—34no—
44#63851—34no—
45#63845—34no—
46#63844—32no—
47#63844—61no—
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×Dic3C2312yes—
57C2×C32×Dic3C2314yes—
58#6295Q814yescomplemented
59C3×#627—42no—
60C3×#624—44no—
61#5730—91no—
62#5511—92no—
63C2×C4×C3⋊D3C2×D312yes—
64C2×#634—31no—
65C2×#634—32no—
66C2×#634—61no—
67C2×#634—62no—
68C2×#634—121no—
69C2×C3×C4×D3—32no—
70C2×C3×C4×D3—32no—
71C2×C3×C4×D3—34no—
72C2×C3×C4×D3—34no—
73C2×C3×C4×D3—62no—
74C2×C3×Dic6—34no—
75C2×#627C2×D312yes—
76C2×#626—32no—
77C2×#625—31no—
78C2×#624C2×D314yes—
79C32×C4⋊C4—32no—
80C32×C4⋊C4—31no—
81#3752—34no—
82C4⋊(C3⋊Dic3)—34no—
83#3750—32no—
84C4×C3⋊Dic3—32no—
85C4×C3⋊Dic3C2×D311yescharacteristic
86C3×C4⋊Dic3—21no—
87C3×C4⋊Dic3—22no—
88C3×C4⋊Dic3C2×D312yes—
89C3×C4⋊Dic3C2×D312yes—
90C3×C4⋊Dic3C2×D312yes—
91C3×Dic3⋊C4—34no—
92C3×Dic3⋊C4—34no—
93C3×Dic3⋊C4—34no—
94C3×C4×Dic3—32no—
95C3×C4×Dic3—61no—
96#3724—31no—
97#3724—34no—
98#3724—62no—
99#3724C2×D314yes—
100#3724C2×D312yes—
101Dic3⋊Dic3—32no—
102Dic3⋊Dic3—34no—
103Dic3⋊Dic3—34no—
104Dic3⋊Dic3—64no—
105#3722—32no—
106(C2×D3)⋊Dic3—32no—
107(C2×D3)⋊Dic3—32no—
108(C2×D3)⋊Dic3—34no—
109(C2×D3)⋊Dic3—34no—
110(C2×D3)⋊Dic3—64no—
111Dic32—34no—
112Dic32—62no—
113C22×C33C2411yescharacteristic
114C2×C3×C3⋊D3—22no—
115C2×C3×C3⋊D3—22no—
116C2×C3×C3⋊D3C2×Q814yes—
117C33×C4#4112yes—
118C3⋊(C3⋊Dic3)—22no—
119C3×C3⋊Dic3—22no—
120C3×C3⋊Dic3—21no—
121C3×C3⋊Dic3C2×Q812yes—
122C32×Dic3—22no—
123C32×Dic3—24no—
124C3×C4⋊Q8—91no—
125C2×C3×C4⋊C4—92no—
126#946—94no—
127C4×#67—92no—
128C4×#67—94no—
129#924—94no—
130#924—92no—
131#897—92no—
132#897—92no—
133#897—182no—
134#896—94no—
135#893—92no—
136D3×C4⋊C4—32no—
137D3×C4⋊C4—32no—
138#890—92no—
139#889—94no—
140#888—32no—
141#887—92no—
142#887—181no—
143C4×D12—91no—
144C4×Dic6—91no—
145C22×C3⋊D3C22×D312yes—
146C22×C3×D3—32no—
147C22×C3×D3—32no—
148C22×C3×D3—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—34no—
157C2×C3⋊Dic3—32no—
158C2×C3⋊Dic3—62no—
159C2×C3⋊Dic3—64no—
160C2×C3⋊Dic3—122no—
161C2×C3⋊Dic3C22×D311yescharacteristic
162C2×C3⋊Dic3C22×D312yes—
163C2×C3⋊Dic3C22×D311yescharacteristic
164C4×C3⋊D3—24no—
165#634—62no—
166#634—64no—
167#634—64no—
168#634—124no—
169#634—124no—
170C2×C3×Dic3—22no—
171C2×C3×Dic3—24no—
172C2×C3×Dic3—32no—
173C2×C3×Dic3—32no—
174C2×C3×Dic3—32no—
175C2×C3×Dic3—32no—
176C2×C3×Dic3—61no—
177C2×C3×Dic3—62no—
178C2×C3×Dic3—61no—
179C2×C3×Dic3C22×D314yes—
180C2×C3×Dic3C22×D314yes—
181C2×C3×Dic3C22×D314yes—
182C3×C4×D3—64no—
183C3×C4×D3—64no—
184C3×C4×D3—124no—
185C3×C4×D3—124no—
186C3×C4×D3—124no—
187C3×Dic6—124no—
188#627—42no—
189#626—38no—
190#625—34no—
191#624—44no—
192#554—271noSylow
193C2×C33C22×Q811yescharacteristic
194C2×C33C2×#4111yescharacteristic
195C2×C33H(5,𝔽2)11yescharacteristic
196C3×C3⋊D3—24no—
197C2×C3×Q8—92no—
198C22×C3×C4—94no—
199C22×C3×C4—92no—
200C2×#67—92no—
201C2×D12—91no—
202C2×C4×D3—32no—
203C2×C4×D3—32no—
204C2×C4×D3—34no—
205C2×C4×D3—34no—
206C2×C4×D3—62no—
207C2×Dic6—34no—
208C2×Dic6—91no—
209C2×Dic6—92no—
210C2×Dic6—181no—
211C2×Dic6—182no—
212C2×Dic6—361no—
213C3×C4⋊C4—32no—
214C3×C4⋊C4—32no—
215C3×C4⋊C4—32no—
216C3×C4⋊C4—61no—
217C3×C4⋊C4—62no—
218C3×C4⋊C4—94no—
219C3×C4⋊C4—92no—
220C3×C42—91no—
221#217—92no—
222#217—94no—
223(C2×D3)⋊C4—92no—
224(C2×D3)⋊C4—92no—
225(C2×D3)⋊C4—94no—
226(C2×D3)⋊C4—94no—
227(C2×D3)⋊C4—184no—
228C4⋊Dic3—91no—
229C4⋊Dic3—94no—
230C4⋊Dic3—94no—
231C4⋊Dic3—184no—
232C4⋊Dic3D3211yescharacteristic
233C4⋊Dic3D3212yes—
234Dic3⋊C4—34no—
235Dic3⋊C4—34no—
236Dic3⋊C4—34no—
237Dic3⋊C4—92no—
238Dic3⋊C4—94no—
239Dic3⋊C4—92no—
240Dic3⋊C4—94no—
241Dic3⋊C4—182no—
242C4×Dic3—32no—
243C4×Dic3—61no—
244C4×Dic3—94no—
245C4×Dic3—92no—
246C4×Dic3—182no—
247C22×C32—21no—
248C22×C32—22no—
249C22×C32—41no—
250C22×C32C23×D312yes—
251C22×C32C23×D311yescharacteristic
252C2×C3⋊D3—22no—
253C2×C3⋊D3—22no—
254C2×C3⋊D3D3×Q814yes—
255C2×C3×D3—34no—
256C2×C3×D3—34no—
257C2×C3×D3—64no—
258C2×C3×D3—62no—
259C2×C3×D3—62no—
260C2×C3×D3—62no—
261C2×C3×D3—62no—
262C2×C3×D3—122no—
263C2×C3×D3—122no—
264C32×C4—22no—
265C32×C4—24no—
266C32×C4—42no—
267C32×C4—64no—
268C32×C4—62no—
269C32×C4#23514yes—
270C32×C4#23912yes—
271C3⋊Dic3—22no—
272C3⋊Dic3—21no—
273C3⋊Dic3—62no—
274C3⋊Dic3—64no—
275C3⋊Dic3—64no—
276C3⋊Dic3—124no—
277C3⋊Dic3—124no—
278C3⋊Dic3D3×Q812yes—
279C3×Dic3—24no—
280C3×Dic3—24no—
281C3×Dic3—24no—
282C3×Dic3—34no—
283C3×Dic3—44no—
284C3×Dic3—42no—
285C3×Dic3—62no—
286C3×Dic3—62no—
287C3×Dic3—62no—
288C3×Dic3—62no—
289C3×Dic3—122no—
290C3×Dic3—121no—
291C4⋊Q8—91no—
292#125—272no—
293C22⋊Q8—272no—
294C22⋊Q8—274no—
295C4×Q8—271no—
296C4×D4—271no—
297C4×D4—272no—
298C2×C4⋊C4—92no—
299C33#55411yescharacteristic, complemented, Sylow
300C23×C3—92no—
301C22×D3—32no—
302C22×D3—32no—
303C22×D3—62no—
304C3×Q8—362no—
305C2×C3×C4—21no—
306C2×C3×C4—22no—
307C2×C3×C4—34no—
308C2×C3×C4—34no—
309C2×C3×C4—34no—
310C2×C3×C4—41no—
311C2×C3×C4—62no—
312C2×C3×C4—64no—
313C2×C3×C4—91no—
314C2×C3×C4—92no—
315C2×C3×C4—91no—
316C2×C3×C4—184no—
317C2×C3×C4—364no—
318C2×C3×C4C2×D3211yescharacteristic
319C2×C3×C4C2×D3212yes—
320#67—98no—
321C2×Dic3—32no—
322C2×Dic3—32no—
323C2×Dic3—32no—
324C2×Dic3—32no—
325C2×Dic3—61no—
326C2×Dic3—61no—
327C2×Dic3—62no—
328C2×Dic3—94no—
329C2×Dic3—92no—
330C2×Dic3—182no—
331C2×Dic3—184no—
332C2×Dic3—362no—
333C2×Dic3C2×D3212yes—
334C2×Dic3C2×D3214yes—
335D12—94no—
336C4×D3—64no—
337C4×D3—64no—
338C4×D3—124no—
339C4×D3—124no—
340C4×D3—124no—
341Dic6—124no—
342Dic6—184no—
343Dic6—182no—
344Dic6—184no—
345Dic6—364no—
346Dic6—364no—
347C2×C32—21no—
348C2×C32—22no—
349C2×C32—21no—
350C2×C32—22no—
351C2×C32—21no—
352C2×C32—22no—
353C2×C32—41no—
354C2×C32—41no—
355C2×C32—41no—
356C2×C32C2×#23512yes—
357C2×C32#100411yescharacteristic
358C2×C32#100412yes—
359C2×C32C2×D3×Q811yescharacteristic
360C2×C32C2×D3×Q812yes—
361C2×C32C2×#23911yescharacteristic
362C3⋊D3—24no—
363C3×D3—64no—
364C3×D3—64no—
365C3×D3—124no—
366C2×Q8—92no—
367C2×Q8—271no—
368C2×D4—271no—
369C22×C4—94no—
370C22×C4—92no—
371C4⋊C4—31no—
372C4⋊C4—32no—
373C4⋊C4—94no—
374C4⋊C4—92no—
375C4⋊C4—272no—
376C4⋊C4—274no—
377C4⋊C4—271no—
378C22⋊C4—274no—
379C22⋊C4—272no—
380C42—91no—
381C42—272no—
382C22×C3—21no—
383C22×C3—22no—
384C22×C3—41no—
385C22×C3—94no—
386C22×C3—182no—
387C22×C3—182no—
388C22×C3C22×D3211yescharacteristic
389C22×C3C22×D3212yes—
390C2×D3—34no—
391C2×D3—34no—
392C2×D3—64no—
393C2×D3—62no—
394C2×D3—62no—
395C2×D3—62no—
396C2×D3—62no—
397C2×D3—122no—
398C2×D3—122no—
399C3×C4—22no—
400C3×C4—24no—
401C3×C4—42no—
402C3×C4—64no—
403C3×C4—64no—
404C3×C4—64no—
405C3×C4—92no—
406C3×C4—122no—
407C3×C4—124no—
408C3×C4—182no—
409C3×C4—181no—
410C3×C4#379812yes—
411C3×C4#379914yes—
412Dic3—24no—
413Dic3—22no—
414Dic3—34no—
415Dic3—62no—
416Dic3—62no—
417Dic3—62no—
418Dic3—62no—
419Dic3—122no—
420Dic3—121no—
421Dic3—184no—
422Dic3—182no—
423Dic3—184no—
424Dic3—364no—
425Dic3—364no—
426C32—21no—
427C32—22no—
428C32—41no—
429C32#551112yescomplemented
430C32#573011yescharacteristic, complemented
431C23—92no—
432Q8—362no—
433Q8—542no—
434D4—274no—
435C2×C4—32no—
436C2×C4—34no—
437C2×C4—91no—
438C2×C4—91no—
439C2×C4—92no—
440C2×C4—184no—
441C2×C4—272no—
442C2×C4—364no—
443C2×C4D3311yescharacteristic
444C2×C3—21no—
445C2×C3—22no—
446C2×C3—21no—
447C2×C3—22no—
448C2×C3—22no—
449C2×C3—21no—
450C2×C3—41no—
451C2×C3—41no—
452C2×C3—41no—
453C2×C3—184no—
454C2×C3C2×#379811yescharacteristic
455C2×C3C2×#379912yes—
456C2×C3#6431311yescharacteristic
457C2×C3#6431312yes—
458C2×C3D32×Q811yescharacteristic
459C2×C3D32×Q812yes—
460D3—64no—
461D3—64no—
462D3—124no—
463C22—94no—
464C22—182no—
465C22—182no—
466C22C2×D3311yescharacteristic
467C4—62no—
468C4—64no—
469C4—92no—
470C4—181no—
471C4—182no—
472C4—542no—
473C4#8896512yes—
474C3—22no—
475C3—21no—
476C3—41no—
477C3#1059453111yescharacteristic, complemented
478C3#1059453712yescomplemented
479C2—184no—
480C2#1172437811yescharacteristic
481C2#1172438211yescharacteristic
482C2C2×#8896511yescharacteristic
subgroupG/H
center, FrattiniC22C2×D33
commutator, socle, cosocleC22×C33C24
FittingC2×C33×C4C23
prime radicalC2×#6295C22
hypercenterC2×C4D33
hypocenterC33#554
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×#63900C2semidirect
6. #11723018C2semidirect
7. #11722970C2semidirect
32. #63919D3semidirect
37. #63900D3semidirect
58. #6295Q8semidirect
299. C33#554semidirect
429. C32#5511semidirect
430. C32#5730semidirect
477. C3#10594531semidirect
478. C3#10594537semidirect

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×#639002
#117230182
#117229702
#117229312
#105945373
#105945313