Finite Group Databasebeta

← #13178693

Subgroups — #13178693

Order 1728.

1148 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
1C3×#64145C212yesmaximal
2C3×D4×C3⋊Dic3C211yesmaximal, characteristic
3#11723721C212yesmaximal, complemented
4#11723716C214yesmaximal, complemented
5#11723715C212yesmaximal, complemented
6C4×#6296C211yesmaximal, characteristic, complemented
7#11723678C211yesmaximal, characteristic, complemented
8#11723668C212yesmaximal, complemented
9#10595273—32nomaximal
10#10595119—31nomaximal
11C2×C33×D4C2211yescharacteristic
12C22×C3×C3⋊Dic3C2212yes—
13C2×C3×#634C2212yes—
14C2×#6296C2211yescharacteristic
15C3×#3757C2212yes—
16C3×#3757C2214yes—
17C3×#3757C2214yes—
18C3×C4⋊(C3⋊Dic3)C2211yescharacteristic
19C3×C4×C3⋊Dic3C2211yescharacteristic
20C3×C4×C3⋊Dic3C2212yes—
21#88815C2214yes—
22#88815C2212yes—
23#88815C2212yes—
24#88815C2211yescharacteristic
25#88815C2212yes—
26#88813C2214yes—
27#64145D312yes—
28D4×C3⋊Dic3D311yescharacteristic
29C3×#936—32no—
30C3×#936—32no—
31C3×#936—62no—
32C3×D4×Dic3—32no—
33C3×D4×Dic3—61no—
34#63982—32no—
35#63964—34no—
36#63957—32no—
37#63956—34no—
38#63956—34no—
39#63956—34no—
40#63955—34no—
41C4×#627—31no—
42C4×#627—32no—
43#63873—31no—
44#63866—32no—
45#63844—32no—
46#63843—32no—
47Dic3×Dic6—32no—
48C23×C33C2312yes—
49C33×D4—22no—
50C2×C33×C4C2311yescharacteristic
51C2×C3×C3⋊Dic3—24no—
52C2×C3×C3⋊Dic3C2312yes—
53C2×C3×C3⋊Dic3C2312yes—
54C2×C3×C3⋊Dic3C2311yescharacteristic
55C2×C3×C3⋊Dic3C2312yes—
56C2×C3×C3⋊Dic3C2314yes—
57C2×C3×C3⋊Dic3C2311yescharacteristic
58C3×#634—24no—
59#6296—22no—
60#5671—92no—
61#5542—91no—
62C2×C32×D4—21no—
63C2×C32×D4—22no—
64C2×C32×D4—41no—
65C2×C32×D4C2×D312yes—
66C2×C32×D4C2×D311yescharacteristic
67C22×C3⋊Dic3C2×D312yes—
68C2×#634C2×D312yes—
69C22×C3×Dic3—34no—
70C22×C3×Dic3—62no—
71C2×C3×Dic6—32no—
72C2×C3×Dic6—32no—
73C2×C3×Dic6—62no—
74C2×#627—32no—
75C2×#627—31no—
76#3757C2×D312yes—
77#3757C2×D314yes—
78#3757C2×D314yes—
79C4⋊(C3⋊Dic3)C2×D311yescharacteristic
80C4×C3⋊Dic3C2×D312yes—
81C4×C3⋊Dic3C2×D311yescharacteristic
82C3×#217—34no—
83C3×#217—34no—
84C3×#217—34no—
85C3×#217—34no—
86C3×#217—34no—
87C3×#217—62no—
88C3×#217—64no—
89C3×#217—64no—
90C3×C4⋊Dic3—32no—
91C3×C4⋊Dic3—61no—
92C3×C4×Dic3—32no—
93C3×C4×Dic3—32no—
94C3×C4×Dic3—32no—
95C3×C4×Dic3—61no—
96C3×C4×Dic3—62no—
97#3724—34no—
98#3724—32no—
99#3724—32no—
100#3724—32no—
101#3724—31no—
102Dic3⋊Dic3—34no—
103Dic3⋊Dic3—34no—
104Dic3⋊Dic3—34no—
105Dic3⋊Dic3—34no—
106Dic3⋊Dic3—32no—
107Dic3⋊Dic3—32no—
108Dic3⋊Dic3—32no—
109Dic32—34no—
110Dic32—34no—
111Dic32—34no—
112C22×C33—22no—
113C22×C33—22no—
114C22×C33—22no—
115C22×C33C2411yescharacteristic
116C33×C4#4112yes—
117C3×C3⋊Dic3—22no—
118C3×C3⋊Dic3—24no—
119C3×C3⋊Dic3—22no—
120C3×C3⋊Dic3—22no—
121C3×C3⋊Dic3—21no—
122C3×C3⋊Dic3#4112yes—
123C3×#124—92no—
124C3×C4×D4—91no—
125Q8×Dic3—92no—
126#944—92no—
127#936—32no—
128#936—32no—
129#936—62no—
130D4×Dic3—32no—
131D4×Dic3—61no—
132#926—92no—
133#924—92no—
134#887—92no—
135#879—94no—
136#879—94no—
137#879—94no—
138#878—94no—
139#877—94no—
140#869—91no—
141C4×Dic6—92no—
142C4×Dic6—91no—
143C4×Dic6—92no—
144C23×C32—22no—
145C23×C32—24no—
146C23×C32—42no—
147C23×C32C22×D312yes—
148C23×C32C22×D314yes—
149C32×D4—22no—
150C32×D4—24no—
151C32×D4—24no—
152C32×D4—44no—
153C32×D4—44no—
154C2×C32×C4—21no—
155C2×C32×C4—22no—
156C2×C32×C4—41no—
157C2×C32×C4C22×D311yescharacteristic
158C2×C32×C4C22×D312yes—
159C2×C3⋊Dic3—24no—
160C2×C3⋊Dic3C22×D312yes—
161C2×C3⋊Dic3C22×D312yes—
162C2×C3⋊Dic3C22×D314yes—
163C2×C3⋊Dic3C22×D311yescharacteristic
164C2×C3⋊Dic3C22×D312yes—
165C2×C3⋊Dic3C22×D311yescharacteristic
166#634—24no—
167C2×C3×Dic3—32no—
168C2×C3×Dic3—32no—
169C2×C3×Dic3—32no—
170C2×C3×Dic3—34no—
171C2×C3×Dic3—34no—
172C2×C3×Dic3—34no—
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×Dic3—62no—
180C2×C3×Dic3—62no—
181C2×C3×Dic3—64no—
182C2×C3×Dic3—68no—
183C2×C3×Dic3—124no—
184C3×Dic6—64no—
185C3×Dic6—64no—
186C3×Dic6—124no—
187#627—64no—
188#627—62no—
189#556—271noSylow
190C2×C33—42no—
191C2×C33C2×#4111yescharacteristic
192C2×C33C2×#4111yescharacteristic
193C2×C332−1+411yescharacteristic
194C2×C3×Q8—92no—
195C2×C3×D4—21no—
196C2×C3×D4—22no—
197C2×C3×D4—41no—
198C2×C3×D4D3212yes—
199C2×C3×D4D3211yescharacteristic
200C22×C3×C4—92no—
201C22×Dic3—34no—
202C22×Dic3—62no—
203C2×Dic6—32no—
204C2×Dic6—32no—
205C2×Dic6—62no—
206C2×Dic6—92no—
207C2×Dic6—91no—
208C3×C4⋊C4—91no—
209C3×C22⋊C4—94no—
210C3×C22⋊C4—94no—
211C3×C22⋊C4—92no—
212C3×C42—92no—
213C3×C42—91no—
214#217—34no—
215#217—34no—
216#217—34no—
217#217—34no—
218#217—34no—
219#217—62no—
220#217—64no—
221#217—64no—
222C4⋊Dic3—32no—
223C4⋊Dic3—61no—
224C4⋊Dic3—92no—
225C4⋊Dic3—91no—
226C4⋊Dic3—92no—
227C4⋊Dic3—92no—
228C4⋊Dic3—94no—
229Dic3⋊C4—92no—
230Dic3⋊C4—94no—
231Dic3⋊C4—92no—
232Dic3⋊C4—92no—
233Dic3⋊C4—94no—
234Dic3⋊C4—94no—
235Dic3⋊C4—94no—
236C4×Dic3—32no—
237C4×Dic3—32no—
238C4×Dic3—32no—
239C4×Dic3—61no—
240C4×Dic3—62no—
241C4×Dic3—94no—
242C4×Dic3—94no—
243C4×Dic3—94no—
244C22×C32—22no—
245C22×C32—24no—
246C22×C32—22no—
247C22×C32—24no—
248C22×C32—24no—
249C22×C32—22no—
250C22×C32—24no—
251C22×C32—22no—
252C22×C32—21no—
253C22×C32—41no—
254C22×C32—44no—
255C22×C32—44no—
256C22×C32—42no—
257C22×C32—42no—
258C22×C32—44no—
259C22×C32—44no—
260C22×C32—82no—
261C22×C32—82no—
262C22×C32C23×D311yescharacteristic
263C22×C32C23×D312yes—
264C32×C4—24no—
265C32×C4—22no—
266C32×C4—42no—
267C32×C4#23512yes—
268C32×C4#23714yes—
269C3⋊Dic3—24no—
270C3⋊Dic3—22no—
271C3⋊Dic3—22no—
272C3⋊Dic3—22no—
273C3⋊Dic3—21no—
274C3⋊Dic3#23712yes—
275C3×Dic3—34no—
276C3×Dic3—62no—
277C3×Dic3—64no—
278C3×Dic3—62no—
279C3×Dic3—64no—
280C3×Dic3—62no—
281C3×Dic3—64no—
282C3×Dic3—62no—
283C3×Dic3—62no—
284C3×Dic3—62no—
285C3×Dic3—124no—
286C3×Dic3—121no—
287C3×Dic3—122no—
288C3×Dic3—122no—
289C3×Dic3—122no—
290C4⋊Q8—271no—
291#126—274no—
292#124—92no—
293C22⋊Q8—272no—
294C4×Q8—271no—
295C4×Q8—272no—
296C4×D4—91no—
297#117—272no—
298C33#55611yescharacteristic, complemented, Sylow
299C23×C3—22no—
300C23×C3—24no—
301C23×C3—42no—
302C23×C3C2×D3212yes—
303C23×C3C2×D3214yes—
304C3×Q8—184no—
305C3×D4—24no—
306C3×D4—24no—
307C3×D4—22no—
308C3×D4—44no—
309C3×D4—44no—
310C2×C3×C4—21no—
311C2×C3×C4—22no—
312C2×C3×C4—41no—
313C2×C3×C4—91no—
314C2×C3×C4—92no—
315C2×C3×C4—92no—
316C2×C3×C4—94no—
317C2×C3×C4—91no—
318C2×C3×C4—92no—
319C2×C3×C4—184no—
320C2×C3×C4C2×D3212yes—
321C2×C3×C4C2×D3211yescharacteristic
322C2×Dic3—32no—
323C2×Dic3—32no—
324C2×Dic3—32no—
325C2×Dic3—32no—
326C2×Dic3—32no—
327C2×Dic3—32no—
328C2×Dic3—34no—
329C2×Dic3—34no—
330C2×Dic3—34no—
331C2×Dic3—61no—
332C2×Dic3—62no—
333C2×Dic3—61no—
334C2×Dic3—62no—
335C2×Dic3—62no—
336C2×Dic3—64no—
337C2×Dic3—68no—
338C2×Dic3—124no—
339Dic6—64no—
340Dic6—64no—
341Dic6—124no—
342Dic6—184no—
343Dic6—182no—
344C2×C32—21no—
345C2×C32—22no—
346C2×C32—22no—
347C2×C32—21no—
348C2×C32—22no—
349C2×C32—21no—
350C2×C32—41no—
351C2×C32—41no—
352C2×C32—41no—
353C2×C32—42no—
354C2×C32—44no—
355C2×C32—44no—
356C2×C32—84no—
357C2×C32—84no—
358C2×C32C2×#23511yescharacteristic
359C2×C32C2×#23711yescharacteristic
360C2×C32C2×#23712yes—
361C2×C32D3×#4112yes—
362C2×C32#101012yes—
363C2×C32#101011yescharacteristic
364C2×Q8—92no—
365C2×Q8—271no—
366C2×D4#111311yescharacteristic
367C22×C4—92no—
368C4⋊C4—91no—
369C4⋊C4—271no—
370C4⋊C4—272no—
371C4⋊C4—274no—
372C4⋊C4—272no—
373C4⋊C4—272no—
374C22⋊C4—94no—
375C22⋊C4—92no—
376C22⋊C4—94no—
377C42—91no—
378C42—92no—
379C42—274no—
380C22×C3—24no—
381C22×C3—22no—
382C22×C3—24no—
383C22×C3—24no—
384C22×C3—22no—
385C22×C3—24no—
386C22×C3—22no—
387C22×C3—21no—
388C22×C3—22no—
389C22×C3—42no—
390C22×C3—42no—
391C22×C3—44no—
392C22×C3—44no—
393C22×C3—44no—
394C22×C3—44no—
395C22×C3—41no—
396C22×C3—82no—
397C22×C3—82no—
398C22×C3C22×D3212yes—
399C22×C3C22×D3211yescharacteristic
400C3×C4—24no—
401C3×C4—22no—
402C3×C4—42no—
403C3×C4—92no—
404C3×C4—181no—
405C3×C4—182no—
406C3×C4—182no—
407C3×C4—182no—
408C3×C4—184no—
409C3×C4#380414yes—
410C3×C4#380512yes—
411Dic3—34no—
412Dic3—62no—
413Dic3—62no—
414Dic3—62no—
415Dic3—62no—
416Dic3—62no—
417Dic3—64no—
418Dic3—62no—
419Dic3—64no—
420Dic3—64no—
421Dic3—124no—
422Dic3—122no—
423Dic3—122no—
424Dic3—121no—
425Dic3—122no—
426C32—21no—
427C32—22no—
428C32—41no—
429C32#554211yescharacteristic, complemented
430C32#567112yescomplemented
431C23C2×#111312yes—
432Q8—184no—
433Q8—542no—
434D4—22no—
435C2×C4—91no—
436C2×C4—92no—
437C2×C4—91no—
438C2×C4—92no—
439C2×C4—92no—
440C2×C4—94no—
441C2×C4—184no—
442C2×C4C2×#111311yescharacteristic
443C2×C3—21no—
444C2×C3—22no—
445C2×C3—21no—
446C2×C3—22no—
447C2×C3—22no—
448C2×C3—21no—
449C2×C3—44no—
450C2×C3—42no—
451C2×C3—44no—
452C2×C3—41no—
453C2×C3—41no—
454C2×C3—41no—
455C2×C3—84no—
456C2×C3—84no—
457C2×C3#6430911yescharacteristic
458C2×C3#6430912yes—
459C2×C3D3×#23712yes—
460C2×C3#6431211yescharacteristic
461C2×C3C2×#380412yes—
462C2×C3C2×#380511yescharacteristic
463C22—22no—
464C22—22no—
465C22—22no—
466C22C22×#111311yescharacteristic
467C4—92no—
468C4—182no—
469C4—181no—
470C4—182no—
471C4—182no—
472C4—184no—
473C4#8904612yes—
474C3—21no—
475C3—22no—
476C3—41no—
477C3#1059511911yescharacteristic, complemented
478C3#1059527312yescomplemented
479C2—42no—
480C2#1172462011yescharacteristic
481C2#1172462211yescharacteristic
482C2C2×#8904611yescharacteristic
subgroupG/H
center, FrattiniC22C22×#1113
commutator, socle, cosocleC22×C33C24
Fitting, prime radicalC2×C33×D4C22
hypercenterC2×D4#1113
hypocenterC33#556
LCSG ▷ C22×C33 ▷ C33
UCSC1 ◁ C22 ◁ C2×D4

Complements

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

NH ≅ G/Ntype
3. #11723721C2semidirect
4. #11723716C2semidirect
5. #11723715C2semidirect
6. C4×#6296C2semidirect
7. #11723678C2semidirect
8. #11723668C2semidirect
298. C33#556semidirect
429. C32#5542semidirect
430. C32#5671semidirect
477. C3#10595119semidirect
478. C3#10595273semidirect

Maximal subgroups

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

subgroupindex
C3×#641452
C3×D4×C3⋊Dic32
#117237212
#117237162
#117237152
C4×#62962
#117236782
#117236682
#105952733
#105951193