Finite Group Databasebeta

← #13172270

Subgroups — #13172270

Order 1728.

1426 conjugacy classes of subgroups.

The subgroup lattice has 464 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#6293C211yesmaximal, characteristic
2#11723715C211yesmaximal, characteristic
3#11723598C212yesmaximal
4C3×#63957C211yesmaximal, characteristic
5C3×#63955C212yesmaximal
6#11722943C214yesmaximal, complemented
7#11722942C214yesmaximal, complemented
8#10594828—31nomaximal
9#10594245—32nomaximal
10C22×C3⋊(C3⋊Dic3)C2211yescharacteristic
11C22×C3×C3⋊Dic3C2211yescharacteristic
12C22×C32×Dic3C2211yescharacteristic
13C2×#6293C418yescomplemented
14C2×#6293C2212yes—
15C2×#6293C2212yes—
16C3×#3757C2212yes—
17C32×#217C2212yes—
18#88815C2212yes—
19#88813C2212yes—
20#88806C2214yes—
21Dic3×C3⋊Dic3C2214yes—
22C3×#3724C2212yes—
23C3×Dic3⋊Dic3C2214yes—
24C3×Dic32C2214yes—
25C3×Dic32C2212yes—
26C22×#634—31no—
27C22×#627—32no—
28C22×#627—61no—
29#64086—32no—
30C3×#926—32no—
31C3×#877—32no—
32C3×#877—32no—
33#63957—31no—
34#63957—32no—
35#63957—32no—
36#63957—62no—
37#63957D311yescharacteristic
38#63955—32no—
39#63955D312yes—
40C4×#627—34no—
41#63844—34no—
42#63844—34no—
43#63838—34no—
44#63837—34no—
45#63837—34no—
46C23×C33C2311yescharacteristic
47C2×C3⋊(C3⋊Dic3)C2×C414yes—
48C2×C3⋊(C3⋊Dic3)C2312yes—
49C2×C3×C3⋊Dic3C2×C414yes—
50C2×C3×C3⋊Dic3C2314yes—
51C2×C3×C3⋊Dic3C2311yescharacteristic
52C2×C3×C3⋊Dic3C2311yescharacteristic
53C2×C32×Dic3C2×C414yes—
54C2×C32×Dic3C2314yes—
55C2×C32×Dic3C2312yes—
56#6293C2×C418yes—
57#6293C2×C418yes—
58#5516—91no—
59#5516—92no—
60C22×C32×C4—31no—
61C22×C3⋊Dic3—32no—
62C22×C3⋊Dic3—31no—
63C22×C3⋊Dic3—61no—
64C22×C3⋊Dic3—62no—
65C22×C3⋊Dic3—121no—
66C22×C3⋊Dic3C2×D311yescharacteristic
67C2×#634—38no—
68C2×#634—32no—
69C2×#634—32no—
70C22×C3×Dic3—21no—
71C22×C3×Dic3—32no—
72C22×C3×Dic3—61no—
73C22×C3×Dic3C2×D312yes—
74C2×#627—316no—
75C2×#627—32no—
76C2×#627—32no—
77C2×#627—32no—
78C2×#627—32no—
79C2×#627—68no—
80C2×#627—62no—
81C2×#627—62no—
82C32×C22⋊C4—32no—
83#3757C2×D312yes—
84#3750—34no—
85C4×C3⋊Dic3—34no—
86C3×#217—22no—
87C3×#217—32no—
88C3×#217—32no—
89C3×#217—62no—
90C3×#217C2×D312yes—
91C3×#217C2×D312yes—
92C3×C4⋊Dic3—34no—
93C3×Dic3⋊C4—34no—
94C3×Dic3⋊C4—34no—
95C3×C4×Dic3—34no—
96C3×C4×Dic3—34no—
97C3×C4×Dic3—34no—
98#3724—32no—
99#3724—34no—
100#3724—34no—
101#3724—64no—
102#3724C2×D312yes—
103Dic3⋊Dic3—34no—
104Dic3⋊Dic3C2×D314yes—
105Dic32—34no—
106Dic32—32no—
107Dic32—34no—
108Dic32—34no—
109Dic32—64no—
110Dic32C2×D314yes—
111Dic32C2×D312yes—
112C22×C33—22no—
113C22×C33C22×C412yes—
114C22×C33C2411yescharacteristic
115C3⋊(C3⋊Dic3)C22×C414yes—
116C3×C3⋊Dic3—24no—
117C3×C3⋊Dic3C22×C412yes—
118C3×C3⋊Dic3C22×C412yes—
119C32×Dic3—24no—
120C32×Dic3C22×C414yes—
121C22×Dic6—92no—
122C22×Dic6—91no—
123C22×Dic6—181no—
124C3×#117—91no—
125C3×#117—92no—
126#926—32no—
127#926—91no—
128#887—94no—
129#887—94no—
130#887—94no—
131#877—32no—
132#877—32no—
133#877—92no—
134#877—92no—
135#877—92no—
136#877—92no—
137#877—182no—
138C4×Dic6—94no—
139C4×Dic6—94no—
140C4×Dic6—94no—
141C23×C32—21no—
142C23×C32—22no—
143C23×C32—41no—
144C23×C32C22×D311yescharacteristic
145C23×C32C22×D312yes—
146C2×C32×C4—34no—
147C2×C32×C4—32no—
148C2×C32×C4—34no—
149C2×C3⋊Dic3—38no—
150C2×C3⋊Dic3—34no—
151C2×C3⋊Dic3—32no—
152C2×C3⋊Dic3—32no—
153C2×C3⋊Dic3—32no—
154C2×C3⋊Dic3—62no—
155C2×C3⋊Dic3—62no—
156C2×C3⋊Dic3—64no—
157C2×C3⋊Dic3—68no—
158C2×C3⋊Dic3—62no—
159C2×C3⋊Dic3—124no—
160C2×C3⋊Dic3—122no—
161C2×C3⋊Dic3C4×D314yes—
162C2×C3⋊Dic3C22×D314yes—
163C2×C3⋊Dic3C22×D311yescharacteristic
164C2×C3⋊Dic3C22×D311yescharacteristic
165#634—38no—
166#634—38no—
167C2×C3×Dic3—24no—
168C2×C3×Dic3—22no—
169C2×C3×Dic3—24no—
170C2×C3×Dic3—38no—
171C2×C3×Dic3—32no—
172C2×C3×Dic3—32no—
173C2×C3×Dic3—34no—
174C2×C3×Dic3—34no—
175C2×C3×Dic3—64no—
176C2×C3×Dic3—61no—
177C2×C3×Dic3—61no—
178C2×C3×Dic3—64no—
179C2×C3×Dic3C4×D318yes—
180C2×C3×Dic3C22×D314yes—
181C2×C3×Dic3C22×D314yes—
182C2×C3×Dic3C22×D312yes—
183C2×C3×Dic3C22×D312yes—
184#627—38no—
185#627—38no—
186#627—38no—
187#627—38no—
188#627—68no—
189#627—68no—
190#519—271noSylow
191C2×C33—22no—
192C2×C33C23×C411yescharacteristic
193C2×C332−1+412yes—
194C22×C3×C4—32no—
195C22×C3×C4—61no—
196C22×C3×C4—91no—
197C22×Dic3—32no—
198C22×Dic3—61no—
199C22×Dic3—92no—
200C22×Dic3—91no—
201C22×Dic3—182no—
202C22×Dic3—181no—
203C22×Dic3—361no—
204C22×Dic3D3211yescharacteristic
205C2×Dic6—92no—
206C2×Dic6—92no—
207C2×Dic6—92no—
208C2×Dic6—92no—
209C2×Dic6—916no—
210C2×Dic6—98no—
211C2×Dic6—92no—
212C2×Dic6—92no—
213C2×Dic6—182no—
214C2×Dic6—182no—
215C2×Dic6—188no—
216C3×C4⋊C4—92no—
217C3×C4⋊C4—94no—
218C3×C22⋊C4—32no—
219C3×C22⋊C4—32no—
220C3×C22⋊C4—62no—
221C3×C22⋊C4—92no—
222C3×C42—94no—
223C3×C42—92no—
224#217—32no—
225#217—32no—
226#217—62no—
227#217D3212yes—
228C4⋊Dic3—34no—
229C4⋊Dic3—92no—
230Dic3⋊C4—34no—
231Dic3⋊C4—34no—
232Dic3⋊C4—94no—
233Dic3⋊C4—94no—
234Dic3⋊C4—94no—
235Dic3⋊C4—94no—
236Dic3⋊C4—184no—
237C4×Dic3—34no—
238C4×Dic3—34no—
239C4×Dic3—34no—
240C4×Dic3—94no—
241C4×Dic3—92no—
242C4×Dic3—94no—
243C4×Dic3—94no—
244C4×Dic3—94no—
245C4×Dic3—184no—
246C22×C32—22no—
247C22×C32—24no—
248C22×C32—24no—
249C22×C32—21no—
250C22×C32—22no—
251C22×C32—24no—
252C22×C32—22no—
253C22×C32—42no—
254C22×C32—41no—
255C22×C32—44no—
256C22×C32—44no—
257C22×C32C2×C4×D312yes—
258C22×C32C2×C4×D314yes—
259C22×C32C23×D312yes—
260C22×C32C23×D311yescharacteristic
261C32×C4—34no—
262C32×C4—64no—
263C3⋊Dic3—24no—
264C3⋊Dic3—34no—
265C3⋊Dic3—34no—
266C3⋊Dic3—34no—
267C3⋊Dic3—64no—
268C3⋊Dic3—64no—
269C3⋊Dic3—64no—
270C3⋊Dic3—124no—
271C3⋊Dic3C2×C4×D312yes—
272C3⋊Dic3C2×C4×D312yes—
273C3×Dic3—24no—
274C3×Dic3—24no—
275C3×Dic3—24no—
276C3×Dic3—34no—
277C3×Dic3—34no—
278C3×Dic3—44no—
279C3×Dic3—62no—
280C3×Dic3—62no—
281C3×Dic3—64no—
282C3×Dic3—64no—
283C3×Dic3—124no—
284C3×Dic3C2×C4×D314yes—
285C3×Dic3C2×C4×D314yes—
286C22×Q8—271no—
287C4×Q8—274no—
288C4×Q8—274no—
289#117—91no—
290#117—92no—
291#117—272no—
292#117—271no—
293C33#51911yescharacteristic, complemented, Sylow
294C23×C3—21no—
295C23×C3—22no—
296C23×C3—41no—
297C23×C3C2×D3212yes—
298C23×C3C2×D3211yescharacteristic
299C2×C3×C4—32no—
300C2×C3×C4—32no—
301C2×C3×C4—38no—
302C2×C3×C4—34no—
303C2×C3×C4—34no—
304C2×C3×C4—62no—
305C2×C3×C4—64no—
306C2×C3×C4—64no—
307C2×C3×C4—91no—
308C2×C3×C4—91no—
309C2×C3×C4—94no—
310C2×C3×C4—94no—
311C2×Dic3—32no—
312C2×Dic3—32no—
313C2×Dic3—38no—
314C2×Dic3—34no—
315C2×Dic3—34no—
316C2×Dic3—64no—
317C2×Dic3—61no—
318C2×Dic3—61no—
319C2×Dic3—64no—
320C2×Dic3—92no—
321C2×Dic3—92no—
322C2×Dic3—92no—
323C2×Dic3—98no—
324C2×Dic3—94no—
325C2×Dic3—182no—
326C2×Dic3—182no—
327C2×Dic3—182no—
328C2×Dic3—184no—
329C2×Dic3—188no—
330C2×Dic3—362no—
331C2×Dic3—364no—
332C2×Dic3#62414yes—
333C2×Dic3C2×D3214yes—
334C2×Dic3C2×D3212yes—
335Dic6—98no—
336Dic6—98no—
337Dic6—98no—
338Dic6—98no—
339Dic6—98no—
340Dic6—98no—
341Dic6—188no—
342Dic6—188no—
343C2×C32—22no—
344C2×C32—24no—
345C2×C32—24no—
346C2×C32—22no—
347C2×C32—22no—
348C2×C32—24no—
349C2×C32—21no—
350C2×C32—41no—
351C2×C32—42no—
352C2×C32—44no—
353C2×C32—44no—
354C2×C32C22×C4×D311yescharacteristic
355C2×C32C22×C4×D312yes—
356C2×C32#101014yes—
357C2×C32#101012yes—
358C2×Q8—272no—
359C2×Q8—272no—
360C2×Q8—278no—
361C22×C4—31no—
362C22×C4—91no—
363C22×C4—271no—
364C4⋊C4—92no—
365C4⋊C4—94no—
366C4⋊C4—274no—
367C4⋊C4—272no—
368C22⋊C4—32no—
369C22⋊C4—92no—
370C42—94no—
371C42—92no—
372C42—274no—
373C42—272no—
374C22×C3—24no—
375C22×C3—22no—
376C22×C3—24no—
377C22×C3—21no—
378C22×C3—22no—
379C22×C3—22no—
380C22×C3—24no—
381C22×C3—44no—
382C22×C3—44no—
383C22×C3—41no—
384C22×C3—42no—
385C22×C3C4×D3214yes—
386C22×C3C2×#62412yes—
387C22×C3C22×D3211yescharacteristic
388C22×C3C22×D3212yes—
389C3×C4—34no—
390C3×C4—34no—
391C3×C4—64no—
392C3×C4—64no—
393C3×C4—64no—
394C3×C4—92no—
395C3×C4—92no—
396C3×C4—124no—
397C3×C4—184no—
398Dic3—24no—
399Dic3—34no—
400Dic3—34no—
401Dic3—62no—
402Dic3—62no—
403Dic3—64no—
404Dic3—64no—
405Dic3—94no—
406Dic3—94no—
407Dic3—94no—
408Dic3—124no—
409Dic3—184no—
410Dic3—184no—
411Dic3—184no—
412Dic3—364no—
413Dic3C2×#62414yes—
414C32—21no—
415C32—22no—
416C32—41no—
417C32#551612yescomplemented
418C32#551611yescharacteristic, complemented
419C23D3311yescharacteristic
420Q8—278no—
421Q8—278no—
422C2×C4—32no—
423C2×C4—34no—
424C2×C4—34no—
425C2×C4—91no—
426C2×C4—91no—
427C2×C4—94no—
428C2×C4—94no—
429C2×C4—272no—
430C2×C4—274no—
431C2×C3—24no—
432C2×C3—22no—
433C2×C3—24no—
434C2×C3—21no—
435C2×C3—22no—
436C2×C3—24no—
437C2×C3—22no—
438C2×C3—44no—
439C2×C3—44no—
440C2×C3—41no—
441C2×C3—42no—
442C2×C3#6429812yes—
443C2×C3C2×C4×D3212yes—
444C2×C3#6430914yes—
445C2×C3C22×#62411yescharacteristic
446C22—22no—
447C22D3×#62412yes—
448C22C2×D3311yescharacteristic
449C4—34no—
450C4—64no—
451C4—92no—
452C4—92no—
453C4—184no—
454C4—274no—
455C3—21no—
456C3—22no—
457C3—41no—
458C3#1059424512yescomplemented
459C3#1059482811yescharacteristic, complemented
460C2—22no—
461C2C2×D3×#62411yescharacteristic
462C2#1172442712yes—
subgroupG/H
center, FrattiniC22C2×D33
commutatorC2×C33C23×C4
Fitting, prime radicalC23×C33C23
socle, cosocleC22×C33C24
hypercenterC23D33
hypocenterC33#519
LCSG ▷ C2×C33 ▷ C33
UCSC1 ◁ C22 ◁ C23

Complements

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

NH ≅ G/Ntype
6. #11722943C2semidirect
7. #11722942C2semidirect
13. C2×#6293C4semidirect
293. C33#519semidirect
417. C32#5516semidirect
418. C32#5516semidirect
458. C3#10594245semidirect
459. C3#10594828semidirect

Maximal subgroups

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

subgroupindex
C22×#62932
#117237152
#117235982
C3×#639572
C3×#639552
#117229432
#117229422
#105948283
#105942453