Finite Group Databasebeta

← #13169394

Subgroups — #13169394

Order 1728.

1202 conjugacy classes of subgroups; only the 204 normal subgroups are shown.

normal subgroups only

This group has 1202 conjugacy classes of subgroups — too many to draw. Only its 204 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
1#11723680C211yesmaximal, characteristic, complemented
2C4×#6289C211yesmaximal, characteristic
3#11723413C212yesmaximal
4C3×#63918C212yesmaximal, complemented
5C3×#63872C211yesmaximal, characteristic, complemented
6#11722973C214yesmaximal
7#11722941C214yesmaximal, complemented
8#10594448—32nomaximal
9#10593866—31nomaximal
10C2×C4×C3⋊(C3⋊D3)C2211yescharacteristic
11C2×#6289C2212yes—
12C2×C3×#627C2214yes—
13C3×C4⋊(C3⋊Dic3)C2212yes—
14C3×C4×C3⋊Dic3C2211yescharacteristic
15C32×C4⋊Dic3C2212yes—
16C32×C4×Dic3C2211yescharacteristic
17#88815C2214yes—
18#88804C2214yes—
19Dic3×C3⋊Dic3C2214yes—
20Dic3×C3⋊Dic3C2212yes—
21C3×Dic3⋊Dic3C2214yes—
22C3×Dic3⋊Dic3C2214yes—
23#64101—32no—
24C42×C3⋊D3—31no—
25C3×#890—32no—
26C3×#888—32no—
27C3×#869—32no—
28#63918D312yescomplemented
29#63895—34no—
30#63888—34no—
31C4×#624—32no—
32C4×#624—61no—
33#63872—31no—
34#63872D311yescharacteristic, complemented
35#63871—34no—
36#63869—32no—
37#63849—32no—
38#63849—32no—
39#63849—62no—
40#63843—34no—
41#63843—34no—
42#63838—34no—
43C22×C3⋊(C3⋊D3)C2311yescharacteristic
44C2×C33×C4C2311yescharacteristic
45C2×C3⋊(C3⋊Dic3)C2311yescharacteristic
46C4×C3⋊(C3⋊D3)Q814yescomplemented
47C2×C3×C3⋊Dic3C2312yes—
48C2×C3×C3⋊Dic3C2314yes—
49C2×C32×Dic3C2314yes—
50C2×C32×Dic3C2312yes—
51#6289—24no—
52C3×#627—28no—
53#5728—92no—
54#5688—91no—
55C2×C4×C3⋊D3—31no—
56C2×C4×C3⋊D3—32no—
57C2×C4×C3⋊D3—32no—
58C2×C4×C3⋊D3—62no—
59C2×C4×C3⋊D3—61no—
60C2×C4×C3⋊D3—121no—
61C2×C3×Dic6—34no—
62C2×C3×Dic6—34no—
63C2×#627C2×D314yes—
64C2×#624—32no—
65C2×#624—32no—
66C2×#624—62no—
67C32×C4⋊C4—32no—
68C32×C42—31no—
69#3752—34no—
70C4⋊(C3⋊Dic3)C2×D312yes—
71C4×C3⋊Dic3—32no—
72C4×C3⋊Dic3—34no—
73C4×C3⋊Dic3C2×D311yescharacteristic
74C3×C4⋊Dic3—22no—
75C3×C4⋊Dic3—32no—
76C3×C4⋊Dic3—32no—
77C3×C4⋊Dic3—34no—
78C3×C4⋊Dic3—34no—
79C3×C4⋊Dic3—62no—
80C3×C4⋊Dic3C2×D312yes—
81C3×C4⋊Dic3C2×D312yes—
82C3×Dic3⋊C4—34no—
83C3×Dic3⋊C4—34no—
84C3×C4×Dic3—21no—
85C3×C4×Dic3—32no—
86C3×C4×Dic3—61no—
87C3×C4×Dic3C2×D312yes—
88#3724—34no—
89Dic3⋊Dic3—34no—
90Dic3⋊Dic3—34no—
91Dic3⋊Dic3C2×D314yes—
92Dic3⋊Dic3C2×D314yes—
93#3722—34no—
94#3722—34no—
95#3722—64no—
96Dic32—32no—
97Dic32—34no—
98Dic32—34no—
99Dic32—32no—
100Dic32—64no—
101Dic32—62no—
102C22×C33C2411yescharacteristic
103C2×C3⋊(C3⋊D3)—22no—
104C2×C3⋊(C3⋊D3)C2×Q812yes—
105C33×C4C2×Q812yes—
106C3⋊(C3⋊Dic3)C2×Q812yes—
107C3×C3⋊Dic3—22no—
108C3×C3⋊Dic3—24no—
109C32×Dic3—24no—
110C32×Dic3#4114yes—
111C3×C4⋊Q8—92no—
112C3×#125—91no—
113#946—94no—
114Q8×Dic3—94no—
115#897—94no—
116#896—94no—
117#892—92no—
118#892—92no—
119#892—92no—
120#892—182no—
121#890—32no—
122#890—92no—
123#888—32no—
124#887—94no—
125#887—94no—
126C42×D3—92no—
127C42×D3—91no—
128C42×D3—181no—
129#870—91no—
130#869—32no—
131C22×C3⋊D3—31no—
132C22×C3⋊D3—32no—
133C22×C3⋊D3—61no—
134C22×C3⋊D3—62no—
135C22×C3⋊D3—121no—
136C2×C32×C4—21no—
137C2×C32×C4—22no—
138C2×C32×C4—32no—
139C2×C32×C4—34no—
140C2×C32×C4—41no—
141C2×C32×C4C22×D312yes—
142C2×C32×C4C22×D311yescharacteristic
143C2×C3⋊Dic3—32no—
144C2×C3⋊Dic3—31no—
145C2×C3⋊Dic3—62no—
146C2×C3⋊Dic3—61no—
147C2×C3⋊Dic3—121no—
148C2×C3⋊Dic3C22×D312yes—
149C2×C3⋊Dic3C22×D314yes—
150C4×C3⋊D3—34no—
151C4×C3⋊D3—38no—
152C4×C3⋊D3—64no—
153C4×C3⋊D3—68no—
154C4×C3⋊D3—64no—
155C4×C3⋊D3—124no—
156C2×C3×Dic3—22no—
157C2×C3×Dic3—24no—
158C2×C3×Dic3—34no—
159C2×C3×Dic3—34no—
160C2×C3×Dic3—32no—
161C2×C3×Dic3—32no—
162C2×C3×Dic3—62no—
163C2×C3×Dic3—64no—
164C2×C3×Dic3C22×D314yes—
165C2×C3×Dic3C22×D314yes—
166C2×C3×Dic3C22×D312yes—
167C2×C3×Dic3C22×D312yes—
168C3×Dic6—68no—
169C3×Dic6—68no—
170#627—28no—
171#624—68no—
172#624—64no—
173#624—64no—
174#533—271noSylow
175C2×C33C22×Q811yescharacteristic
176C2×C33C2×#4112yes—
177C3⋊(C3⋊D3)—22no—
178C2×C3×Q8—94no—
179C2×C4×D3—92no—
180C2×C4×D3—91no—
181C2×C4×D3—92no—
182C2×C4×D3—92no—
183C2×C4×D3—92no—
184C2×C4×D3—182no—
185C2×C4×D3—181no—
186C2×C4×D3—182no—
187C2×C4×D3—361no—
188C2×Dic6—34no—
189C2×Dic6—34no—
190C3×C4⋊C4—32no—
191C3×C4⋊C4—32no—
192C3×C4⋊C4—62no—
193C3×C4⋊C4—94no—
194C3×C4⋊C4—94no—
195C3×C4⋊C4—92no—
196C3×C42—32no—
197C3×C42—61no—
198C3×C42—91no—
199(C2×D3)⋊C4—94no—
200(C2×D3)⋊C4—94no—
201(C2×D3)⋊C4—94no—
202(C2×D3)⋊C4—184no—
203C4⋊Dic3—32no—
204C4⋊Dic3—32no—
205C4⋊Dic3—34no—
206C4⋊Dic3—34no—
207C4⋊Dic3—62no—
208C4⋊Dic3—94no—
209C4⋊Dic3D3212yes—
210Dic3⋊C4—34no—
211Dic3⋊C4—34no—
212Dic3⋊C4—94no—
213Dic3⋊C4—94no—
214C4×Dic3—32no—
215C4×Dic3—61no—
216C4×Dic3—92no—
217C4×Dic3—92no—
218C4×Dic3—94no—
219C4×Dic3—94no—
220C4×Dic3—92no—
221C4×Dic3—94no—
222C4×Dic3—182no—
223C4×Dic3—184no—
224C4×Dic3D3211yescharacteristic
225C22×C32—21no—
226C22×C32—22no—
227C22×C32—41no—
228C22×C32C23×D312yes—
229C22×C32C23×D311yescharacteristic
230C2×C3⋊D3—32no—
231C2×C3⋊D3—34no—
232C2×C3⋊D3—62no—
233C2×C3⋊D3—64no—
234C2×C3⋊D3—62no—
235C2×C3⋊D3—64no—
236C2×C3⋊D3—62no—
237C2×C3⋊D3—62no—
238C2×C3⋊D3—122no—
239C2×C3⋊D3—122no—
240C2×C3⋊D3—122no—
241C2×C3⋊D3—124no—
242C32×C4—22no—
243C32×C4—24no—
244C32×C4—34no—
245C32×C4—42no—
246C32×C4—64no—
247C32×C4D3×Q812yes—
248C32×C4D3×Q814yes—
249C3⋊Dic3—22no—
250C3⋊Dic3—24no—
251C3⋊Dic3—32no—
252C3⋊Dic3—34no—
253C3⋊Dic3—62no—
254C3⋊Dic3—64no—
255C3⋊Dic3—122no—
256C3×Dic3—24no—
257C3×Dic3—24no—
258C3×Dic3—24no—
259C3×Dic3—44no—
260C3×Dic3—64no—
261C3×Dic3—62no—
262C3×Dic3—62no—
263C3×Dic3—64no—
264C3×Dic3—64no—
265C3×Dic3—124no—
266C3×Dic3#23714yes—
267C3×Dic3#23914yes—
268C4⋊Q8—92no—
269#125—91no—
270#125—271no—
271C22⋊Q8—274no—
272C4×Q8—274no—
273#117—272no—
274C2×C42—271no—
275C33#53311yescharacteristic, complemented, Sylow
276C22×D3—91no—
277C22×D3—92no—
278C22×D3—181no—
279C22×D3—182no—
280C22×D3—361no—
281C3×Q8—188no—
282C2×C3×C4—21no—
283C2×C3×C4—22no—
284C2×C3×C4—32no—
285C2×C3×C4—32no—
286C2×C3×C4—34no—
287C2×C3×C4—34no—
288C2×C3×C4—41no—
289C2×C3×C4—62no—
290C2×C3×C4—64no—
291C2×C3×C4—92no—
292C2×C3×C4—94no—
293C2×C3×C4C2×D3212yes—
294C2×C3×C4C2×D3211yescharacteristic
295C2×Dic3—32no—
296C2×Dic3—32no—
297C2×Dic3—34no—
298C2×Dic3—34no—
299C2×Dic3—62no—
300C2×Dic3—64no—
301C2×Dic3—91no—
302C2×Dic3—92no—
303C2×Dic3—181no—
304C2×Dic3—182no—
305C2×Dic3—361no—
306C2×Dic3C2×D3214yes—
307C2×Dic3C2×D3212yes—
308C4×D3—94no—
309C4×D3—98no—
310C4×D3—184no—
311C4×D3—188no—
312C4×D3—184no—
313C4×D3—184no—
314C4×D3—188no—
315C4×D3—184no—
316C4×D3—364no—
317Dic6—68no—
318Dic6—68no—
319C2×C32—22no—
320C2×C32—21no—
321C2×C32—22no—
322C2×C32—22no—
323C2×C32—22no—
324C2×C32—41no—
325C2×C32—42no—
326C2×C32C2×#23712yes—
327C2×C32C2×D3×Q811yescharacteristic
328C2×C32C2×D3×Q812yes—
329C2×C32C2×#23912yes—
330C2×C32D3×#4112yes—
331C3⋊D3—62no—
332C3⋊D3—64no—
333C3⋊D3—64no—
334C3⋊D3—124no—
335C3⋊D3—124no—
336C2×Q8—94no—
337C22×C4—272no—
338C22×C4—271no—
339C4⋊C4—32no—
340C4⋊C4—94no—
341C4⋊C4—94no—
342C4⋊C4—92no—
343C4⋊C4—274no—
344C22⋊C4—274no—
345C42—31no—
346C42—91no—
347C42—272no—
348C42—274no—
349C22×C3—21no—
350C22×C3—22no—
351C22×C3—41no—
352C22×C3C22×D3211yescharacteristic
353C22×C3C22×D3212yes—
354C2×D3—92no—
355C2×D3—94no—
356C2×D3—182no—
357C2×D3—184no—
358C2×D3—184no—
359C2×D3—182no—
360C2×D3—182no—
361C2×D3—182no—
362C2×D3—362no—
363C2×D3—362no—
364C2×D3—362no—
365C2×D3—364no—
366C3×C4—22no—
367C3×C4—24no—
368C3×C4—34no—
369C3×C4—34no—
370C3×C4—42no—
371C3×C4—64no—
372C3×C4—64no—
373C3×C4—64no—
374C3×C4—124no—
375C3×C4—182no—
376C3×C4—184no—
377C3×C4#379712yes—
378C3×C4#379714yes—
379Dic3—24no—
380Dic3—64no—
381Dic3—64no—
382Dic3—64no—
383Dic3—62no—
384Dic3—62no—
385Dic3—92no—
386Dic3—94no—
387Dic3—124no—
388Dic3—182no—
389Dic3—184no—
390Dic3—362no—
391Dic3#379614yes—
392C32—21no—
393C32—22no—
394C32—41no—
395C32#568811yescharacteristic, complemented
396C32#572812yescomplemented
397C23—271no—
398Q8—188no—
399C2×C4—32no—
400C2×C4—34no—
401C2×C4—92no—
402C2×C4—94no—
403C2×C4—271no—
404C2×C4—274no—
405C2×C4—544no—
406C2×C4D3311yescharacteristic
407C2×C3—22no—
408C2×C3—22no—
409C2×C3—22no—
410C2×C3—21no—
411C2×C3—22no—
412C2×C3—42no—
413C2×C3—41no—
414C2×C3C2×#379612yes—
415C2×C3C2×#379711yescharacteristic
416C2×C3C2×#379712yes—
417C2×C3D3×#23712yes—
418C2×C3D3×#23912yes—
419D3—184no—
420D3—182no—
421D3—184no—
422D3—364no—
423D3—364no—
424C22—272no—
425C22—542no—
426C22C2×D3311yescharacteristic
427C4—34no—
428C4—64no—
429C4—182no—
430C4—184no—
431C4—272no—
432C4#8895812yes—
433C3—21no—
434C3—22no—
435C3—41no—
436C3#1059386611yescharacteristic, complemented
437C3#1059444812yescomplemented
438C2—542no—
439C2D3×#379612yes—
440C2C2×#8895811yescharacteristic
subgroupG/H
center, FrattiniC22C2×D33
commutator, socle, cosocleC22×C33C24
FittingC2×C33×C4C23
prime radicalC22×C3⋊(C3⋊D3)C23
hypercenterC2×C4D33
hypocenterC33#533
LCSG ▷ C22×C33 ▷ C33
UCSC1 ◁ C22 ◁ C2×C4

Complements

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

NH ≅ G/Ntype
1. #11723680C2semidirect
4. C3×#63918C2semidirect
5. C3×#63872C2semidirect
7. #11722941C2semidirect
28. #63918D3semidirect
34. #63872D3semidirect
46. C4×C3⋊(C3⋊D3)Q8semidirect
275. C33#533semidirect
395. C32#5688semidirect
396. C32#5728semidirect
436. C3#10593866semidirect
437. C3#10594448semidirect

Maximal subgroups

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

subgroupindex
#117236802
C4×#62892
#117234132
C3×#639182
C3×#638722
#117229732
#117229412
#105944483
#105938663