Finite Group Databasebeta

← #13167514

Subgroups — #13167514

Order 1728.

476 conjugacy classes of subgroups; only the 59 normal subgroups are shown.

normal subgroups only

This group has 476 conjugacy classes of subgroups — too many to draw. Only its 59 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
1C2×C3×#3787C211yesmaximal, characteristic, complemented
2C2×#88942C211yesmaximal, characteristic, complemented
3#11723695C211yesmaximal, characteristic
4#11723671C211yesmaximal, characteristic, complemented
5#11722796C211yesmaximal, characteristic, complemented
6#11722794C211yesmaximal, characteristic, complemented
7#11722793C211yesmaximal, characteristic, complemented
8#10593435—31nomaximal
9C2×C3×C4×C3⋊D3C2211yescharacteristic
10C2×#6296C2211yescharacteristic
11C2×#6295—21no—
12C2×#6295C2211yescharacteristic
13C2×C3×#622C2211yescharacteristic
14C2×C3×#622C2211yescharacteristic
15C3×#3787—22no—
16#88942—22no—
17C3×#3752—21no—
18C3×#3750—21no—
19#88815C2211yescharacteristic
20#88815C2211yescharacteristic
21#88814—21no—
22C2×#3787D311yescharacteristic, complemented
23C2×#3775—31no—
24#63912—31no—
25#63906—61no—
26#63865—31no—
27#63851—61no—
28#63742—31no—
29#63740—31no—
30#63739—31no—
31C22×C3×C3⋊D3—21no—
32C22×C3×C3⋊D3D411yescharacteristic
33C2×C33×C4D411yescharacteristic
34C2×C3×C3⋊Dic3—21no—
35C2×C3×C3⋊Dic3—21no—
36C2×C3×C3⋊Dic3—21no—
37C2×C3×C3⋊Dic3C2311yescharacteristic
38C3×C4×C3⋊D3—21no—
39C3×C4×C3⋊D3—21no—
40#6296—21no—
41#6296—21no—
42#6295—21no—
43#6295—21no—
44#6295—42no—
45C3×#622—21no—
46C3×#622D412yescomplemented
47#5114—91nomaximal
48C2×C4×C3⋊D3C2×D311yescharacteristic
49C2×C3×C4×D3—61no—
50C2×C3×C4×D3—61no—
51C2×#627—31no—
52C2×#627—61no—
53C2×#626—61no—
54C2×#626—61no—
55C2×#626—61no—
56C2×#625—31no—
57C2×#625—61no—
58C2×#622C2×D311yescharacteristic
59C2×#622C2×D311yescharacteristic
60#3787—22no—
61#3775—62no—
62#3752—21no—
63#3750—21no—
64C3×(C2×D3)⋊C4—61no—
65C3×(C2×D3)⋊C4—61no—
66C3×(C2×D3)⋊C4—121no—
67C3×Dic3⋊C4—61no—
68C3×Dic3⋊C4—61no—
69C3×Dic3⋊C4—121no—
70#3724—31no—
71#3724—31no—
72Dic3⋊Dic3—61no—
73Dic3⋊Dic3—61no—
74#3722—61no—
75(C2×D3)⋊Dic3—61no—
76(C2×D3)⋊Dic3—61no—
77C22×C33C2×D411yescharacteristic
78C2×C3×C3⋊D3—21no—
79C2×C3×C3⋊D3—21no—
80C2×C3×C3⋊D3—41no—
81C2×C3×C3⋊D3—41no—
82C2×C3×C3⋊D3—41no—
83C2×C3×C3⋊D3SD1612yescomplemented
84C33×C4—21no—
85C3×C3⋊Dic3—41no—
86C3×C3⋊Dic3—41no—
87C3×C3⋊Dic3—41no—
88C3×C3⋊Dic3C2×D411yescharacteristic
89C3×C3⋊Dic3#4111yescharacteristic
90C22×C3×C8—91no—
91#930—91no—
92#924—91no—
93C2×#204—91no—
94#896—181no—
95#884—181no—
96#821—91no—
97#817—91no—
98#816—91no—
99C22×C3⋊D3—21no—
100C22×C3⋊D3D1211yescharacteristic
101C22×C3×D3—61no—
102C22×C3×D3—61no—
103C22×C3×D3—61no—
104C22×C3×D3—61no—
105C22×C3×D3—121no—
106C2×C32×C4—21no—
107C2×C32×C4—21no—
108C2×C32×C4—41no—
109C2×C32×C4—41no—
110C2×C32×C4D1211yescharacteristic
111C2×C3⋊Dic3—21no—
112C2×C3⋊Dic3—21no—
113C2×C3⋊Dic3—21no—
114C2×C3⋊Dic3C22×D311yescharacteristic
115C4×C3⋊D3—21no—
116C4×C3⋊D3—21no—
117C2×C3×Dic3—61no—
118C2×C3×Dic3—61no—
119C2×C3×Dic3—61no—
120C2×C3×Dic3—61no—
121C2×C3×Dic3—61no—
122C2×C3×Dic3—61no—
123C2×C3×Dic3—61no—
124C2×C3×Dic3—61no—
125C2×C3×Dic3—121no—
126C2×C3×Dic3—121no—
127C2×C3×Dic3—121no—
128C3×C4×D3—121no—
129C3×C4×D3—121no—
130C3×C4×D3—121no—
131C3×C4×D3—121no—
132#627—61no—
133#627—61no—
134#627—121no—
135#627—121no—
136#626—121no—
137#626—121no—
138#626—122no—
139#626—122no—
140#625—61no—
141#625—61no—
142#625—122no—
143#622—21no—
144#622#6712yescomplemented
145#465—271noSylow
146C2×C33#12111yescharacteristic
147C2×C33C2×SD1611yescharacteristic
148C2×C33#13511yescharacteristic
149C3×C3⋊D3—22no—
150C3×C3⋊D3—81no—
151C22×C3×C4—91no—
152C2×#67—181no—
153C2×#67—181no—
154C2×#67—181no—
155C2×D12—91no—
156C2×D12—181no—
157C2×C4×D3—61no—
158C2×C4×D3—61no—
159C2×Dic6—91no—
160C2×Dic6—181no—
161C2×C3×C8—91no—
162C2×C3×C8—91no—
163C2×C3×C8—182no—
164C3×C4⋊C4—181no—
165C3×C22⋊C4—181no—
166#217—181no—
167(C2×D3)⋊C4—61no—
168(C2×D3)⋊C4—61no—
169(C2×D3)⋊C4—121no—
170(C2×D3)⋊C4—181no—
171(C2×D3)⋊C4—181no—
172C4⋊Dic3—91no—
173C4⋊Dic3—91no—
174Dic3⋊C4—61no—
175Dic3⋊C4—61no—
176Dic3⋊C4—121no—
177Dic3⋊C4—181no—
178Dic3⋊C4—181no—
179#204—182no—
180C22×C32—21no—
181C22×C32—21no—
182C22×C32—41no—
183C22×C32—41no—
184C22×C32C2×D1211yescharacteristic
185C2×C3⋊D3—21no—
186C2×C3⋊D3—21no—
187C2×C3⋊D3—41no—
188C2×C3⋊D3—41no—
189C2×C3⋊D3—41no—
190C2×C3⋊D3#20412yescomplemented
191C2×C3×D3—62no—
192C2×C3×D3—62no—
193C2×C3×D3—121no—
194C2×C3×D3—121no—
195C2×C3×D3—121no—
196C2×C3×D3—121no—
197C2×C3×D3—121no—
198C2×C3×D3—121no—
199C2×C3×D3—121no—
200C2×C3×D3—122no—
201C2×C3×D3—121no—
202C2×C3×D3—121no—
203C2×C3×D3—121no—
204C2×C3×D3—241no—
205C2×C3×D3—241no—
206C32×C4—21no—
207C32×C4—41no—
208C32×C4—41no—
209C32×C4—81no—
210C32×C4—81no—
211C3⋊Dic3—41no—
212C3⋊Dic3—41no—
213C3⋊Dic3—41no—
214C3⋊Dic3#23511yescharacteristic
215C3⋊Dic3C2×#6711yescharacteristic
216C3×Dic3—61no—
217C3×Dic3—61no—
218C3×Dic3—61no—
219C3×Dic3—61no—
220C3×Dic3—121no—
221C3×Dic3—121no—
222C3×Dic3—121no—
223C3×Dic3—122no—
224C3×Dic3—121no—
225C3×Dic3—121no—
226C3×Dic3—121no—
227C3×Dic3—241no—
228C3×Dic3—241no—
229C2×SD16—271no—
230C22×C8—91no—
231C22⋊Q8—271no—
232#121—271no—
233#106—271no—
234#103—271no—
235#102—271no—
236C33#46511yescharacteristic, complemented, Sylow
237C23×C3—91no—
238C23×C3—181no—
239C22×D3—61no—
240C22×D3—61no—
241C22×D3—61no—
242C22×D3—61no—
243C22×D3—121no—
244C2×C3×C4—21no—
245C2×C3×C4—21no—
246C2×C3×C4—41no—
247C2×C3×C4—41no—
248C2×C3×C4—91no—
249C2×C3×C4—181no—
250C2×C3×C4—181no—
251C2×C3×C4—181no—
252C2×C3×C4—181no—
253C2×C3×C4—181no—
254C2×C3×C4GO+(4,𝔽2)11yescharacteristic
255#67—361no—
256#67—361no—
257#67—362no—
258#67—362no—
259C2×Dic3—61no—
260C2×Dic3—61no—
261C2×Dic3—61no—
262C2×Dic3—61no—
263C2×Dic3—61no—
264C2×Dic3—61no—
265C2×Dic3—61no—
266C2×Dic3—61no—
267C2×Dic3—121no—
268C2×Dic3—121no—
269C2×Dic3—121no—
270D12—181no—
271D12—181no—
272D12—362no—
273C4×D3—121no—
274C4×D3—121no—
275C4×D3—121no—
276C4×D3—121no—
277Dic6—181no—
278Dic6—181no—
279Dic6—361no—
280Dic6—361no—
281C3×C8—92no—
282C3×C8—181no—
283C2×C32—21no—
284C2×C32—21no—
285C2×C32—21no—
286C2×C32—21no—
287C2×C32—21no—
288C2×C32—21no—
289C2×C32—41no—
290C2×C32—41no—
291C2×C32—41no—
292C2×C32—41no—
293C2×C32—41no—
294C2×C32—41no—
295C2×C32C2×#20411yescharacteristic
296C2×C32#90411yescharacteristic
297C2×C32#93011yescharacteristic
298C3⋊D3—22no—
299C3⋊D3—81no—
300C3×D3—122no—
301C3×D3—122no—
302C3×D3—241no—
303C3×D3—242no—
304C3×D3—241no—
305C2×Q8—271no—
306C2×D4—271no—
307C2×D4—541no—
308C22×C4—91no—
309SD16—542no—
310C2×C8—91no—
311C2×C8—91no—
312C2×C8—182no—
313C4⋊C4—181no—
314C4⋊C4—271no—
315C4⋊C4—271no—
316C22⋊C4—181no—
317C22⋊C4—541no—
318C22×C3—21no—
319C22×C3—21no—
320C22×C3—41no—
321C22×C3—41no—
322C22×C3—92no—
323C22×C3—181no—
324C22×C3—181no—
325C22×C3—361no—
326C22×C3—361no—
327C22×C3—361no—
328C22×C3C2×GO+(4,𝔽2)11yescharacteristic
329C2×D3—62no—
330C2×D3—62no—
331C2×D3—121no—
332C2×D3—121no—
333C2×D3—121no—
334C2×D3—121no—
335C2×D3—121no—
336C2×D3—121no—
337C2×D3—121no—
338C2×D3—122no—
339C2×D3—121no—
340C2×D3—121no—
341C2×D3—121no—
342C2×D3—241no—
343C2×D3—241no—
344C3×C4—21no—
345C3×C4—41no—
346C3×C4—41no—
347C3×C4—81no—
348C3×C4—81no—
349C3×C4—91no—
350C3×C4—91no—
351C3×C4—361no—
352C3×C4—361no—
353C3×C4—361no—
354Dic3—61no—
355Dic3—61no—
356Dic3—61no—
357Dic3—61no—
358Dic3—121no—
359Dic3—121no—
360Dic3—121no—
361Dic3—122no—
362Dic3—121no—
363Dic3—121no—
364Dic3—121no—
365Dic3—241no—
366Dic3—241no—
367C32—21no—
368C32—21no—
369C32—41no—
370C32—41no—
371C32#511411yescharacteristic, complemented
372C23—91no—
373C23—181no—
374Q8—541no—
375Q8—541no—
376D4—541no—
377D4—541no—
378D4—1082no—
379C2×C4—91no—
380C2×C4—181no—
381C2×C4—181no—
382C2×C4—181no—
383C2×C4—181no—
384C2×C4—181no—
385C2×C4#157⋊D311yescharacteristic
386C8—92no—
387C8—181no—
388C2×C3—21no—
389C2×C3—21no—
390C2×C3—21no—
391C2×C3—21no—
392C2×C3—21no—
393C2×C3—21no—
394C2×C3—41no—
395C2×C3—41no—
396C2×C3—41no—
397C2×C3—41no—
398C2×C3—41no—
399C2×C3—41no—
400C2×C3—182no—
401C2×C3—721no—
402C2×C3#6422311yescharacteristic
403C2×C3#6422711yescharacteristic
404C2×C3#6424211yescharacteristic
405D3—122no—
406D3—122no—
407D3—241no—
408D3—242no—
409D3—241no—
410C22—92no—
411C22—181no—
412C22—181no—
413C22—361no—
414C22—361no—
415C22—361no—
416C22C2×#157⋊D311yescharacteristic
417C4—21no—
418C4—91no—
419C4—91no—
420C4—361no—
421C4—361no—
422C4—361no—
423C3—21no—
424C3—21no—
425C3—41no—
426C3—41no—
427C3#1059343511yescharacteristic, complemented
428C2—182no—
429C2—721no—
430C2#1172433711yescharacteristic
431C2#1172433911yescharacteristic
432C2#1172435511yescharacteristic
subgroupG/H
center, FrattiniC22C2×#157⋊D3
commutator, cosocleC2×C3×C3⋊Dic3C23
FittingC2×C33×C4D4
socleC22×C33C2×D4
prime radical#11723695C2
hypercenterC2×C4#157⋊D3
hypocenterC33#465
LCSG ▷ C2×C3×C3⋊Dic3 ▷ C2×C33 ▷ C33
UCSC1 ◁ C22 ◁ C2×C4

Complements

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

NH ≅ G/Ntype
1. C2×C3×#3787C2semidirect
2. C2×#88942C2semidirect
4. #11723671C2semidirect
5. #11722796C2semidirect
6. #11722794C2semidirect
7. #11722793C2semidirect
22. C2×#3787D3semidirect
46. C3×#622D4semidirect
83. C2×C3×C3⋊D3SD16semidirect
144. #622#67semidirect
190. C2×C3⋊D3#204semidirect
236. C33#465semidirect
371. C32#5114semidirect
427. C3#10593435semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#37872
C2×#889422
#117236952
#117236712
#117227962
#117227942
#117227932
#105934353
#51149