Finite Group Databasebeta

← #13167600

Subgroups — #13167600

Order 1728.

496 conjugacy classes of subgroups; only the 55 normal subgroups are shown.

normal subgroups only

This group has 496 conjugacy classes of subgroups — too many to draw. Only its 55 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×#3792C211yesmaximal, characteristic, complemented
2C2×#88942C211yesmaximal, characteristic, complemented
3#11723721C211yesmaximal, characteristic, complemented
4#11723717C211yesmaximal, characteristic
5#11722797C211yesmaximal, characteristic, complemented
6#11722794C211yesmaximal, characteristic, complemented
7#11722793C211yesmaximal, characteristic, complemented
8#10593495—31nomaximal
9C2×C3×#638—21no—
10C22×C3×C3⋊Dic3C2211yescharacteristic
11C2×#6296C2211yescharacteristic
12C2×#6295C2211yescharacteristic
13C3×#3792—22no—
14C2×C3×#622C2211yescharacteristic
15C2×C3×#622C2211yescharacteristic
16#88942—22no—
17C3×#3757—21no—
18#88815—21no—
19#88815C2211yescharacteristic
20#88815C2211yescharacteristic
21#88814—21no—
22C2×#3792D311yescharacteristic, complemented
23C2×#3775—31no—
24#63982—31no—
25#63980—31no—
26#63978—61no—
27#63964—61no—
28#63743—31no—
29#63740—31no—
30#63739—31no—
31C23×C33D411yescharacteristic
32C22×C3×C3⋊D3—21no—
33C3×#638—42no—
34C2×C3×C3⋊Dic3—21no—
35C2×C3×C3⋊Dic3—21no—
36C2×C3×C3⋊Dic3—21no—
37C2×C3×C3⋊Dic3—21no—
38C2×C3×C3⋊Dic3—21no—
39C2×C3×C3⋊Dic3D411yescharacteristic
40C2×C3×C3⋊Dic3C2311yescharacteristic
41#6296—21no—
42#6296—21no—
43#6295—21no—
44#6295—21no—
45C3×#622—21no—
46C3×#622D412yescomplemented
47#5134—91nomaximal
48C2×#638—21no—
49C22×C3⋊Dic3C2×D311yescharacteristic
50C2×C3×#67—61no—
51C2×C3×#67—61no—
52C2×C3×#67—121no—
53C22×C3×Dic3—61no—
54C22×C3×Dic3—61no—
55C2×#627—31no—
56C2×#627—61no—
57C2×#626—61no—
58C2×#625—31no—
59#3792—22no—
60C2×#622C2×D311yescharacteristic
61C2×#622C2×D311yescharacteristic
62#3775—62no—
63#3757—21no—
64C3×#217—61no—
65C3×#217—61no—
66C3×#217—121no—
67#3724—31no—
68#3724—31no—
69#3724—61no—
70Dic3⋊Dic3—61no—
71Dic3⋊Dic3—61no—
72Dic3⋊Dic3—61no—
73Dic3⋊Dic3—61no—
74#3722—61no—
75(C2×D3)⋊Dic3—61no—
76(C2×D3)⋊Dic3—61no—
77C22×C33—21no—
78C22×C33—21no—
79C22×C33—21no—
80C22×C33C2×D411yescharacteristic
81C2×C3×C3⋊D3—41no—
82C2×C3×C3⋊D3—41no—
83C2×C3×C3⋊D3—41no—
84C3×C3⋊Dic3—21no—
85C3×C3⋊Dic3—41no—
86C3×C3⋊Dic3—41no—
87C3×C3⋊Dic3—41no—
88C3×C3⋊Dic3C2×D411yescharacteristic
89C3×C3⋊Dic3#4111yescharacteristic
90C2×C3×M16—91no—
91#939—181no—
92#930—91no—
93#924—91no—
94C2×#204—91no—
95#878—181no—
96#821—91no—
97#818—91no—
98#816—91no—
99C23×C32—21no—
100C23×C32—21no—
101C23×C32—41no—
102C23×C32—41no—
103C23×C32D1211yescharacteristic
104C22×C3⋊D3—21no—
105C22×C3×D3—61no—
106C22×C3×D3—61no—
107C22×C3×D3—121no—
108#638—42no—
109C2×C3⋊Dic3—21no—
110C2×C3⋊Dic3—21no—
111C2×C3⋊Dic3—21no—
112C2×C3⋊Dic3—21no—
113C2×C3⋊Dic3—21no—
114C2×C3⋊Dic3D1211yescharacteristic
115C2×C3⋊Dic3C22×D311yescharacteristic
116C3×#67—122no—
117C3×#67—122no—
118C3×#67—124no—
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—61no—
126C2×C3×Dic3—61no—
127C2×C3×Dic3—61no—
128C2×C3×Dic3—61no—
129C2×C3×Dic3—121no—
130C2×C3×Dic3—121no—
131C2×C3×Dic3—121no—
132C2×C3×Dic3—121no—
133C2×C3×Dic3—121no—
134C2×C3×Dic3—121no—
135C2×C3×Dic3—121no—
136#627—61no—
137#627—61no—
138#627—121no—
139#627—121no—
140#626—121no—
141#626—121no—
142#625—61no—
143#625—61no—
144#622—21no—
145#622#6712yescomplemented
146#468—271noSylow
147C2×C33—41no—
148C2×C33#12111yescharacteristic
149C2×C33AGL(1,ℤ/8)11yescharacteristic
150C2×C33#13711yescharacteristic
151C3×C3⋊D3—81no—
152C2×C3×D4—181no—
153C22×C3×C4—91no—
154C2×#67—61no—
155C2×#67—61no—
156C2×#67—121no—
157C2×#67—181no—
158C22×Dic3—61no—
159C22×Dic3—61no—
160C2×D12—91no—
161C2×Dic6—91no—
162C2×Dic6—181no—
163C3×M16—182no—
164C2×C3×C8—91no—
165C2×C3×C8—91no—
166C3×C22⋊C4—181no—
167#217—61no—
168#217—61no—
169#217—121no—
170#217—181no—
171(C2×D3)⋊C4—181no—
172(C2×D3)⋊C4—181no—
173C4⋊Dic3—91no—
174C4⋊Dic3—91no—
175C4⋊Dic3—181no—
176Dic3⋊C4—181no—
177Dic3⋊C4—181no—
178Dic3⋊C4—181no—
179Dic3⋊C4—181no—
180#204—182no—
181C22×C32—22no—
182C22×C32—22no—
183C22×C32—21no—
184C22×C32—21no—
185C22×C32—21no—
186C22×C32—21no—
187C22×C32—21no—
188C22×C32—41no—
189C22×C32—41no—
190C22×C32—41no—
191C22×C32—41no—
192C22×C32—41no—
193C22×C32—41no—
194C22×C32—42no—
195C22×C32—42no—
196C22×C32—81no—
197C22×C32—81no—
198C22×C32—81no—
199C22×C32—81no—
200C22×C32C2×D1211yescharacteristic
201C2×C3⋊D3—41no—
202C2×C3⋊D3—41no—
203C2×C3⋊D3—41no—
204C2×C3×D3—121no—
205C2×C3×D3—121no—
206C2×C3×D3—121no—
207C2×C3×D3—122no—
208C2×C3×D3—121no—
209C2×C3×D3—121no—
210C2×C3×D3—121no—
211C2×C3×D3—241no—
212C2×C3×D3—241no—
213C3⋊Dic3—21no—
214C3⋊Dic3—41no—
215C3⋊Dic3—41no—
216C3⋊Dic3—41no—
217C3⋊Dic3#23511yescharacteristic
218C3⋊Dic3C2×#6711yescharacteristic
219C3×Dic3—61no—
220C3×Dic3—61no—
221C3×Dic3—61no—
222C3×Dic3—61no—
223C3×Dic3—121no—
224C3×Dic3—121no—
225C3×Dic3—121no—
226C3×Dic3—121no—
227C3×Dic3—121no—
228C3×Dic3—122no—
229C3×Dic3—121no—
230C3×Dic3—121no—
231C3×Dic3—121no—
232C3×Dic3—241no—
233C3×Dic3—241no—
234C2×SD16—271no—
235C2×M16—91no—
236C22⋊Q8—271no—
237#121—271no—
238#107—271no—
239#103—271no—
240#102—271no—
241C33#46811yescharacteristic, complemented, Sylow
242C23×C3—21no—
243C23×C3—21no—
244C23×C3—41no—
245C23×C3—41no—
246C23×C3—181no—
247C23×C3GO+(4,𝔽2)11yescharacteristic
248C22×D3—61no—
249C22×D3—61no—
250C22×D3—121no—
251C3×D4—362no—
252C2×C3×C4—91no—
253C2×C3×C4—91no—
254C2×C3×C4—181no—
255C2×C3×C4—181no—
256C2×C3×C4—181no—
257C2×C3×C4—181no—
258C2×C3×C4—181no—
259#67—122no—
260#67—122no—
261#67—124no—
262#67—361no—
263#67—361no—
264C2×Dic3—61no—
265C2×Dic3—61no—
266C2×Dic3—61no—
267C2×Dic3—61no—
268C2×Dic3—61no—
269C2×Dic3—61no—
270C2×Dic3—61no—
271C2×Dic3—61no—
272C2×Dic3—61no—
273C2×Dic3—61no—
274C2×Dic3—121no—
275C2×Dic3—121no—
276C2×Dic3—121no—
277C2×Dic3—121no—
278C2×Dic3—121no—
279C2×Dic3—121no—
280C2×Dic3—121no—
281D12—181no—
282D12—181no—
283Dic6—181no—
284Dic6—181no—
285Dic6—361no—
286Dic6—361no—
287C3×C8—92no—
288C3×C8—181no—
289C2×C32—21no—
290C2×C32—21no—
291C2×C32—21no—
292C2×C32—21no—
293C2×C32—21no—
294C2×C32—21no—
295C2×C32—41no—
296C2×C32—41no—
297C2×C32—41no—
298C2×C32—41no—
299C2×C32—41no—
300C2×C32—41no—
301C2×C32—41no—
302C2×C32—42no—
303C2×C32—42no—
304C2×C32—82no—
305C2×C32—82no—
306C2×C32#90811yescharacteristic
307C2×C32#90911yescharacteristic
308C2×C32#93011yescharacteristic
309C3⋊D3—81no—
310C3×D3—241no—
311C3×D3—242no—
312C3×D3—241no—
313C2×Q8—271no—
314C2×D4—181no—
315C2×D4—271no—
316C22×C4—91no—
317SD16—542no—
318M16—182no—
319C2×C8—91no—
320C2×C8—91no—
321C4⋊C4—271no—
322C4⋊C4—271no—
323C4⋊C4—541no—
324C22⋊C4—181no—
325C22⋊C4—541no—
326C22×C3—22no—
327C22×C3—22no—
328C22×C3—21no—
329C22×C3—21no—
330C22×C3—21no—
331C22×C3—21no—
332C22×C3—21no—
333C22×C3—41no—
334C22×C3—41no—
335C22×C3—41no—
336C22×C3—41no—
337C22×C3—42no—
338C22×C3—42no—
339C22×C3—41no—
340C22×C3—41no—
341C22×C3—81no—
342C22×C3—81no—
343C22×C3—81no—
344C22×C3—81no—
345C22×C3—361no—
346C22×C3—361no—
347C22×C3—361no—
348C22×C3C2×GO+(4,𝔽2)11yescharacteristic
349C2×D3—121no—
350C2×D3—121no—
351C2×D3—121no—
352C2×D3—122no—
353C2×D3—121no—
354C2×D3—121no—
355C2×D3—121no—
356C2×D3—241no—
357C2×D3—241no—
358C3×C4—91no—
359C3×C4—91no—
360C3×C4—181no—
361C3×C4—361no—
362C3×C4—361no—
363C3×C4—361no—
364Dic3—61no—
365Dic3—61no—
366Dic3—61no—
367Dic3—61no—
368Dic3—121no—
369Dic3—121no—
370Dic3—121no—
371Dic3—121no—
372Dic3—121no—
373Dic3—122no—
374Dic3—121no—
375Dic3—121no—
376Dic3—121no—
377Dic3—241no—
378Dic3—241no—
379C32—21no—
380C32—21no—
381C32—41no—
382C32—41no—
383C32#513411yescharacteristic, complemented
384C23—181no—
385C23#157⋊D311yescharacteristic
386Q8—541no—
387Q8—541no—
388D4—362no—
389D4—541no—
390D4—541no—
391C2×C4—91no—
392C2×C4—91no—
393C2×C4—181no—
394C2×C4—181no—
395C2×C4—181no—
396C2×C4—181no—
397C2×C4—181no—
398C8—92no—
399C8—181no—
400C2×C3—21no—
401C2×C3—21no—
402C2×C3—21no—
403C2×C3—21no—
404C2×C3—21no—
405C2×C3—21no—
406C2×C3—42no—
407C2×C3—42no—
408C2×C3—41no—
409C2×C3—41no—
410C2×C3—41no—
411C2×C3—41no—
412C2×C3—41no—
413C2×C3—41no—
414C2×C3—41no—
415C2×C3—82no—
416C2×C3—82no—
417C2×C3—721no—
418C2×C3#6423611yescharacteristic
419C2×C3#6423911yescharacteristic
420C2×C3#6424211yescharacteristic
421D3—241no—
422D3—242no—
423D3—241no—
424C22—21no—
425C22—21no—
426C22—21no—
427C22—361no—
428C22—361no—
429C22—361no—
430C22C2×#157⋊D311yescharacteristic
431C4—91no—
432C4—91no—
433C4—181no—
434C4—361no—
435C4—361no—
436C4—361no—
437C3—21no—
438C3—21no—
439C3—41no—
440C3—41no—
441C3#1059349511yescharacteristic, complemented
442C2—41no—
443C2—721no—
444C2#1172435011yescharacteristic
445C2#1172435211yescharacteristic
446C2#1172435511yescharacteristic
subgroupG/H
center, FrattiniC22C2×#157⋊D3
commutator, cosocleC2×C3×C3⋊Dic3C23
FittingC23×C33D4
socleC22×C33C2×D4
prime radical#11723717C2
hypercenterC23#157⋊D3
hypocenterC33#468
LCSG ▷ C2×C3×C3⋊Dic3 ▷ C2×C33 ▷ C33
UCSC1 ◁ C22 ◁ C23

Complements

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

NH ≅ G/Ntype
1. C2×C3×#3792C2semidirect
2. C2×#88942C2semidirect
3. #11723721C2semidirect
5. #11722797C2semidirect
6. #11722794C2semidirect
7. #11722793C2semidirect
22. C2×#3792D3semidirect
46. C3×#622D4semidirect
145. #622#67semidirect
241. C33#468semidirect
383. C32#5134semidirect
441. C3#10593495semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#37922
C2×#889422
#117237212
#117237172
#117227972
#117227942
#117227932
#105934953
#51349