Finite Group Databasebeta

← #13165593

Subgroups — #13165593

Order 1728.

486 conjugacy classes of subgroups; only the 52 normal subgroups are shown.

normal subgroups only

This group has 486 conjugacy classes of subgroups — too many to draw. Only its 52 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×#88935C211yesmaximal, characteristic
2C2×C3×#3775C211yesmaximal, characteristic, complemented
3#11723007C211yesmaximal, characteristic, complemented
4Dic3×#625C211yesmaximal, characteristic, complemented
5#11722793C211yesmaximal, characteristic, complemented
6#11722775C211yesmaximal, characteristic, complemented
7#11722732C211yesmaximal, characteristic, complemented
8#10593422—31nomaximal
9C2×#6295C2211yescharacteristic
10C2×C3×#627C2211yescharacteristic
11C2×C3×#625C2211yescharacteristic
12C2×C3×D3×Dic3—21no—
13C2×C3⋊#622C2211yescharacteristic
14C2×C3×#622C2211yescharacteristic
15#88935—22no—
16C3×#3775—22no—
17#88815C2211yescharacteristic
18#88803—21no—
19(C2×D3)⋊(C3⋊Dic3)—21no—
20Dic3×C3⋊Dic3C2211yescharacteristic
21C3×#3722—21no—
22C2×#3775D311yescharacteristic, complemented
23C2×#3774—31no—
24Dic3×#67—61no—
25#63915—31no—
26#63907—61no—
27C4×#625—31no—
28#63779—31no—
29#63740—31no—
30#63739—31no—
31C22×C3×C3⋊D3—21no—
32C22×C32×D3—21no—
33C2×C3⋊(C3⋊Dic3)D411yescharacteristic
34C2×C3×C3⋊Dic3—21no—
35C2×C3×C3⋊Dic3C2311yescharacteristic
36C2×C32×Dic3—21no—
37C2×C32×Dic3D411yescharacteristic
38#6295—21no—
39#6295—21no—
40C3×#627—21no—
41C3×#627—21no—
42C3×#625—21no—
43C3×#625D412yes—
44C3×D3×Dic3—42no—
45C3⋊#622—21no—
46C3×#622—21no—
47#5165—91nomaximal
48C2×C3×#67—61no—
49C22×C3×Dic3—61no—
50C2×C3×C4×D3—61no—
51C2×C3×Dic6—61no—
52C2×#627C2×D311yescharacteristic
53C2×#626—61no—
54C2×#625—31no—
55C2×#625C2×D311yescharacteristic
56C2×D3×Dic3—21no—
57C2×#622—31no—
58C2×#622C2×D311yescharacteristic
59#3775—22no—
60#3774—62no—
61#3757—61no—
62#3752—61no—
63C4×C3⋊Dic3—31no—
64C3×(C2×D3)⋊C4—61no—
65C3×(C2×D3)⋊C4—61no—
66#3724—31no—
67Dic3⋊Dic3—61no—
68#3722—21no—
69(C2×D3)⋊Dic3—61no—
70(C2×D3)⋊Dic3—61no—
71(C2×D3)⋊Dic3—61no—
72(C2×D3)⋊Dic3—61no—
73(C2×D3)⋊Dic3—121no—
74(C2×D3)⋊Dic3—121no—
75Dic32—61no—
76Dic32—61no—
77C22×C33C2×D411yescharacteristic
78C2×C3×C3⋊D3—41no—
79C2×C3×C3⋊D3—41no—
80C2×C3×C3⋊D3—41no—
81C2×C32×D3—22no—
82C2×C32×D3—41no—
83C2×C32×D3—41no—
84C3⋊(C3⋊Dic3)—21no—
85C3×C3⋊Dic3—41no—
86C3×C3⋊Dic3C2×D411yescharacteristic
87C3×C3⋊Dic3#4111yescharacteristic
88C32×Dic3—21no—
89C32×Dic3—41no—
90C2×C3×SD16—91no—
91#947—91no—
92D4×Dic3—91no—
93C2×#213—91no—
94C4×#67—181no—
95#885—181no—
96#835—91no—
97#821—91no—
98Dic3⋊C8—91no—
99C23×C32—61no—
100C22×C3⋊D3—21no—
101C22×C3×D3—21no—
102C22×C3×D3—21no—
103C22×C3×D3—41no—
104C22×C3×D3—61no—
105C22×C3×D3—61no—
106C22×C3×D3—121no—
107C2×C32×C4—31no—
108C2×C32×C4—61no—
109C2×C3⋊Dic3—21no—
110C2×C3⋊Dic3—31no—
111C2×C3⋊Dic3—61no—
112C2×C3⋊Dic3—61no—
113C2×C3⋊Dic3—121no—
114C2×C3⋊Dic3—121no—
115C2×C3⋊Dic3C22×D311yescharacteristic
116C3×#67—62no—
117C3×#67—62no—
118C2×C3×Dic3—21no—
119C2×C3×Dic3—21no—
120C2×C3×Dic3—21no—
121C2×C3×Dic3—21no—
122C2×C3×Dic3—41no—
123C2×C3×Dic3—61no—
124C2×C3×Dic3—61no—
125C2×C3×Dic3—61no—
126C2×C3×Dic3—61no—
127C2×C3×Dic3—121no—
128C2×C3×Dic3—122no—
129C3×C4×D3—122no—
130C3×Dic6—121no—
131C3×Dic6—121no—
132#627—21no—
133#627—21no—
134#626—121no—
135#626—121no—
136#625—21no—
137#625—61no—
138#625—61no—
139#625#6712yes—
140D3×Dic3—42no—
141#622—21no—
142#622—61no—
143#463—271noSylow
144C2×C33#12111yescharacteristic
145C2×C33#13511yescharacteristic
146C2×C33AGL(1,ℤ/8)11yescharacteristic
147C3×C3⋊D3—81no—
148C32×D3—42no—
149C2×C3×Q8—91no—
150C2×C3×D4—91no—
151C22×C3×C4—181no—
152C2×#67—61no—
153C2×#67—181no—
154C22×Dic3—61no—
155C2×D12—91no—
156C2×C4×D3—61no—
157C2×Dic6—61no—
158C3×SD16—182no—
159C2×C3×C8—91no—
160C3×C22⋊C4—181no—
161#217—181no—
162#217—181no—
163#217—181no—
164#217—361no—
165#213—182no—
166(C2×D3)⋊C4—61no—
167(C2×D3)⋊C4—61no—
168(C2×D3)⋊C4—181no—
169(C2×D3)⋊C4—181no—
170(C2×D3)⋊C4—181no—
171(C2×D3)⋊C4—361no—
172C4⋊Dic3—91no—
173Dic3⋊C4—181no—
174C4×Dic3—91no—
175C4×Dic3—181no—
176C4×Dic3—181no—
177C2×C3⋊C8—91no—
178C22×C32—21no—
179C22×C32—21no—
180C22×C32—41no—
181C22×C32—41no—
182C22×C32—62no—
183C22×C32—121no—
184C22×C32—121no—
185C22×C32D3×D411yescharacteristic
186C2×C3⋊D3—41no—
187C2×C3⋊D3—41no—
188C2×C3⋊D3—41no—
189C2×C3×D3—22no—
190C2×C3×D3—22no—
191C2×C3×D3—42no—
192C2×C3×D3—41no—
193C2×C3×D3—41no—
194C2×C3×D3—41no—
195C2×C3×D3—41no—
196C2×C3×D3—81no—
197C2×C3×D3—81no—
198C2×C3×D3—121no—
199C2×C3×D3—121no—
200C2×C3×D3—121no—
201C2×C3×D3—122no—
202C2×C3×D3—121no—
203C2×C3×D3—121no—
204C2×C3×D3—121no—
205C2×C3×D3—241no—
206C2×C3×D3—241no—
207C32×C4—61no—
208C32×C4—121no—
209C3⋊Dic3—41no—
210C3⋊Dic3—61no—
211C3⋊Dic3—121no—
212C3⋊Dic3—121no—
213C3⋊Dic3—241no—
214C3⋊Dic3—241no—
215C3⋊Dic3#23711yescharacteristic
216C3⋊Dic3C2×#6711yescharacteristic
217C3×Dic3—22no—
218C3×Dic3—41no—
219C3×Dic3—41no—
220C3×Dic3—41no—
221C3×Dic3—61no—
222C3×Dic3—61no—
223C3×Dic3—61no—
224C3×Dic3—61no—
225C3×Dic3—81no—
226C3×Dic3—121no—
227C3×Dic3—121no—
228C3×Dic3—241no—
229C2×SD16—91no—
230C2×D8—271no—
231#124—271no—
232C4×D4—271no—
233C4⋊C8—271no—
234#103—271no—
235#102—271no—
236C33#46311yescharacteristic, complemented, Sylow
237C23×C3—61no—
238C23×C3—61no—
239C23×C3—121no—
240C23×C3—181no—
241C22×D3—21no—
242C22×D3—61no—
243C22×D3—61no—
244C22×D3—121no—
245C3×Q8—181no—
246C3×Q8—181no—
247C3×D4—92no—
248C3×D4—181no—
249C2×C3×C4—61no—
250C2×C3×C4—61no—
251C2×C3×C4—61no—
252C2×C3×C4—61no—
253C2×C3×C4—91no—
254C2×C3×C4—121no—
255C2×C3×C4—181no—
256C2×C3×C4—362no—
257#67—62no—
258#67—62no—
259#67—361no—
260#67—361no—
261C2×Dic3—21no—
262C2×Dic3—61no—
263C2×Dic3—61no—
264C2×Dic3—61no—
265C2×Dic3—61no—
266C2×Dic3—91no—
267C2×Dic3—121no—
268C2×Dic3—122no—
269C2×Dic3—181no—
270C2×Dic3—181no—
271C2×Dic3—361no—
272C2×Dic3—361no—
273C2×Dic3GO+(4,𝔽2)11yescharacteristic
274D12—181no—
275D12—181no—
276C4×D3—122no—
277Dic6—121no—
278Dic6—121no—
279C3×C8—181no—
280C3⋊C8—181no—
281C2×C32—21no—
282C2×C32—21no—
283C2×C32—21no—
284C2×C32—21no—
285C2×C32—21no—
286C2×C32—21no—
287C2×C32—41no—
288C2×C32—41no—
289C2×C32—41no—
290C2×C32—41no—
291C2×C32—41no—
292C2×C32—41no—
293C2×C32—122no—
294C2×C32#91411yescharacteristic
295C2×C32#91611yescharacteristic
296C2×C32#93911yescharacteristic
297C3⋊D3—81no—
298C3×D3—42no—
299C3×D3—42no—
300C3×D3—82no—
301C3×D3—241no—
302C3×D3—242no—
303C3×D3—241no—
304C2×Q8—91no—
305C2×D4—91no—
306C2×D4—271no—
307C22×C4—181no—
308SD16—182no—
309D8—542no—
310C2×C8—91no—
311C2×C8—271no—
312C4⋊C4—271no—
313C22⋊C4—181no—
314C22⋊C4—541no—
315C22⋊C4—541no—
316C42—271no—
317C22×C3—21no—
318C22×C3—21no—
319C22×C3—41no—
320C22×C3—41no—
321C22×C3—62no—
322C22×C3—62no—
323C22×C3—122no—
324C22×C3—121no—
325C22×C3—121no—
326C22×C3—121no—
327C22×C3—121no—
328C22×C3—241no—
329C22×C3—241no—
330C22×C3—361no—
331C22×C3—361no—
332C22×C3—361no—
333C22×C3C2×GO+(4,𝔽2)11yescharacteristic
334C2×D3—22no—
335C2×D3—41no—
336C2×D3—41no—
337C2×D3—121no—
338C2×D3—121no—
339C2×D3—121no—
340C2×D3—122no—
341C2×D3—121no—
342C2×D3—121no—
343C2×D3—121no—
344C2×D3—241no—
345C2×D3—241no—
346C3×C4—62no—
347C3×C4—91no—
348C3×C4—91no—
349C3×C4—121no—
350C3×C4—121no—
351C3×C4—121no—
352C3×C4—241no—
353C3×C4—361no—
354Dic3—21no—
355Dic3—41no—
356Dic3—61no—
357Dic3—61no—
358Dic3—61no—
359Dic3—61no—
360Dic3—121no—
361Dic3—121no—
362Dic3—181no—
363Dic3—241no—
364Dic3—361no—
365Dic3—361no—
366Dic3—721no—
367Dic3—721no—
368C32—21no—
369C32—21no—
370C32—41no—
371C32—41no—
372C32#516511yescharacteristic, complemented
373C23—61no—
374C23—181no—
375Q8—181no—
376Q8—181no—
377D4—92no—
378D4—181no—
379D4—541no—
380D4—541no—
381C2×C4—31no—
382C2×C4—61no—
383C2×C4—91no—
384C2×C4—181no—
385C2×C4—271no—
386C2×C4—362no—
387C8—181no—
388C8—541no—
389C2×C3—21no—
390C2×C3—21no—
391C2×C3—21no—
392C2×C3—21no—
393C2×C3—21no—
394C2×C3—21no—
395C2×C3—41no—
396C2×C3—41no—
397C2×C3—41no—
398C2×C3—41no—
399C2×C3—41no—
400C2×C3—41no—
401C2×C3—122no—
402C2×C3—122no—
403C2×C3—242no—
404C2×C3—721no—
405C2×C3#6422311yescharacteristic
406C2×C3#6423011yescharacteristic
407C2×C3#6423611yescharacteristic
408D3—42no—
409D3—241no—
410D3—242no—
411D3—241no—
412C22—62no—
413C22—121no—
414C22—121no—
415C22—361no—
416C22—361no—
417C22—361no—
418C22D3×GO+(4,𝔽2)11yescharacteristic
419C4—61no—
420C4—91no—
421C4—91no—
422C4—121no—
423C4—361no—
424C4—541no—
425C3—21no—
426C3—21no—
427C3—41no—
428C3—41no—
429C3#1059342211yescharacteristic, complemented
430C2—122no—
431C2—721no—
432C2#1172411711yescharacteristic
433C2#1172413611yescharacteristic
434C2#1172413911yescharacteristic
subgroupG/H
center, Frattini, hypercenterC22D3×GO+(4,𝔽2)
commutator, cosocleC2×C3×C3⋊Dic3C23
Fitting, socleC22×C33C2×D4
prime radicalC2×#88935C2
hypocenterC33#463
LCSG ▷ C2×C3×C3⋊Dic3 ▷ C2×C33 ▷ C33
UCSC1 ◁ C22

Complements

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

NH ≅ G/Ntype
2. C2×C3×#3775C2semidirect
3. #11723007C2semidirect
4. Dic3×#625C2semidirect
5. #11722793C2semidirect
6. #11722775C2semidirect
7. #11722732C2semidirect
22. C2×#3775D3semidirect
236. C33#463semidirect
372. C32#5165semidirect
429. C3#10593422semidirect

Maximal subgroups

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

subgroupindex
C2×#889352
C2×C3×#37752
#117230072
Dic3×#6252
#117227932
#117227752
#117227322
#105934223
#51659