Finite Group Databasebeta

← #13167434

Subgroups — #13167434

Order 1728.

614 conjugacy classes of subgroups; only the 71 normal subgroups are shown.

normal subgroups only

This group has 614 conjugacy classes of subgroups — too many to draw. Only its 71 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×#89042C211yesmaximal, characteristic
2C3×#64286C211yesmaximal, characteristic, complemented
3#11723656C211yesmaximal, characteristic, complemented
4#11722785C214yesmaximal, complemented
5#10593323—31nomaximal
6C2×C3×C4×C3⋊D3—21no—
7C2×C3×C4×C3⋊D3C2211yescharacteristic
8C2×#6296C411yescharacteristic, complemented
9C2×#6295—21no—
10C2×#6295C411yescharacteristic, complemented
11#89042—21no—
12#89042—21no—
13#89042C412yescomplemented
14#89042C2212yes—
15C3×C22⋊#157—21no—
16C3×C4⋊#157—21no—
17C3×C4×#157C2212yescomplemented
18C3×#3748—21no—
19#88809C2212yescomplemented
20#88808—21no—
21C2×#3798—31no—
22C2×#3798—61no—
23#64286D311yescharacteristic, complemented
24#63818—31no—
25D3×#208—61no—
26#63731—34no—
27C22×C3×C3⋊D3—21no—
28C22×C3×C3⋊D3D411yescharacteristic
29C2×C3×#157—22no—
30C2×C33×C4D411yescharacteristic
31C2×C3×C3⋊Dic3—21no—
32C2×C3×C3⋊Dic3C2×C411yescharacteristic
33C3×C4×C3⋊D3—22no—
34C3×C4×C3⋊D3—22no—
35C3×C4×C3⋊D3C2×C411yescharacteristic
36C3×C4×C3⋊D3D411yescharacteristic
37C3×C4×C3⋊D3D411yescharacteristic
38C3×C4×C3⋊D3C2311yescharacteristic
39#6296—21no—
40#6296C2×C412yes—
41#6295—22no—
42#6295—22no—
43#6295—21no—
44#6295C2×C412yes—
45C3×C3⋊(C3⋊C8)—22no—
46#5005—91nomaximal
47C2×C4×C3⋊D3—21no—
48C2×C4×C3⋊D3C2×D311yescharacteristic
49C2×C3×C4×D3—61no—
50C2×C3×C4×D3—61no—
51C2×C3×C4×D3—61no—
52C2×C3×C4×D3—61no—
53C2×C3×C4×D3—121no—
54C2×#627—31no—
55C2×#627—61no—
56C2×#626—61no—
57C2×#626—61no—
58C2×#626—61no—
59C2×#625—31no—
60C2×#625—61no—
61#3798—32no—
62#3798—32no—
63#3798—62no—
64#3798—62no—
65#3798—62no—
66#3798—62no—
67#3798—61no—
68#3798—61no—
69C22⋊#157—21no—
70C4⋊#157—21no—
71C4×#157C2×D312yescomplemented
72#3748—21no—
73C3×#208—61no—
74C3×#208—61no—
75C3×#208—121no—
76#3712—32no—
77#3711—62no—
78#3710—61no—
79D3×C3⋊C8—62no—
80C22×C33C22⋊C411yescharacteristic
81C2×C3×C3⋊D3—22no—
82C2×C3×C3⋊D3—41no—
83C2×C3×C3⋊D3—42no—
84C2×C3×C3⋊D3C22⋊C411yescharacteristic
85C2×C3×C3⋊D3C2×D411yescharacteristic
86C3×#157—42no—
87C33×C4C22⋊C411yescharacteristic
88C33×C4C2×D411yescharacteristic
89C3×C3⋊Dic3—22no—
90C3×C3⋊Dic3C22⋊C411yescharacteristic
91C3×C3⋊Dic3C22×C411yescharacteristic
92C2×#235—91no—
93C2×#235—181no—
94C3×#117—91no—
95C2×#208—91no—
96D3×M16—181no—
97#805—94no—
98C22×C3⋊D3—21no—
99C22×C3⋊D3#6711yescharacteristic
100C22×C3×D3—61no—
101C22×C3×D3—61no—
102C22×C3×D3—61no—
103C22×C3×D3—61no—
104C22×C3×D3—121no—
105C2×#157—22no—
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⋊Dic3C4×D311yescharacteristic
113C4×C3⋊D3—22no—
114C4×C3⋊D3—22no—
115C4×C3⋊D3C4×D311yescharacteristic
116C4×C3⋊D3D1211yescharacteristic
117C4×C3⋊D3#6711yescharacteristic
118C4×C3⋊D3C22×D311yescharacteristic
119C2×C3×Dic3—61no—
120C2×C3×Dic3—61no—
121C2×C3×Dic3—61no—
122C2×C3×Dic3—61no—
123C2×C3×Dic3—121no—
124C3×C4×D3—61no—
125C3×C4×D3—61no—
126C3×C4×D3—61no—
127C3×C4×D3—61no—
128C3×C4×D3—61no—
129C3×C4×D3—61no—
130C3×C4×D3—61no—
131C3×C4×D3—61no—
132C3×C4×D3—62no—
133C3×C4×D3—62no—
134C3×C4×D3—62no—
135C3×C4×D3—62no—
136C3×C4×D3—122no—
137C3×C4×D3—122no—
138#627—32no—
139#627—62no—
140#627—62no—
141#627—61no—
142#626—62no—
143#626—62no—
144#626—62no—
145#626—62no—
146#626—62no—
147#626—62no—
148#625—32no—
149#625—62no—
150#625—62no—
151#625—61no—
152C3⋊(C3⋊C8)—22no—
153C3×C3⋊C8—62no—
154C3×C3⋊C8—62no—
155C3×C3⋊C8—122no—
156#421—271noSylow
157C2×C33—21no—
158C2×C33C2×C22⋊C411yescharacteristic
159C3×C3⋊D3—21no—
160C3×C3⋊D3—21no—
161C3×C3⋊D3—42no—
162C22×C3×C4—91no—
163C22×C3×C4—181no—
164C2×#67—181no—
165C2×#67—181no—
166C2×#67—181no—
167#235—92no—
168#235—92no—
169#235—182no—
170#235—182no—
171#235—181no—
172#235—181no—
173#235—182no—
174#235—182no—
175C2×D12—91no—
176C2×D12—181no—
177C2×C4×D3—61no—
178C2×C4×D3—61no—
179C2×C4×D3—61no—
180C2×C4×D3—61no—
181C2×C4×D3—121no—
182C2×Dic6—91no—
183C2×Dic6—181no—
184C3×M16—181no—
185C3×C4⋊C4—181no—
186C3×C22⋊C4—181no—
187C3×C42—92no—
188#208—61no—
189#208—61no—
190#208—92no—
191#208—121no—
192C2×C3⋊C8—181no—
193ΓL(1,𝔽25)—182no—
194D3×C8—182no—
195C22×C32—21no—
196C22×C32—21no—
197C22×C32—41no—
198C22×C32—41no—
199C22×C32(C2×D3)⋊C411yescharacteristic
200C2×C3⋊D3—22no—
201C2×C3⋊D3—41no—
202C2×C3⋊D3—42no—
203C2×C3⋊D3(C2×D3)⋊C411yescharacteristic
204C2×C3⋊D3C2×#6711yescharacteristic
205C2×C3×D3—61no—
206C2×C3×D3—61no—
207C2×C3×D3—61no—
208C2×C3×D3—61no—
209C2×C3×D3—62no—
210C2×C3×D3—62no—
211C2×C3×D3—122no—
212C2×C3×D3—122no—
213C2×C3×D3—122no—
214C2×C3×D3—122no—
215C2×C3×D3—122no—
216C2×C3×D3—124no—
217#157—42no—
218C32×C4—21no—
219C32×C4—21no—
220C32×C4—21no—
221C32×C4—21no—
222C32×C4—41no—
223C32×C4—41no—
224C32×C4—41no—
225C32×C4—41no—
226C32×C4(C2×D3)⋊C411yescharacteristic
227C32×C4C2×D1211yescharacteristic
228C3⋊Dic3—22no—
229C3⋊Dic3(C2×D3)⋊C411yescharacteristic
230C3⋊Dic3C2×C4×D311yescharacteristic
231C3×Dic3—61no—
232C3×Dic3—61no—
233C3×Dic3—61no—
234C3×Dic3—61no—
235C3×Dic3—62no—
236C3×Dic3—62no—
237C3×Dic3—122no—
238C2×#41—271no—
239C2×M16—271no—
240#117—91no—
241#104—274no—
242C33#42111yescharacteristic, complemented, Sylow
243C23×C3—91no—
244C23×C3—181no—
245C22×D3—61no—
246C22×D3—61no—
247C22×D3—61no—
248C22×D3—61no—
249C22×D3—121no—
250C2×C3×C4—21no—
251C2×C3×C4—21no—
252C2×C3×C4—41no—
253C2×C3×C4—41no—
254C2×C3×C4—91no—
255C2×C3×C4—91no—
256C2×C3×C4—91no—
257C2×C3×C4—91no—
258C2×C3×C4—91no—
259C2×C3×C4—182no—
260C2×C3×C4—182no—
261C2×C3×C4—181no—
262C2×C3×C4—182no—
263C2×C3×C4GO+(4,𝔽2)11yescharacteristic
264#67—182no—
265#67—182no—
266#67—182no—
267#67—182no—
268#67—182no—
269#67—182no—
270C2×Dic3—61no—
271C2×Dic3—61no—
272C2×Dic3—61no—
273C2×Dic3—61no—
274C2×Dic3—121no—
275D12—92no—
276D12—181no—
277D12—182no—
278D12—182no—
279C4×D3—61no—
280C4×D3—61no—
281C4×D3—61no—
282C4×D3—61no—
283C4×D3—61no—
284C4×D3—61no—
285C4×D3—61no—
286C4×D3—61no—
287C4×D3—62no—
288C4×D3—62no—
289C4×D3—62no—
290C4×D3—62no—
291C4×D3—122no—
292C4×D3—122no—
293Dic6—92no—
294Dic6—182no—
295Dic6—181no—
296Dic6—182no—
297C3×C8—182no—
298C3⋊C8—62no—
299C3⋊C8—62no—
300C3⋊C8—122no—
301C2×C32—22no—
302C2×C32—21no—
303C2×C32—21no—
304C2×C32—21no—
305C2×C32—41no—
306C2×C32—41no—
307C2×C32—41no—
308C2×C32—42no—
309C2×C32—81no—
310C2×C32C2×(C2×D3)⋊C411yescharacteristic
311C3⋊D3—21no—
312C3⋊D3—21no—
313C3⋊D3—42no—
314C3×D3—62no—
315C3×D3—121no—
316C3×D3—121no—
317C3×D3—121no—
318C3×D3—122no—
319C3×D3—124no—
320C3×D3—122no—
321#41—272no—
322#41—272no—
323#41—541no—
324#41—541no—
325C2×Q8—271no—
326C2×D4—271no—
327C2×D4—541no—
328C22×C4—91no—
329C22×C4—181no—
330M16—181no—
331M16—272no—
332C2×C8—541no—
333C4⋊C4—181no—
334C22⋊C4—181no—
335C42—92no—
336C22×C3—21no—
337C22×C3—21no—
338C22×C3—41no—
339C22×C3—41no—
340C22×C3—91no—
341C22×C3—91no—
342C22×C3—182no—
343C22×C3—361no—
344C22×C3—362no—
345C22×C3D32⋊C411yescharacteristic
346C2×D3—61no—
347C2×D3—61no—
348C2×D3—61no—
349C2×D3—61no—
350C2×D3—62no—
351C2×D3—62no—
352C2×D3—122no—
353C2×D3—122no—
354C2×D3—122no—
355C2×D3—122no—
356C2×D3—122no—
357C2×D3—124no—
358C3×C4—21no—
359C3×C4—21no—
360C3×C4—21no—
361C3×C4—21no—
362C3×C4—41no—
363C3×C4—41no—
364C3×C4—41no—
365C3×C4—41no—
366C3×C4—91no—
367C3×C4—91no—
368C3×C4—182no—
369C3×C4—362no—
370C3×C4D32⋊C411yescharacteristic
371C3×C4C2×GO+(4,𝔽2)11yescharacteristic
372Dic3—61no—
373Dic3—61no—
374Dic3—61no—
375Dic3—61no—
376Dic3—62no—
377Dic3—62no—
378Dic3—122no—
379C32—21no—
380C32—21no—
381C32—41no—
382C32—41no—
383C32#500511yescharacteristic, complemented
384C23—91no—
385C23—181no—
386Q8—272no—
387Q8—541no—
388D4—272no—
389D4—541no—
390D4—542no—
391D4—542no—
392C2×C4—91no—
393C2×C4—91no—
394C2×C4—91no—
395C2×C4—91no—
396C2×C4—91no—
397C2×C4—181no—
398C2×C4—182no—
399C2×C4—182no—
400C2×C4—182no—
401C2×C4#157⋊D311yescharacteristic
402C8—182no—
403C2×C3—22no—
404C2×C3—21no—
405C2×C3—21no—
406C2×C3—21no—
407C2×C3—41no—
408C2×C3—42no—
409C2×C3—41no—
410C2×C3—41no—
411C2×C3—81no—
412C2×C3—181no—
413C2×C3—181no—
414C2×C3—362no—
415C2×C3C2×D32⋊C411yescharacteristic
416D3—62no—
417D3—121no—
418D3—121no—
419D3—121no—
420D3—122no—
421D3—124no—
422D3—122no—
423C22—91no—
424C22—91no—
425C22—182no—
426C22—361no—
427C22—362no—
428C22#8893911yescharacteristic
429C4—91no—
430C4—91no—
431C4—182no—
432C4—362no—
433C4#8893911yescharacteristic
434C4C2×#157⋊D311yescharacteristic
435C3—21no—
436C3—21no—
437C3—41no—
438C3—41no—
439C3#1059332311yescharacteristic, complemented
440C2—21no—
441C2—181no—
442C2—181no—
443C2—362no—
444C2C2×#8893911yescharacteristic
subgroupG/H
center, FrattiniC4C2×#157⋊D3
commutatorC3×C3⋊Dic3C22×C4
FittingC2×C33×C4D4
socleC2×C33C2×C22⋊C4
prime radicalC2×#89042C2
cosocleC3×C4×C3⋊D3C23
hypercenterC2×C4#157⋊D3
hypocenterC33#421
LCSG ▷ C3×C3⋊Dic3 ▷ C2×C33 ▷ C33
UCSC1 ◁ C4 ◁ C2×C4

Complements

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

NH ≅ G/Ntype
2. C3×#64286C2semidirect
3. #11723656C2semidirect
4. #11722785C2semidirect
8. C2×#6296C4semidirect
10. C2×#6295C4semidirect
13. #89042C4semidirect
17. C3×C4×#157C22semidirect
19. #88809C22semidirect
23. #64286D3semidirect
71. C4×#157C2×D3semidirect
242. C33#421semidirect
383. C32#5005semidirect
439. C3#10593323semidirect

Maximal subgroups

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

subgroupindex
C2×#890422
C3×#642862
#117236562
#117227852
#105933233
#50059