Finite Group Databasebeta

← #13155235

Subgroups — #13155235

Order 1728.

500 conjugacy classes of subgroups; only the 78 normal subgroups are shown.

normal subgroups only

This group has 500 conjugacy classes of subgroups — too many to draw. Only its 78 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#11723337C211yesmaximal, characteristic, complemented
2C3×D3×D24C211yesmaximal, characteristic, complemented
3#11722068C211yesmaximal, characteristic, complemented
4#11722006C211yesmaximal, characteristic, complemented
5#11722005C211yesmaximal, characteristic, complemented
6C3×#63548C211yesmaximal, characteristic, complemented
7C3×#63543C211yesmaximal, characteristic, complemented
8#10591033—31nomaximal
9#10590633—31nomaximal
10#10590233—31nomaximal
11#89022—21no—
12C3×D3×D12—21no—
13#88852C2211yescharacteristic, complemented
14C3×#3746C2211yescharacteristic, complemented
15C3×#3745C2211yescharacteristic
16C32×D24C2211yescharacteristic
17C32×D3×C8C2211yescharacteristic
18#88797—21no—
19C3×#3714—21no—
20C3×C3⋊(C3⋊C16)C2211yescharacteristic, complemented
21C32×C3⋊C16C2211yescharacteristic, complemented
22#64113—31no—
23C3×D3×D8—31no—
24C2×C3×D24—31no—
25#63808—31no—
26#63808—31no—
27#63808—61no—
28D3×D24D311yescharacteristic, complemented
29#63653—31no—
30#63626—31no—
31C3×#827—31no—
32C3×#812—31no—
33C3×#800—31no—
34C3×#798—31no—
35#63549—31no—
36#63548—31no—
37#63548D311yescharacteristic, complemented
38#63547—31no—
39#63546—31no—
40#63546—31no—
41#63546—31no—
42#63546—31no—
43#63546—61no—
44#63546—61no—
45#63543D311yescharacteristic, complemented
46C2×C3×D32—41no—
47C4⋊(C3⋊(C3⋊D3))—21no—
48C3×C4⋊(C3⋊D3)—21no—
49C3×#634—21no—
50C32×D12—21no—
51C32×C4×D3D411yescharacteristic
52#6291—41no—
53#6289—41no—
54C3×#626—41no—
55C33×C8C2311yescharacteristic
56C32×C3⋊C8D411yescharacteristic
57#5147—91no—
58#4914—91no—
59#4908—91no—
60#3828—61no—
61C3×D3×D4—61no—
62C2×C3×D12—61no—
63D3×D12—21no—
64#3799—62no—
65#3799—61no—
66#3799—61no—
67C32×D8—31no—
68C2×C32×C8—31no—
69#3746C2×D311yescharacteristic, complemented
70#3745—31no—
71#3745—31no—
72#3745—31no—
73#3745—61no—
74#3745—61no—
75#3745—61no—
76#3745—121no—
77#3745C2×D311yescharacteristic
78#3744—61no—
79C3×#213—61no—
80C3×Dic12—31no—
81C3×Dic12—31no—
82C3×Dic12—61no—
83C3×D24—21no—
84C3×D24—31no—
85C3×D24—31no—
86C3×D24—61no—
87C3×D24—61no—
88C3×D24C2×D311yescharacteristic
89C3×D24C2×D311yescharacteristic
90C3×D3×C8—21no—
91C3×D3×C8C2×D311yescharacteristic
92C3×D3×C8C2×D311yescharacteristic
93#3717—62no—
94#3717—61no—
95#3717—61no—
96#3714—21no—
97C32×C16—31no—
98C3⋊(C3⋊C16)C2×D311yescharacteristic, complemented
99C3×C3⋊C16—21no—
100C3×C3⋊C16—31no—
101C3×C3⋊C16—31no—
102C3×C3⋊C16—61no—
103C3×C3⋊C16C2×D311yescharacteristic, complemented
104C3×C3⋊C16C2×D311yescharacteristic, complemented
105C2×C3⋊(C3⋊D3)—41no—
106C2×C3×C3⋊D3—41no—
107C2×C32×D3—41no—
108C2×C32×D3D811yescharacteristic
109C3×D32—82no—
110C33×C4C2×D411yescharacteristic
111C3×C3⋊Dic3—41no—
112C32×Dic3D811yescharacteristic
113C2×C3×D8—91no—
114#919—91no—
115D3×D8—31no—
116#904—91no—
117#904—91no—
118#904—181no—
119C2×D24—31no—
120C3×SD32—91no—
121C3×M32—91no—
122#828—91no—
123#827—31no—
124#827—91no—
125#826—91no—
126#826—91no—
127#826—91no—
128#826—181no—
129#812—31no—
130#800—31no—
131#800—91no—
132D48—91no—
133D48—91no—
134D48—91no—
135D48—181no—
136#798—31no—
137C22×C3×D3—121no—
138C22×C3×D3—121no—
139C2×D32—41no—
140C32×D4—61no—
141C2×C32×C4—31no—
142#638—121no—
143C4⋊(C3⋊D3)—21no—
144C4⋊(C3⋊D3)—61no—
145C4⋊(C3⋊D3)—62no—
146C4⋊(C3⋊D3)—61no—
147C4⋊(C3⋊D3)—61no—
148C4⋊(C3⋊D3)—121no—
149C4⋊(C3⋊D3)—121no—
150C4⋊(C3⋊D3)—122no—
151C4×C3⋊D3—121no—
152#634—21no—
153C3×#67—121no—
154C3×D12—21no—
155C3×D12—21no—
156C3×D12—41no—
157C3×D12—61no—
158C3×D12—61no—
159C3×D12—121no—
160C3×D12—121no—
161C3×C4×D3—21no—
162C3×C4×D3D1211yescharacteristic
163C3×C4×D3#6711yescharacteristic
164C3×Dic6—62no—
165C3×Dic6—61no—
166C3×Dic6—61no—
167#626—41no—
168#626—122no—
169#626—121no—
170#626—121no—
171#624—122no—
172#624—121no—
173#624—121no—
174C32×C8—21no—
175C32×C8—21no—
176C32×C8—21no—
177C32×C8—31no—
178C32×C8—41no—
179C32×C8C22×D311yescharacteristic
180C32×C8C22×D311yescharacteristic
181C32×C8C22×D311yescharacteristic
182C3×C3⋊C8—21no—
183C3×C3⋊C8D1211yescharacteristic
184C3×C3⋊C8#6711yescharacteristic
185#509—271noSylow
186C2×C33C2×D811yescharacteristic
187C3⋊(C3⋊D3)—81no—
188C3×C3⋊D3—81no—
189C32×D3—21no—
190C32×D3—81no—
191C2×C3×D4—181no—
192#239—181no—
193D3×D4—61no—
194#235—181no—
195#235—181no—
196#235—182no—
197C2×D12—61no—
198C3×Q16—91no—
199C3×D8—31no—
200C3×D8—31no—
201C3×D8—61no—
202C3×D8—91no—
203C3×D8—181no—
204C2×C3×C8—31no—
205C2×C3×C8—31no—
206C2×C3×C8—61no—
207#215—181no—
208#213—61no—
209Dic12—31no—
210Dic12—31no—
211Dic12—61no—
212D24—31no—
213D24—31no—
214D24—61no—
215D24—61no—
216D24—91no—
217D24—91no—
218D24—91no—
219D24—181no—
220D24—181no—
221D24—181no—
222D24—361no—
223D24D3211yescharacteristic
224#204—181no—
225#204—181no—
226#204—182no—
227D3×C8D3211yescharacteristic
228C3×C16—31no—
229C3×C16—31no—
230C3×C16—61no—
231C3×C16—91no—
232C3⋊C16—31no—
233C3⋊C16—31no—
234C3⋊C16—61no—
235C3⋊C16D3211yescharacteristic, complemented
236C22×C32—31no—
237C22×C32—121no—
238C2×C3⋊D3—41no—
239C2×C3⋊D3—121no—
240C2×C3⋊D3—122no—
241C2×C3⋊D3—121no—
242C2×C3⋊D3—121no—
243C2×C3⋊D3—241no—
244C2×C3⋊D3—241no—
245C2×C3⋊D3—242no—
246C2×C3×D3—21no—
247C2×C3×D3—41no—
248C2×C3×D3—41no—
249C2×C3×D3—81no—
250C2×C3×D3—121no—
251C2×C3×D3—121no—
252C2×C3×D3—241no—
253C2×C3×D3—242no—
254C2×C3×D3—242no—
255C2×C3×D3D2411yescharacteristic
256C2×C3×D3#21311yescharacteristic
257D32—82no—
258C32×C4—21no—
259C32×C4—21no—
260C32×C4—21no—
261C32×C4—31no—
262C32×C4—41no—
263C32×C4C2×D1211yescharacteristic
264C32×C4D3×D411yescharacteristic
265C32×C4C2×#6711yescharacteristic
266C3⋊Dic3—41no—
267C3×Dic3—21no—
268C3×Dic3—122no—
269C3×Dic3—121no—
270C3×Dic3—121no—
271C3×Dic3D2411yescharacteristic
272C3×Dic3#21311yescharacteristic
273#135—271no—
274C2×D8—91no—
275SD32—91no—
276SD32—271no—
277D16—271no—
278D16—271no—
279M32—91no—
280C33#50911yescharacteristic, complemented, Sylow
281C23×C3—361no—
282C22×D3—121no—
283C22×D3—121no—
284C3×Q8—181no—
285C3×D4—61no—
286C3×D4—61no—
287C3×D4—121no—
288C3×D4—181no—
289C3×D4—361no—
290C2×C3×C4—31no—
291C2×C3×C4—31no—
292C2×C3×C4—61no—
293#67—121no—
294#67—361no—
295#67—361no—
296#67—362no—
297D12—21no—
298D12—61no—
299D12—61no—
300D12—121no—
301D12—121no—
302D12—182no—
303D12—181no—
304D12—181no—
305D12—181no—
306D12—361no—
307D12—361no—
308D12—362no—
309D12—361no—
310C4×D3—361no—
311C4×D3—361no—
312C4×D3—362no—
313C4×D3—361no—
314C4×D3#62611yescharacteristic
315Dic6—62no—
316Dic6—61no—
317Dic6—61no—
318C3×C8—21no—
319C3×C8—21no—
320C3×C8—21no—
321C3×C8—31no—
322C3×C8—31no—
323C3×C8—41no—
324C3×C8—61no—
325C3×C8C2×D3211yescharacteristic
326C3×C8C2×D3211yescharacteristic
327C3×C8C2×D3211yescharacteristic
328C3⋊C8#62611yescharacteristic
329C2×C32—21no—
330C2×C32—21no—
331C2×C32—21no—
332C2×C32—41no—
333C2×C32—61no—
334C2×C32—241no—
335C2×C32C2×D2411yescharacteristic
336C2×C32D3×D811yescharacteristic
337C2×C32C2×#21311yescharacteristic
338C3⋊D3—81no—
339C3⋊D3—241no—
340C3⋊D3—242no—
341C3⋊D3—241no—
342C3⋊D3—241no—
343C3⋊D3—481no—
344C3⋊D3—481no—
345C3⋊D3—482no—
346C3×D3—21no—
347C3×D3—21no—
348C3×D3—41no—
349C3×D3—81no—
350C3×D3—81no—
351C3×D3—161no—
352C3×D3—241no—
353C3×D3—241no—
354C3×D3—481no—
355#41—541no—
356C2×D4—181no—
357Q16—91no—
358SD16—541no—
359D8—31no—
360D8—91no—
361D8—181no—
362D8—271no—
363C2×C8—31no—
364C16—31no—
365C16—91no—
366C22×C3—31no—
367C22×C3—31no—
368C22×C3—61no—
369C22×C3—121no—
370C22×C3—121no—
371C22×C3—241no—
372C22×C3—361no—
373C22×C3—722no—
374C2×D3—41no—
375C2×D3—121no—
376C2×D3—121no—
377C2×D3—241no—
378C2×D3—242no—
379C2×D3—242no—
380C2×D3—362no—
381C2×D3—361no—
382C2×D3—361no—
383C2×D3—361no—
384C2×D3—721no—
385C2×D3—721no—
386C2×D3—722no—
387C2×D3#371411yescharacteristic
388C3×C4—21no—
389C3×C4—21no—
390C3×C4—21no—
391C3×C4—31no—
392C3×C4—31no—
393C3×C4—41no—
394C3×C4—61no—
395C3×C4—361no—
396C3×C4D3×D1211yescharacteristic
397C3×C4C2×#62611yescharacteristic
398C3×C4D3×#6711yescharacteristic
399Dic3—122no—
400Dic3—121no—
401Dic3—121no—
402Dic3#371411yescharacteristic
403C32—21no—
404C32—21no—
405C32—21no—
406C32—41no—
407C32#490811yescharacteristic, complemented
408C32#491411yescharacteristic, complemented
409C32#514711yescharacteristic, complemented
410C23—361no—
411Q8—181no—
412D4—61no—
413D4—181no—
414D4—361no—
415D4—541no—
416D4—1081no—
417C2×C4—31no—
418C2×C4—1081no—
419C8—31no—
420C8D3311yescharacteristic
421C2×C3—21no—
422C2×C3—21no—
423C2×C3—21no—
424C2×C3—41no—
425C2×C3—61no—
426C2×C3—61no—
427C2×C3—121no—
428C2×C3—241no—
429C2×C3—241no—
430C2×C3—481no—
431C2×C3—721no—
432C2×C3D3×D2411yescharacteristic
433C2×C3C2×#371411yescharacteristic
434C2×C3D3×#21311yescharacteristic
435D3—21no—
436D3—81no—
437D3—241no—
438D3—241no—
439D3—481no—
440D3—722no—
441D3—721no—
442D3—721no—
443D3—721no—
444D3—1441no—
445D3—1441no—
446D3—1442no—
447C22—31no—
448C22—121no—
449C22—361no—
450C22—722no—
451C22—1081no—
452C4—31no—
453C4—361no—
454C4D3×#62611yescharacteristic
455C3—21no—
456C3—21no—
457C3—21no—
458C3—41no—
459C3#1059023311yescharacteristic, complemented
460C3#1059063311yescharacteristic, complemented
461C3#1059103311yescharacteristic, complemented
462C2—61no—
463C2—241no—
464C2—721no—
465C2—2161no—
466C2D3×#371411yescharacteristic
subgroupG/H
centerC2D3×#3714
commutator, Fitting, cosocleC33×C8C23
Frattini, hypercenterC8D33
socleC2×C33C2×D8
hypocenterC33#509
LCSG ▷ C33×C8 ▷ C33×C4 ▷ C2×C33 ▷ C33
UCSC1 ◁ C2 ◁ C4 ◁ C8

Complements

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

NH ≅ G/Ntype
1. #11723337C2semidirect
2. C3×D3×D24C2semidirect
3. #11722068C2semidirect
4. #11722006C2semidirect
5. #11722005C2semidirect
6. C3×#63548C2semidirect
7. C3×#63543C2semidirect
13. #88852C22semidirect
14. C3×#3746C22semidirect
20. C3×C3⋊(C3⋊C16)C22semidirect
21. C32×C3⋊C16C22semidirect
28. D3×D24D3semidirect
37. #63548D3semidirect
45. #63543D3semidirect
69. #3746C2×D3semidirect
98. C3⋊(C3⋊C16)C2×D3semidirect
103. C3×C3⋊C16C2×D3semidirect
104. C3×C3⋊C16C2×D3semidirect
235. C3⋊C16D32semidirect
280. C33#509semidirect
407. C32#4908semidirect
408. C32#4914semidirect
409. C32#5147semidirect
459. C3#10590233semidirect
460. C3#10590633semidirect
461. C3#10591033semidirect

Maximal subgroups

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

subgroupindex
#117233372
C3×D3×D242
#117220682
#117220062
#117220052
C3×#635482
C3×#635432
#105910333
#105906333
#105902333