Finite Group Databasebeta

← #13166526

Subgroups — #13166526

Order 1728.

732 conjugacy classes of subgroups; only the 68 normal subgroups are shown.

normal subgroups only

This group has 732 conjugacy classes of subgroups — too many to draw. Only its 68 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
1C3⋊D3×#237C211yesmaximal, characteristic
2D3×C4⋊#157C211yesmaximal, characteristic, complemented
3#11724254C211yesmaximal, characteristic, complemented
4#11723107C211yesmaximal, characteristic, complemented
5C3×#63782C211yesmaximal, characteristic, complemented
6#11722709C211yesmaximal, characteristic, complemented
7#11722708C211yesmaximal, characteristic, complemented
8#10593840—31nomaximal
9C2×D3×#157—21no—
10#89078C411yescharacteristic, complemented
11C3×D4×C3⋊D3C2211yescharacteristic
12C32×#237C411yescharacteristic, complemented
13C3⋊D3×#67—21no—
14#89031—21no—
15C2×Dic3×C3⋊D3—21no—
16#89029—21no—
17C4×D3×C3⋊D3C2211yescharacteristic
18#89021C411yescharacteristic, complemented
19#89017C411yescharacteristic, complemented
20C3⋊D3×Dic6C2211yescharacteristic
21C4⋊(C3⋊#157)C2211yescharacteristic
22#88988C2211yescharacteristic
23C3×C4⋊#157C2211yescharacteristic
24C3×#3787C2211yescharacteristic
25#88926—21no—
26#88925—21no—
27Dic3⋊#157—21no—
28#41×C3⋊D3—31no—
29D3×#237—62no—
30C2×C4⋊#157—31no—
31#64283—31no—
32#63784—31no—
33#63784—31no—
34#63782—31no—
35#63782D311yescharacteristic, complemented
36C22×C3×C3⋊D3—21no—
37C2×D3×C3⋊D3D411yescharacteristic
38C2×C3⋊#157—21no—
39C2×C3×#157—21no—
40D3×#157—42no—
41C33×D4C2×C411yescharacteristic, complemented
42#6312—21no—
43C2×C3⋊(C3⋊Dic3)—21no—
44C4×C3⋊(C3⋊D3)C2×C411yescharacteristic
45#6308C2×C411yescharacteristic, complemented
46C3×#638—21no—
47C3×C4⋊(C3⋊D3)C2×C411yescharacteristic, complemented
48C3×C4×C3⋊D3C2311yescharacteristic
49C32×#67—21no—
50C2×C32×Dic3—21no—
51C32×C4×D3C2×C411yescharacteristic
52C32×Dic6C2×C411yescharacteristic, complemented
53#6293—21no—
54#6292—21no—
55#6290—21no—
56#6289D411yescharacteristic
57Dic3×C3⋊D3—21no—
58Dic3×C3⋊D3—21no—
59Dic3×C3⋊D3—21no—
60Dic3×C3⋊D3D411yescharacteristic
61D3×C3⋊Dic3D411yescharacteristic
62C3⋊#622—21no—
63C3×#622—21no—
64#4771—91nomaximal
65C22×#157—61no—
66C32×#41—31no—
67#3832—31no—
68Q8×C3⋊D3—31no—
69#3830—31no—
70#3830—61no—
71#3830—62no—
72#3830—122no—
73D4×C3⋊D3—61no—
74D4×C3⋊D3C2×D311yescharacteristic
75#3828—31no—
76#3828—61no—
77C2×C4×C3⋊D3—31no—
78C2×C4×C3⋊D3—61no—
79C3×#237—22no—
80C3×D3×D4—62no—
81D3×#67—64no—
82#3805—64no—
83#3804—64no—
84C2×D3×Dic3—64no—
85C4×D32—62no—
86#3796—62no—
87#3795—62no—
88D3×Dic6—62no—
89#3792—61no—
90C4⋊#157—31no—
91C4⋊#157—61no—
92C4⋊#157C2×D311yescharacteristic
93#3788—61no—
94#3787—31no—
95#3787C2×D311yescharacteristic
96C22×C33—21no—
97C2×C3⋊(C3⋊D3)C22⋊C411yescharacteristic
98C2×C3×C3⋊D3—21no—
99C2×C3×C3⋊D3—41no—
100C2×C3×C3⋊D3—41no—
101C2×C3×C3⋊D3C2×D411yescharacteristic
102C2×C32×D3C22⋊C411yescharacteristic
103D3×C3⋊D3—21no—
104D3×C3⋊D3—21no—
105C3⋊#157—41no—
106C3×#157—41no—
107C33×C4C22×C411yescharacteristic
108C3⋊(C3⋊Dic3)—21no—
109C3⋊(C3⋊Dic3)C22⋊C411yescharacteristic
110C3×C3⋊Dic3C2×D411yescharacteristic
111C32×Dic3—21no—
112C32×Dic3C22⋊C411yescharacteristic
113D3×#41—182no—
114C2×#237—91no—
115C2×ΓL(1,𝔽25)—91no—
116D3×C4⋊C4—91no—
117C3×#102—91no—
118#832—91no—
119#816—91no—
120#810—91no—
121C22×C3⋊D3—21no—
122C22×C3⋊D3—31no—
123C22×C3×D3—64no—
124C2×D32—62no—
125C2×#157—21no—
126C2×#157—61no—
127C2×#157—122no—
128C32×Q8—31no—
129C32×D4—22no—
130C32×D4—42no—
131C32×D4—61no—
132C32×D4C4×D311yescharacteristic, complemented
133C2×C32×C4—31no—
134C2×C32×C4—61no—
135#638—21no—
136#638—61no—
137#638—61no—
138#638—64no—
139#638—124no—
140C2×C3⋊Dic3—31no—
141C2×C3⋊Dic3—61no—
142C2×C3⋊Dic3—64no—
143C2×C3⋊Dic3—124no—
144C4⋊(C3⋊D3)—61no—
145C4⋊(C3⋊D3)C4×D311yescharacteristic, complemented
146C4×C3⋊D3—31no—
147C4×C3⋊D3—31no—
148C4×C3⋊D3—31no—
149C4×C3⋊D3—62no—
150C4×C3⋊D3—61no—
151C4×C3⋊D3—61no—
152C4×C3⋊D3—61no—
153C4×C3⋊D3—122no—
154C4×C3⋊D3C22×D311yescharacteristic
155#634—31no—
156#634—62no—
157#634—61no—
158#634—122no—
159C3×#67—24no—
160C3×#67—64no—
161C2×C3×Dic3—24no—
162C3×D12—62no—
163C3×C4×D3—22no—
164C3×C4×D3—62no—
165C3×Dic6—22no—
166#627—64no—
167#626—64no—
168#625—64no—
169#624—62no—
170D3×Dic3—62no—
171D3×Dic3—62no—
172D3×Dic3—64no—
173D3×Dic3—64no—
174D3×Dic3—64no—
175#622—21no—
176#622—61no—
177#417—271noSylow
178C2×C33—41no—
179C2×C33C2×C22⋊C411yescharacteristic
180C3⋊(C3⋊D3)—21no—
181C3×C3⋊D3—41no—
182C3×C3⋊D3AGL(1,ℤ/8)11yescharacteristic
183C3×C3⋊D3#13711yescharacteristic
184C32×D3—21no—
185C3×#41—62no—
186C2×C3×D4—91no—
187C2×#67—181no—
188C22×Dic3—181no—
189#239—182no—
190D3×Q8—182no—
191#237—91no—
192#237—91no—
193#237—91no—
194#237—182no—
195#237—181no—
196#237—181no—
197#237—184no—
198#237—362no—
199#237#15711yescharacteristic, complemented
200D3×D4—62no—
201D3×D4—184no—
202#235—182no—
203#235—184no—
204C2×C4×D3—91no—
205C2×C4×D3—182no—
206C2×C4×D3—184no—
207C2×C4×D3—181no—
208C2×Dic6—91no—
209C2×C3×C8—91no—
210C3×C4⋊C4—91no—
211C4⋊Dic3—91no—
212Dic3⋊C4—181no—
213C2×C3⋊C8—91no—
214ΓL(1,𝔽25)—181no—
215ΓL(1,𝔽25)—181no—
216C22×C32—21no—
217C22×C32—24no—
218C22×C32—31no—
219C22×C32—44no—
220C2×C3⋊D3—21no—
221C2×C3⋊D3—31no—
222C2×C3⋊D3—41no—
223C2×C3⋊D3—41no—
224C2×C3⋊D3—62no—
225C2×C3⋊D3—61no—
226C2×C3⋊D3—61no—
227C2×C3⋊D3—122no—
228C2×C3⋊D3D3×D411yescharacteristic
229C2×C3×D3—22no—
230C2×C3×D3—62no—
231C2×C3×D3—64no—
232C2×C3×D3—124no—
233C2×C3×D3—124no—
234D32—122no—
235D32—122no—
236#157—41no—
237#157—121no—
238C32×C4—22no—
239C32×C4—31no—
240C32×C4—42no—
241C32×C4—61no—
242C32×C4C2×C4×D311yescharacteristic
243C3⋊Dic3—31no—
244C3⋊Dic3—62no—
245C3⋊Dic3—61no—
246C3⋊Dic3—64no—
247C3⋊Dic3—124no—
248C3⋊Dic3—122no—
249C3⋊Dic3D3×D411yescharacteristic
250C3×Dic3—24no—
251C3×Dic3—22no—
252C3×Dic3—62no—
253C2×#41—271no—
254C2×M16—271no—
255C2×C4⋊C4—271no—
256#103—271no—
257#103—271no—
258#102—91no—
259#102—271no—
260C33#41711yescharacteristic, complemented, Sylow
261C23×C3—181no—
262C22×D3—64no—
263C22×D3—91no—
264C22×D3—182no—
265C3×Q8—62no—
266C3×D4—22no—
267C3×D4—42no—
268C3×D4—64no—
269C3×D4—91no—
270C3×D4—181no—
271C3×D4C2×#15711yescharacteristic, complemented
272C2×C3×C4—62no—
273C2×C3×C4—64no—
274C2×C3×C4—91no—
275C2×C3×C4—181no—
276#67—21no—
277#67—64no—
278#67—181no—
279#67—184no—
280#67—181no—
281#67—184no—
282#67—181no—
283#67—364no—
284C2×Dic3—21no—
285C2×Dic3—91no—
286C2×Dic3—182no—
287C2×Dic3—181no—
288C2×Dic3—181no—
289C2×Dic3—181no—
290C2×Dic3—184no—
291C2×Dic3—181no—
292C2×Dic3—181no—
293C2×Dic3—364no—
294D12—62no—
295D12—184no—
296C4×D3—62no—
297C4×D3—91no—
298C4×D3—91no—
299C4×D3—91no—
300C4×D3—182no—
301C4×D3—182no—
302C4×D3—182no—
303C4×D3—184no—
304C4×D3—184no—
305C4×D3—184no—
306C4×D3—362no—
307C4×D3—362no—
308C4×D3C2×#15711yescharacteristic
309Dic6—91no—
310Dic6—182no—
311Dic6—181no—
312Dic6—184no—
313Dic6—362no—
314Dic6C2×#15711yescharacteristic, complemented
315C3×C8—181no—
316C3⋊C8—181no—
317C2×C32—22no—
318C2×C32—42no—
319C2×C32—41no—
320C2×C32—44no—
321C2×C32—61no—
322C2×C32—84no—
323C2×C32D3×C22⋊C411yescharacteristic
324C3⋊D3—41no—
325C3⋊D3—61no—
326C3⋊D3—122no—
327C3⋊D3—242no—
328C3⋊D3#91111yescharacteristic
329C3⋊D3#91511yescharacteristic
330C3×D3—42no—
331C3×D3—62no—
332C3×D3—62no—
333C3×D3—124no—
334#41—31no—
335#41—271no—
336#41—271no—
337#41—271no—
338#41—541no—
339#41—541no—
340C2×Q8—271no—
341C2×D4—91no—
342C2×D4—541no—
343C22×C4—271no—
344C22×C4—541no—
345C22×C4—541no—
346M16—541no—
347M16—541no—
348C2×C8—91no—
349C2×C8—271no—
350C4⋊C4—91no—
351C4⋊C4—271no—
352C4⋊C4—541no—
353C22×C3—24no—
354C22×C3—21no—
355C22×C3—44no—
356C22×C3—62no—
357C22×C3—91no—
358C22×C3—181no—
359C22×C3—361no—
360C22×C3—361no—
361C2×D3—62no—
362C2×D3—64no—
363C2×D3—91no—
364C2×D3—124no—
365C2×D3—124no—
366C2×D3—182no—
367C2×D3—181no—
368C2×D3—181no—
369C2×D3—362no—
370C2×D3—362no—
371C2×D3—362no—
372C2×D3C22⋊#15711yescharacteristic
373C3×C4—22no—
374C3×C4—42no—
375C3×C4—62no—
376C3×C4—64no—
377C3×C4—91no—
378C3×C4—361no—
379C3×C4C22×#15711yescharacteristic
380Dic3—21no—
381Dic3—62no—
382Dic3—91no—
383Dic3—182no—
384Dic3—184no—
385Dic3—181no—
386Dic3—362no—
387Dic3—364no—
388Dic3—361no—
389Dic3C22⋊#15711yescharacteristic
390C32—22no—
391C32—42no—
392C32#477111yescharacteristic, complemented
393C23—181no—
394C23—271no—
395Q8—31no—
396Q8—271no—
397Q8—541no—
398D4—61no—
399D4—91no—
400D4—181no—
401D4—541no—
402D4—541no—
403D4—541no—
404D4D3×#15711yescharacteristic, complemented
405C2×C4—31no—
406C2×C4—61no—
407C2×C4—91no—
408C2×C4—181no—
409C2×C4—271no—
410C2×C4—271no—
411C2×C4—271no—
412C2×C4—271no—
413C2×C4—541no—
414C2×C4—541no—
415C2×C4—541no—
416C2×C4—541no—
417C2×C4—541no—
418C2×C4—1082no—
419C8—181no—
420C8—541no—
421C2×C3—22no—
422C2×C3—44no—
423C2×C3—41no—
424C2×C3—42no—
425C2×C3—84no—
426C2×C3—91no—
427C2×C3—91no—
428C2×C3—122no—
429C2×C3—361no—
430C2×C3C2×C22⋊#15711yescharacteristic
431D3—21no—
432D3—62no—
433D3—62no—
434D3—124no—
435D3—181no—
436D3—362no—
437D3—722no—
438C22—21no—
439C22—31no—
440C22—91no—
441C22—181no—
442C22—271no—
443C22—361no—
444C22—361no—
445C22—541no—
446C22—541no—
447C4—31no—
448C4—61no—
449C4—91no—
450C4—271no—
451C4—361no—
452C4—541no—
453C4—1081no—
454C4C2×D3×#15711yescharacteristic
455C3—22no—
456C3—42no—
457C3#1059384011yescharacteristic, complemented
458C2—41no—
459C2—61no—
460C2—91no—
461C2—91no—
462C2—361no—
463C2—541no—
464C2D3×C22⋊#15711yescharacteristic
subgroupG/H
centerC2D3×C22⋊#157
commutatorC33×C4C22×C4
FrattiniC4C2×D3×#157
FittingC33×D4C2×C4
socleC2×C33C2×C22⋊C4
prime radicalC3⋊D3×#237C2
cosocleC3×C4×C3⋊D3C23
hypercenterD4D3×#157
hypocenterC33#417
LCSG ▷ C33×C4 ▷ C2×C33 ▷ C33
UCSC1 ◁ C2 ◁ C4 ◁ D4

Complements

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

NH ≅ G/Ntype
2. D3×C4⋊#157C2semidirect
3. #11724254C2semidirect
4. #11723107C2semidirect
5. C3×#63782C2semidirect
6. #11722709C2semidirect
7. #11722708C2semidirect
10. #89078C4semidirect
12. C32×#237C4semidirect
18. #89021C4semidirect
19. #89017C4semidirect
35. #63782D3semidirect
41. C33×D4C2×C4semidirect
45. #6308C2×C4semidirect
47. C3×C4⋊(C3⋊D3)C2×C4semidirect
52. C32×Dic6C2×C4semidirect
132. C32×D4C4×D3semidirect
145. C4⋊(C3⋊D3)C4×D3semidirect
199. #237#157semidirect
260. C33#417semidirect
271. C3×D4C2×#157semidirect
314. Dic6C2×#157semidirect
392. C32#4771semidirect
404. D4D3×#157semidirect
457. C3#10593840semidirect

Maximal subgroups

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

subgroupindex
C3⋊D3×#2372
D3×C4⋊#1572
#117242542
#117231072
C3×#637822
#117227092
#117227082
#105938403
#47719