Finite Group Databasebeta

← #13167420

Subgroups — #13167420

Order 1728.

616 conjugacy classes of subgroups; only the 75 normal subgroups are shown.

normal subgroups only

This group has 616 conjugacy classes of subgroups — too many to draw. Only its 75 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#11724618C211yesmaximal, characteristic
2C2×C3×C4⋊#157C211yesmaximal, characteristic, complemented
3#11723656C211yesmaximal, characteristic, complemented
4#11722784C212yesmaximal, complemented
5#11722783C212yesmaximal, complemented
6#10593319—31nomaximal
7C22×C3×#157—21no—
8C3×#3828—21no—
9C2×C3×C4×C3⋊D3C2211yescharacteristic
10#89049—21no—
11#89046C412yescomplemented
12C4⋊#1113C2211yescharacteristic
13C4×#1113—21no—
14#89042C412yescomplemented
15#89041—21no—
16#89040C2211yescharacteristic
17C3×C4⋊#157—21no—
18C3×C4⋊#157C2212yes—
19C3×#3748—21no—
20#88809—21no—
21#88808C2212yes—
22#64306—31no—
23D3×#235—61no—
24C2×C4⋊#157D311yescharacteristic, complemented
25#63818—31no—
26D3×#208—61no—
27#63730—32no—
28#63729—32no—
29C22×C3×C3⋊D3D411yescharacteristic
30C2×#1113—21no—
31C2×C3×#157—22no—
32C2×C3×#157—42no—
33C2×C33×C4D411yescharacteristic
34C3×#638—22no—
35C2×C3×C3⋊Dic3C2×C411yescharacteristic
36C3×C4⋊(C3⋊D3)—21no—
37C3×C4×C3⋊D3—22no—
38C3×C4×C3⋊D3C2×C411yescharacteristic
39C3×C4×C3⋊D3D411yescharacteristic
40C3×C4×C3⋊D3D411yescharacteristic
41C3×C4×C3⋊D3C2311yescharacteristic
42C3×#634—21no—
43#6296C2×C412yescomplemented
44#6295—22no—
45#6295C2×C412yescomplemented
46#6294—21no—
47#6294—22no—
48C3×C3⋊(C3⋊C8)—22no—
49#4966—91nomaximal
50C22×#157—21no—
51#3828—21no—
52C2×C4×C3⋊D3C2×D311yescharacteristic
53C3×#235—61no—
54C3×#235—61no—
55C3×#235—121no—
56C2×C3×C4×D3—61no—
57C2×C3×C4×D3—61no—
58#3811—61no—
59D3×#67—62no—
60#3805—32no—
61#3804—62no—
62#3802—31no—
63D3×D12—61no—
64C4×D32—62no—
65C4×D32—61no—
66#3799—61no—
67#3798—32no—
68#3798—62no—
69#3797—31no—
70#3796—61no—
71#3795—61no—
72D3×Dic6—61no—
73C4⋊#157—21no—
74C4⋊#157C2×D312yes—
75#3748—21no—
76C3×#208—61no—
77C3×#208—61no—
78C3×#208—121no—
79#3712—61no—
80#3711—62no—
81#3710—32no—
82D3×C3⋊C8—62no—
83C22×C33C22⋊C411yescharacteristic
84C2×C3×C3⋊D3—21no—
85C2×C3×C3⋊D3—21no—
86C2×C3×C3⋊D3—22no—
87C2×C3×C3⋊D3C22⋊C411yescharacteristic
88C2×C3×C3⋊D3C2×D411yescharacteristic
89#1113—41no—
90#1113—41no—
91C3×#157—42no—
92C33×C4C22⋊C411yescharacteristic
93C33×C4C2×D411yescharacteristic
94C3×C3⋊Dic3—22no—
95C3×C3⋊Dic3C22⋊C411yescharacteristic
96C3×C3⋊Dic3C22×C411yescharacteristic
97D3×#41—181no—
98C2×#235—91no—
99C2×C3×C4⋊C4—91no—
100C2×#208—91no—
101D3×M16—181no—
102#810—92no—
103#809—92no—
104C22×C3⋊D3#6711yescharacteristic
105C22×C3×D3—61no—
106C22×C3×D3—61no—
107C2×D32—62no—
108C2×D32—61no—
109C2×#157—22no—
110C2×#157—42no—
111C2×C32×C4—21no—
112C2×C32×C4—21no—
113C2×C32×C4—41no—
114C2×C32×C4—41no—
115C2×C32×C4D1211yescharacteristic
116#638—22no—
117C2×C3⋊Dic3C4×D311yescharacteristic
118C4⋊(C3⋊D3)—21no—
119C4×C3⋊D3—22no—
120C4×C3⋊D3C4×D311yescharacteristic
121C4×C3⋊D3D1211yescharacteristic
122C4×C3⋊D3#6711yescharacteristic
123C4×C3⋊D3C22×D311yescharacteristic
124#634—21no—
125C3×#67—62no—
126C3×#67—62no—
127C3×#67—122no—
128C2×C3×Dic3—61no—
129C2×C3×Dic3—61no—
130C3×D12—61no—
131C3×D12—61no—
132C3×D12—121no—
133C3×C4×D3—61no—
134C3×C4×D3—61no—
135C3×C4×D3—61no—
136C3×C4×D3—61no—
137C3×C4×D3—61no—
138C3×C4×D3—61no—
139C3×C4×D3—61no—
140C3×C4×D3—61no—
141C3×C4×D3—62no—
142C3×C4×D3—62no—
143C3×C4×D3—122no—
144C3×Dic6—61no—
145C3×Dic6—61no—
146C3×Dic6—121no—
147#627—32no—
148#627—62no—
149#626—62no—
150#626—62no—
151#626—62no—
152#625—32no—
153#625—62no—
154#624—62no—
155#624—61no—
156D3×Dic3—62no—
157D3×Dic3—62no—
158D3×Dic3—62no—
159C3⋊(C3⋊C8)—22no—
160C3×C3⋊C8—62no—
161C3×C3⋊C8—62no—
162C3×C3⋊C8—122no—
163#417—271noSylow
164C2×C33—21no—
165C2×C33C2×C22⋊C411yescharacteristic
166C3×C3⋊D3—21no—
167C3×C3⋊D3—42no—
168C3×C3⋊D3AGL(1,ℤ/8)11yescharacteristic
169C3×C3⋊D3#13711yescharacteristic
170C3×#41—181no—
171C22×C3×C4—91no—
172C22×C3×C4—181no—
173C2×#67—181no—
174#239—181no—
175D3×Q8—181no—
176#237—182no—
177#237—181no—
178D3×D4—181no—
179D3×D4—182no—
180#235—61no—
181#235—61no—
182#235—92no—
183#235—92no—
184#235—121no—
185#235—182no—
186#235—181no—
187C2×D12—91no—
188C2×C4×D3—61no—
189C2×C4×D3—61no—
190C2×C4×D3—181no—
191C2×C4×D3—182no—
192C2×Dic6—91no—
193C3×M16—181no—
194C3×C4⋊C4—92no—
195C3×C4⋊C4—181no—
196#208—61no—
197#208—61no—
198#208—121no—
199#208—181no—
200C2×C3⋊C8—92no—
201ΓL(1,𝔽25)—182no—
202D3×C8—182no—
203C22×C32—21no—
204C22×C32—21no—
205C22×C32—41no—
206C22×C32—41no—
207C22×C32(C2×D3)⋊C411yescharacteristic
208C2×C3⋊D3—22no—
209C2×C3⋊D3—21no—
210C2×C3⋊D3—21no—
211C2×C3⋊D3(C2×D3)⋊C411yescharacteristic
212C2×C3⋊D3C2×#6711yescharacteristic
213C2×C3×D3—61no—
214C2×C3×D3—61no—
215C2×C3×D3—61no—
216C2×C3×D3—61no—
217C2×C3×D3—62no—
218C2×C3×D3—62no—
219C2×C3×D3—121no—
220C2×C3×D3—121no—
221C2×C3×D3—121no—
222C2×C3×D3—121no—
223C2×C3×D3—122no—
224D32—122no—
225D32—122no—
226D32—121no—
227D32—121no—
228#157—42no—
229C32×C4—21no—
230C32×C4—21no—
231C32×C4—21no—
232C32×C4—21no—
233C32×C4—41no—
234C32×C4—41no—
235C32×C4—41no—
236C32×C4—41no—
237C32×C4(C2×D3)⋊C411yescharacteristic
238C32×C4C2×D1211yescharacteristic
239C3⋊Dic3—22no—
240C3⋊Dic3(C2×D3)⋊C411yescharacteristic
241C3⋊Dic3C2×C4×D311yescharacteristic
242C3×Dic3—61no—
243C3×Dic3—61no—
244C3×Dic3—61no—
245C3×Dic3—61no—
246C3×Dic3—62no—
247C3×Dic3—62no—
248C3×Dic3—122no—
249C2×#41—271no—
250C2×M16—271no—
251C2×C4⋊C4—91no—
252#103—272no—
253#102—272no—
254C33#41711yescharacteristic, complemented, Sylow
255C23×C3—91no—
256C22×D3—61no—
257C22×D3—61no—
258C22×D3—181no—
259C22×D3—182no—
260C3×Q8—181no—
261C3×D4—181no—
262C3×D4—182no—
263C2×C3×C4—21no—
264C2×C3×C4—21no—
265C2×C3×C4—41no—
266C2×C3×C4—41no—
267C2×C3×C4—91no—
268C2×C3×C4—91no—
269C2×C3×C4—91no—
270C2×C3×C4—91no—
271C2×C3×C4—91no—
272C2×C3×C4—182no—
273C2×C3×C4—182no—
274C2×C3×C4—362no—
275C2×C3×C4GO+(4,𝔽2)11yescharacteristic
276#67—62no—
277#67—62no—
278#67—122no—
279#67—182no—
280#67—182no—
281#67—182no—
282C2×Dic3—61no—
283C2×Dic3—61no—
284C2×Dic3—181no—
285C2×Dic3—182no—
286D12—61no—
287D12—61no—
288D12—92no—
289D12—121no—
290D12—182no—
291C4×D3—61no—
292C4×D3—61no—
293C4×D3—61no—
294C4×D3—61no—
295C4×D3—61no—
296C4×D3—61no—
297C4×D3—61no—
298C4×D3—61no—
299C4×D3—62no—
300C4×D3—62no—
301C4×D3—122no—
302C4×D3—182no—
303C4×D3—182no—
304C4×D3—182no—
305Dic6—61no—
306Dic6—61no—
307Dic6—92no—
308Dic6—121no—
309Dic6—182no—
310C3×C8—182no—
311C3⋊C8—62no—
312C3⋊C8—62no—
313C3⋊C8—122no—
314C2×C32—21no—
315C2×C32—21no—
316C2×C32—21no—
317C2×C32—41no—
318C2×C32—41no—
319C2×C32—41no—
320C2×C32—41no—
321C2×C32—81no—
322C2×C32—81no—
323C2×C32C2×(C2×D3)⋊C411yescharacteristic
324C3⋊D3—21no—
325C3⋊D3—42no—
326C3⋊D3#93211yescharacteristic
327C3⋊D3#94211yescharacteristic
328C3×D3—61no—
329C3×D3—61no—
330C3×D3—61no—
331C3×D3—61no—
332C3×D3—121no—
333C3×D3—121no—
334C3×D3—122no—
335C3×D3—122no—
336C3×D3—242no—
337#41—181no—
338#41—272no—
339#41—272no—
340#41—541no—
341C2×Q8—271no—
342C2×D4—271no—
343C2×D4—541no—
344C22×C4—91no—
345C22×C4—181no—
346C22×C4—541no—
347M16—181no—
348M16—541no—
349C2×C8—272no—
350C4⋊C4—92no—
351C4⋊C4—181no—
352C22×C3—21no—
353C22×C3—21no—
354C22×C3—41no—
355C22×C3—41no—
356C22×C3—91no—
357C22×C3—91no—
358C22×C3—181no—
359C22×C3—181no—
360C22×C3—182no—
361C22×C3D32⋊C411yescharacteristic
362C2×D3—61no—
363C2×D3—61no—
364C2×D3—61no—
365C2×D3—61no—
366C2×D3—62no—
367C2×D3—62no—
368C2×D3—121no—
369C2×D3—121no—
370C2×D3—121no—
371C2×D3—121no—
372C2×D3—122no—
373C2×D3—362no—
374C2×D3—362no—
375C2×D3—361no—
376C2×D3—361no—
377C3×C4—21no—
378C3×C4—21no—
379C3×C4—21no—
380C3×C4—21no—
381C3×C4—41no—
382C3×C4—41no—
383C3×C4—41no—
384C3×C4—41no—
385C3×C4—91no—
386C3×C4—91no—
387C3×C4—182no—
388C3×C4—362no—
389C3×C4D32⋊C411yescharacteristic
390C3×C4C2×GO+(4,𝔽2)11yescharacteristic
391Dic3—61no—
392Dic3—61no—
393Dic3—61no—
394Dic3—61no—
395Dic3—62no—
396Dic3—62no—
397Dic3—122no—
398C32—21no—
399C32—21no—
400C32—41no—
401C32—41no—
402C32#496611yescharacteristic, complemented
403C23—91no—
404C23—541no—
405Q8—181no—
406Q8—272no—
407D4—182no—
408D4—181no—
409D4—272no—
410D4—542no—
411C2×C4—91no—
412C2×C4—91no—
413C2×C4—91no—
414C2×C4—91no—
415C2×C4—91no—
416C2×C4—182no—
417C2×C4—182no—
418C2×C4—362no—
419C2×C4—542no—
420C2×C4—541no—
421C2×C4#157⋊D311yescharacteristic
422C8—182no—
423C2×C3—21no—
424C2×C3—21no—
425C2×C3—21no—
426C2×C3—41no—
427C2×C3—41no—
428C2×C3—41no—
429C2×C3—41no—
430C2×C3—81no—
431C2×C3—81no—
432C2×C3—91no—
433C2×C3—91no—
434C2×C3—181no—
435C2×C3—362no—
436C2×C3C2×D32⋊C411yescharacteristic
437D3—61no—
438D3—61no—
439D3—61no—
440D3—61no—
441D3—121no—
442D3—121no—
443D3—122no—
444D3—122no—
445D3—242no—
446C22—91no—
447C22—91no—
448C22—181no—
449C22—181no—
450C22—182no—
451C22—1081no—
452C22—1081no—
453C22#8893911yescharacteristic
454C4—91no—
455C4—91no—
456C4—182no—
457C4—362no—
458C4#8893911yescharacteristic
459C4C2×#157⋊D311yescharacteristic
460C3—21no—
461C3—21no—
462C3—41no—
463C3—41no—
464C3#1059331911yescharacteristic, complemented
465C2—21no—
466C2—91no—
467C2—91no—
468C2—181no—
469C2—362no—
470C2C2×#8893911yescharacteristic
subgroupG/H
centerC2C2×#88939
commutatorC3×C3⋊Dic3C22×C4
FrattiniC4C2×#157⋊D3
FittingC2×C33×C4D4
socleC2×C33C2×C22⋊C4
prime radical#11724618C2
cosocleC3×C4×C3⋊D3C23
hypercenterC2×C4#157⋊D3
hypocenterC33#417
LCSG ▷ C3×C3⋊Dic3 ▷ C2×C33 ▷ C33
UCSC1 ◁ C2 ◁ C2×C4

Complements

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

NH ≅ G/Ntype
2. C2×C3×C4⋊#157C2semidirect
3. #11723656C2semidirect
4. #11722784C2semidirect
5. #11722783C2semidirect
11. #89046C4semidirect
14. #89042C4semidirect
24. C2×C4⋊#157D3semidirect
43. #6296C2×C4semidirect
45. #6295C2×C4semidirect
254. C33#417semidirect
402. C32#4966semidirect
464. C3#10593319semidirect

Maximal subgroups

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

subgroupindex
#117246182
C2×C3×C4⋊#1572
#117236562
#117227842
#117227832
#105933193
#49669