Finite Group Databasebeta

← #13167236

Subgroups — #13167236

Order 1728.

476 conjugacy classes of subgroups; only the 51 normal subgroups are shown.

normal subgroups only

This group has 476 conjugacy classes of subgroups — too many to draw. Only its 51 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
1D4×C3⋊#157C211yesmaximal, characteristic, complemented
2#11724315C211yesmaximal, characteristic, complemented
3#11723702C211yesmaximal, characteristic, complemented
4C3×#63926C211yesmaximal, characteristic, complemented
5#11722763C211yesmaximal, characteristic, complemented
6#11722758C211yesmaximal, characteristic, complemented
7#11722757C211yesmaximal, characteristic, complemented
8#10593464—31nomaximal
9C22×C3⋊#157—21no—
10C2×D32⋊D3—21no—
11C2×D32⋊D3—21no—
12C3×D4×C3⋊D3C2211yescharacteristic
13C4⋊#1113C2211yescharacteristic
14C3×#3802C2211yescharacteristic
15#88994—21no—
16C4⋊(C3⋊#157)C2211yescharacteristic
17C4×C3⋊#157C2211yescharacteristic
18C3×#3753—21no—
19C32×#213—21no—
20#88810—21no—
21#88808C2211yescharacteristic
22C3×#3714—21no—
23C3×#3710C2211yescharacteristic
24D4×#157—31no—
25#64231—31no—
26C3×D3×D8—61no—
27#63926—31no—
28#63926D311yescharacteristic, complemented
29D3×#213—61no—
30#63732—31no—
31#63729—31no—
32#63729—31no—
33C22×C3×C3⋊D3—21no—
34C2×#1113—21no—
35C2×C3×D32—21no—
36C2×C3⋊#157—21no—
37C2×C3⋊#157—42no—
38C2×C3⋊#157D411yescharacteristic
39C2×C3⋊#157D411yescharacteristic
40D32⋊D3—42no—
41D32⋊D3—42no—
42C33×D4D411yescharacteristic, complemented
43C3×#638—21no—
44C3×C4⋊(C3⋊D3)—21no—
45C3×C4⋊(C3⋊D3)D411yescharacteristic, complemented
46C3×C4×C3⋊D3C2311yescharacteristic
47C32×D12—21no—
48#6295—21no—
49C3×#625—21no—
50C3×C3⋊(C3⋊C8)—21no—
51C32×C3⋊C8—21no—
52#5150—91nomaximal
53C22×#157—61no—
54C2×GO+(4,𝔽2)—61no—
55C2×GO+(4,𝔽2)—61no—
56D4×C3⋊D3C2×D311yescharacteristic
57C3×D3×D4—61no—
58C3×D3×D4—61no—
59C3×D3×D4—61no—
60#3802—31no—
61#3802C2×D311yescharacteristic
62D3×D12—61no—
63C22⋊#157—61no—
64C4⋊#157—31no—
65C4×#157—31no—
66C32×D8—61no—
67#3753—21no—
68C3×#213—21no—
69C3×#213—21no—
70C3×#213—41no—
71C3×#213—61no—
72C3×#213—61no—
73C3×#213—61no—
74C3×#213—121no—
75C3×D24—61no—
76C3×D3×C8—61no—
77#3714—21no—
78#3714—61no—
79#3714—61no—
80#3713—61no—
81#3710—31no—
82#3710C2×D311yescharacteristic
83D3×C3⋊C8—61no—
84C22×C33—21no—
85C2×C3×C3⋊D3—21no—
86C2×C3×C3⋊D3—41no—
87C2×C3×C3⋊D3—41no—
88C2×C3×C3⋊D3—41no—
89C2×C3×C3⋊D3C2×D411yescharacteristic
90C2×C32×D3—41no—
91#1113—41no—
92#1113—41no—
93C3×D32—41no—
94C3×D32—41no—
95C3⋊#157—21no—
96C3⋊#157—41no—
97C3⋊#157D812yescomplemented
98C33×C4C2×D411yescharacteristic
99C3×C3⋊Dic3#4111yescharacteristic
100C2×C3×D8—91no—
101#940—91no—
102D4×Dic3—91no—
103C2×#213—91no—
104D3×D8—61no—
105D3×D8—181no—
106#832—91no—
107#821—91no—
108Dic3⋊C8—91no—
109C22×C3⋊D3—21no—
110C22×C3×D3—121no—
111C22×C3×D3—121no—
112C22×C3×D3—121no—
113C2×D32—21no—
114C2×D32—61no—
115C2×D32—121no—
116C2×#157—31no—
117C2×#157—31no—
118C2×#157—61no—
119C2×#157—122no—
120GO+(4,𝔽2)—122no—
121GO+(4,𝔽2)—122no—
122C32×D4—21no—
123C32×D4—21no—
124C32×D4—41no—
125C32×D4—41no—
126C32×D4—61no—
127C32×D4#6711yescharacteristic, complemented
128#638—21no—
129C4⋊(C3⋊D3)—21no—
130C4⋊(C3⋊D3)#6711yescharacteristic, complemented
131C4×C3⋊D3C22×D311yescharacteristic
132C3×#67—121no—
133C3×#67—121no—
134C3×#67—121no—
135C3×D12—21no—
136C3×D12—21no—
137C3×D12—41no—
138C3×D12—61no—
139C3×D12—61no—
140C3×D12—61no—
141C3×D12—61no—
142C3×D12—121no—
143C3×C4×D3—61no—
144C3×C4×D3—61no—
145#626—121no—
146#625—21no—
147#625—61no—
148C32×C8—61no—
149C3⋊(C3⋊C8)—21no—
150C3×C3⋊C8—21no—
151C3×C3⋊C8—21no—
152C3×C3⋊C8—41no—
153C3×C3⋊C8—61no—
154C3×C3⋊C8—61no—
155C3×C3⋊C8—121no—
156#459—271noSylow
157C2×C33—41no—
158C2×C33#12111yescharacteristic
159C3×C3⋊D3—41no—
160C3×C3⋊D3—81no—
161C3×C3⋊D3C2×D811yescharacteristic
162C3×C3⋊D3AGL(1,ℤ/8)11yescharacteristic
163C32×D3—81no—
164C2×C3×D4—91no—
165C2×C3×D4—91no—
166C2×#67—181no—
167C2×#67—181no—
168C22×Dic3—181no—
169D3×D4—61no—
170D3×D4—61no—
171D3×D4—61no—
172D3×D4—181no—
173C2×D12—91no—
174C3×D8—61no—
175C3×D8—61no—
176C3×D8—121no—
177C3×D8—181no—
178C3×D8—181no—
179C2×C3×C8—91no—
180#217—181no—
181#213—21no—
182#213—61no—
183#213—61no—
184#213—61no—
185#213—121no—
186#213—181no—
187#213—181no—
188C4⋊Dic3—91no—
189C4×Dic3—91no—
190C2×C3⋊C8—91no—
191D24—61no—
192D24—181no—
193D3×C8—61no—
194D3×C8—181no—
195C22×C32—21no—
196C22×C32—41no—
197C22×C32—41no—
198C22×C32—81no—
199C22×C32—81no—
200C22×C32—121no—
201C2×C3⋊D3—21no—
202C2×C3⋊D3—41no—
203C2×C3⋊D3—41no—
204C2×C3⋊D3—41no—
205C2×C3⋊D3D3×D411yescharacteristic
206C2×C3×D3—41no—
207C2×C3×D3—41no—
208C2×C3×D3—61no—
209C2×C3×D3—61no—
210C2×C3×D3—81no—
211C2×C3×D3—121no—
212C2×C3×D3—121no—
213C2×C3×D3—121no—
214C2×C3×D3—121no—
215C2×C3×D3—241no—
216C2×C3×D3—241no—
217C2×C3×D3—241no—
218C2×C3×D3—241no—
219C2×C3×D3—241no—
220C2×C3×D3—241no—
221C2×C3×D3—241no—
222D32—41no—
223D32—41no—
224D32—121no—
225D32—121no—
226D32—241no—
227D32—241no—
228#157—32no—
229#157—61no—
230#157—121no—
231C32×C4—21no—
232C32×C4—21no—
233C32×C4—41no—
234C32×C4—41no—
235C32×C4C2×#6711yescharacteristic
236C3⋊Dic3#23711yescharacteristic
237C3×Dic3—61no—
238C3×Dic3—61no—
239C2×D8—91no—
240C2×D8—271no—
241#127—271no—
242C4×D4—271no—
243C4⋊C8—271no—
244#102—271no—
245#102—271no—
246C33#45911yescharacteristic, complemented, Sylow
247C23×C3—181no—
248C23×C3—181no—
249C22×D3—121no—
250C22×D3—121no—
251C22×D3—121no—
252C22×D3—181no—
253C22×D3—361no—
254C3×D4—21no—
255C3×D4—21no—
256C3×D4—41no—
257C3×D4—41no—
258C3×D4—61no—
259C3×D4—61no—
260C3×D4—91no—
261C3×D4—121no—
262C3×D4—181no—
263C3×D4—181no—
264C3×D4—181no—
265C3×D4GO+(4,𝔽2)11yescharacteristic, complemented
266C2×C3×C4—91no—
267#67—121no—
268#67—121no—
269#67—121no—
270#67—361no—
271#67—362no—
272#67—362no—
273C2×Dic3—91no—
274C2×Dic3—91no—
275C2×Dic3—181no—
276C2×Dic3—362no—
277D12—21no—
278D12—61no—
279D12—61no—
280D12—61no—
281D12—61no—
282D12—121no—
283D12—181no—
284C4×D3—61no—
285C4×D3—61no—
286C3×C8—61no—
287C3×C8—61no—
288C3×C8—121no—
289C3×C8—181no—
290C3⋊C8—21no—
291C3⋊C8—61no—
292C3⋊C8—61no—
293C3⋊C8—121no—
294C2×C32—21no—
295C2×C32—21no—
296C2×C32—41no—
297C2×C32—41no—
298C2×C32—41no—
299C2×C32—81no—
300C2×C32—81no—
301C2×C32—161no—
302C2×C32—161no—
303C2×C32—241no—
304C2×C32#93911yescharacteristic
305C3⋊D3—41no—
306C3⋊D3—81no—
307C3⋊D3D3×D811yescharacteristic
308C3⋊D3#91111yescharacteristic
309C3×D3—61no—
310C3×D3—61no—
311C3×D3—61no—
312C3×D3—61no—
313C3×D3—81no—
314C3×D3—81no—
315C3×D3—161no—
316C3×D3—241no—
317C3×D3—241no—
318C3×D3—241no—
319C3×D3—241no—
320C3×D3—481no—
321C2×D4—91no—
322C2×D4—91no—
323C2×D4—271no—
324C2×D4—541no—
325C2×D4—541no—
326C22×C4—541no—
327D8—61no—
328D8—181no—
329D8—181no—
330D8—541no—
331C2×C8—91no—
332C2×C8—271no—
333C4⋊C4—271no—
334C22⋊C4—541no—
335C42—271no—
336C22×C3—21no—
337C22×C3—41no—
338C22×C3—41no—
339C22×C3—81no—
340C22×C3—81no—
341C22×C3—91no—
342C22×C3—121no—
343C22×C3—121no—
344C22×C3—181no—
345C22×C3—241no—
346C22×C3—361no—
347C22×C3—361no—
348C22×C3—361no—
349C22×C3—361no—
350C22×C3—361no—
351C2×D3—41no—
352C2×D3—61no—
353C2×D3—61no—
354C2×D3—121no—
355C2×D3—121no—
356C2×D3—121no—
357C2×D3—121no—
358C2×D3—241no—
359C2×D3—241no—
360C2×D3—241no—
361C2×D3—241no—
362C2×D3—241no—
363C2×D3—241no—
364C2×D3—241no—
365C2×D3—361no—
366C2×D3—361no—
367C2×D3—721no—
368C2×D3—721no—
369C3×C4—21no—
370C3×C4—21no—
371C3×C4—41no—
372C3×C4—41no—
373C3×C4—91no—
374C3×C4C2×GO+(4,𝔽2)11yescharacteristic
375Dic3—61no—
376Dic3—61no—
377Dic3—92no—
378Dic3—181no—
379Dic3—361no—
380C32—21no—
381C32—21no—
382C32—41no—
383C32—41no—
384C32#515011yescharacteristic, complemented
385C23—181no—
386C23—181no—
387C23—541no—
388D4—61no—
389D4—91no—
390D4—181no—
391D4—181no—
392D4—181no—
393D4—541no—
394D4—1082no—
395D4—1082no—
396D4D32⋊D311yescharacteristic, complemented
397C2×C4—91no—
398C2×C4—271no—
399C2×C4—271no—
400C2×C4—541no—
401C2×C4—1082no—
402C8—61no—
403C8—181no—
404C2×C3—21no—
405C2×C3—21no—
406C2×C3—41no—
407C2×C3—41no—
408C2×C3—41no—
409C2×C3—81no—
410C2×C3—81no—
411C2×C3—91no—
412C2×C3—91no—
413C2×C3—161no—
414C2×C3—161no—
415C2×C3—241no—
416C2×C3—241no—
417C2×C3—361no—
418C2×C3—481no—
419C2×C3—721no—
420C2×C3#6424211yescharacteristic
421D3—61no—
422D3—61no—
423D3—61no—
424D3—61no—
425D3—81no—
426D3—241no—
427D3—241no—
428D3—241no—
429D3—241no—
430D3—481no—
431C22—21no—
432C22—91no—
433C22—121no—
434C22—181no—
435C22—361no—
436C22—361no—
437C22—361no—
438C22—361no—
439C22—361no—
440C22—1081no—
441C22—1081no—
442C4—91no—
443C4—272no—
444C4—541no—
445C4—1081no—
446C4C2×D32⋊D311yescharacteristic
447C3—21no—
448C3—21no—
449C3—41no—
450C3—41no—
451C3#1059346411yescharacteristic, complemented
452C2—41no—
453C2—91no—
454C2—91no—
455C2—241no—
456C2—361no—
457C2—721no—
458C2#1172433211yescharacteristic
subgroupG/H
centerC2#11724332
commutator, cosocleC3×C4×C3⋊D3C23
FrattiniC4C2×D32⋊D3
FittingC33×D4D4
socleC2×C33#121
hypercenterD4D32⋊D3
hypocenterC33#459
LCSG ▷ C3×C4×C3⋊D3 ▷ C2×C33 ▷ C33
UCSC1 ◁ C2 ◁ C4 ◁ D4

Complements

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

NH ≅ G/Ntype
1. D4×C3⋊#157C2semidirect
2. #11724315C2semidirect
3. #11723702C2semidirect
4. C3×#63926C2semidirect
5. #11722763C2semidirect
6. #11722758C2semidirect
7. #11722757C2semidirect
28. #63926D3semidirect
42. C33×D4D4semidirect
45. C3×C4⋊(C3⋊D3)D4semidirect
97. C3⋊#157D8semidirect
127. C32×D4#67semidirect
130. C4⋊(C3⋊D3)#67semidirect
246. C33#459semidirect
265. C3×D4GO+(4,𝔽2)semidirect
384. C32#5150semidirect
396. D4D32⋊D3semidirect
451. C3#10593464semidirect

Maximal subgroups

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

subgroupindex
D4×C3⋊#1572
#117243152
#117237022
C3×#639262
#117227632
#117227582
#117227572
#105934643
#51509