Finite Group Databasebeta

← #13140526

Subgroups — #13140526

Order 1728.

556 conjugacy classes of subgroups; only the 93 normal subgroups are shown.

normal subgroups only

This group has 556 conjugacy classes of subgroups — too many to draw. Only its 93 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×#1084C211yesmaximal, characteristic, complemented
2C2×#3753⋊C3C211yesmaximal, characteristic
3#11721136C211yesmaximal, characteristic, complemented
4#63659⋊C3C211yesmaximal, characteristic, complemented
5#11720308C211yesmaximal, characteristic, complemented
6#11720307C211yesmaximal, characteristic, complemented
7#11720303C211yesmaximal, characteristic, complemented
8#10591430—31nomaximal
9#10591404—31nomaximal
10C2×D4×H(3,𝔽3)C2211yescharacteristic
11C22×#1084—21no—
12C2×C4⋊((C3⋊D3)⋊C3)C2211yescharacteristic
13C2×((C3⋊D3)⋊C3)⋊C4—21no—
14#88552—21no—
15C4⋊#1084C2211yescharacteristic
16C4×#1084C2211yescharacteristic
17#3753⋊C3C414yescomplemented
18C4⋊((C3⋊Dic3)⋊C3)C2211yescharacteristic
19C2×(C3⋊(C3⋊C8))⋊C3C2211yescharacteristic
20C2×(C3⋊(C3⋊C8))⋊C3C2211yescharacteristic
21#88450—21no—
22C2×#3753D311yescharacteristic
23C3×D4×Dic3—31no—
24C3×D4×Dic3—31no—
25C3×D4×Dic3—61no—
26C2×C3×#213—31no—
27#63894—31no—
28Dic3×D12—31no—
29#63659D311yescharacteristic, complemented
30C3×#832—31no—
31#63574—31no—
32#63573—31no—
33#63564—31no—
34#63562—31no—
35#63554—31no—
36#63554—31no—
37C23×H(3,𝔽3)—21no—
38C22×(C3⋊D3)⋊C3—21no—
39D4×H(3,𝔽3)—21no—
40D4×H(3,𝔽3)C2×C412yescomplemented
41C2×C4×H(3,𝔽3)C2311yescharacteristic
42C2×#1084—21no—
43C2×#1084—42no—
44C2×#1084D411yescharacteristic
45C2×#1084D411yescharacteristic
46C2×(C3⋊Dic3)⋊C3—21no—
47C4⋊((C3⋊D3)⋊C3)—21no—
48C4⋊((C3⋊D3)⋊C3)C2×C412yescomplemented
49((C3⋊D3)⋊C3)⋊C4—42no—
50(C3⋊(C3⋊C8))⋊C3—21no—
51(C3⋊(C3⋊C8))⋊C3C2×C412yescomplemented
52#5152—91no—
53#5014—91no—
54#4785—91no—
55C2×C32×D4—21no—
56C2×C32×D4C2×D311yescharacteristic
57C2×C32×D4C2×D311yescharacteristic
58C2×C4⋊(C3⋊D3)C2×D311yescharacteristic
59C22×C3×Dic3—61no—
60C22×C3×Dic3—61no—
61C22×C3×Dic3—62no—
62C2×C3×D12—31no—
63C2×#624—61no—
64C2×D3×Dic3—61no—
65#3753Dic314yescomplemented
66C4⋊(C3⋊Dic3)C2×D311yescharacteristic
67C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
68C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
69C3×#217—61no—
70C3×#217—61no—
71C3×#217—62no—
72C3×#213—34no—
73C3×C4⋊Dic3—31no—
74C3×C4⋊Dic3—31no—
75C3×C4⋊Dic3—31no—
76C3×C4⋊Dic3—61no—
77C3×C4×Dic3—31no—
78C3×C4×Dic3—31no—
79C3×C4×Dic3—61no—
80C2×C3×C3⋊C8—31no—
81C2×C3×C3⋊C8—31no—
82#3722—61no—
83(C2×D3)⋊Dic3—61no—
84C22×H(3,𝔽3)—22no—
85C22×H(3,𝔽3)—41no—
86C22×H(3,𝔽3)—41no—
87C22×H(3,𝔽3)C2×D411yescharacteristic
88C2×(C3⋊D3)⋊C3—22no—
89C2×(C3⋊D3)⋊C3—41no—
90C2×(C3⋊D3)⋊C3—41no—
91C4×H(3,𝔽3)C22×C411yescharacteristic
92C4×H(3,𝔽3)#4111yescharacteristic
93#1084—21no—
94#1084—21no—
95#1084—41no—
96(C3⋊Dic3)⋊C3—41no—
97C2×C3×D8—91no—
98C3×C4×D4—91no—
99D4×Dic3—31no—
100D4×Dic3—31no—
101D4×Dic3—61no—
102D4×Dic3—91no—
103C2×#213—31no—
104C2×#213—91no—
105#893—91no—
106C4×D12—91no—
107C3×#102—91no—
108#832—31no—
109#832—91no—
110#832—91no—
111#821—91no—
112#817—91no—
113#815—91no—
114#815—91no—
115#809—91no—
116#808—91no—
117#808—91no—
118#803—91no—
119C23×C32—22no—
120C23×C32—21no—
121C23×C32—21no—
122C22×C3⋊D3—21no—
123C22×C3×D3—61no—
124C32×D4—22no—
125C32×D4—21no—
126C32×D4—21no—
127C32×D4—22no—
128C32×D4C4×D312yescomplemented
129C32×D4C2×Dic312yescomplemented
130C2×C32×C4—21no—
131C2×C32×C4C22×D311yescharacteristic
132C2×C32×C4C22×D311yescharacteristic
133C2×C3⋊Dic3—21no—
134C4⋊(C3⋊D3)—21no—
135C4⋊(C3⋊D3)C2×Dic312yescomplemented
136C2×C3×Dic3—31no—
137C2×C3×Dic3—31no—
138C2×C3×Dic3—31no—
139C2×C3×Dic3—31no—
140C2×C3×Dic3—61no—
141C2×C3×Dic3—61no—
142C2×C3×Dic3—62no—
143C2×C3×Dic3—61no—
144C2×C3×Dic3—61no—
145C2×C3×Dic3—61no—
146C2×C3×Dic3—124no—
147C2×C3×Dic3—122no—
148C2×C3×Dic3—122no—
149C3×D12—32no—
150C3×D12—61no—
151#624—122no—
152D3×Dic3—122no—
153C3⋊(C3⋊C8)—21no—
154C3⋊(C3⋊C8)C2×Dic312yescomplemented
155C3×C3⋊C8—32no—
156C3×C3⋊C8—61no—
157#442—271noSylow
158C2×H(3,𝔽3)—42no—
159C2×H(3,𝔽3)C4×D411yescharacteristic
160C2×H(3,𝔽3)AGL(1,ℤ/8)11yescharacteristic
161C2×H(3,𝔽3)AGL(1,ℤ/8)11yescharacteristic
162(C3⋊D3)⋊C3—42no—
163C2×C3×D4—31no—
164C2×C3×D4—31no—
165C2×C3×D4—61no—
166C2×C3×D4—91no—
167C2×C3×D4D3211yescharacteristic
168C22×C3×C4—181no—
169C22×Dic3—61no—
170C22×Dic3—61no—
171C22×Dic3—62no—
172C22×Dic3—181no—
173C2×D12—31no—
174C2×D12—91no—
175C2×C4×D3—181no—
176C2×C4×D3—181no—
177C3×D8—94no—
178C2×C3×C8—91no—
179C2×C3×C8—91no—
180C3×C4⋊C4—91no—
181C3×C4⋊C4—91no—
182C3×C22⋊C4—181no—
183C3×C42—91no—
184#217—61no—
185#217—61no—
186#217—62no—
187#217—181no—
188#213—34no—
189#213—94no—
190(C2×D3)⋊C4—181no—
191(C2×D3)⋊C4—181no—
192C4⋊Dic3—31no—
193C4⋊Dic3—31no—
194C4⋊Dic3—31no—
195C4⋊Dic3—61no—
196C4⋊Dic3—91no—
197C4×Dic3—31no—
198C4×Dic3—31no—
199C4×Dic3—61no—
200C2×C3⋊C8—31no—
201C2×C3⋊C8—31no—
202C2×C3⋊C8—91no—
203C2×C3⋊C8—91no—
204C22×C32—24no—
205C22×C32—22no—
206C22×C32—22no—
207C22×C32—21no—
208C22×C32—42no—
209C22×C32—42no—
210C22×C32—41no—
211C22×C32—41no—
212C22×C32—41no—
213C22×C32—41no—
214C22×C32D3×D411yescharacteristic
215C22×C32D3×D411yescharacteristic
216C2×C3⋊D3—22no—
217C2×C3⋊D3—41no—
218C2×C3⋊D3—41no—
219C2×C3×D3—62no—
220C2×C3×D3—121no—
221C2×C3×D3—121no—
222C32×C4—21no—
223C32×C4—21no—
224C32×C4C2×C4×D311yescharacteristic
225C32×C4#23711yescharacteristic
226C32×C4#23711yescharacteristic
227C32×C4C22×Dic311yescharacteristic
228C3⋊Dic3—41no—
229C3×Dic3—62no—
230C3×Dic3—61no—
231C3×Dic3—61no—
232C3×Dic3—61no—
233C3×Dic3—61no—
234C3×Dic3—121no—
235C3×Dic3—122no—
236C3×Dic3—121no—
237C3×Dic3—121no—
238C3×Dic3—121no—
239C2×D8—91no—
240C4×D4—91no—
241C4×D4—271no—
242#106—271no—
243#102—91no—
244#102—271no—
245#97—271no—
246H(3,𝔽3)#44211yescharacteristic, complemented, Sylow
247C23×C3—21no—
248C23×C3—61no—
249C23×C3—61no—
250C23×C3—62no—
251C23×C3—181no—
252C22×D3—61no—
253C22×D3—181no—
254C3×D4—21no—
255C3×D4—32no—
256C3×D4—32no—
257C3×D4—61no—
258C3×D4—61no—
259C3×D4—62no—
260C3×D4—62no—
261C3×D4—92no—
262C3×D4—181no—
263C3×D4D3×Dic312yes—
264C2×C3×C4—31no—
265C2×C3×C4—31no—
266C2×C3×C4—61no—
267C2×C3×C4—91no—
268C2×C3×C4—91no—
269C2×C3×C4—181no—
270C2×C3×C4—181no—
271C2×C3×C4—362no—
272C2×C3×C4C2×D3211yescharacteristic
273C2×Dic3—31no—
274C2×Dic3—31no—
275C2×Dic3—31no—
276C2×Dic3—31no—
277C2×Dic3—61no—
278C2×Dic3—61no—
279C2×Dic3—61no—
280C2×Dic3—61no—
281C2×Dic3—62no—
282C2×Dic3—61no—
283C2×Dic3—122no—
284C2×Dic3—122no—
285C2×Dic3—124no—
286C2×Dic3—181no—
287C2×Dic3—362no—
288D12—32no—
289D12—61no—
290D12—92no—
291D12—181no—
292C4×D3—362no—
293C4×D3—362no—
294C3×C8—92no—
295C3×C8—181no—
296C3⋊C8—32no—
297C3⋊C8—61no—
298C3⋊C8—92no—
299C3⋊C8—181no—
300C2×C32—21no—
301C2×C32—21no—
302C2×C32—21no—
303C2×C32—44no—
304C2×C32—42no—
305C2×C32—42no—
306C2×C32#67⋊C411yescharacteristic
307C2×C32#91111yescharacteristic
308C2×C32#91111yescharacteristic
309C2×C32#91111yescharacteristic
310C2×C32#91411yescharacteristic
311C2×C32D4×Dic311yescharacteristic
312C3⋊D3—42no—
313C3×D3—122no—
314C2×D4—91no—
315C2×D4((C3⋊D3)⋊C3)⋊C211yescharacteristic
316C22×C4—181no—
317C22×C4—541no—
318D8—94no—
319C2×C8—91no—
320C2×C8—91no—
321C4⋊C4—91no—
322C4⋊C4—91no—
323C22⋊C4—181no—
324C22⋊C4—541no—
325C42—91no—
326C22×C3—22no—
327C22×C3—31no—
328C22×C3—31no—
329C22×C3—41no—
330C22×C3—41no—
331C22×C3—62no—
332C22×C3—62no—
333C22×C3—64no—
334C22×C3—61no—
335C22×C3—121no—
336C22×C3—121no—
337C22×C3—121no—
338C22×C3—121no—
339C22×C3—122no—
340C22×C3—122no—
341C22×C3—182no—
342C22×C3—361no—
343C22×C3—361no—
344C22×C3#381111yescharacteristic
345C2×D3—62no—
346C2×D3—121no—
347C2×D3—121no—
348C2×D3—182no—
349C2×D3—361no—
350C2×D3—361no—
351C3×C4—31no—
352C3×C4—31no—
353C3×C4—31no—
354C3×C4—31no—
355C3×C4—61no—
356C3×C4—61no—
357C3×C4—181no—
358C3×C4—181no—
359C3×C4—361no—
360C3×C4—361no—
361C3×C4C2×D3×Dic311yescharacteristic
362C3×C4#380511yescharacteristic
363Dic3—61no—
364Dic3—61no—
365Dic3—61no—
366Dic3—61no—
367Dic3—62no—
368Dic3—121no—
369Dic3—121no—
370Dic3—121no—
371Dic3—122no—
372Dic3—121no—
373Dic3—361no—
374C32—21no—
375C32#478511yescharacteristic, complemented
376C32#515211yescharacteristic, complemented
377C23—21no—
378C23—181no—
379D4—21no—
380D4—92no—
381D4—181no—
382D4((C3⋊D3)⋊C3)⋊C412yescomplemented
383C2×C4—91no—
384C2×C4—91no—
385C2×C4—181no—
386C2×C4—181no—
387C2×C4—362no—
388C2×C4—1082no—
389C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
390C8—92no—
391C8—181no—
392C2×C3—31no—
393C2×C3—31no—
394C2×C3—31no—
395C2×C3—31no—
396C2×C3—31no—
397C2×C3—31no—
398C2×C3—42no—
399C2×C3—61no—
400C2×C3—61no—
401C2×C3—61no—
402C2×C3—122no—
403C2×C3—122no—
404C2×C3—124no—
405C2×C3—362no—
406C2×C3#6392711yescharacteristic
407C2×C3#6393711yescharacteristic
408C2×C3#67⋊Dic311yescharacteristic
409D3—122no—
410D3—362no—
411C22—22no—
412C22—41no—
413C22—41no—
414C22—182no—
415C22—361no—
416C22—361no—
417C22#8867711yescharacteristic
418C4—181no—
419C4—181no—
420C4—361no—
421C4—361no—
422C4C2×((C3⋊D3)⋊C3)⋊C411yescharacteristic
423C4#8867211yescharacteristic
424C3—31no—
425C3—31no—
426C3—61no—
427C3#1059143611yescharacteristic
428C2—42no—
429C2—362no—
430C2#1172119111yescharacteristic
431C2#1172119311yescharacteristic
432C2#1172131511yescharacteristic
subgroupG/H
centerC22#88677
commutatorC4×H(3,𝔽3)C22×C4
FrattiniC2×C3×C4C2×D32
FittingC2×D4×H(3,𝔽3)C22
socleC22×C3#3811
prime radicalC2×#3753⋊C3C2
cosocleC2×C4×H(3,𝔽3)C23
hypercenterC2×D4((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)#442
LCSG ▷ C4×H(3,𝔽3) ▷ C2×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C22 ◁ C2×C4 ◁ C2×D4

Complements

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

NH ≅ G/Ntype
1. D4×#1084C2semidirect
3. #11721136C2semidirect
4. #63659⋊C3C2semidirect
5. #11720308C2semidirect
6. #11720307C2semidirect
7. #11720303C2semidirect
17. #3753⋊C3C4semidirect
29. #63659D3semidirect
40. D4×H(3,𝔽3)C2×C4semidirect
48. C4⋊((C3⋊D3)⋊C3)C2×C4semidirect
51. (C3⋊(C3⋊C8))⋊C3C2×C4semidirect
65. #3753Dic3semidirect
128. C32×D4C4×D3semidirect
129. C32×D4C2×Dic3semidirect
135. C4⋊(C3⋊D3)C2×Dic3semidirect
154. C3⋊(C3⋊C8)C2×Dic3semidirect
246. H(3,𝔽3)#442semidirect
375. C32#4785semidirect
376. C32#5152semidirect
382. D4((C3⋊D3)⋊C3)⋊C4semidirect

Maximal subgroups

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

subgroupindex
D4×#10842
C2×#3753⋊C32
#117211362
#63659⋊C32
#117203082
#117203072
#117203032
#105914303
#105914043