Finite Group Databasebeta

← #13140625

Subgroups — #13140625

Order 1728.

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

normal subgroups only

This group has 516 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
1C2×#88546C211yesmaximal, characteristic, complemented
2#64149⋊C3C211yesmaximal, characteristic
3#11721083C211yesmaximal, characteristic, complemented
4C2×#88438C211yesmaximal, characteristic, complemented
5#63659⋊C3C211yesmaximal, characteristic, complemented
6#11720318C211yesmaximal, characteristic, complemented
7#11720311C211yesmaximal, characteristic, complemented
8#10591444—31nomaximal
9#10591438—31nomaximal
10C2×D4×H(3,𝔽3)C2211yescharacteristic
11C2×#6229C2211yescharacteristic
12C2×#638⋊C3—21no—
13C2×#638⋊C3—21no—
14C2×C4⋊((C3⋊D3)⋊C3)C2211yescharacteristic
15#88546—22no—
16C2×#6180C2211yescharacteristic
17C4⋊((C3⋊Dic3)⋊C3)C2211yescharacteristic
18C4×(C3⋊Dic3)⋊C3C2211yescharacteristic
19C2×(C3⋊(C3⋊C8))⋊C3C2211yescharacteristic
20#88449—21no—
21#88447—21no—
22#88438—22no—
23#64149D311yescharacteristic
24C3×#940—31no—
25C2×C3×#214—31no—
26C2×C3×#214—31no—
27C2×C3×#214—61no—
28#63843—31no—
29Dic3×Dic6—31no—
30C2×#3717—31no—
31C2×#3715—31no—
32#63659D311yescharacteristic, complemented
33C3×#832—31no—
34#63572—31no—
35Dic3⋊(C3⋊C8)—31no—
36#63564—31no—
37#63563—31no—
38C23×H(3,𝔽3)—21no—
39C22×(C3⋊D3)⋊C3—21no—
40D4×H(3,𝔽3)—21no—
41D4×H(3,𝔽3)—21no—
42C2×C4×H(3,𝔽3)C2311yescharacteristic
43C2×#1084—21no—
44#6229—21no—
45#6229D412yescomplemented
46#638⋊C3—42no—
47#638⋊C3—42no—
48C2×(C3⋊Dic3)⋊C3—21no—
49C2×(C3⋊Dic3)⋊C3D411yescharacteristic
50C2×(C3⋊Dic3)⋊C3D411yescharacteristic
51C4⋊((C3⋊D3)⋊C3)—21no—
52C4⋊((C3⋊D3)⋊C3)—21no—
53#6180—21no—
54(C3⋊(C3⋊C8))⋊C3—21no—
55#5163—91no—
56#5075—91no—
57#4762—91no—
58C2×C32×D4—21no—
59C2×C32×D4C2×D311yescharacteristic
60C2×C32×D4C2×D311yescharacteristic
61C2×#638—21no—
62C2×#638—21no—
63C2×C4⋊(C3⋊D3)C2×D311yescharacteristic
64C2×C3×#67—61no—
65C2×C3×#67—61no—
66C2×C3×D12—31no—
67C2×C3×Dic6—31no—
68C2×C3×Dic6—31no—
69C2×C3×Dic6—61no—
70C4⋊(C3⋊Dic3)C2×D311yescharacteristic
71C4×C3⋊Dic3C2×D311yescharacteristic
72C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
73C3×#214—62no—
74C3×#214—62no—
75C3×#214—64no—
76C3×C4⋊Dic3—31no—
77C3×C4×Dic3—31no—
78C2×C3×C3⋊C8—31no—
79C2×C3×C3⋊C8—31no—
80C2×C3×C3⋊C8—31no—
81C2×C3×C3⋊C8—61no—
82Dic3⋊Dic3—61no—
83Dic3⋊Dic3—61no—
84Dic32—61no—
85Dic32—61no—
86#3717—62no—
87#3715—62no—
88C22×H(3,𝔽3)—41no—
89C22×H(3,𝔽3)—41no—
90C22×H(3,𝔽3)—41no—
91C22×H(3,𝔽3)C2×D411yescharacteristic
92C2×(C3⋊D3)⋊C3—41no—
93C2×(C3⋊D3)⋊C3—41no—
94C2×(C3⋊D3)⋊C3—41no—
95C4×H(3,𝔽3)C2×D411yescharacteristic
96C4×H(3,𝔽3)#4111yescharacteristic
97#1084—22no—
98(C3⋊Dic3)⋊C3—21no—
99(C3⋊Dic3)⋊C3—41no—
100(C3⋊Dic3)⋊C3SD1612yescomplemented
101C2×C3×SD16—91no—
102C3×#127—91no—
103Q8×Dic3—91no—
104C2×#215—91no—
105#940—31no—
106#940—91no—
107C2×#214—31no—
108C2×#214—31no—
109C2×#214—61no—
110C2×#214—91no—
111C2×#204—91no—
112#887—91no—
113C4×Dic6—91no—
114C3×#102—91no—
115#832—31no—
116#832—91no—
117#832—91no—
118#821—91no—
119Dic3⋊C8—91no—
120Dic3⋊C8—91no—
121#809—91no—
122C4⋊(C3⋊C8)—91no—
123C23×C32—21no—
124C23×C32—21no—
125C23×C32—41no—
126C22×C3⋊D3—21no—
127C22×C3×D3—61no—
128C32×D4—22no—
129C32×D4—21no—
130C32×D4—21no—
131C32×D4—21no—
132C32×D4—21no—
133C32×D4—41no—
134C2×C32×C4—21no—
135C2×C32×C4C22×D311yescharacteristic
136C2×C32×C4C22×D311yescharacteristic
137#638—42no—
138#638—42no—
139C2×C3⋊Dic3—21no—
140C2×C3⋊Dic3#6711yescharacteristic
141C2×C3⋊Dic3#6711yescharacteristic
142C4⋊(C3⋊D3)—21no—
143C4⋊(C3⋊D3)—21no—
144C3×#67—122no—
145C3×#67—122no—
146C2×C3×Dic3—31no—
147C2×C3×Dic3—31no—
148C2×C3×Dic3—61no—
149C2×C3×Dic3—61no—
150C2×C3×Dic3—61no—
151C2×C3×Dic3—121no—
152C3×D12—61no—
153C3×D12—61no—
154C3×Dic6—32no—
155C3×Dic6—32no—
156C3×Dic6—62no—
157C3×Dic6—61no—
158C3×Dic6—61no—
159C3×Dic6—121no—
160C3⋊(C3⋊C8)—21no—
161C3×C3⋊C8—62no—
162C3×C3⋊C8—61no—
163C3×C3⋊C8—61no—
164C3×C3⋊C8—61no—
165#460—271noSylow
166C2×H(3,𝔽3)—81no—
167C2×H(3,𝔽3)#12111yescharacteristic
168C2×H(3,𝔽3)C2×SD1611yescharacteristic
169C2×H(3,𝔽3)AGL(1,ℤ/8)11yescharacteristic
170(C3⋊D3)⋊C3—81no—
171C2×C3×Q8—91no—
172C2×C3×D4—31no—
173C2×C3×D4—31no—
174C2×C3×D4—61no—
175C2×C3×D4—91no—
176C2×C3×D4—181no—
177C2×C3×D4—181no—
178C2×C3×D4D3211yescharacteristic
179C2×#67—61no—
180C2×#67—61no—
181C2×#67—181no—
182C2×#67—181no—
183C2×D12—31no—
184C2×D12—91no—
185C2×Dic6—31no—
186C2×Dic6—31no—
187C2×Dic6—61no—
188C3×SD16—182no—
189C2×C3×C8—91no—
190C2×C3×C8—91no—
191C3×C4⋊C4—91no—
192C3×C42—91no—
193#215—182no—
194#214—62no—
195#214—62no—
196#214—64no—
197#214—182no—
198C4⋊Dic3—31no—
199C4⋊Dic3—91no—
200C4⋊Dic3—181no—
201Dic3⋊C4—181no—
202Dic3⋊C4—181no—
203C4×Dic3—31no—
204C4×Dic3—91no—
205C4×Dic3—181no—
206C4×Dic3—181no—
207C4×Dic3—181no—
208C2×C3⋊C8—31no—
209C2×C3⋊C8—31no—
210C2×C3⋊C8—31no—
211C2×C3⋊C8—61no—
212C2×C3⋊C8—91no—
213#204—182no—
214C22×C32—21no—
215C22×C32—42no—
216C22×C32—41no—
217C22×C32—41no—
218C22×C32—41no—
219C22×C32—41no—
220C22×C32—41no—
221C22×C32—41no—
222C22×C32—81no—
223C22×C32—81no—
224C22×C32D3×D411yescharacteristic
225C22×C32C2×#6711yescharacteristic
226C2×C3⋊D3—41no—
227C2×C3⋊D3—41no—
228C2×C3⋊D3—41no—
229C2×C3×D3—121no—
230C2×C3×D3—121no—
231C2×C3×D3—121no—
232C32×C4—21no—
233C32×C4—21no—
234C32×C4#23511yescharacteristic
235C32×C4D3×D411yescharacteristic
236C32×C4D3×D411yescharacteristic
237C32×C4#23711yescharacteristic
238C3⋊Dic3—21no—
239C3⋊Dic3—41no—
240C3⋊Dic3#21412yescomplemented
241C3×Dic3—32no—
242C3×Dic3—62no—
243C3×Dic3—61no—
244C3×Dic3—62no—
245C3×Dic3—122no—
246C3×Dic3—121no—
247C2×SD16—91no—
248C2×SD16—271no—
249#127—91no—
250C4×Q8—271no—
251C4⋊C8—271no—
252#102—91no—
253#102—271no—
254H(3,𝔽3)#46011yescharacteristic, complemented, Sylow
255C23×C3—21no—
256C23×C3—61no—
257C23×C3—61no—
258C23×C3—121no—
259C23×C3—181no—
260C22×D3—61no—
261C22×D3—181no—
262C3×Q8—92no—
263C3×Q8—181no—
264C3×D4—21no—
265C3×D4—21no—
266C3×D4—61no—
267C3×D4—61no—
268C3×D4—61no—
269C3×D4—61no—
270C3×D4—62no—
271C3×D4—121no—
272C3×D4—181no—
273C3×D4—181no—
274C3×D4—362no—
275C3×D4—362no—
276C2×C3×C4—31no—
277C2×C3×C4—31no—
278C2×C3×C4—61no—
279C2×C3×C4—91no—
280C2×C3×C4—91no—
281C2×C3×C4—181no—
282C2×C3×C4—181no—
283C2×C3×C4C2×D3211yescharacteristic
284#67—122no—
285#67—122no—
286#67—362no—
287#67—362no—
288C2×Dic3—31no—
289C2×Dic3—31no—
290C2×Dic3—61no—
291C2×Dic3—61no—
292C2×Dic3—61no—
293C2×Dic3—91no—
294C2×Dic3—91no—
295C2×Dic3—121no—
296C2×Dic3—181no—
297D12—61no—
298D12—61no—
299D12—181no—
300D12—181no—
301Dic6—32no—
302Dic6—32no—
303Dic6—62no—
304Dic6—61no—
305Dic6—61no—
306Dic6—121no—
307C3×C8—181no—
308C3×C8—181no—
309C3⋊C8—61no—
310C3⋊C8—61no—
311C3⋊C8—62no—
312C3⋊C8—61no—
313C3⋊C8—181no—
314C2×C32—21no—
315C2×C32—21no—
316C2×C32—21no—
317C2×C32—82no—
318C2×C32—81no—
319C2×C32—81no—
320C2×C32#88411yescharacteristic
321C2×C32#91111yescharacteristic
322C2×C32D3×SD1611yescharacteristic
323C2×C32#93211yescharacteristic
324C2×C32C2×#21411yescharacteristic
325C2×C32#93811yescharacteristic
326C3⋊D3—81no—
327C3×D3—241no—
328C2×Q8—91no—
329C2×D4—91no—
330C2×D4—181no—
331C2×D4—181no—
332C2×D4((C3⋊D3)⋊C3)⋊C211yescharacteristic
333SD16—182no—
334SD16—542no—
335C2×C8—91no—
336C2×C8—91no—
337C4⋊C4—91no—
338C4⋊C4—541no—
339C42—91no—
340C42—541no—
341C22×C3—31no—
342C22×C3—31no—
343C22×C3—41no—
344C22×C3—41no—
345C22×C3—41no—
346C22×C3—61no—
347C22×C3—121no—
348C22×C3—121no—
349C22×C3—121no—
350C22×C3—121no—
351C22×C3—121no—
352C22×C3—121no—
353C22×C3—122no—
354C22×C3—241no—
355C22×C3—241no—
356C22×C3—361no—
357C22×C3—361no—
358C22×C3—361no—
359C22×C3D3×#6711yescharacteristic
360C2×D3—121no—
361C2×D3—121no—
362C2×D3—121no—
363C2×D3—361no—
364C2×D3—361no—
365C2×D3—361no—
366C3×C4—31no—
367C3×C4—31no—
368C3×C4—31no—
369C3×C4—31no—
370C3×C4—61no—
371C3×C4—61no—
372C3×C4—92no—
373C3×C4—181no—
374C3×C4—182no—
375C3×C4—361no—
376C3×C4#380411yescharacteristic
377C3×C4#381111yescharacteristic
378Dic3—32no—
379Dic3—62no—
380Dic3—62no—
381Dic3—61no—
382Dic3—92no—
383Dic3—122no—
384Dic3—121no—
385Dic3—181no—
386Dic3—361no—
387C32—21no—
388C32#476211yescharacteristic, complemented
389C32#507511yescharacteristic, complemented
390C23—21no—
391C23—181no—
392Q8—92no—
393Q8—181no—
394D4—21no—
395D4—21no—
396D4—181no—
397D4—181no—
398D4—362no—
399D4—362no—
400C2×C4—91no—
401C2×C4—91no—
402C2×C4—181no—
403C2×C4—181no—
404C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
405C8—181no—
406C8—181no—
407C2×C3—31no—
408C2×C3—31no—
409C2×C3—31no—
410C2×C3—31no—
411C2×C3—31no—
412C2×C3—31no—
413C2×C3—61no—
414C2×C3—61no—
415C2×C3—61no—
416C2×C3—81no—
417C2×C3—241no—
418C2×C3—241no—
419C2×C3—242no—
420C2×C3—721no—
421C2×C3D3×#21411yescharacteristic
422C2×C3#6393211yescharacteristic
423C2×C3#6395811yescharacteristic
424D3—241no—
425D3—721no—
426C22—41no—
427C22—41no—
428C22—41no—
429C22—361no—
430C22—361no—
431C22—361no—
432C22#8867611yescharacteristic
433C4—92no—
434C4—182no—
435C4—181no—
436C4—361no—
437C4#8867111yescharacteristic
438C4#8867711yescharacteristic
439C3—31no—
440C3—31no—
441C3—61no—
442C3#1059143911yescharacteristic
443C2—81no—
444C2—721no—
445C2#1172119811yescharacteristic
446C2#1172120011yescharacteristic
447C2#1172130311yescharacteristic
subgroupG/H
centerC22#88676
commutator, cosocleC2×C4×H(3,𝔽3)C23
FrattiniC2×C3×C4C2×D32
FittingC2×D4×H(3,𝔽3)C22
socleC22×C3D3×#67
prime radical#64149⋊C3C2
hypercenterC2×D4((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)#460
LCSG ▷ C2×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. C2×#88546C2semidirect
3. #11721083C2semidirect
4. C2×#88438C2semidirect
5. #63659⋊C3C2semidirect
6. #11720318C2semidirect
7. #11720311C2semidirect
32. #63659D3semidirect
45. #6229D4semidirect
100. (C3⋊Dic3)⋊C3SD16semidirect
240. C3⋊Dic3#214semidirect
254. H(3,𝔽3)#460semidirect
388. C32#4762semidirect
389. C32#5075semidirect

Maximal subgroups

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

subgroupindex
C2×#885462
#64149⋊C32
#117210832
C2×#884382
#63659⋊C32
#117203182
#117203112
#105914443
#105914383