Finite Group Databasebeta

← #13140563

Subgroups — #13140563

Order 1728.

603 conjugacy classes of subgroups; only the 107 normal subgroups are shown.

normal subgroups only

This group has 603 conjugacy classes of subgroups — too many to draw. Only its 107 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×#3830⋊C3C211yesmaximal, characteristic
2#11721081C211yesmaximal, characteristic, complemented
3C2×((C3⋊D3)⋊C3)⋊C8C211yesmaximal, characteristic, complemented
4#11720578C211yesmaximal, characteristic, complemented
5#63659⋊C3C211yesmaximal, characteristic, complemented
6#11720313C211yesmaximal, characteristic, complemented
7#11720309C211yesmaximal, characteristic, complemented
8#10591424—31nomaximal
9#10591420—31nomaximal
10C2×D4×H(3,𝔽3)C2211yescharacteristic
11C2×#638⋊C3—21no—
12C22×(C3⋊Dic3)⋊C3—21no—
13#3830⋊C3—22no—
14#3830⋊C3C414yescomplemented
15C2×C4×(C3⋊D3)⋊C3C2211yescharacteristic
16C2×#634⋊C3C2211yescharacteristic
17C4⋊#1084C2211yescharacteristic
18C2×#6180C2211yescharacteristic
19C4⋊((C3⋊Dic3)⋊C3)C2211yescharacteristic
20C2×(C3⋊(C3⋊C8))⋊C3C2211yescharacteristic
21#88450—21no—
22#88447—21no—
23((C3⋊D3)⋊C3)⋊C8—22no—
24C2×#3830D311yescharacteristic
25C2×C3×#237—31no—
26#63849—31no—
27#63841—31no—
28C2×#3710—31no—
29C2×D3×C3⋊C8—31no—
30#63659D311yescharacteristic, complemented
31C3×#832—31no—
32C3×#832—31no—
33C3×#832—31no—
34C3×#832—61no—
35#63567—31no—
36#63567—31no—
37#63566—31no—
38#63565—31no—
39C23×H(3,𝔽3)—21no—
40C22×(C3⋊D3)⋊C3D411yescharacteristic
41D4×H(3,𝔽3)—21no—
42D4×H(3,𝔽3)C2×C412yes—
43C2×C4×H(3,𝔽3)C2311yescharacteristic
44C2×#1084—21no—
45#638⋊C3—24no—
46C2×(C3⋊Dic3)⋊C3—21no—
47C2×(C3⋊Dic3)⋊C3—24no—
48C2×(C3⋊Dic3)⋊C3—21no—
49C2×(C3⋊Dic3)⋊C3D411yescharacteristic
50C4×(C3⋊D3)⋊C3C2×C412yes—
51C4×(C3⋊D3)⋊C3D412yes—
52#634⋊C3—21no—
53#634⋊C3C2×C412yes—
54#6180—21no—
55(C3⋊(C3⋊C8))⋊C3—21no—
56#5233—91no—
57#4772—91no—
58#4772—91no—
59C2×C32×D4—21no—
60C2×C32×D4C2×D311yescharacteristic
61C2×C32×D4C2×D311yescharacteristic
62C2×#638—21no—
63C22×C3⋊Dic3—21no—
64#3830—22no—
65#3830Dic314yescomplemented
66C2×C4×C3⋊D3C2×D311yescharacteristic
67C2×#634C2×D311yescharacteristic
68C2×C3×#67—61no—
69C22×C3×Dic3—61no—
70C3×#237—34no—
71C3×#237—62no—
72C2×C3×C4×D3—31no—
73C2×C3×Dic6—31no—
74C4⋊(C3⋊Dic3)C2×D311yescharacteristic
75C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
76C3×C4⋊Dic3—31no—
77C3×C4⋊Dic3—31no—
78C3×C4⋊Dic3—31no—
79C3×C4⋊Dic3—61no—
80C2×C3×C3⋊C8—31no—
81C2×C3×C3⋊C8—31no—
82C2×C3×C3⋊C8—31no—
83C2×C3×C3⋊C8—61no—
84#3722—61no—
85(C2×D3)⋊Dic3—61no—
86Dic32—61no—
87Dic32—61no—
88#3710—62no—
89D3×C3⋊C8—62no—
90C22×H(3,𝔽3)—22no—
91C22×H(3,𝔽3)—41no—
92C22×H(3,𝔽3)—41no—
93C22×H(3,𝔽3)C2×D411yescharacteristic
94C2×(C3⋊D3)⋊C3—21no—
95C2×(C3⋊D3)⋊C3—21no—
96C2×(C3⋊D3)⋊C3C22⋊C412yes—
97C4×H(3,𝔽3)C22×C411yescharacteristic
98C4×H(3,𝔽3)C2×D411yescharacteristic
99#1084—41no—
100(C3⋊Dic3)⋊C3—22no—
101(C3⋊Dic3)⋊C3—41no—
102(C3⋊Dic3)⋊C3C22⋊C412yes—
103C2×C3×#41—91no—
104C2×#237—31no—
105C2×#237—91no—
106#926—91no—
107C22×C3⋊C8—91no—
108C2×D3×C8—91no—
109C2×D3×C8—91no—
110#892—91no—
111#892—91no—
112C3×#102—91no—
113C3×#102—91no—
114#835—91no—
115#835—91no—
116#832—31no—
117#832—31no—
118#832—31no—
119#832—61no—
120#832—91no—
121#816—91no—
122#816—91no—
123#810—91no—
124#810—91no—
125C23×C32—21no—
126C23×C32—21no—
127C23×C32—41no—
128C22×C3⋊D3#6711yescharacteristic
129C22×C3×D3—31no—
130C32×D4—21no—
131C32×D4—21no—
132C32×D4—22no—
133C32×D4—41no—
134C32×D4C4×D312yes—
135C32×D4C2×Dic312yes—
136C2×C32×C4—21no—
137C2×C32×C4C22×D311yescharacteristic
138C2×C32×C4C22×D311yescharacteristic
139#638—24no—
140C2×C3⋊Dic3—21no—
141C2×C3⋊Dic3—24no—
142C2×C3⋊Dic3—21no—
143C2×C3⋊Dic3#6711yescharacteristic
144C4×C3⋊D3C2×Dic312yes—
145C4×C3⋊D3#6712yes—
146#634—21no—
147#634C2×Dic312yes—
148C3×#67—64no—
149C2×C3×Dic3—31no—
150C2×C3×Dic3—64no—
151C2×C3×Dic3—61no—
152C2×C3×Dic3—61no—
153C2×C3×Dic3—61no—
154C2×C3×Dic3—61no—
155C2×C3×Dic3—121no—
156C3×C4×D3—32no—
157C3×C4×D3—32no—
158C3×Dic6—32no—
159C3×Dic6—61no—
160C3⋊(C3⋊C8)—21no—
161C3×C3⋊C8—61no—
162C3×C3⋊C8—61no—
163C3×C3⋊C8—61no—
164C3×C3⋊C8—121no—
165#416—271noSylow
166C2×H(3,𝔽3)—42no—
167C2×H(3,𝔽3)C2×C22⋊C411yescharacteristic
168C2×H(3,𝔽3)#13511yescharacteristic
169C2×H(3,𝔽3)#13511yescharacteristic
170(C3⋊D3)⋊C3—22no—
171C3×#41—94no—
172C3×#41—182no—
173C2×C3×Q8—91no—
174C2×C3×D4—31no—
175C2×C3×D4—31no—
176C2×C3×D4—61no—
177C2×C3×D4—181no—
178C2×C3×D4D3211yescharacteristic
179C22×C3×C4—91no—
180C22×C3×C4—181no—
181C2×#67—61no—
182C2×#67—181no—
183C22×Dic3—61no—
184C22×Dic3—181no—
185#237—34no—
186#237—62no—
187#237—94no—
188#237—182no—
189C2×C4×D3—31no—
190C2×C4×D3—91no—
191C2×Dic6—31no—
192C2×Dic6—91no—
193C2×C3×C8—91no—
194C2×C3×C8—91no—
195C3×C4⋊C4—91no—
196C3×C4⋊C4—91no—
197#217—181no—
198(C2×D3)⋊C4—181no—
199(C2×D3)⋊C4—181no—
200C4⋊Dic3—31no—
201C4⋊Dic3—31no—
202C4⋊Dic3—31no—
203C4⋊Dic3—61no—
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—
213C2×C3⋊C8—182no—
214D3×C8—182no—
215D3×C8—182no—
216C22×C32—22no—
217C22×C32—22no—
218C22×C32—21no—
219C22×C32—42no—
220C22×C32—41no—
221C22×C32—41no—
222C22×C32—41no—
223C22×C32—41no—
224C22×C32—81no—
225C22×C32—81no—
226C22×C32D3×D411yescharacteristic
227C22×C32C2×#6711yescharacteristic
228C2×C3⋊D3—21no—
229C2×C3⋊D3—21no—
230C2×C3⋊D3#21712yes—
231C2×C3×D3—32no—
232C2×C3×D3—61no—
233C2×C3×D3—61no—
234C32×C4—21no—
235C32×C4—21no—
236C32×C4C2×C4×D311yescharacteristic
237C32×C4D3×D411yescharacteristic
238C32×C4C22×Dic311yescharacteristic
239C32×C4C2×#6711yescharacteristic
240C3⋊Dic3—22no—
241C3⋊Dic3—41no—
242C3⋊Dic3#21712yes—
243C3×Dic3—32no—
244C3×Dic3—62no—
245C3×Dic3—121no—
246C3×Dic3—121no—
247C3×Dic3—121no—
248C3×Dic3—241no—
249C2×#41—91no—
250C22×C8—271no—
251#117—271no—
252#103—271no—
253#103—271no—
254#102—91no—
255#102—91no—
256H(3,𝔽3)#41611yescharacteristic, complemented, Sylow
257C23×C3—21no—
258C23×C3—61no—
259C23×C3—61no—
260C23×C3—91no—
261C23×C3—121no—
262C22×D3—31no—
263C22×D3—91no—
264C3×Q8—92no—
265C3×Q8—181no—
266C3×D4—21no—
267C3×D4—32no—
268C3×D4—32no—
269C3×D4—61no—
270C3×D4—61no—
271C3×D4—62no—
272C3×D4—121no—
273C3×D4—184no—
274C3×D4D3×Dic312yes—
275C2×C3×C4—31no—
276C2×C3×C4—31no—
277C2×C3×C4—61no—
278C2×C3×C4—92no—
279C2×C3×C4—92no—
280C2×C3×C4—91no—
281C2×C3×C4—184no—
282C2×C3×C4—181no—
283C2×C3×C4—181no—
284C2×C3×C4—181no—
285C2×C3×C4C2×D3211yescharacteristic
286#67—64no—
287#67—184no—
288C2×Dic3—31no—
289C2×Dic3—64no—
290C2×Dic3—61no—
291C2×Dic3—61no—
292C2×Dic3—61no—
293C2×Dic3—61no—
294C2×Dic3—91no—
295C2×Dic3—121no—
296C2×Dic3—184no—
297C2×Dic3—181no—
298C2×Dic3—181no—
299C4×D3—32no—
300C4×D3—32no—
301C4×D3—92no—
302C4×D3—92no—
303Dic6—32no—
304Dic6—61no—
305Dic6—92no—
306Dic6—181no—
307C3×C8—181no—
308C3×C8—181no—
309C3⋊C8—61no—
310C3⋊C8—61no—
311C3⋊C8—61no—
312C3⋊C8—121no—
313C3⋊C8—181no—
314C2×C32—21no—
315C2×C32—21no—
316C2×C32—21no—
317C2×C32—42no—
318C2×C32—42no—
319C2×C32—82no—
320C2×C32D3×C22⋊C411yescharacteristic
321C2×C32#91211yescharacteristic
322C2×C32#91611yescharacteristic
323C2×C32#95011yescharacteristic
324C2×C32#95011yescharacteristic
325C2×C32C2×#21711yescharacteristic
326C3⋊D3—22no—
327C3×D3—62no—
328#41—94no—
329#41—182no—
330C2×Q8—91no—
331C2×D4—181no—
332C2×D4((C3⋊D3)⋊C3)⋊C211yescharacteristic
333C22×C4—91no—
334C22×C4—181no—
335C2×C8—91no—
336C2×C8—91no—
337C2×C8—542no—
338C4⋊C4—91no—
339C4⋊C4—91no—
340C22⋊C4—541no—
341C42—541no—
342C22×C3—22no—
343C22×C3—31no—
344C22×C3—31no—
345C22×C3—41no—
346C22×C3—41no—
347C22×C3—62no—
348C22×C3—62no—
349C22×C3—61no—
350C22×C3—92no—
351C22×C3—122no—
352C22×C3—121no—
353C22×C3—121no—
354C22×C3—121no—
355C22×C3—121no—
356C22×C3—181no—
357C22×C3—181no—
358C22×C3—241no—
359C22×C3—241no—
360C22×C3D3×#6711yescharacteristic
361C2×D3—32no—
362C2×D3—61no—
363C2×D3—61no—
364C2×D3—92no—
365C2×D3—181no—
366C2×D3—181no—
367C3×C4—31no—
368C3×C4—31no—
369C3×C4—31no—
370C3×C4—31no—
371C3×C4—61no—
372C3×C4—61no—
373C3×C4—92no—
374C3×C4—182no—
375C3×C4—361no—
376C3×C4—361no—
377C3×C4C2×D3×Dic311yescharacteristic
378C3×C4D3×#6711yescharacteristic
379Dic3—32no—
380Dic3—62no—
381Dic3—92no—
382Dic3—121no—
383Dic3—121no—
384Dic3—121no—
385Dic3—182no—
386Dic3—241no—
387Dic3—361no—
388C32—21no—
389C32#477211yescharacteristic, complemented
390C32#523311yescharacteristic, complemented
391C23—21no—
392C23—91no—
393Q8—92no—
394Q8—181no—
395D4—21no—
396D4—184no—
397D4((C3⋊D3)⋊C3)⋊C412yes—
398C2×C4—91no—
399C2×C4—92no—
400C2×C4—92no—
401C2×C4—184no—
402C2×C4—181no—
403C2×C4—181no—
404C2×C4—181no—
405C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
406C8—181no—
407C8—181no—
408C2×C3—31no—
409C2×C3—31no—
410C2×C3—31no—
411C2×C3—31no—
412C2×C3—31no—
413C2×C3—31no—
414C2×C3—42no—
415C2×C3—61no—
416C2×C3—61no—
417C2×C3—61no—
418C2×C3—122no—
419C2×C3—122no—
420C2×C3—182no—
421C2×C3—242no—
422C2×C3#6393311yescharacteristic
423C2×C3#6393511yescharacteristic
424C2×C3D3×#21711yescharacteristic
425D3—62no—
426D3—182no—
427C22—22no—
428C22—41no—
429C22—41no—
430C22—92no—
431C22—181no—
432C22—181no—
433C22#8867611yescharacteristic
434C4—92no—
435C4—182no—
436C4—361no—
437C4—361no—
438C4C2×((C3⋊D3)⋊C3)⋊C411yescharacteristic
439C4#8867611yescharacteristic
440C3—31no—
441C3—31no—
442C3—61no—
443C3#1059142011yescharacteristic
444C2—42no—
445C2—182no—
446C2#1172119711yescharacteristic
447C2#1172120111yescharacteristic
448C2#1172131011yescharacteristic
subgroupG/H
centerC22#88676
commutatorC4×H(3,𝔽3)C22×C4
FrattiniC2×C3×C4C2×D32
FittingC2×D4×H(3,𝔽3)C22
socleC22×C3D3×#67
prime radicalC2×#3830⋊C3C2
cosocleC2×C4×H(3,𝔽3)C23
hypercenterC2×D4((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)#416
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
2. #11721081C2semidirect
3. C2×((C3⋊D3)⋊C3)⋊C8C2semidirect
4. #11720578C2semidirect
5. #63659⋊C3C2semidirect
6. #11720313C2semidirect
7. #11720309C2semidirect
14. #3830⋊C3C4semidirect
30. #63659D3semidirect
65. #3830Dic3semidirect
256. H(3,𝔽3)#416semidirect
389. C32#4772semidirect
390. C32#5233semidirect

Maximal subgroups

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

subgroupindex
C2×#3830⋊C32
#117210812
C2×((C3⋊D3)⋊C3)⋊C82
#117205782
#63659⋊C32
#117203132
#117203092
#105914243
#105914203