Finite Group Databasebeta

← #13147814

Subgroups — #13147814

Order 1728.

896 conjugacy classes of subgroups; only the 149 normal subgroups are shown.

normal subgroups only

This group has 896 conjugacy classes of subgroups — too many to draw. Only its 149 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
1C4×#6229C211yesmaximal, characteristic, complemented
2#11721385C211yesmaximal, characteristic
3C4×#634⋊C3C211yesmaximal, characteristic, complemented
4#11721136C212yesmaximal, complemented
5#11721132C212yesmaximal, complemented
6#11721125C214yesmaximal, complemented
7#11721113C212yesmaximal, complemented
8#11721110C212yesmaximal, complemented
9#10593963—31nomaximal
10#10593951—31nomaximal
11C2×#6229C2211yescharacteristic
12C2×C4⋊((C3⋊D3)⋊C3)C2212yes—
13C2×C4⋊((C3⋊D3)⋊C3)C2212yes—
14C2×C4⋊((C3⋊D3)⋊C3)C2212yes—
15C2×#634⋊C3C2211yescharacteristic
16C2×((C3⋊D3)⋊C3)⋊C4C2214yes—
17C42×H(3,𝔽3)C2211yescharacteristic
18C4⋊#1084C2211yescharacteristic
19#88540C2212yes—
20C4×#1084C2212yes—
21C4⋊((C3⋊Dic3)⋊C3)C2211yescharacteristic
22#3750⋊C3C2212yes—
23C4×(C3⋊Dic3)⋊C3C2212yes—
24#88450C2214yes—
25#88450C2214yes—
26#88450C2214yes—
27#64083D311yescharacteristic
28C4×#634D311yescharacteristic, complemented
29C3×#874—31no—
30C3×C4×Dic6—31no—
31C3×C4×Dic6—31no—
32C3×C4×Dic6—31no—
33C3×C4×Dic6—61no—
34#63898—32no—
35#63897—34no—
36#63894—32no—
37#63894—32no—
38Dic3×D12—32no—
39#63884—34no—
40#63864—32no—
41#63864—32no—
42#63863—32no—
43#63860—32no—
44C22×(C3⋊D3)⋊C3C2314yes—
45C2×C4×H(3,𝔽3)C2311yescharacteristic
46C2×C4×H(3,𝔽3)C2311yescharacteristic
47C2×C4×H(3,𝔽3)C2311yescharacteristic
48C2×#1084C2312yes—
49C2×#1084C2312yes—
50#6229—22no—
51C2×(C3⋊Dic3)⋊C3C2312yes—
52C2×(C3⋊Dic3)⋊C3C2312yes—
53C4⋊((C3⋊D3)⋊C3)—24no—
54C4⋊((C3⋊D3)⋊C3)—24no—
55C4⋊((C3⋊D3)⋊C3)—42no—
56#634⋊C3—22no—
57((C3⋊D3)⋊C3)⋊C4—44no—
58#5713—91no—
59#5585—91no—
60#5585—91no—
61C2×C4⋊(C3⋊D3)C2×D312yes—
62C2×C4⋊(C3⋊D3)C2×D312yes—
63C2×C4⋊(C3⋊D3)C2×D312yes—
64C2×#634C2×D311yescharacteristic
65C2×C3×D12—32no—
66C2×C3×D12—32no—
67C2×C3×D12—32no—
68C2×C3×Dic6—31no—
69C2×C3×Dic6—31no—
70C2×C3×Dic6—31no—
71C2×C3×Dic6—61no—
72C2×#624—34no—
73C2×D3×Dic3—34no—
74C32×C42—21no—
75C32×C42C2×D311yescharacteristic
76C32×C42C2×D311yescharacteristic
77C4⋊(C3⋊Dic3)C2×D311yescharacteristic
78#3750C2×D312yes—
79C4×C3⋊Dic3C2×D312yes—
80C3×C4⋊Dic3—31no—
81C3×C4⋊Dic3—31no—
82C3×C4⋊Dic3—31no—
83C3×C4⋊Dic3—61no—
84C3×Dic3⋊C4—32no—
85C3×Dic3⋊C4—32no—
86C3×Dic3⋊C4—32no—
87C3×Dic3⋊C4—62no—
88C3×C4×Dic3—32no—
89C3×C4×Dic3—32no—
90C3×C4×Dic3—32no—
91C3×C4×Dic3—62no—
92#3722—34no—
93#3722—34no—
94#3722—34no—
95(C2×D3)⋊Dic3—34no—
96(C2×D3)⋊Dic3—34no—
97(C2×D3)⋊Dic3—34no—
98C22×H(3,𝔽3)C2411yescharacteristic
99C2×(C3⋊D3)⋊C3—24no—
100C2×(C3⋊D3)⋊C3—24no—
101C2×(C3⋊D3)⋊C3—24no—
102C4×H(3,𝔽3)—21no—
103C4×H(3,𝔽3)#4112yes—
104C4×H(3,𝔽3)#4112yes—
105#1084—22no—
106#1084—22no—
107(C3⋊Dic3)⋊C3—22no—
108(C3⋊Dic3)⋊C3—22no—
109C3×#127—91no—
110C3×C4×Q8—91no—
111C3×C4×Q8—91no—
112#947—92no—
113#947—92no—
114#936—92no—
115#936—92no—
116#935—94no—
117D4×Dic3—92no—
118D4×Dic3—92no—
119#894—94no—
120#894—94no—
121#893—92no—
122#893—92no—
123#875—92no—
124#875—92no—
125#874—31no—
126#874—91no—
127C4×D12—92no—
128C4×D12—92no—
129C4×Dic6—31no—
130C4×Dic6—31no—
131C4×Dic6—31no—
132C4×Dic6—61no—
133C4×Dic6—91no—
134C22×C3⋊D3C22×D314yes—
135C22×C3×D3—34no—
136C2×C32×C4—21no—
137C2×C32×C4—21no—
138C2×C32×C4—21no—
139C2×C32×C4C22×D311yescharacteristic
140C2×C32×C4C22×D311yescharacteristic
141C2×C32×C4C22×D311yescharacteristic
142C2×C32×C4C22×D311yescharacteristic
143C2×C32×C4C22×D311yescharacteristic
144C2×C32×C4C22×D311yescharacteristic
145C2×C3⋊Dic3C22×D312yes—
146C2×C3⋊Dic3C22×D312yes—
147C4⋊(C3⋊D3)—24no—
148C4⋊(C3⋊D3)—24no—
149C4⋊(C3⋊D3)—42no—
150#634—22no—
151C2×C3×Dic3—32no—
152C2×C3×Dic3—32no—
153C2×C3×Dic3—32no—
154C2×C3×Dic3—32no—
155C2×C3×Dic3—32no—
156C2×C3×Dic3—32no—
157C2×C3×Dic3—62no—
158C2×C3×Dic3—62no—
159C3×D12—64no—
160C3×D12—64no—
161C3×D12—122no—
162C3×Dic6—64no—
163C3×Dic6—62no—
164C3×Dic6—62no—
165C3×Dic6—62no—
166#624—124no—
167D3×Dic3—124no—
168#559—271noSylow
169C2×H(3,𝔽3)C2×#4111yescharacteristic
170C2×H(3,𝔽3)C2×#4111yescharacteristic
171C2×H(3,𝔽3)H(5,𝔽2)11yescharacteristic
172(C3⋊D3)⋊C3—44no—
173C2×C3×Q8—91no—
174C2×C3×Q8—91no—
175C2×C3×D4—92no—
176C2×C3×D4—92no—
177C2×C3×D4—92no—
178C22×Dic3—94no—
179C2×D12—32no—
180C2×D12—32no—
181C2×D12—32no—
182C2×D12—92no—
183C2×D12—92no—
184C2×D12—92no—
185C2×C4×D3—94no—
186C2×C4×D3—94no—
187C2×Dic6—31no—
188C2×Dic6—31no—
189C2×Dic6—31no—
190C2×Dic6—61no—
191C2×Dic6—91no—
192C3×C4⋊C4—91no—
193C3×C4⋊C4—92no—
194C3×C4⋊C4—91no—
195C3×C4⋊C4—92no—
196C3×C42—31no—
197C3×C42—31no—
198C3×C42—61no—
199C3×C42—92no—
200C3×C42—92no—
201C3×C42D3211yescharacteristic
202#217—94no—
203#217—94no—
204#217—94no—
205(C2×D3)⋊C4—94no—
206(C2×D3)⋊C4—94no—
207(C2×D3)⋊C4—94no—
208(C2×D3)⋊C4—94no—
209(C2×D3)⋊C4—94no—
210(C2×D3)⋊C4—94no—
211C4⋊Dic3—31no—
212C4⋊Dic3—31no—
213C4⋊Dic3—31no—
214C4⋊Dic3—61no—
215C4⋊Dic3—91no—
216Dic3⋊C4—32no—
217Dic3⋊C4—32no—
218Dic3⋊C4—32no—
219Dic3⋊C4—62no—
220Dic3⋊C4—92no—
221C4×Dic3—32no—
222C4×Dic3—32no—
223C4×Dic3—32no—
224C4×Dic3—62no—
225C4×Dic3—92no—
226C22×C32—21no—
227C22×C32C23×D311yescharacteristic
228C22×C32C23×D311yescharacteristic
229C2×C3⋊D3—24no—
230C2×C3⋊D3—24no—
231C2×C3⋊D3—24no—
232C2×C3×D3—64no—
233C2×C3×D3—64no—
234C2×C3×D3—64no—
235C32×C4—21no—
236C32×C4—21no—
237C32×C4—22no—
238C32×C4—22no—
239C32×C4—41no—
240C32×C4#23512yes—
241C32×C4#23712yes—
242C32×C4#23712yes—
243C32×C4#23912yes—
244C3⋊Dic3—22no—
245C3⋊Dic3—22no—
246C3×Dic3—64no—
247C3×Dic3—62no—
248C3×Dic3—62no—
249C3×Dic3—62no—
250C3×Dic3—62no—
251C3×Dic3—62no—
252C3×Dic3—62no—
253C3×Dic3—122no—
254#127—91no—
255#124—272no—
256#124—272no—
257#123—274no—
258C4×Q8—91no—
259C4×Q8—91no—
260C4×D4—272no—
261C4×D4—272no—
262H(3,𝔽3)#55911yescharacteristic, complemented, Sylow
263C23×C3—94no—
264C22×D3—34no—
265C22×D3—94no—
266C3×Q8—182no—
267C3×Q8—182no—
268C3×D4—184no—
269C3×D4—184no—
270C3×D4—362no—
271C2×C3×C4—31no—
272C2×C3×C4—31no—
273C2×C3×C4—31no—
274C2×C3×C4—31no—
275C2×C3×C4—31no—
276C2×C3×C4—31no—
277C2×C3×C4—61no—
278C2×C3×C4—61no—
279C2×C3×C4—61no—
280C2×C3×C4—92no—
281C2×C3×C4—92no—
282C2×C3×C4—92no—
283C2×C3×C4—92no—
284C2×C3×C4C2×D3211yescharacteristic
285C2×C3×C4C2×D3211yescharacteristic
286C2×C3×C4C2×D3211yescharacteristic
287C2×Dic3—32no—
288C2×Dic3—32no—
289C2×Dic3—32no—
290C2×Dic3—32no—
291C2×Dic3—32no—
292C2×Dic3—32no—
293C2×Dic3—62no—
294C2×Dic3—62no—
295C2×Dic3—92no—
296C2×Dic3—92no—
297C2×Dic3—364no—
298D12—64no—
299D12—64no—
300D12—122no—
301D12—184no—
302D12—184no—
303D12—362no—
304C4×D3—364no—
305C4×D3—364no—
306Dic6—62no—
307Dic6—62no—
308Dic6—64no—
309Dic6—62no—
310Dic6—182no—
311C2×C32—21no—
312C2×C32—21no—
313C2×C32—21no—
314C2×C32C2×#23511yescharacteristic
315C2×C32C2×#23711yescharacteristic
316C2×C32C2×#23711yescharacteristic
317C2×C32#100411yescharacteristic
318C2×C32C2×#23911yescharacteristic
319C2×C32#100911yescharacteristic
320C3⋊D3—44no—
321C3×D3—124no—
322C2×Q8—91no—
323C2×Q8—91no—
324C2×D4—92no—
325C2×D4—92no—
326C2×D4—92no—
327C22×C4—274no—
328C4⋊C4—91no—
329C4⋊C4—91no—
330C4⋊C4—92no—
331C4⋊C4—92no—
332C22⋊C4—274no—
333C22⋊C4—274no—
334C22⋊C4—274no—
335C42—92no—
336C42—92no—
337C42((C3⋊D3)⋊C3)⋊C211yescharacteristic
338C22×C3—31no—
339C22×C3—31no—
340C22×C3—61no—
341C22×C3—184no—
342C22×C3—184no—
343C22×C3—184no—
344C22×C3C22×D3211yescharacteristic
345C2×D3—64no—
346C2×D3—64no—
347C2×D3—64no—
348C2×D3—184no—
349C2×D3—184no—
350C2×D3—184no—
351C3×C4—21no—
352C3×C4—32no—
353C3×C4—32no—
354C3×C4—32no—
355C3×C4—32no—
356C3×C4—61no—
357C3×C4—61no—
358C3×C4—62no—
359C3×C4—62no—
360C3×C4—121no—
361C3×C4—182no—
362C3×C4—182no—
363C3×C4—182no—
364C3×C4—182no—
365C3×C4#379512yes—
366C3×C4#379612yes—
367Dic3—62no—
368Dic3—62no—
369Dic3—62no—
370Dic3—62no—
371Dic3—64no—
372Dic3—62no—
373Dic3—62no—
374Dic3—122no—
375Dic3—182no—
376Dic3—182no—
377C32—21no—
378C32#558511yescharacteristic, complemented
379C32#571311yescharacteristic, complemented
380C23—94no—
381Q8—182no—
382Q8—182no—
383D4—184no—
384D4—184no—
385D4—362no—
386C2×C4—92no—
387C2×C4—92no—
388C2×C4—92no—
389C2×C4—92no—
390C2×C4—1084no—
391C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
392C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
393C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
394C2×C3—31no—
395C2×C3—31no—
396C2×C3—31no—
397C2×C3—31no—
398C2×C3—31no—
399C2×C3—31no—
400C2×C3—61no—
401C2×C3—61no—
402C2×C3—61no—
403C2×C3—364no—
404C2×C3C2×#379511yescharacteristic
405C2×C3C2×#379611yescharacteristic
406C2×C3#6430711yescharacteristic
407D3—124no—
408D3—364no—
409C22—184no—
410C22—184no—
411C22—184no—
412C22C22×((C3⋊D3)⋊C3)⋊C211yescharacteristic
413C4—21no—
414C4—182no—
415C4—182no—
416C4—182no—
417C4—182no—
418C4#8864812yes—
419C4#8864912yes—
420C3—31no—
421C3—31no—
422C3—61no—
423C3#1059395111yescharacteristic
424C2—364no—
425C2C2×#8864811yescharacteristic
426C2C2×#8864911yescharacteristic
427C2#1172249211yescharacteristic
subgroupG/H
centerC22C22×((C3⋊D3)⋊C3)⋊C2
commutator, cosocleC22×H(3,𝔽3)C24
Frattini, socleC22×C3C22×D32
FittingC42×H(3,𝔽3)C22
prime radical#11721385C2
hypercenterC42((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)#559
LCSG ▷ C22×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C22 ◁ C42

Complements

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

NH ≅ G/Ntype
1. C4×#6229C2semidirect
3. C4×#634⋊C3C2semidirect
4. #11721136C2semidirect
5. #11721132C2semidirect
6. #11721125C2semidirect
7. #11721113C2semidirect
8. #11721110C2semidirect
28. C4×#634D3semidirect
262. H(3,𝔽3)#559semidirect
378. C32#5585semidirect
379. C32#5713semidirect

Maximal subgroups

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

subgroupindex
C4×#62292
#117213852
C4×#634⋊C32
#117211362
#117211322
#117211252
#117211132
#117211102
#105939633
#105939513