Finite Group Databasebeta

← #13147761

Subgroups — #13147761

Order 1728.

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

normal subgroups only

This group has 810 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
1#11721715C211yesmaximal, characteristic, complemented
2C4×C4⋊((C3⋊D3)⋊C3)C211yesmaximal, characteristic
3C4×#634⋊C3C211yesmaximal, characteristic, complemented
4#11721111C214yesmaximal, complemented
5#11721110C212yesmaximal
6#11721106C212yesmaximal, complemented
7#11721083C212yesmaximal, complemented
8#11721081C212yesmaximal, complemented
9#10593865—31nomaximal
10#10593863—31nomaximal
11C2×#6229C2212yes—
12C2×C4⋊((C3⋊D3)⋊C3)C2211yescharacteristic
13C2×C4×(C3⋊D3)⋊C3C2212yes—
14C2×#634⋊C3C2211yescharacteristic
15C42×H(3,𝔽3)C2211yescharacteristic
16C4⋊#1084C2212yes—
17C4⋊#1084C2212yes—
18#3752⋊C3C2212yes—
19C4⋊((C3⋊Dic3)⋊C3)C2211yescharacteristic
20C4⋊((C3⋊Dic3)⋊C3)C2211yescharacteristic
21#3750⋊C3C2212yes—
22C4×(C3⋊Dic3)⋊C3C2212yes—
23#88450C2214yes—
24#88450C2214yes—
25#88449C2214yes—
26#88447C2214yes—
27C4×C4⋊(C3⋊D3)D311yescharacteristic
28C4×#634D311yescharacteristic, complemented
29C3×C4×D12—31no—
30C3×#869—31no—
31C3×#869—31no—
32C3×#869—61no—
33C3×C4×Dic6—31no—
34#63871—32no—
35#63864—32no—
36#63863—32no—
37#63858—34no—
38#63857—32no—
39#63849—32no—
40#63848—34no—
41#63843—32no—
42Dic3×Dic6—32no—
43#63841—32no—
44C22×(C3⋊D3)⋊C3C2312yes—
45C2×C4×H(3,𝔽3)C2311yescharacteristic
46C2×C4×H(3,𝔽3)C2311yescharacteristic
47C2×C4×H(3,𝔽3)C2311yescharacteristic
48C2×#1084C2314yes—
49#6229—24no—
50C2×(C3⋊Dic3)⋊C3C2312yes—
51C2×(C3⋊Dic3)⋊C3C2312yes—
52C2×(C3⋊Dic3)⋊C3C2312yes—
53C4⋊((C3⋊D3)⋊C3)—22no—
54C4×(C3⋊D3)⋊C3—24no—
55#634⋊C3—22no—
56#5732—91no—
57#5584—91no—
58#5542—91no—
59C2×C4⋊(C3⋊D3)C2×D311yescharacteristic
60C2×C4×C3⋊D3C2×D312yes—
61C2×#634C2×D311yescharacteristic
62C2×C3×D12—31no—
63C2×C3×C4×D3—32no—
64C2×C3×Dic6—32no—
65C2×C3×Dic6—32no—
66C2×C3×Dic6—31no—
67C2×C3×Dic6—62no—
68C32×C42—21no—
69C32×C42C2×D311yescharacteristic
70C32×C42C2×D311yescharacteristic
71#3752C2×D312yes—
72C4⋊(C3⋊Dic3)C2×D311yescharacteristic
73C4⋊(C3⋊Dic3)C2×D311yescharacteristic
74#3750C2×D312yes—
75C4×C3⋊Dic3C2×D312yes—
76C3×(C2×D3)⋊C4—32no—
77C3×C4⋊Dic3—32no—
78C3×C4⋊Dic3—31no—
79C3×C4⋊Dic3—32no—
80C3×C4⋊Dic3—31no—
81C3×C4⋊Dic3—32no—
82C3×C4⋊Dic3—32no—
83C3×C4⋊Dic3—62no—
84C3×C4⋊Dic3—62no—
85C3×Dic3⋊C4—32no—
86C3×C4×Dic3—32no—
87Dic3⋊Dic3—34no—
88Dic3⋊Dic3—34no—
89#3722—34no—
90#3722—34no—
91(C2×D3)⋊Dic3—34no—
92(C2×D3)⋊Dic3—34no—
93Dic32—34no—
94Dic32—34no—
95C22×H(3,𝔽3)C2411yescharacteristic
96C2×(C3⋊D3)⋊C3—22no—
97C2×(C3⋊D3)⋊C3—22no—
98C2×(C3⋊D3)⋊C3—22no—
99C4×H(3,𝔽3)—21no—
100C4×H(3,𝔽3)#4112yes—
101C4×H(3,𝔽3)#4112yes—
102#1084—24no—
103(C3⋊Dic3)⋊C3—22no—
104(C3⋊Dic3)⋊C3—22no—
105(C3⋊Dic3)⋊C3—22no—
106C3×C4⋊Q8—91no—
107C3×C4×Q8—91no—
108C3×C4×D4—91no—
109#947—92no—
110#946—92no—
111Q8×Dic3—92no—
112#936—92no—
113#926—92no—
114#924—92no—
115#898—94no—
116#898—94no—
117#897—92no—
118#892—92no—
119#892—92no—
120#887—92no—
121#879—94no—
122#875—92no—
123C4×D12—31no—
124C4×D12—91no—
125#869—31no—
126#869—31no—
127#869—61no—
128C4×Dic6—31no—
129C4×Dic6—91no—
130C4×Dic6—92no—
131C22×C3⋊D3C22×D312yes—
132C22×C3×D3—32no—
133C2×C32×C4—21no—
134C2×C32×C4—21no—
135C2×C32×C4—21no—
136C2×C32×C4C22×D311yescharacteristic
137C2×C32×C4C22×D311yescharacteristic
138C2×C32×C4C22×D311yescharacteristic
139C2×C32×C4C22×D311yescharacteristic
140C2×C32×C4C22×D311yescharacteristic
141C2×C32×C4C22×D311yescharacteristic
142C2×C3⋊Dic3C22×D312yes—
143C2×C3⋊Dic3C22×D312yes—
144C2×C3⋊Dic3C22×D312yes—
145C4⋊(C3⋊D3)—22no—
146C4×C3⋊D3—24no—
147#634—22no—
148C2×C3×Dic3—32no—
149C2×C3×Dic3—32no—
150C2×C3×Dic3—32no—
151C2×C3×Dic3—34no—
152C2×C3×Dic3—34no—
153C2×C3×Dic3—64no—
154C3×D12—62no—
155C3×C4×D3—64no—
156C3×Dic6—64no—
157C3×Dic6—62no—
158C3×Dic6—64no—
159C3×Dic6—124no—
160#556—271noSylow
161C2×H(3,𝔽3)C2×#4111yescharacteristic
162C2×H(3,𝔽3)C2×#4111yescharacteristic
163C2×H(3,𝔽3)2−1+411yescharacteristic
164(C3⋊D3)⋊C3—42no—
165C2×C3×Q8—91no—
166C2×C3×Q8—92no—
167C2×C3×D4—91no—
168C22×C3×C4—92no—
169C2×D12—31no—
170C2×D12—91no—
171C2×C4×D3—32no—
172C2×C4×D3—92no—
173C2×Dic6—32no—
174C2×Dic6—32no—
175C2×Dic6—31no—
176C2×Dic6—62no—
177C2×Dic6—91no—
178C3×C4⋊C4—91no—
179C3×C4⋊C4—91no—
180C3×C4⋊C4—92no—
181C3×C4⋊C4—92no—
182C3×C4⋊C4—92no—
183C3×C22⋊C4—92no—
184C3×C42—31no—
185C3×C42—31no—
186C3×C42—61no—
187C3×C42—92no—
188C3×C42D3211yescharacteristic
189#217—94no—
190#217—94no—
191(C2×D3)⋊C4—32no—
192(C2×D3)⋊C4—92no—
193(C2×D3)⋊C4—94no—
194(C2×D3)⋊C4—94no—
195(C2×D3)⋊C4—94no—
196(C2×D3)⋊C4—94no—
197C4⋊Dic3—32no—
198C4⋊Dic3—32no—
199C4⋊Dic3—32no—
200C4⋊Dic3—32no—
201C4⋊Dic3—31no—
202C4⋊Dic3—31no—
203C4⋊Dic3—62no—
204C4⋊Dic3—62no—
205C4⋊Dic3—91no—
206C4⋊Dic3—91no—
207C4⋊Dic3—94no—
208Dic3⋊C4—32no—
209Dic3⋊C4—92no—
210Dic3⋊C4—94no—
211Dic3⋊C4—94no—
212C4×Dic3—32no—
213C4×Dic3—92no—
214C4×Dic3—94no—
215C4×Dic3—94no—
216C4×Dic3—94no—
217C22×C32—21no—
218C22×C32C23×D311yescharacteristic
219C22×C32C23×D311yescharacteristic
220C2×C3⋊D3—22no—
221C2×C3⋊D3—22no—
222C2×C3⋊D3—22no—
223C2×C3×D3—62no—
224C2×C3×D3—62no—
225C2×C3×D3—62no—
226C32×C4—22no—
227C32×C4—21no—
228C32×C4—21no—
229C32×C4—22no—
230C32×C4—22no—
231C32×C4#23512yes—
232C32×C4#23512yes—
233C32×C4#23712yes—
234C32×C4#23912yes—
235C3⋊Dic3—22no—
236C3⋊Dic3—22no—
237C3⋊Dic3—22no—
238C3×Dic3—62no—
239C3×Dic3—64no—
240C3×Dic3—62no—
241C3×Dic3—64no—
242C3×Dic3—62no—
243C3×Dic3—124no—
244C4⋊Q8—91no—
245#126—274no—
246#124—272no—
247C22⋊Q8—272no—
248C4×Q8—91no—
249C4×Q8—272no—
250C4×D4—91no—
251#117—272no—
252H(3,𝔽3)#55611yescharacteristic, complemented, Sylow
253C23×C3—92no—
254C22×D3—32no—
255C22×D3—92no—
256C3×Q8—182no—
257C3×Q8—184no—
258C3×D4—182no—
259C2×C3×C4—31no—
260C2×C3×C4—31no—
261C2×C3×C4—31no—
262C2×C3×C4—31no—
263C2×C3×C4—31no—
264C2×C3×C4—31no—
265C2×C3×C4—61no—
266C2×C3×C4—61no—
267C2×C3×C4—61no—
268C2×C3×C4—92no—
269C2×C3×C4—92no—
270C2×C3×C4—92no—
271C2×C3×C4—94no—
272C2×C3×C4—184no—
273C2×C3×C4C2×D3211yescharacteristic
274C2×C3×C4C2×D3211yescharacteristic
275C2×C3×C4C2×D3211yescharacteristic
276C2×Dic3—34no—
277C2×Dic3—34no—
278C2×Dic3—32no—
279C2×Dic3—32no—
280C2×Dic3—32no—
281C2×Dic3—64no—
282C2×Dic3—92no—
283C2×Dic3—92no—
284C2×Dic3—92no—
285D12—62no—
286D12—182no—
287C4×D3—64no—
288C4×D3—184no—
289Dic6—64no—
290Dic6—64no—
291Dic6—62no—
292Dic6—124no—
293Dic6—182no—
294C2×C32—21no—
295C2×C32—21no—
296C2×C32—21no—
297C2×C32C2×#23511yescharacteristic
298C2×C32C2×#23511yescharacteristic
299C2×C32C2×#23711yescharacteristic
300C2×C32C2×#23911yescharacteristic
301C2×C32#101011yescharacteristic
302C2×C32#101011yescharacteristic
303C3⋊D3—42no—
304C3×D3—122no—
305C2×Q8—91no—
306C2×Q8—92no—
307C2×D4—91no—
308C22×C4—92no—
309C4⋊C4—91no—
310C4⋊C4—92no—
311C4⋊C4—91no—
312C4⋊C4—92no—
313C4⋊C4—92no—
314C4⋊C4—274no—
315C22⋊C4—92no—
316C22⋊C4—274no—
317C22⋊C4—274no—
318C42—92no—
319C42—274no—
320C42((C3⋊D3)⋊C3)⋊C211yescharacteristic
321C22×C3—31no—
322C22×C3—31no—
323C22×C3—61no—
324C22×C3—182no—
325C22×C3—182no—
326C22×C3—182no—
327C22×C3C22×D3211yescharacteristic
328C2×D3—62no—
329C2×D3—62no—
330C2×D3—62no—
331C2×D3—182no—
332C2×D3—182no—
333C2×D3—182no—
334C3×C4—21no—
335C3×C4—32no—
336C3×C4—32no—
337C3×C4—32no—
338C3×C4—32no—
339C3×C4—61no—
340C3×C4—61no—
341C3×C4—62no—
342C3×C4—62no—
343C3×C4—62no—
344C3×C4—182no—
345C3×C4—182no—
346C3×C4—182no—
347C3×C4—184no—
348C3×C4#379512yes—
349C3×C4#379912yes—
350Dic3—64no—
351Dic3—64no—
352Dic3—62no—
353Dic3—62no—
354Dic3—62no—
355Dic3—124no—
356Dic3—182no—
357Dic3—182no—
358Dic3—182no—
359C32—21no—
360C32#554211yescharacteristic, complemented
361C32#558411yescharacteristic, complemented
362C23—92no—
363Q8—182no—
364Q8—184no—
365D4—182no—
366C2×C4—92no—
367C2×C4—92no—
368C2×C4—92no—
369C2×C4—94no—
370C2×C4—184no—
371C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
372C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
373C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
374C2×C3—31no—
375C2×C3—31no—
376C2×C3—31no—
377C2×C3—31no—
378C2×C3—31no—
379C2×C3—31no—
380C2×C3—61no—
381C2×C3—61no—
382C2×C3—61no—
383C2×C3—362no—
384C2×C3C2×#379511yescharacteristic
385C2×C3#6429811yescharacteristic
386C2×C3C2×#379911yescharacteristic
387D3—122no—
388D3—362no—
389C22—182no—
390C22—182no—
391C22—182no—
392C22C22×((C3⋊D3)⋊C3)⋊C211yescharacteristic
393C4—21no—
394C4—182no—
395C4—182no—
396C4—182no—
397C4—184no—
398C4#8864812yes—
399C4#8865212yes—
400C3—31no—
401C3—31no—
402C3—61no—
403C3#1059387911yescharacteristic
404C2—362no—
405C2C2×#8864811yescharacteristic
406C2#1172248311yescharacteristic
407C2C2×#8865211yescharacteristic
subgroupG/H
centerC22C22×((C3⋊D3)⋊C3)⋊C2
commutator, cosocleC22×H(3,𝔽3)C24
Frattini, socleC22×C3C22×D32
FittingC42×H(3,𝔽3)C22
prime radicalC2×C4⋊((C3⋊D3)⋊C3)C22
hypercenterC42((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)#556
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. #11721715C2semidirect
3. C4×#634⋊C3C2semidirect
4. #11721111C2semidirect
6. #11721106C2semidirect
7. #11721083C2semidirect
8. #11721081C2semidirect
28. C4×#634D3semidirect
252. H(3,𝔽3)#556semidirect
360. C32#5542semidirect
361. C32#5584semidirect

Maximal subgroups

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

subgroupindex
#117217152
C4×C4⋊((C3⋊D3)⋊C3)2
C4×#634⋊C32
#117211112
#117211102
#117211062
#117210832
#117210812
#105938653
#105938633