Finite Group Databasebeta

← #13147758

Subgroups — #13147758

Order 1728.

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

normal subgroups only

This group has 812 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
2C4×C4⋊((C3⋊D3)⋊C3)C211yesmaximal, characteristic
3#64078⋊C3C211yesmaximal, characteristic, complemented
4#11721113C212yesmaximal
5#11721108C214yesmaximal, complemented
6#11721107C212yesmaximal, complemented
7#11721084C212yesmaximal, complemented
8#11721081C212yesmaximal, complemented
9#10593879—31nomaximal
10#10593865—31nomaximal
11C2×#6229C2211yescharacteristic
12C2×C4⋊((C3⋊D3)⋊C3)C2211yescharacteristic
13C2×C4×(C3⋊D3)⋊C3C2212yes—
14C2×#634⋊C3C2212yes—
15C42×H(3,𝔽3)C2211yescharacteristic
16C4⋊#1084C2211yescharacteristic
17#88540C2212yes—
18C4×#1084C2212yes—
19#3752⋊C3C2212yes—
20C4⋊((C3⋊Dic3)⋊C3)C2212yes—
21C4⋊((C3⋊Dic3)⋊C3)C2211yescharacteristic
22C4⋊((C3⋊Dic3)⋊C3)C2212yes—
23#88450C2214yes—
24#88450C2214yes—
25#88448C2214yes—
26#88447C2214yes—
27C4×C4⋊(C3⋊D3)D311yescharacteristic
28#64078D311yescharacteristic, complemented
29C3×C4×D12—31no—
30C3×#869—31no—
31C3×C4×Dic6—31no—
32C3×C4×Dic6—31no—
33C3×C4×Dic6—61no—
34#63871—32no—
35#63864—32no—
36#63861—34no—
37#63860—32no—
38#63856—32no—
39#63849—32no—
40#63848—34no—
41#63844—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×#1084C2312yes—
49C2×#1084C2312yes—
50#6229—22no—
51C2×(C3⋊Dic3)⋊C3C2314yes—
52C2×(C3⋊Dic3)⋊C3C2312yes—
53C4⋊((C3⋊D3)⋊C3)—22no—
54C4×(C3⋊D3)⋊C3—24no—
55#634⋊C3—24no—
56#5732—91no—
57#5584—91no—
58#5542—91no—
59C2×C4⋊(C3⋊D3)C2×D311yescharacteristic
60C2×C4×C3⋊D3C2×D312yes—
61C2×#634C2×D312yes—
62C2×C3×D12—31no—
63C2×C3×C4×D3—32no—
64C2×C3×Dic6—32no—
65C2×C3×Dic6—31no—
66C2×C3×Dic6—31no—
67C2×C3×Dic6—61no—
68C32×C42—21no—
69C32×C42C2×D311yescharacteristic
70C32×C42C2×D311yescharacteristic
71#3752C2×D312yes—
72C4⋊(C3⋊Dic3)C2×D312yes—
73C4⋊(C3⋊Dic3)C2×D312yes—
74C4⋊(C3⋊Dic3)C2×D311yescharacteristic
75C3×(C2×D3)⋊C4—32no—
76C3×C4⋊Dic3—31no—
77C3×C4⋊Dic3—31no—
78C3×C4⋊Dic3—32no—
79C3×C4⋊Dic3—31no—
80C3×C4⋊Dic3—32no—
81C3×C4⋊Dic3—61no—
82C3×Dic3⋊C4—32no—
83C3×Dic3⋊C4—32no—
84C3×Dic3⋊C4—62no—
85C3×C4×Dic3—32no—
86C3×C4×Dic3—32no—
87C3×C4×Dic3—62no—
88#3724—34no—
89Dic3⋊Dic3—34no—
90#3722—34no—
91#3722—34no—
92(C2×D3)⋊Dic3—34no—
93(C2×D3)⋊Dic3—34no—
94Dic32—34no—
95Dic32—34no—
96C22×H(3,𝔽3)C2411yescharacteristic
97C2×(C3⋊D3)⋊C3—22no—
98C2×(C3⋊D3)⋊C3—22no—
99C2×(C3⋊D3)⋊C3—22no—
100C4×H(3,𝔽3)—21no—
101C4×H(3,𝔽3)#4112yes—
102C4×H(3,𝔽3)#4112yes—
103#1084—22no—
104#1084—22no—
105(C3⋊Dic3)⋊C3—24no—
106(C3⋊Dic3)⋊C3—22no—
107C3×C4⋊Q8—91no—
108C3×C4×Q8—91no—
109C3×C4×D4—91no—
110#947—92no—
111#946—92no—
112Q8×Dic3—92no—
113#936—92no—
114#926—92no—
115#924—92no—
116#898—94no—
117#898—94no—
118#897—92no—
119#892—92no—
120#892—92no—
121#887—92no—
122#879—94no—
123#875—92no—
124C4×D12—31no—
125C4×D12—91no—
126#869—31no—
127#869—91no—
128C4×Dic6—31no—
129C4×Dic6—31no—
130C4×Dic6—61no—
131C4×Dic6—92no—
132C22×C3⋊D3C22×D312yes—
133C22×C3×D3—32no—
134C2×C32×C4—21no—
135C2×C32×C4—21no—
136C2×C32×C4—21no—
137C2×C32×C4C22×D311yescharacteristic
138C2×C32×C4C22×D311yescharacteristic
139C2×C32×C4C22×D311yescharacteristic
140C2×C32×C4C22×D311yescharacteristic
141C2×C32×C4C22×D311yescharacteristic
142C2×C32×C4C22×D311yescharacteristic
143C2×C3⋊Dic3C22×D312yes—
144C2×C3⋊Dic3C22×D314yes—
145C4⋊(C3⋊D3)—22no—
146C4×C3⋊D3—24no—
147#634—24no—
148C2×C3×Dic3—32no—
149C2×C3×Dic3—34no—
150C2×C3×Dic3—32no—
151C2×C3×Dic3—32no—
152C2×C3×Dic3—32no—
153C2×C3×Dic3—32no—
154C2×C3×Dic3—62no—
155C2×C3×Dic3—62no—
156C3×D12—62no—
157C3×C4×D3—64no—
158C3×Dic6—64no—
159C3×Dic6—64no—
160C3×Dic6—62no—
161C3×Dic6—62no—
162#556—271noSylow
163C2×H(3,𝔽3)C2×#4111yescharacteristic
164C2×H(3,𝔽3)C2×#4111yescharacteristic
165C2×H(3,𝔽3)2−1+411yescharacteristic
166(C3⋊D3)⋊C3—42no—
167C2×C3×Q8—92no—
168C2×C3×Q8—91no—
169C2×C3×D4—91no—
170C22×C3×C4—92no—
171C2×D12—31no—
172C2×D12—91no—
173C2×C4×D3—32no—
174C2×C4×D3—92no—
175C2×Dic6—31no—
176C2×Dic6—31no—
177C2×Dic6—32no—
178C2×Dic6—61no—
179C2×Dic6—92no—
180C3×C4⋊C4—91no—
181C3×C4⋊C4—92no—
182C3×C4⋊C4—92no—
183C3×C4⋊C4—91no—
184C3×C4⋊C4—92no—
185C3×C22⋊C4—92no—
186C3×C42—31no—
187C3×C42—31no—
188C3×C42—61no—
189C3×C42—92no—
190C3×C42D3211yescharacteristic
191#217—94no—
192#217—94no—
193(C2×D3)⋊C4—32no—
194(C2×D3)⋊C4—92no—
195(C2×D3)⋊C4—94no—
196(C2×D3)⋊C4—94no—
197(C2×D3)⋊C4—94no—
198(C2×D3)⋊C4—94no—
199C4⋊Dic3—31no—
200C4⋊Dic3—31no—
201C4⋊Dic3—31no—
202C4⋊Dic3—32no—
203C4⋊Dic3—32no—
204C4⋊Dic3—61no—
205C4⋊Dic3—92no—
206C4⋊Dic3—92no—
207C4⋊Dic3—91no—
208C4⋊Dic3—94no—
209Dic3⋊C4—32no—
210Dic3⋊C4—32no—
211Dic3⋊C4—62no—
212Dic3⋊C4—94no—
213Dic3⋊C4—94no—
214C4×Dic3—32no—
215C4×Dic3—32no—
216C4×Dic3—62no—
217C4×Dic3—94no—
218C4×Dic3—94no—
219C4×Dic3—94no—
220C22×C32—21no—
221C22×C32C23×D311yescharacteristic
222C22×C32C23×D311yescharacteristic
223C2×C3⋊D3—22no—
224C2×C3⋊D3—22no—
225C2×C3⋊D3—22no—
226C2×C3×D3—62no—
227C2×C3×D3—62no—
228C2×C3×D3—62no—
229C32×C4—21no—
230C32×C4—21no—
231C32×C4—22no—
232C32×C4—22no—
233C32×C4—41no—
234C32×C4#23512yes—
235C32×C4#23712yes—
236C32×C4#23712yes—
237C32×C4#23912yes—
238C3⋊Dic3—24no—
239C3⋊Dic3—22no—
240C3×Dic3—64no—
241C3×Dic3—64no—
242C3×Dic3—62no—
243C3×Dic3—62no—
244C3×Dic3—62no—
245C3×Dic3—62no—
246C3×Dic3—62no—
247C3×Dic3—122no—
248C4⋊Q8—91no—
249#126—274no—
250#124—272no—
251C22⋊Q8—272no—
252C4×Q8—91no—
253C4×Q8—272no—
254C4×D4—91no—
255#117—272no—
256H(3,𝔽3)#55611yescharacteristic, complemented, Sylow
257C23×C3—92no—
258C22×D3—32no—
259C22×D3—92no—
260C3×Q8—184no—
261C3×Q8—182no—
262C3×D4—182no—
263C2×C3×C4—31no—
264C2×C3×C4—31no—
265C2×C3×C4—31no—
266C2×C3×C4—31no—
267C2×C3×C4—31no—
268C2×C3×C4—31no—
269C2×C3×C4—61no—
270C2×C3×C4—61no—
271C2×C3×C4—61no—
272C2×C3×C4—92no—
273C2×C3×C4—94no—
274C2×C3×C4—92no—
275C2×C3×C4—92no—
276C2×C3×C4—184no—
277C2×C3×C4C2×D3211yescharacteristic
278C2×C3×C4C2×D3211yescharacteristic
279C2×C3×C4C2×D3211yescharacteristic
280C2×Dic3—32no—
281C2×Dic3—32no—
282C2×Dic3—32no—
283C2×Dic3—32no—
284C2×Dic3—32no—
285C2×Dic3—34no—
286C2×Dic3—62no—
287C2×Dic3—62no—
288C2×Dic3—92no—
289C2×Dic3—94no—
290D12—62no—
291D12—182no—
292C4×D3—64no—
293C4×D3—184no—
294Dic6—62no—
295Dic6—62no—
296Dic6—64no—
297Dic6—64no—
298Dic6—184no—
299C2×C32—21no—
300C2×C32—21no—
301C2×C32—21no—
302C2×C32C2×#23511yescharacteristic
303C2×C32C2×#23711yescharacteristic
304C2×C32C2×#23711yescharacteristic
305C2×C32C2×#23911yescharacteristic
306C2×C32#100711yescharacteristic
307C2×C32#101011yescharacteristic
308C3⋊D3—42no—
309C3×D3—122no—
310C2×Q8—91no—
311C2×Q8—92no—
312C2×D4—91no—
313C22×C4—92no—
314C4⋊C4—91no—
315C4⋊C4—92no—
316C4⋊C4—91no—
317C4⋊C4—92no—
318C4⋊C4—92no—
319C4⋊C4—274no—
320C22⋊C4—92no—
321C22⋊C4—274no—
322C22⋊C4—274no—
323C42—92no—
324C42—274no—
325C42((C3⋊D3)⋊C3)⋊C211yescharacteristic
326C22×C3—31no—
327C22×C3—31no—
328C22×C3—61no—
329C22×C3—182no—
330C22×C3—182no—
331C22×C3—182no—
332C22×C3C22×D3211yescharacteristic
333C2×D3—62no—
334C2×D3—62no—
335C2×D3—62no—
336C2×D3—182no—
337C2×D3—182no—
338C2×D3—182no—
339C3×C4—21no—
340C3×C4—32no—
341C3×C4—32no—
342C3×C4—32no—
343C3×C4—32no—
344C3×C4—61no—
345C3×C4—61no—
346C3×C4—62no—
347C3×C4—62no—
348C3×C4—121no—
349C3×C4—182no—
350C3×C4—184no—
351C3×C4—182no—
352C3×C4—182no—
353C3×C4#379512yes—
354C3×C4#379612yes—
355Dic3—62no—
356Dic3—62no—
357Dic3—62no—
358Dic3—62no—
359Dic3—64no—
360Dic3—64no—
361Dic3—62no—
362Dic3—122no—
363Dic3—184no—
364Dic3—182no—
365C32—21no—
366C32#554211yescharacteristic, complemented
367C32#573211yescharacteristic, complemented
368C23—92no—
369Q8—184no—
370Q8—182no—
371D4—182no—
372C2×C4—92no—
373C2×C4—92no—
374C2×C4—92no—
375C2×C4—94no—
376C2×C4—184no—
377C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
378C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
379C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
380C2×C3—31no—
381C2×C3—31no—
382C2×C3—31no—
383C2×C3—31no—
384C2×C3—31no—
385C2×C3—31no—
386C2×C3—61no—
387C2×C3—61no—
388C2×C3—61no—
389C2×C3—362no—
390C2×C3C2×#379511yescharacteristic
391C2×C3C2×#379611yescharacteristic
392C2×C3#6429711yescharacteristic
393D3—122no—
394D3—362no—
395C22—182no—
396C22—182no—
397C22—182no—
398C22C22×((C3⋊D3)⋊C3)⋊C211yescharacteristic
399C4—21no—
400C4—184no—
401C4—182no—
402C4—182no—
403C4—182no—
404C4#8864812yes—
405C4#8864912yes—
406C3—31no—
407C3—31no—
408C3—61no—
409C3#1059386311yescharacteristic
410C2—362no—
411C2C2×#8864811yescharacteristic
412C2C2×#8864911yescharacteristic
413C2#1172248211yescharacteristic
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. C4×#6229C2semidirect
3. #64078⋊C3C2semidirect
5. #11721108C2semidirect
6. #11721107C2semidirect
7. #11721084C2semidirect
8. #11721081C2semidirect
28. #64078D3semidirect
256. H(3,𝔽3)#556semidirect
366. C32#5542semidirect
367. C32#5732semidirect

Maximal subgroups

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

subgroupindex
C4×#62292
C4×C4⋊((C3⋊D3)⋊C3)2
#64078⋊C32
#117211132
#117211082
#117211072
#117210842
#117210812
#105938793
#105938653