Finite Group Databasebeta

← #13147751

Subgroups — #13147751

Order 1728.

770 conjugacy classes of subgroups; only the 157 normal subgroups are shown.

normal subgroups only

This group has 770 conjugacy classes of subgroups — too many to draw. Only its 157 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
2#64078⋊C3C211yesmaximal, characteristic
3C4×#634⋊C3C211yesmaximal, characteristic
4#11721099C212yesmaximal
5#11721093C212yesmaximal
6#11721086C214yesmaximal
7#11721084C212yesmaximal
8#11721083C212yesmaximal
9#10593861—31nomaximal
10#10593860—31nomaximal
11C2×#6229C2211yescharacteristic
12C2×#634⋊C3C2212yes—
13C2×#634⋊C3C2211yescharacteristic
14C42×H(3,𝔽3)C2211yescharacteristic
15C4⋊#1084C2211yescharacteristic
16#88540C2212yes—
17C4×#1084C2212yes—
18C4⋊((C3⋊Dic3)⋊C3)C2212yes—
19C4⋊((C3⋊Dic3)⋊C3)C2211yescharacteristic
20C4⋊((C3⋊Dic3)⋊C3)C2212yes—
21#3750⋊C3C2212yes—
22C4×(C3⋊Dic3)⋊C3C2212yes—
23#88449C2214yes—
24#88449C2214yes—
25#88448C2214yes—
26#88447C2214yes—
27#64078D311yescharacteristic
28C4×#634D311yescharacteristic
29C3×#869—31no—
30C3×C4×Dic6—31no—
31C3×C4×Dic6—31no—
32C3×C4×Dic6—31no—
33C3×C4×Dic6—61no—
34#63872—32no—
35#63869—32no—
36#63866—32no—
37#63866—32no—
38#63846—34no—
39#63845—34no—
40#63844—32no—
41#63843—32no—
42Dic3×Dic6—32no—
43Dic3×Dic6—32no—
44C2×C4×H(3,𝔽3)C2311yescharacteristic
45C2×C4×H(3,𝔽3)C2311yescharacteristic
46C2×C4×H(3,𝔽3)C2311yescharacteristic
47C2×#1084C2312yes—
48C2×#1084C2312yes—
49#6229Q814yescomplemented
50C2×(C3⋊Dic3)⋊C3C2314yes—
51C2×(C3⋊Dic3)⋊C3C2312yes—
52C2×(C3⋊Dic3)⋊C3C2312yes—
53#634⋊C3—24no—
54#634⋊C3—22no—
55#5721—91no—
56#5570—91no—
57#5567—91no—
58C2×#634C2×D311yescharacteristic
59C2×#634C2×D312yes—
60C2×C3×Dic6—31no—
61C2×C3×Dic6—31no—
62C2×C3×Dic6—32no—
63C2×C3×Dic6—31no—
64C2×C3×Dic6—61no—
65C32×C42—21no—
66C32×C42C2×D311yescharacteristic
67C32×C42C2×D311yescharacteristic
68C4⋊(C3⋊Dic3)C2×D311yescharacteristic
69C4⋊(C3⋊Dic3)C2×D312yes—
70C4⋊(C3⋊Dic3)C2×D312yes—
71#3750C2×D312yes—
72C4×C3⋊Dic3C2×D312yes—
73C3×C4⋊Dic3—32no—
74C3×C4⋊Dic3—31no—
75C3×C4⋊Dic3—32no—
76C3×C4⋊Dic3—31no—
77C3×C4⋊Dic3—31no—
78C3×C4⋊Dic3—61no—
79C3×Dic3⋊C4—32no—
80C3×Dic3⋊C4—32no—
81C3×Dic3⋊C4—32no—
82C3×Dic3⋊C4—62no—
83C3×C4×Dic3—32no—
84C3×C4×Dic3—32no—
85C3×C4×Dic3—32no—
86C3×C4×Dic3—62no—
87#3724—34no—
88Dic3⋊Dic3—34no—
89Dic3⋊Dic3—34no—
90Dic3⋊Dic3—34no—
91Dic3⋊Dic3—34no—
92Dic3⋊Dic3—34no—
93Dic32—34no—
94Dic32—34no—
95C22×H(3,𝔽3)C2411yescharacteristic
96C4×H(3,𝔽3)—21no—
97C4×H(3,𝔽3)C2×Q812yes—
98C4×H(3,𝔽3)#4112yes—
99#1084—22no—
100#1084C2×Q814yes—
101(C3⋊Dic3)⋊C3—24no—
102(C3⋊Dic3)⋊C3—22no—
103(C3⋊Dic3)⋊C3—22no—
104C3×C4⋊Q8—91no—
105C3×C4×Q8—91no—
106C3×C4×Q8—91no—
107Q8×Dic3—92no—
108Q8×Dic3—92no—
109#944—92no—
110#944—92no—
111#890—92no—
112#890—94no—
113#890—92no—
114#889—94no—
115#889—94no—
116#887—92no—
117#887—92no—
118#870—92no—
119#869—31no—
120#869—91no—
121#869—92no—
122C4×Dic6—31no—
123C4×Dic6—31no—
124C4×Dic6—31no—
125C4×Dic6—61no—
126C4×Dic6—91no—
127C4×Dic6—92no—
128C4×Dic6—92no—
129C2×C32×C4—21no—
130C2×C32×C4—21no—
131C2×C32×C4—21no—
132C2×C32×C4C22×D311yescharacteristic
133C2×C32×C4C22×D311yescharacteristic
134C2×C32×C4C22×D311yescharacteristic
135C2×C32×C4C22×D311yescharacteristic
136C2×C32×C4C22×D311yescharacteristic
137C2×C32×C4C22×D311yescharacteristic
138C2×C3⋊Dic3C22×D312yes—
139C2×C3⋊Dic3C22×D314yes—
140C2×C3⋊Dic3C22×D312yes—
141#634—22no—
142#634—24no—
143C2×C3×Dic3—32no—
144C2×C3×Dic3—32no—
145C2×C3×Dic3—32no—
146C2×C3×Dic3—34no—
147C2×C3×Dic3—32no—
148C2×C3×Dic3—32no—
149C2×C3×Dic3—32no—
150C2×C3×Dic3—62no—
151C2×C3×Dic3—62no—
152C3×Dic6—34no—
153C3×Dic6—34no—
154C3×Dic6—64no—
155C3×Dic6—64no—
156C3×Dic6—62no—
157#557—271noSylow
158C2×H(3,𝔽3)C22×Q811yescharacteristic
159C2×H(3,𝔽3)C2×#4111yescharacteristic
160C2×H(3,𝔽3)2−1+411yescharacteristic
161C2×C3×Q8—91no—
162C2×C3×Q8—92no—
163C2×C3×Q8—91no—
164C2×Dic6—31no—
165C2×Dic6—31no—
166C2×Dic6—31no—
167C2×Dic6—32no—
168C2×Dic6—61no—
169C2×Dic6—91no—
170C2×Dic6—92no—
171C3×C4⋊C4—91no—
172C3×C4⋊C4—92no—
173C3×C4⋊C4—92no—
174C3×C4⋊C4—92no—
175C3×C4⋊C4—92no—
176C3×C4⋊C4—91no—
177C3×C42—31no—
178C3×C42—31no—
179C3×C42—61no—
180C3×C42—92no—
181C3×C42—92no—
182C3×C42D3211yescharacteristic
183C4⋊Dic3—31no—
184C4⋊Dic3—31no—
185C4⋊Dic3—31no—
186C4⋊Dic3—32no—
187C4⋊Dic3—32no—
188C4⋊Dic3—61no—
189C4⋊Dic3—92no—
190C4⋊Dic3—91no—
191C4⋊Dic3—92no—
192C4⋊Dic3—94no—
193C4⋊Dic3—94no—
194C4⋊Dic3—94no—
195Dic3⋊C4—32no—
196Dic3⋊C4—32no—
197Dic3⋊C4—32no—
198Dic3⋊C4—62no—
199Dic3⋊C4—92no—
200Dic3⋊C4—94no—
201Dic3⋊C4—94no—
202Dic3⋊C4—94no—
203Dic3⋊C4—94no—
204Dic3⋊C4—94no—
205Dic3⋊C4—94no—
206C4×Dic3—32no—
207C4×Dic3—32no—
208C4×Dic3—32no—
209C4×Dic3—62no—
210C4×Dic3—92no—
211C4×Dic3—94no—
212C4×Dic3—94no—
213C4×Dic3—94no—
214C22×C32—21no—
215C22×C32C23×D311yescharacteristic
216C22×C32C23×D311yescharacteristic
217C32×C4—21no—
218C32×C4—21no—
219C32×C4—22no—
220C32×C4—22no—
221C32×C4—41no—
222C32×C4#23512yes—
223C32×C4#23712yes—
224C32×C4D3×Q812yes—
225C32×C4D3×Q812yes—
226C3⋊Dic3—22no—
227C3⋊Dic3—24no—
228C3⋊Dic3—22no—
229C3×Dic3—34no—
230C3×Dic3—34no—
231C3×Dic3—64no—
232C3×Dic3—64no—
233C3×Dic3—62no—
234C3×Dic3—62no—
235C3×Dic3—62no—
236C3×Dic3—62no—
237C3×Dic3—122no—
238C4⋊Q8—91no—
239C4⋊Q8—272no—
240#125—272no—
241#125—274no—
242C4×Q8—91no—
243C4×Q8—91no—
244C4×Q8—272no—
245C4×Q8—272no—
246H(3,𝔽3)#55711yescharacteristic, complemented, Sylow
247C3×Q8—94no—
248C3×Q8—182no—
249C3×Q8—184no—
250C2×C3×C4—31no—
251C2×C3×C4—31no—
252C2×C3×C4—31no—
253C2×C3×C4—31no—
254C2×C3×C4—31no—
255C2×C3×C4—31no—
256C2×C3×C4—61no—
257C2×C3×C4—61no—
258C2×C3×C4—61no—
259C2×C3×C4—92no—
260C2×C3×C4—92no—
261C2×C3×C4—94no—
262C2×C3×C4—92no—
263C2×C3×C4—92no—
264C2×C3×C4C2×D3211yescharacteristic
265C2×C3×C4C2×D3211yescharacteristic
266C2×C3×C4C2×D3211yescharacteristic
267C2×Dic3—32no—
268C2×Dic3—32no—
269C2×Dic3—32no—
270C2×Dic3—32no—
271C2×Dic3—32no—
272C2×Dic3—32no—
273C2×Dic3—34no—
274C2×Dic3—62no—
275C2×Dic3—62no—
276C2×Dic3—92no—
277C2×Dic3—92no—
278C2×Dic3—94no—
279Dic6—34no—
280Dic6—34no—
281Dic6—64no—
282Dic6—64no—
283Dic6—62no—
284Dic6—184no—
285Dic6—182no—
286C2×C32—21no—
287C2×C32—21no—
288C2×C32—21no—
289C2×C32C2×#23511yescharacteristic
290C2×C32C2×#23711yescharacteristic
291C2×C32C2×D3×Q811yescharacteristic
292C2×C32C2×D3×Q811yescharacteristic
293C2×C32#100711yescharacteristic
294C2×C32#101011yescharacteristic
295C2×Q8—91no—
296C2×Q8—91no—
297C2×Q8—92no—
298C4⋊C4—91no—
299C4⋊C4—92no—
300C4⋊C4—92no—
301C4⋊C4—91no—
302C4⋊C4—92no—
303C4⋊C4—92no—
304C4⋊C4—274no—
305C4⋊C4—274no—
306C4⋊C4—274no—
307C42—92no—
308C42—92no—
309C42—274no—
310C42((C3⋊D3)⋊C3)⋊C211yescharacteristic
311C22×C3—31no—
312C22×C3—31no—
313C22×C3—61no—
314C22×C3C22×D3211yescharacteristic
315C3×C4—21no—
316C3×C4—32no—
317C3×C4—32no—
318C3×C4—32no—
319C3×C4—32no—
320C3×C4—61no—
321C3×C4—61no—
322C3×C4—62no—
323C3×C4—62no—
324C3×C4—94no—
325C3×C4—121no—
326C3×C4—182no—
327C3×C4—184no—
328C3×C4—182no—
329C3×C4—182no—
330C3×C4#379512yes—
331C3×C4#379712yes—
332Dic3—34no—
333Dic3—34no—
334Dic3—64no—
335Dic3—62no—
336Dic3—62no—
337Dic3—62no—
338Dic3—64no—
339Dic3—62no—
340Dic3—122no—
341Dic3—184no—
342Dic3—182no—
343Dic3—182no—
344C32—21no—
345C32#556711yescharacteristic, complemented
346C32#572111yescharacteristic, complemented
347Q8—94no—
348Q8—182no—
349Q8—184no—
350C2×C4—92no—
351C2×C4—92no—
352C2×C4—92no—
353C2×C4—92no—
354C2×C4—94no—
355C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
356C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
357C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
358C2×C3—31no—
359C2×C3—31no—
360C2×C3—31no—
361C2×C3—31no—
362C2×C3—31no—
363C2×C3—31no—
364C2×C3—61no—
365C2×C3—61no—
366C2×C3—61no—
367C2×C3C2×#379511yescharacteristic
368C2×C3#6429711yescharacteristic
369C2×C3C2×#379711yescharacteristic
370C22C22×((C3⋊D3)⋊C3)⋊C211yescharacteristic
371C4—21no—
372C4—94no—
373C4—182no—
374C4—184no—
375C4—182no—
376C4—182no—
377C4#8864812yes—
378C4#8865112yes—
379C3—31no—
380C3—31no—
381C3—61no—
382C3#1059386411yescharacteristic
383C2C2×#8864811yescharacteristic
384C2#1172248211yescharacteristic
385C2C2×#8865111yescharacteristic
subgroupG/H
centerC22C22×((C3⋊D3)⋊C3)⋊C2
commutator, prime radical, cosocleC22×H(3,𝔽3)C24
Frattini, socleC22×C3C22×D32
FittingC42×H(3,𝔽3)C22
hypercenterC42((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)#557
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
49. #6229Q8semidirect
246. H(3,𝔽3)#557semidirect
345. C32#5567semidirect
346. C32#5721semidirect

Maximal subgroups

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

subgroupindex
C4×#62292
#64078⋊C32
C4×#634⋊C32
#117210992
#117210932
#117210862
#117210842
#117210832
#105938613
#105938603