Finite Group Databasebeta

← #13140974

Subgroups — #13140974

Order 1728.

446 conjugacy classes of subgroups; only the 73 normal subgroups are shown.

normal subgroups only

This group has 446 conjugacy classes of subgroups — too many to draw. Only its 73 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#11721861C211yesmaximal, characteristic
2C2×#3756⋊C3C211yesmaximal, characteristic
3#11721084C211yesmaximal, characteristic
4C2×#88439C211yesmaximal, characteristic
5#63662⋊C3C211yesmaximal, characteristic
6#11720326C211yesmaximal, characteristic
7#11720304C211yesmaximal, characteristic
8#10591575—31nomaximal
9#10591574—31nomaximal
10C2×Q8×H(3,𝔽3)C2211yescharacteristic
11C2×#6229C2211yescharacteristic
12C2×#634⋊C3C2211yescharacteristic
13#88540—21no—
14#88540—21no—
15C4×#1084C2211yescharacteristic
16#3756⋊C3—22no—
17C4⋊((C3⋊Dic3)⋊C3)C2211yescharacteristic
18C2×(C3⋊(C3⋊C8))⋊C3C2211yescharacteristic
19C2×(C3⋊(C3⋊C8))⋊C3C2211yescharacteristic
20#88448—21no—
21#88447—21no—
22#88439—22no—
23C2×#3756D311yescharacteristic
24C3×#944—31no—
25C3×#944—31no—
26C3×#944—61no—
27C2×C3×#216—31no—
28#63844—31no—
29Dic3×Dic6—31no—
30C2×#3719—31no—
31C2×#3718—31no—
32#63662D311yescharacteristic
33C3×#835—31no—
34Dic3⋊(C3⋊C8)—31no—
35Dic3⋊(C3⋊C8)—31no—
36#63566—31no—
37#63565—31no—
38Q8×H(3,𝔽3)—21no—
39Q8×H(3,𝔽3)—21no—
40C2×C4×H(3,𝔽3)—21no—
41C2×C4×H(3,𝔽3)C2311yescharacteristic
42C2×#1084—21no—
43C2×#1084D411yescharacteristic
44C2×#1084D411yescharacteristic
45#6229—21no—
46#6229—21no—
47C2×(C3⋊Dic3)⋊C3—21no—
48C2×(C3⋊Dic3)⋊C3—21no—
49#634⋊C3—21no—
50#634⋊C3D412yes—
51(C3⋊(C3⋊C8))⋊C3—21no—
52(C3⋊(C3⋊C8))⋊C3—21no—
53#5182—91no—
54#5090—91no—
55#4795—91no—
56C2×C32×Q8—21no—
57C2×C32×Q8C2×D311yescharacteristic
58C2×C32×Q8C2×D311yescharacteristic
59C2×#634C2×D311yescharacteristic
60C2×C3×Dic6—31no—
61C2×C3×Dic6—31no—
62C2×C3×Dic6—31no—
63C2×C3×Dic6—61no—
64#3756—22no—
65C4⋊(C3⋊Dic3)C2×D311yescharacteristic
66C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
67C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
68C3×#216—62no—
69C3×C4⋊Dic3—31no—
70C3×Dic3⋊C4—61no—
71C3×Dic3⋊C4—61no—
72C3×Dic3⋊C4—61no—
73C3×Dic3⋊C4—61no—
74C3×Dic3⋊C4—62no—
75C3×Dic3⋊C4—62no—
76C3×C4×Dic3—31no—
77C3×C4×Dic3—31no—
78C3×C4×Dic3—61no—
79C2×C3×C3⋊C8—31no—
80C2×C3×C3⋊C8—31no—
81#3724—61no—
82Dic3⋊Dic3—61no—
83Dic32—61no—
84Dic32—61no—
85#3719—62no—
86#3718—62no—
87C22×H(3,𝔽3)C2×D411yescharacteristic
88C4×H(3,𝔽3)—41no—
89C4×H(3,𝔽3)C2×D411yescharacteristic
90C4×H(3,𝔽3)#4111yescharacteristic
91#1084—21no—
92#1084—41no—
93#1084Q1612yescomplemented
94(C3⋊Dic3)⋊C3—22no—
95(C3⋊Dic3)⋊C3—41no—
96C2×C3×Q16—91no—
97C3×C4⋊Q8—91no—
98Q8×Dic3—91no—
99#944—31no—
100#944—31no—
101#944—61no—
102C2×#216—31no—
103C2×#216—91no—
104C2×#216—91no—
105C2×#216—91no—
106C2×Dic12—91no—
107#887—91no—
108C4×Dic6—91no—
109C3×#103—91no—
110#835—31no—
111#835—91no—
112#835—91no—
113#816—91no—
114Dic3⋊C8—91no—
115Dic3⋊C8—91no—
116#810—91no—
117C4⋊(C3⋊C8)—91no—
118C32×Q8—21no—
119C32×Q8—21no—
120C32×Q8—21no—
121C32×Q8—21no—
122C32×Q8—41no—
123C32×Q8—41no—
124C2×C32×C4—22no—
125C2×C32×C4—21no—
126C2×C32×C4—21no—
127C2×C32×C4—21no—
128C2×C32×C4C22×D311yescharacteristic
129C2×C32×C4C22×D311yescharacteristic
130C2×C3⋊Dic3—21no—
131C2×C3⋊Dic3—21no—
132#634—21no—
133#634#6712yes—
134C2×C3×Dic3—31no—
135C2×C3×Dic3—31no—
136C2×C3×Dic3—31no—
137C2×C3×Dic3—31no—
138C2×C3×Dic3—61no—
139C2×C3×Dic3—61no—
140C2×C3×Dic3—61no—
141C2×C3×Dic3—62no—
142C2×C3×Dic3—61no—
143C2×C3×Dic3—61no—
144C2×C3×Dic3—61no—
145C3×Dic6—32no—
146C3×Dic6—61no—
147C3×Dic6—61no—
148C3×Dic6—61no—
149C3×Dic6—61no—
150C3×Dic6—61no—
151C3×Dic6—121no—
152C3×Dic6—121no—
153C3⋊(C3⋊C8)—21no—
154C3⋊(C3⋊C8)—21no—
155C3×C3⋊C8—61no—
156C3×C3⋊C8—61no—
157#462—271noSylow
158C2×H(3,𝔽3)#12111yescharacteristic
159C2×H(3,𝔽3)C2×Q1611yescharacteristic
160C2×H(3,𝔽3)#13711yescharacteristic
161C2×C3×Q8—31no—
162C2×C3×Q8—31no—
163C2×C3×Q8—61no—
164C2×C3×Q8—91no—
165C2×C3×Q8—91no—
166C2×C3×Q8D3211yescharacteristic
167C2×Dic6—31no—
168C2×Dic6—31no—
169C2×Dic6—31no—
170C2×Dic6—61no—
171C2×Dic6—91no—
172C3×Q16—182no—
173C2×C3×C8—91no—
174C2×C3×C8—91no—
175C3×C4⋊C4—91no—
176C3×C4⋊C4—181no—
177C3×C4⋊C4—181no—
178C3×C42—91no—
179#216—62no—
180#216—182no—
181#216—182no—
182#216—182no—
183C4⋊Dic3—31no—
184C4⋊Dic3—91no—
185C4⋊Dic3—181no—
186Dic3⋊C4—61no—
187Dic3⋊C4—61no—
188Dic3⋊C4—61no—
189Dic3⋊C4—61no—
190Dic3⋊C4—62no—
191Dic3⋊C4—62no—
192Dic3⋊C4—181no—
193Dic3⋊C4—181no—
194C4×Dic3—31no—
195C4×Dic3—31no—
196C4×Dic3—61no—
197C4×Dic3—181no—
198C4×Dic3—181no—
199C4×Dic3—181no—
200C2×C3⋊C8—31no—
201C2×C3⋊C8—31no—
202C2×C3⋊C8—91no—
203C2×C3⋊C8—91no—
204Dic12—182no—
205C22×C32—21no—
206C22×C32D3×D411yescharacteristic
207C22×C32D3×D411yescharacteristic
208C32×C4—21no—
209C32×C4—21no—
210C32×C4—42no—
211C32×C4—41no—
212C32×C4—41no—
213C32×C4#23511yescharacteristic
214C32×C4D3×D411yescharacteristic
215C32×C4#23711yescharacteristic
216C32×C4C2×#6711yescharacteristic
217C3⋊Dic3—22no—
218C3⋊Dic3—41no—
219C3×Dic3—32no—
220C3×Dic3—32no—
221C3×Dic3—62no—
222C3×Dic3—62no—
223C3×Dic3—62no—
224C3×Dic3—61no—
225C3×Dic3—61no—
226C3×Dic3—122no—
227C3×Dic3—121no—
228C3×Dic3—121no—
229C3×Dic3—121no—
230C2×Q16—91no—
231C2×Q16—271no—
232C4⋊Q8—91no—
233C4×Q8—271no—
234C4⋊C8—271no—
235#103—91no—
236#103—271no—
237H(3,𝔽3)#46211yescharacteristic, complemented, Sylow
238C3×Q8—21no—
239C3×Q8—21no—
240C3×Q8—61no—
241C3×Q8—61no—
242C3×Q8—61no—
243C3×Q8—61no—
244C3×Q8—92no—
245C3×Q8—121no—
246C3×Q8—121no—
247C3×Q8—181no—
248C3×Q8—181no—
249C3×Q8—181no—
250C2×C3×C4—21no—
251C2×C3×C4—31no—
252C2×C3×C4—31no—
253C2×C3×C4—61no—
254C2×C3×C4—61no—
255C2×C3×C4—62no—
256C2×C3×C4—61no—
257C2×C3×C4—91no—
258C2×C3×C4—91no—
259C2×C3×C4—181no—
260C2×C3×C4—181no—
261C2×C3×C4—181no—
262C2×C3×C4C2×D3211yescharacteristic
263C2×Dic3—31no—
264C2×Dic3—31no—
265C2×Dic3—31no—
266C2×Dic3—31no—
267C2×Dic3—61no—
268C2×Dic3—61no—
269C2×Dic3—61no—
270C2×Dic3—61no—
271C2×Dic3—61no—
272C2×Dic3—62no—
273C2×Dic3—61no—
274C2×Dic3—181no—
275C2×Dic3—181no—
276Dic6—32no—
277Dic6—61no—
278Dic6—61no—
279Dic6—61no—
280Dic6—61no—
281Dic6—61no—
282Dic6—92no—
283Dic6—121no—
284Dic6—121no—
285Dic6—181no—
286C3×C8—181no—
287C3×C8—181no—
288C3⋊C8—61no—
289C3⋊C8—61no—
290C3⋊C8—181no—
291C3⋊C8—181no—
292C2×C32—21no—
293C2×C32—21no—
294C2×C32—21no—
295C2×C32#88411yescharacteristic
296C2×C32#91511yescharacteristic
297C2×C32D3×Q1611yescharacteristic
298C2×C32D3×Q1611yescharacteristic
299C2×C32#91811yescharacteristic
300C2×C32#93911yescharacteristic
301C2×Q8—91no—
302C2×Q8—91no—
303C2×Q8((C3⋊D3)⋊C3)⋊C211yescharacteristic
304Q16—182no—
305Q16—542no—
306C2×C8—91no—
307C2×C8—91no—
308C4⋊C4—91no—
309C4⋊C4—181no—
310C4⋊C4—181no—
311C4⋊C4—541no—
312C42—91no—
313C42—541no—
314C22×C3—31no—
315C22×C3—31no—
316C22×C3—61no—
317C22×C3#381111yescharacteristic
318C3×C4—31no—
319C3×C4—31no—
320C3×C4—31no—
321C3×C4—31no—
322C3×C4—41no—
323C3×C4—61no—
324C3×C4—61no—
325C3×C4—92no—
326C3×C4—121no—
327C3×C4—121no—
328C3×C4—122no—
329C3×C4—182no—
330C3×C4—181no—
331C3×C4—361no—
332C3×C4—361no—
333C3×C4#380411yescharacteristic
334C3×C4D3×#6711yescharacteristic
335Dic3—32no—
336Dic3—32no—
337Dic3—62no—
338Dic3—61no—
339Dic3—61no—
340Dic3—62no—
341Dic3—62no—
342Dic3—121no—
343Dic3—121no—
344Dic3—122no—
345Dic3—121no—
346Dic3—182no—
347Dic3—361no—
348C32—21no—
349C32#479511yescharacteristic, complemented
350C32#518211yescharacteristic, complemented
351Q8—21no—
352Q8—21no—
353Q8—92no—
354Q8—181no—
355Q8—181no—
356Q8—181no—
357C2×C4—21no—
358C2×C4—91no—
359C2×C4—91no—
360C2×C4—181no—
361C2×C4—181no—
362C2×C4—181no—
363C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
364C8—181no—
365C8—181no—
366C2×C3—31no—
367C2×C3—31no—
368C2×C3—31no—
369C2×C3—31no—
370C2×C3—31no—
371C2×C3—31no—
372C2×C3—61no—
373C2×C3—61no—
374C2×C3—61no—
375C2×C3#6394411yescharacteristic
376C2×C3#6395111yescharacteristic
377C2×C3#6397111yescharacteristic
378C22#8867711yescharacteristic
379C4—41no—
380C4—92no—
381C4—181no—
382C4—182no—
383C4—361no—
384C4—361no—
385C4#8867111yescharacteristic
386C4#8867611yescharacteristic
387C3—31no—
388C3—31no—
389C3—61no—
390C3#1059158011yescharacteristic
391C2#1172122611yescharacteristic
392C2#1172123111yescharacteristic
393C2#1172131211yescharacteristic
subgroupG/H
centerC22#88677
commutator, cosocleC2×C4×H(3,𝔽3)C23
FrattiniC2×C3×C4C2×D32
FittingC2×Q8×H(3,𝔽3)C22
socleC22×C3#3811
prime radicalC22×H(3,𝔽3)C2×D4
hypercenterC2×Q8((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)#462
LCSG ▷ C2×C4×H(3,𝔽3) ▷ C2×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C22 ◁ C2×C4 ◁ C2×Q8

Complements

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

NH ≅ G/Ntype
93. #1084Q16semidirect
237. H(3,𝔽3)#462semidirect
349. C32#4795semidirect
350. C32#5182semidirect

Maximal subgroups

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

subgroupindex
#117218612
C2×#3756⋊C32
#117210842
C2×#884392
#63662⋊C32
#117203262
#117203042
#105915753
#105915743