Finite Group Databasebeta

← #13140987

Subgroups — #13140987

Order 1728.

452 conjugacy classes of subgroups; only the 79 normal subgroups are shown.

normal subgroups only

This group has 452 conjugacy classes of subgroups — too many to draw. Only its 79 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
1C2×#88550C211yesmaximal, characteristic, complemented
2Q8×(C3⋊Dic3)⋊C3C211yesmaximal, characteristic, complemented
3C2×#3755⋊C3C211yesmaximal, characteristic
4#11721110C211yesmaximal, characteristic
5#11720326C211yesmaximal, characteristic, complemented
6#11720318C211yesmaximal, characteristic
7#11720311C211yesmaximal, characteristic, complemented
8#10591612—31nomaximal
9#10591607—31nomaximal
10C2×Q8×H(3,𝔽3)C2211yescharacteristic
11C2×#6229C2211yescharacteristic
12C2×C4⋊((C3⋊D3)⋊C3)C2211yescharacteristic
13#88550—22no—
14C2×#6180C2211yescharacteristic
15#3755⋊C3—22no—
16C4⋊((C3⋊Dic3)⋊C3)—21no—
17C4⋊((C3⋊Dic3)⋊C3)C2211yescharacteristic
18C4×(C3⋊Dic3)⋊C3—21no—
19C4×(C3⋊Dic3)⋊C3C2211yescharacteristic
20C2×(C3⋊(C3⋊C8))⋊C3C2211yescharacteristic
21#88450—21no—
22#88450—21no—
23Q8×C3⋊Dic3D311yescharacteristic, complemented
24C2×#3755D311yescharacteristic
25C3×Q8×Dic3—31no—
26C2×C3×#216—31no—
27C2×C3×#216—31no—
28C2×C3×#216—61no—
29C2×C3×#215—31no—
30#63864—31no—
31#63863—31no—
32#63572—31no—
33Dic3⋊(C3⋊C8)—31no—
34#63566—31no—
35#63565—31no—
36#63564—31no—
37#63563—31no—
38C22×(C3⋊D3)⋊C3—21no—
39Q8×H(3,𝔽3)—21no—
40Q8×H(3,𝔽3)D412yes—
41C2×C4×H(3,𝔽3)—21no—
42C2×C4×H(3,𝔽3)C2311yescharacteristic
43C2×#1084—21no—
44#6229—21no—
45#6229—21no—
46C2×(C3⋊Dic3)⋊C3—21no—
47C2×(C3⋊Dic3)⋊C3D411yescharacteristic
48C2×(C3⋊Dic3)⋊C3D411yescharacteristic
49C4⋊((C3⋊D3)⋊C3)—21no—
50C4⋊((C3⋊D3)⋊C3)—21no—
51#6180—21no—
52(C3⋊(C3⋊C8))⋊C3—21no—
53#5185—91no—
54#5166—91no—
55#5028—91no—
56C2×C32×Q8—21no—
57C2×C32×Q8C2×D311yescharacteristic
58C2×C32×Q8C2×D311yescharacteristic
59C2×C4⋊(C3⋊D3)C2×D311yescharacteristic
60C2×C3×D12—31no—
61C2×C3×Dic6—31no—
62C2×C3×Dic6—31no—
63C2×C3×Dic6—61no—
64#3755—22no—
65C4⋊(C3⋊Dic3)—21no—
66C4⋊(C3⋊Dic3)C2×D311yescharacteristic
67C4×C3⋊Dic3—21no—
68C4×C3⋊Dic3C2×D311yescharacteristic
69C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
70C3×#216—64no—
71C3×#216—62no—
72C3×#216—62no—
73C3×#215—62no—
74C3×C4⋊Dic3—31no—
75C3×C4⋊Dic3—61no—
76C3×C4×Dic3—31no—
77C3×C4×Dic3—61no—
78C2×C3×C3⋊C8—31no—
79C2×C3×C3⋊C8—31no—
80C2×C3×C3⋊C8—31no—
81C2×C3×C3⋊C8—61no—
82#3722—61no—
83#3722—61no—
84(C2×D3)⋊Dic3—61no—
85(C2×D3)⋊Dic3—61no—
86C22×H(3,𝔽3)C2×D411yescharacteristic
87C2×(C3⋊D3)⋊C3—41no—
88C2×(C3⋊D3)⋊C3—41no—
89C2×(C3⋊D3)⋊C3—41no—
90C4×H(3,𝔽3)—22no—
91C4×H(3,𝔽3)C2×D411yescharacteristic
92C4×H(3,𝔽3)#4111yescharacteristic
93#1084—41no—
94(C3⋊Dic3)⋊C3—21no—
95(C3⋊Dic3)⋊C3—21no—
96(C3⋊Dic3)⋊C3—41no—
97C2×C3×Q16—91no—
98C2×C3×SD16—91no—
99C3×C4×Q8—91no—
100#947—91no—
101Q8×Dic3—31no—
102Q8×Dic3—91no—
103C2×#216—31no—
104C2×#216—31no—
105C2×#216—61no—
106C2×#215—31no—
107C2×#215—91no—
108#936—91no—
109#875—91no—
110#835—91no—
111#832—91no—
112#821—91no—
113#816—91no—
114Dic3⋊C8—91no—
115Dic3⋊C8—91no—
116#810—91no—
117#809—91no—
118C4⋊(C3⋊C8)—91no—
119C22×C3⋊D3—21no—
120C22×C3×D3—61no—
121C32×Q8—21no—
122C32×Q8—21no—
123C32×Q8—22no—
124C32×Q8—41no—
125C32×Q8D1212yes—
126C32×Q8#6712yes—
127C2×C32×C4—21no—
128C2×C32×C4—21no—
129C2×C32×C4—21no—
130C2×C32×C4—41no—
131C2×C32×C4C22×D311yescharacteristic
132C2×C32×C4C22×D311yescharacteristic
133C2×C3⋊Dic3—21no—
134C2×C3⋊Dic3#6711yescharacteristic
135C2×C3⋊Dic3#6711yescharacteristic
136C4⋊(C3⋊D3)—21no—
137C4⋊(C3⋊D3)—21no—
138C2×C3×Dic3—31no—
139C2×C3×Dic3—31no—
140C2×C3×Dic3—61no—
141C2×C3×Dic3—61no—
142C2×C3×Dic3—61no—
143C2×C3×Dic3—121no—
144C3×D12—61no—
145C3×D12—61no—
146C3×Dic6—62no—
147C3×Dic6—61no—
148C3×Dic6—61no—
149C3×Dic6—61no—
150C3×Dic6—61no—
151C3×Dic6—121no—
152C3⋊(C3⋊C8)—21no—
153C3×C3⋊C8—62no—
154C3×C3⋊C8—61no—
155C3×C3⋊C8—61no—
156C3×C3⋊C8—61no—
157#464—271noSylow
158C2×H(3,𝔽3)#12111yescharacteristic
159C2×H(3,𝔽3)#13511yescharacteristic
160C2×H(3,𝔽3)#13711yescharacteristic
161(C3⋊D3)⋊C3—81no—
162C2×C3×Q8—31no—
163C2×C3×Q8—31no—
164C2×C3×Q8—61no—
165C2×C3×Q8—91no—
166C2×C3×Q8D3211yescharacteristic
167C2×C3×D4—91no—
168C2×D12—31no—
169C2×D12—91no—
170C2×Dic6—31no—
171C2×Dic6—31no—
172C2×Dic6—61no—
173C3×Q16—182no—
174C3×SD16—182no—
175C2×C3×C8—91no—
176C2×C3×C8—91no—
177C3×C4⋊C4—91no—
178C3×C4⋊C4—181no—
179C3×C42—91no—
180C3×C42—181no—
181#217—181no—
182#217—181no—
183#216—62no—
184#216—62no—
185#216—64no—
186#215—62no—
187#215—182no—
188(C2×D3)⋊C4—181no—
189(C2×D3)⋊C4—181no—
190(C2×D3)⋊C4—181no—
191(C2×D3)⋊C4—181no—
192C4⋊Dic3—31no—
193C4⋊Dic3—61no—
194C4⋊Dic3—91no—
195C4⋊Dic3—181no—
196C4×Dic3—31no—
197C4×Dic3—61no—
198C4×Dic3—91no—
199C4×Dic3—181no—
200C2×C3⋊C8—31no—
201C2×C3⋊C8—31no—
202C2×C3⋊C8—31no—
203C2×C3⋊C8—61no—
204C2×C3⋊C8—91no—
205C22×C32—21no—
206C22×C32D3×D411yescharacteristic
207C22×C32C2×#6711yescharacteristic
208C2×C3⋊D3—41no—
209C2×C3⋊D3—41no—
210C2×C3⋊D3—41no—
211C2×C3×D3—121no—
212C2×C3×D3—121no—
213C2×C3×D3—121no—
214C32×C4—22no—
215C32×C4—22no—
216C32×C4—21no—
217C32×C4—21no—
218C32×C4—42no—
219C32×C4C2×D1211yescharacteristic
220C32×C4#23511yescharacteristic
221C32×C4#23711yescharacteristic
222C32×C4C2×#6711yescharacteristic
223C3⋊Dic3—21no—
224C3⋊Dic3—21no—
225C3⋊Dic3—41no—
226C3×Dic3—61no—
227C3×Dic3—61no—
228C3×Dic3—122no—
229C3×Dic3—121no—
230C3×Dic3—121no—
231C3×Dic3—121no—
232C2×Q16—91no—
233C2×SD16—91no—
234#124—271no—
235C4×Q8—91no—
236C4⋊C8—271no—
237#103—271no—
238#102—271no—
239H(3,𝔽3)#46411yescharacteristic, complemented, Sylow
240C23×C3—181no—
241C22×D3—61no—
242C22×D3—181no—
243C3×Q8—21no—
244C3×Q8—32no—
245C3×Q8—32no—
246C3×Q8—61no—
247C3×Q8—61no—
248C3×Q8—62no—
249C3×Q8—121no—
250C3×Q8—181no—
251C3×Q8—181no—
252C3×Q8#62612yes—
253C3×D4—181no—
254C3×D4—181no—
255C2×C3×C4—21no—
256C2×C3×C4—31no—
257C2×C3×C4—31no—
258C2×C3×C4—61no—
259C2×C3×C4—61no—
260C2×C3×C4—61no—
261C2×C3×C4—91no—
262C2×C3×C4—91no—
263C2×C3×C4—121no—
264C2×C3×C4—181no—
265C2×C3×C4—181no—
266C2×C3×C4C2×D3211yescharacteristic
267C2×Dic3—31no—
268C2×Dic3—31no—
269C2×Dic3—61no—
270C2×Dic3—61no—
271C2×Dic3—61no—
272C2×Dic3—91no—
273C2×Dic3—91no—
274C2×Dic3—121no—
275C2×Dic3—181no—
276D12—61no—
277D12—61no—
278D12—181no—
279D12—181no—
280Dic6—61no—
281Dic6—61no—
282Dic6—61no—
283Dic6—61no—
284Dic6—62no—
285Dic6—121no—
286C3×C8—181no—
287C3×C8—181no—
288C3⋊C8—61no—
289C3⋊C8—61no—
290C3⋊C8—62no—
291C3⋊C8—61no—
292C3⋊C8—181no—
293C2×C32—21no—
294C2×C32—21no—
295C2×C32—21no—
296C2×C32#91511yescharacteristic
297C2×C32#91611yescharacteristic
298C2×C32#93011yescharacteristic
299C2×C32#93911yescharacteristic
300C2×C32#94211yescharacteristic
301C2×C32#95011yescharacteristic
302C3⋊D3—81no—
303C3×D3—241no—
304C2×Q8—91no—
305C2×Q8((C3⋊D3)⋊C3)⋊C211yescharacteristic
306C2×D4—91no—
307Q16—182no—
308SD16—182no—
309C2×C8—91no—
310C2×C8—91no—
311C4⋊C4—91no—
312C4⋊C4—181no—
313C22⋊C4—541no—
314C22⋊C4—541no—
315C42—91no—
316C42—181no—
317C22×C3—31no—
318C22×C3—31no—
319C22×C3—61no—
320C22×C3—361no—
321C22×C3—361no—
322C22×C3—361no—
323C22×C3D3×#6711yescharacteristic
324C2×D3—121no—
325C2×D3—121no—
326C2×D3—121no—
327C2×D3—361no—
328C2×D3—361no—
329C2×D3—361no—
330C3×C4—22no—
331C3×C4—31no—
332C3×C4—31no—
333C3×C4—31no—
334C3×C4—31no—
335C3×C4—62no—
336C3×C4—62no—
337C3×C4—61no—
338C3×C4—61no—
339C3×C4—122no—
340C3×C4—181no—
341C3×C4—181no—
342C3×C4—361no—
343C3×C4—361no—
344C3×C4#380411yescharacteristic
345C3×C4C2×#62611yescharacteristic
346Dic3—61no—
347Dic3—61no—
348Dic3—121no—
349Dic3—121no—
350Dic3—122no—
351Dic3—121no—
352Dic3—181no—
353Dic3—181no—
354Dic3—361no—
355C32—21no—
356C32#502811yescharacteristic, complemented
357C32#516611yescharacteristic, complemented
358C23—181no—
359Q8—21no—
360Q8—181no—
361Q8—181no—
362Q8#620012yes—
363D4—181no—
364D4—181no—
365C2×C4—21no—
366C2×C4—91no—
367C2×C4—91no—
368C2×C4—181no—
369C2×C4—181no—
370C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
371C8—181no—
372C8—181no—
373C2×C3—31no—
374C2×C3—31no—
375C2×C3—31no—
376C2×C3—31no—
377C2×C3—31no—
378C2×C3—31no—
379C2×C3—61no—
380C2×C3—61no—
381C2×C3—61no—
382C2×C3—721no—
383C2×C3#6394611yescharacteristic
384C2×C3#6394711yescharacteristic
385C2×C3#6397811yescharacteristic
386D3—241no—
387D3—721no—
388C22—361no—
389C22—361no—
390C22—361no—
391C22#8867611yescharacteristic
392C4—22no—
393C4—181no—
394C4—181no—
395C4—361no—
396C4—361no—
397C4#8867111yescharacteristic
398C4C2×#620011yescharacteristic
399C3—31no—
400C3—31no—
401C3—61no—
402C3#1059160411yescharacteristic
403C2—721no—
404C2#1172123811yescharacteristic
405C2#1172123911yescharacteristic
406C2#1172132211yescharacteristic
subgroupG/H
centerC22#88676
commutator, cosocleC2×C4×H(3,𝔽3)C23
FrattiniC2×C3×C4C2×D32
FittingC2×Q8×H(3,𝔽3)C22
socleC22×C3D3×#67
prime radicalC2×C4⋊((C3⋊D3)⋊C3)C22
hypercenterC2×Q8((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)#464
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
1. C2×#88550C2semidirect
2. Q8×(C3⋊Dic3)⋊C3C2semidirect
5. #11720326C2semidirect
7. #11720311C2semidirect
23. Q8×C3⋊Dic3D3semidirect
239. H(3,𝔽3)#464semidirect
356. C32#5028semidirect
357. C32#5166semidirect

Maximal subgroups

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

subgroupindex
C2×#885502
Q8×(C3⋊Dic3)⋊C32
C2×#3755⋊C32
#117211102
#117203262
#117203182
#117203112
#105916123
#105916073