Finite Group Databasebeta

← #13140921

Subgroups — #13140921

Order 1728.

441 conjugacy classes of subgroups; only the 77 normal subgroups are shown.

normal subgroups only

This group has 441 conjugacy classes of subgroups — too many to draw. Only its 77 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#64155⋊C3C211yesmaximal, characteristic
2C2×#3756⋊C3C211yesmaximal, characteristic, complemented
3#11721106C211yesmaximal, characteristic
4C2×((C3⋊D3)⋊C3)⋊C8C211yesmaximal, characteristic
5#11720583C211yesmaximal, characteristic, complemented
6#11720313C211yesmaximal, characteristic, complemented
7#11720306C211yesmaximal, characteristic, complemented
8#10591601—31nomaximal
9#10591596—31nomaximal
10C2×Q8×H(3,𝔽3)C2211yescharacteristic
11C2×C4×(C3⋊D3)⋊C3C2211yescharacteristic
12C2×#634⋊C3C2211yescharacteristic
13C4⋊#1084C2211yescharacteristic
14C2×#6180C2211yescharacteristic
15#3756⋊C3—22no—
16#3752⋊C3—21no—
17C4⋊((C3⋊Dic3)⋊C3)C2211yescharacteristic
18#3750⋊C3—21no—
19C2×(C3⋊(C3⋊C8))⋊C3C2211yescharacteristic
20#88450—21no—
21#88449—21no—
22((C3⋊D3)⋊C3)⋊C8—22no—
23#64155D311yescharacteristic
24C2×#3756D311yescharacteristic, complemented
25C3×#946—31no—
26C2×C3×#216—31no—
27#63871—31no—
28#63857—31no—
29C2×#3710—31no—
30C2×D3×C3⋊C8—31no—
31C3×#835—31no—
32C3×#835—31no—
33C3×#835—61no—
34#63576—31no—
35#63575—31no—
36#63567—31no—
37#63566—31no—
38C22×(C3⋊D3)⋊C3D411yescharacteristic
39Q8×H(3,𝔽3)—21no—
40Q8×H(3,𝔽3)—21no—
41C2×C4×H(3,𝔽3)—21no—
42C2×C4×H(3,𝔽3)C2311yescharacteristic
43C2×#1084—21no—
44C2×(C3⋊Dic3)⋊C3—21no—
45C2×(C3⋊Dic3)⋊C3—21no—
46C2×(C3⋊Dic3)⋊C3D411yescharacteristic
47C4×(C3⋊D3)⋊C3—21no—
48C4×(C3⋊D3)⋊C3—21no—
49#634⋊C3—21no—
50#634⋊C3—21no—
51#6180—21no—
52(C3⋊(C3⋊C8))⋊C3D412yes—
53#5188—91no—
54#5051—91no—
55#4813—91no—
56C2×C32×Q8—21no—
57C2×C32×Q8C2×D311yescharacteristic
58C2×C32×Q8C2×D311yescharacteristic
59C2×C4×C3⋊D3C2×D311yescharacteristic
60C2×#634C2×D311yescharacteristic
61C2×C3×C4×D3—31no—
62C2×C3×Dic6—31no—
63#3756—22no—
64#3752—21no—
65C4⋊(C3⋊Dic3)C2×D311yescharacteristic
66#3750—21no—
67C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
68C3×#216—62no—
69C3×(C2×D3)⋊C4—61no—
70C3×C4⋊Dic3—31no—
71C3×C4⋊Dic3—31no—
72C3×C4⋊Dic3—31no—
73C3×C4⋊Dic3—61no—
74C3×Dic3⋊C4—61no—
75C2×C3×C3⋊C8—31no—
76C2×C3×C3⋊C8—31no—
77C2×C3×C3⋊C8—31no—
78C2×C3×C3⋊C8—61no—
79Dic3⋊Dic3—61no—
80Dic3⋊Dic3—61no—
81#3722—61no—
82(C2×D3)⋊Dic3—61no—
83#3710—62no—
84D3×C3⋊C8—62no—
85C22×H(3,𝔽3)C2×D411yescharacteristic
86C2×(C3⋊D3)⋊C3—21no—
87C2×(C3⋊D3)⋊C3—21no—
88C2×(C3⋊D3)⋊C3Q1612yescomplemented
89C4×H(3,𝔽3)—41no—
90C4×H(3,𝔽3)C2×D411yescharacteristic
91C4×H(3,𝔽3)#4111yescharacteristic
92#1084—41no—
93(C3⋊Dic3)⋊C3—21no—
94(C3⋊Dic3)⋊C3—41no—
95(C3⋊Dic3)⋊C3—41no—
96C2×C3×Q16—91no—
97C3×C22⋊Q8—91no—
98#946—31no—
99#946—91no—
100#946—91no—
101C2×#216—31no—
102C2×#216—91no—
103#924—91no—
104C22×C3⋊C8—91no—
105C2×D3×C8—91no—
106C2×D3×C8—91no—
107#897—91no—
108C3×#103—91no—
109#835—31no—
110#835—31no—
111#835—61no—
112#835—91no—
113#818—91no—
114#816—91no—
115#810—91no—
116#807—91no—
117#807—91no—
118C22×C3⋊D3#6711yescharacteristic
119C22×C3×D3—31no—
120C32×Q8—22no—
121C32×Q8—21no—
122C32×Q8—21no—
123C32×Q8—21no—
124C32×Q8—21no—
125C32×Q8—41no—
126C2×C32×C4—21no—
127C2×C32×C4—21no—
128C2×C32×C4—21no—
129C2×C32×C4—41no—
130C2×C32×C4C22×D311yescharacteristic
131C2×C32×C4C22×D311yescharacteristic
132C2×C3⋊Dic3—21no—
133C2×C3⋊Dic3—21no—
134C2×C3⋊Dic3#6711yescharacteristic
135C4×C3⋊D3—21no—
136C4×C3⋊D3—21no—
137#634—21no—
138#634—21no—
139C2×C3×Dic3—31no—
140C2×C3×Dic3—61no—
141C2×C3×Dic3—61no—
142C2×C3×Dic3—61no—
143C2×C3×Dic3—61no—
144C2×C3×Dic3—121no—
145C3×C4×D3—61no—
146C3×C4×D3—61no—
147C3×Dic6—61no—
148C3×Dic6—61no—
149C3⋊(C3⋊C8)#6712yes—
150C3×C3⋊C8—32no—
151C3×C3⋊C8—61no—
152C3×C3⋊C8—61no—
153C3×C3⋊C8—121no—
154#467—271noSylow
155C2×H(3,𝔽3)#12111yescharacteristic
156C2×H(3,𝔽3)C2×Q1611yescharacteristic
157C2×H(3,𝔽3)#13511yescharacteristic
158(C3⋊D3)⋊C3—22no—
159C2×C3×Q8—31no—
160C2×C3×Q8—31no—
161C2×C3×Q8—61no—
162C2×C3×Q8—91no—
163C2×C3×Q8D3211yescharacteristic
164C22×C3×C4—91no—
165C2×C4×D3—31no—
166C2×C4×D3—91no—
167C2×Dic6—31no—
168C2×Dic6—91no—
169C3×Q16—182no—
170C2×C3×C8—91no—
171C2×C3×C8—91no—
172C3×C4⋊C4—91no—
173C3×C4⋊C4—91no—
174C3×C4⋊C4—181no—
175C3×C22⋊C4—181no—
176#217—181no—
177#216—62no—
178#216—182no—
179(C2×D3)⋊C4—61no—
180(C2×D3)⋊C4—181no—
181(C2×D3)⋊C4—181no—
182(C2×D3)⋊C4—181no—
183C4⋊Dic3—31no—
184C4⋊Dic3—31no—
185C4⋊Dic3—31no—
186C4⋊Dic3—61no—
187C4⋊Dic3—91no—
188C4⋊Dic3—181no—
189Dic3⋊C4—61no—
190Dic3⋊C4—181no—
191Dic3⋊C4—181no—
192Dic3⋊C4—181no—
193C2×C3⋊C8—31no—
194C2×C3⋊C8—31no—
195C2×C3⋊C8—31no—
196C2×C3⋊C8—61no—
197C2×C3⋊C8—91no—
198C2×C3⋊C8—182no—
199D3×C8—182no—
200D3×C8—182no—
201C22×C32—21no—
202C22×C32D3×D411yescharacteristic
203C22×C32C2×#6711yescharacteristic
204C2×C3⋊D3—21no—
205C2×C3⋊D3—21no—
206C2×C3⋊D3#21612yescomplemented
207C2×C3×D3—32no—
208C2×C3×D3—61no—
209C2×C3×D3—61no—
210C32×C4—21no—
211C32×C4—21no—
212C32×C4—42no—
213C32×C4—41no—
214C32×C4—41no—
215C32×C4D3×D411yescharacteristic
216C32×C4#23711yescharacteristic
217C32×C4#23711yescharacteristic
218C32×C4C2×#6711yescharacteristic
219C3⋊Dic3—21no—
220C3⋊Dic3—41no—
221C3⋊Dic3—41no—
222C3×Dic3—61no—
223C3×Dic3—121no—
224C3×Dic3—121no—
225C3×Dic3—121no—
226C3×Dic3—121no—
227C3×Dic3—241no—
228C2×Q16—91no—
229C22×C8—271no—
230C22⋊Q8—91no—
231C22⋊Q8—271no—
232#107—271no—
233#103—91no—
234#103—271no—
235H(3,𝔽3)#46711yescharacteristic, complemented, Sylow
236C23×C3—91no—
237C22×D3—31no—
238C22×D3—91no—
239C3×Q8—21no—
240C3×Q8—21no—
241C3×Q8—61no—
242C3×Q8—61no—
243C3×Q8—61no—
244C3×Q8—61no—
245C3×Q8—62no—
246C3×Q8—121no—
247C3×Q8—181no—
248C3×Q8—181no—
249C2×C3×C4—21no—
250C2×C3×C4—31no—
251C2×C3×C4—31no—
252C2×C3×C4—61no—
253C2×C3×C4—61no—
254C2×C3×C4—61no—
255C2×C3×C4—91no—
256C2×C3×C4—121no—
257C2×C3×C4—181no—
258C2×C3×C4—181no—
259C2×C3×C4—181no—
260C2×C3×C4—181no—
261C2×C3×C4—181no—
262C2×C3×C4C2×D3211yescharacteristic
263C2×Dic3—31no—
264C2×Dic3—61no—
265C2×Dic3—61no—
266C2×Dic3—61no—
267C2×Dic3—61no—
268C2×Dic3—91no—
269C2×Dic3—121no—
270C2×Dic3—181no—
271C2×Dic3—181no—
272C4×D3—61no—
273C4×D3—61no—
274C4×D3—181no—
275C4×D3—181no—
276Dic6—61no—
277Dic6—61no—
278Dic6—181no—
279Dic6—181no—
280C3×C8—92no—
281C3×C8—181no—
282C3⋊C8—32no—
283C3⋊C8—61no—
284C3⋊C8—61no—
285C3⋊C8—92no—
286C3⋊C8—121no—
287C2×C32—21no—
288C2×C32—21no—
289C2×C32—21no—
290C2×C32D3×Q1611yescharacteristic
291C2×C32#91911yescharacteristic
292C2×C32#93811yescharacteristic
293C2×C32#93911yescharacteristic
294C2×C32C2×#21611yescharacteristic
295C2×C32#95011yescharacteristic
296C3⋊D3—22no—
297C3×D3—62no—
298C2×Q8—91no—
299C2×Q8((C3⋊D3)⋊C3)⋊C211yescharacteristic
300C22×C4—91no—
301Q16—182no—
302C2×C8—91no—
303C2×C8—91no—
304C2×C8—542no—
305C4⋊C4—91no—
306C4⋊C4—91no—
307C4⋊C4—181no—
308C4⋊C4—541no—
309C22⋊C4—181no—
310C22⋊C4—541no—
311C22×C3—31no—
312C22×C3—31no—
313C22×C3—61no—
314C22×C3—92no—
315C22×C3—181no—
316C22×C3—181no—
317C22×C3D3×#6711yescharacteristic
318C2×D3—32no—
319C2×D3—61no—
320C2×D3—61no—
321C2×D3—92no—
322C2×D3—181no—
323C2×D3—181no—
324C3×C4—31no—
325C3×C4—31no—
326C3×C4—31no—
327C3×C4—31no—
328C3×C4—41no—
329C3×C4—61no—
330C3×C4—61no—
331C3×C4—121no—
332C3×C4—121no—
333C3×C4—122no—
334C3×C4—181no—
335C3×C4—361no—
336C3×C4—361no—
337C3×C4—361no—
338C3×C4#380511yescharacteristic
339C3×C4D3×#6711yescharacteristic
340Dic3—61no—
341Dic3—121no—
342Dic3—121no—
343Dic3—121no—
344Dic3—121no—
345Dic3—181no—
346Dic3—241no—
347Dic3—361no—
348Dic3—361no—
349C32—21no—
350C32#505111yescharacteristic, complemented
351C32#518811yescharacteristic, complemented
352C23—91no—
353Q8—21no—
354Q8—21no—
355Q8—181no—
356Q8—181no—
357C2×C4—21no—
358C2×C4—91no—
359C2×C4—181no—
360C2×C4—181no—
361C2×C4—181no—
362C2×C4—181no—
363C2×C4—181no—
364C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
365C8—92no—
366C8—181no—
367C2×C3—31no—
368C2×C3—31no—
369C2×C3—31no—
370C2×C3—31no—
371C2×C3—31no—
372C2×C3—31no—
373C2×C3—61no—
374C2×C3—61no—
375C2×C3—61no—
376C2×C3—182no—
377C2×C3D3×#21611yescharacteristic
378C2×C3#6394911yescharacteristic
379C2×C3#6397011yescharacteristic
380D3—62no—
381D3—182no—
382C22—92no—
383C22—181no—
384C22—181no—
385C22#8867611yescharacteristic
386C4—41no—
387C4—181no—
388C4—361no—
389C4—361no—
390C4—361no—
391C4#8867211yescharacteristic
392C4#8867611yescharacteristic
393C3—31no—
394C3—31no—
395C3—61no—
396C3#1059163511yescharacteristic
397C2—182no—
398C2#1172123511yescharacteristic
399C2#1172123611yescharacteristic
400C2#1172131311yescharacteristic
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 radicalC22×(C3⋊D3)⋊C3D4
hypercenterC2×Q8((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)#467
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
2. C2×#3756⋊C3C2semidirect
5. #11720583C2semidirect
6. #11720313C2semidirect
7. #11720306C2semidirect
24. C2×#3756D3semidirect
88. C2×(C3⋊D3)⋊C3Q16semidirect
206. C2×C3⋊D3#216semidirect
235. H(3,𝔽3)#467semidirect
350. C32#5051semidirect
351. C32#5188semidirect

Maximal subgroups

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

subgroupindex
#64155⋊C32
C2×#3756⋊C32
#117211062
C2×((C3⋊D3)⋊C3)⋊C82
#117205832
#117203132
#117203062
#105916013
#105915963