Finite Group Databasebeta

← #13138034

Subgroups — #13138034

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#64106⋊C3C211yesmaximal, characteristic
2#11721107C211yesmaximal, characteristic
3C2×#88439C211yesmaximal, characteristic, complemented
4C2×((C3⋊D3)⋊C3)⋊C8C211yesmaximal, characteristic
5#11720532C211yesmaximal, characteristic, complemented
6#63634⋊C3C211yesmaximal, characteristic, complemented
7#11720313C211yesmaximal, characteristic, complemented
8#10590525—31nomaximal
9#10590518—31nomaximal
10C2×#6229C2211yescharacteristic
11C2×C4×(C3⋊D3)⋊C3C2211yescharacteristic
12C2×#634⋊C3C2211yescharacteristic
13C4⋊C4×H(3,𝔽3)C2211yescharacteristic
14C2×#6180C2211yescharacteristic
15#3752⋊C3—21no—
16C4⋊((C3⋊Dic3)⋊C3)—21no—
17C4⋊((C3⋊Dic3)⋊C3)C2211yescharacteristic
18C2×(C3⋊(C3⋊C8))⋊C3C2211yescharacteristic
19#88450—21no—
20#88448—21no—
21#88439—22no—
22((C3⋊D3)⋊C3)⋊C8—22no—
23#64106D311yescharacteristic
24C3×#897—31no—
25#63871—31no—
26#63856—31no—
27C2×#3719—31no—
28C2×#3718—31no—
29C2×#3710—31no—
30C2×D3×C3⋊C8—31no—
31#63634D311yescharacteristic, complemented
32C3×#810—31no—
33C3×#810—31no—
34C3×#810—61no—
35C3×#807—31no—
36#63567—31no—
37#63566—31no—
38C22×(C3⋊D3)⋊C3D411yescharacteristic
39C2×C4×H(3,𝔽3)—21no—
40C2×C4×H(3,𝔽3)C2311yescharacteristic
41C2×#1084—21no—
42#6229—21no—
43#6229—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#5051—91no—
54#4887—91no—
55#4813—91no—
56C2×C4×C3⋊D3C2×D311yescharacteristic
57C2×#634C2×D311yescharacteristic
58C2×C3×C4×D3—31no—
59C2×C3×Dic6—31no—
60C2×C3×Dic6—31no—
61C2×C3×Dic6—31no—
62C2×C3×Dic6—61no—
63C32×C4⋊C4—21no—
64C32×C4⋊C4C2×D311yescharacteristic
65C32×C4⋊C4C2×D311yescharacteristic
66#3752—21no—
67C4⋊(C3⋊Dic3)—21no—
68C4⋊(C3⋊Dic3)C2×D311yescharacteristic
69C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
70C3×(C2×D3)⋊C4—61no—
71C3×C4⋊Dic3—31no—
72C3×C4⋊Dic3—61no—
73C2×C3×C3⋊C8—31no—
74C2×C3×C3⋊C8—31no—
75C2×C3×C3⋊C8—31no—
76C2×C3×C3⋊C8—61no—
77#3724—61no—
78Dic3⋊Dic3—61no—
79#3722—61no—
80(C2×D3)⋊Dic3—61no—
81#3719—62no—
82#3718—62no—
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—
96C3×C22⋊Q8—91no—
97#946—91no—
98C2×#216—91no—
99C2×#216—91no—
100#924—91no—
101C22×C3⋊C8—91no—
102C2×Dic12—91no—
103C2×D3×C8—91no—
104C2×D3×C8—91no—
105#897—31no—
106#897—91no—
107#897—91no—
108C3×#107—91no—
109C3×#103—91no—
110#835—91no—
111#816—91no—
112#810—31no—
113#810—31no—
114#810—61no—
115#810—91no—
116#807—31no—
117#807—91no—
118C22×C3⋊D3#6711yescharacteristic
119C22×C3×D3—31no—
120C2×C32×C4—21no—
121C2×C32×C4—21no—
122C2×C32×C4—21no—
123C2×C32×C4—41no—
124C2×C32×C4C22×D311yescharacteristic
125C2×C32×C4C22×D311yescharacteristic
126C2×C3⋊Dic3—21no—
127C2×C3⋊Dic3—21no—
128C2×C3⋊Dic3#6711yescharacteristic
129C4×C3⋊D3—21no—
130C4×C3⋊D3—21no—
131#634—21no—
132#634—21no—
133C2×C3×Dic3—31no—
134C2×C3×Dic3—61no—
135C2×C3×Dic3—61no—
136C2×C3×Dic3—61no—
137C2×C3×Dic3—61no—
138C2×C3×Dic3—121no—
139C3×C4×D3—61no—
140C3×C4×D3—61no—
141C3×Dic6—62no—
142C3×Dic6—61no—
143C3×Dic6—61no—
144C3×Dic6—61no—
145C3×Dic6—61no—
146C3×Dic6—61no—
147C3×Dic6—61no—
148C3×Dic6—121no—
149C3⋊(C3⋊C8)D1212yes—
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—91no—
160C2×C3×Q8—91no—
161C22×C3×C4—91no—
162C2×C4×D3—31no—
163C2×C4×D3—91no—
164C2×Dic6—31no—
165C2×Dic6—31no—
166C2×Dic6—31no—
167C2×Dic6—61no—
168C2×Dic6—91no—
169C2×C3×C8—91no—
170C2×C3×C8—91no—
171C3×C4⋊C4—31no—
172C3×C4⋊C4—31no—
173C3×C4⋊C4—61no—
174C3×C4⋊C4—91no—
175C3×C4⋊C4—181no—
176C3×C4⋊C4D3211yescharacteristic
177C3×C22⋊C4—181no—
178#217—181no—
179#216—182no—
180#216—182no—
181(C2×D3)⋊C4—61no—
182(C2×D3)⋊C4—181no—
183(C2×D3)⋊C4—181no—
184(C2×D3)⋊C4—181no—
185C4⋊Dic3—31no—
186C4⋊Dic3—61no—
187C4⋊Dic3—91no—
188C4⋊Dic3—181no—
189C4⋊Dic3—181no—
190Dic3⋊C4—181no—
191Dic3⋊C4—181no—
192C2×C3⋊C8—31no—
193C2×C3⋊C8—31no—
194C2×C3⋊C8—31no—
195C2×C3⋊C8—61no—
196C2×C3⋊C8—91no—
197C2×C3⋊C8—182no—
198Dic12—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—41no—
213C32×C4—41no—
214C32×C4—81no—
215C32×C4C2×D1211yescharacteristic
216C32×C4D3×D411yescharacteristic
217C32×C4#23711yescharacteristic
218C32×C4#23911yescharacteristic
219C3⋊Dic3—21no—
220C3⋊Dic3—41no—
221C3⋊Dic3—41no—
222C3×Dic3—61no—
223C3×Dic3—122no—
224C3×Dic3—121no—
225C3×Dic3—121no—
226C3×Dic3—121no—
227C3×Dic3—121no—
228C2×Q16—271no—
229C22×C8—271no—
230C22⋊Q8—91no—
231C22⋊Q8—271no—
232#107—91no—
233#103—91no—
234#103—271no—
235H(3,𝔽3)#46711yescharacteristic, complemented, Sylow
236C23×C3—91no—
237C22×D3—31no—
238C22×D3—91no—
239C3×Q8—181no—
240C3×Q8—181no—
241C3×Q8—181no—
242C3×Q8—181no—
243C2×C3×C4—21no—
244C2×C3×C4—31no—
245C2×C3×C4—31no—
246C2×C3×C4—61no—
247C2×C3×C4—61no—
248C2×C3×C4—61no—
249C2×C3×C4—91no—
250C2×C3×C4—121no—
251C2×C3×C4—181no—
252C2×C3×C4—181no—
253C2×C3×C4—181no—
254C2×C3×C4—181no—
255C2×C3×C4—181no—
256C2×C3×C4C2×D3211yescharacteristic
257C2×Dic3—31no—
258C2×Dic3—61no—
259C2×Dic3—61no—
260C2×Dic3—61no—
261C2×Dic3—61no—
262C2×Dic3—91no—
263C2×Dic3—121no—
264C2×Dic3—181no—
265C2×Dic3—181no—
266C4×D3—61no—
267C4×D3—61no—
268C4×D3—181no—
269C4×D3—181no—
270Dic6—61no—
271Dic6—61no—
272Dic6—61no—
273Dic6—61no—
274Dic6—62no—
275Dic6—61no—
276Dic6—61no—
277Dic6—121no—
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×C32#89511yescharacteristic
291C2×C32#91211yescharacteristic
292C2×C32D3×Q1611yescharacteristic
293C2×C32#93911yescharacteristic
294C2×C32C2×#21611yescharacteristic
295C2×C32#95011yescharacteristic
296C3⋊D3—22no—
297C3×D3—62no—
298C2×Q8—91no—
299C2×Q8—91no—
300C22×C4—91no—
301Q16—542no—
302C2×C8—91no—
303C2×C8—91no—
304C2×C8—542no—
305C4⋊C4—91no—
306C4⋊C4—181no—
307C4⋊C4—541no—
308C4⋊C4((C3⋊D3)⋊C3)⋊C211yescharacteristic
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—181no—
334C3×C4—241no—
335C3×C4—361no—
336C3×C4—361no—
337C3×C4—361no—
338C3×C4#379611yescharacteristic
339C3×C4D3×D1211yescharacteristic
340Dic3—61no—
341Dic3—121no—
342Dic3—121no—
343Dic3—122no—
344Dic3—121no—
345Dic3—121no—
346Dic3—181no—
347Dic3—361no—
348Dic3—361no—
349C32—21no—
350C32#488711yescharacteristic, complemented
351C32#505111yescharacteristic, complemented
352C23—91no—
353Q8—181no—
354Q8—181no—
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×C3#6388611yescharacteristic
378C2×C3#6393311yescharacteristic
379C2×C3D3×#21611yescharacteristic
380D3—62no—
381D3—182no—
382C22—92no—
383C22—181no—
384C22—181no—
385C22#8867611yescharacteristic
386C4—41no—
387C4—181no—
388C4—361no—
389C4—361no—
390C4—361no—
391C4#8864911yescharacteristic
392C4#8865411yescharacteristic
393C3—31no—
394C3—31no—
395C3—61no—
396C3#1059057911yescharacteristic
397C2—182no—
398C2#1172112711yescharacteristic
399C2#1172120111yescharacteristic
400C2#1172123611yescharacteristic
subgroupG/H
centerC22#88676
commutator, cosocleC2×C4×H(3,𝔽3)C23
FrattiniC2×C3×C4C2×D32
FittingC4⋊C4×H(3,𝔽3)C22
socleC22×C3D3×#67
prime radicalC22×(C3⋊D3)⋊C3D4
hypercenterC4⋊C4((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 ◁ C4⋊C4

Complements

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

NH ≅ G/Ntype
3. C2×#88439C2semidirect
5. #11720532C2semidirect
6. #63634⋊C3C2semidirect
7. #11720313C2semidirect
31. #63634D3semidirect
88. C2×(C3⋊D3)⋊C3Q16semidirect
206. C2×C3⋊D3#216semidirect
235. H(3,𝔽3)#467semidirect
350. C32#4887semidirect
351. C32#5051semidirect

Maximal subgroups

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

subgroupindex
#64106⋊C32
#117211072
C2×#884392
C2×((C3⋊D3)⋊C3)⋊C82
#117205322
#63634⋊C32
#117203132
#105905253
#105905183