Finite Group Databasebeta

← #13140583

Subgroups — #13140583

Order 1728.

465 conjugacy classes of subgroups; only the 67 normal subgroups are shown.

normal subgroups only

This group has 465 conjugacy classes of subgroups — too many to draw. Only its 67 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#11721848C211yesmaximal, characteristic
2#11721089C211yesmaximal, characteristic, complemented
3#63659⋊C3C211yesmaximal, characteristic, complemented
4#63659⋊C3C211yesmaximal, characteristic, complemented
5#11720323C211yesmaximal, characteristic, complemented
6#11720307C211yesmaximal, characteristic, complemented
7#11720306C211yesmaximal, characteristic, complemented
8#10591423—31nomaximal
9#10591422—31nomaximal
10C2×D4×H(3,𝔽3)C2211yescharacteristic
11C2×#6235—21no—
12C2×C4×#272C2211yescharacteristic
13#88552—21no—
14C4⋊#1084C2211yescharacteristic
15C4⋊((C3⋊Dic3)⋊C3)C2211yescharacteristic
16C4⋊((C3⋊Dic3)⋊C3)C2211yescharacteristic
17C2×(C3⋊(C3⋊C8))⋊C3C2211yescharacteristic
18C2×(C3⋊(C3⋊C8))⋊C3C2211yescharacteristic
19#88451—21no—
20#88447—21no—
21C3×#938—31no—
22C3×#938—31no—
23C3×#938—61no—
24#63841—31no—
25#63841—31no—
26#63659D311yescharacteristic, complemented
27#63659D311yescharacteristic, complemented
28C3×#832—31no—
29C3×#832—31no—
30#63576—31no—
31#63575—31no—
32#63574—31no—
33#63573—31no—
34(C2×D3)⋊(C3⋊C8)—31no—
35(C2×D3)⋊(C3⋊C8)—31no—
36C23×H(3,𝔽3)—21no—
37C22×#272D411yescharacteristic
38D4×H(3,𝔽3)—21no—
39D4×H(3,𝔽3)—21no—
40C2×C4×H(3,𝔽3)C2311yescharacteristic
41#6235—42no—
42C2×#1084—21no—
43C2×#1084D411yescharacteristic
44C4×#272—21no—
45C4×#272—21no—
46C2×(C3⋊Dic3)⋊C3—21no—
47C2×(C3⋊Dic3)⋊C3—21no—
48(C3⋊(C3⋊C8))⋊C3—21no—
49(C3⋊(C3⋊C8))⋊C3—21no—
50#5035—91no—
51#4784—91no—
52#4779—91no—
53C2×C32×D4—21no—
54C2×C32×D4C2×D311yescharacteristic
55C2×C32×D4C2×D311yescharacteristic
56C2×C3×#67—61no—
57C2×C3×#67—61no—
58C2×C3×#67—62no—
59C2×C3×C4×D3—31no—
60C2×C3×C4×D3—31no—
61C2×C3×C4×D3—61no—
62C4⋊(C3⋊Dic3)C2×D311yescharacteristic
63C4⋊(C3⋊Dic3)C2×D311yescharacteristic
64C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
65C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
66C3×#217—61no—
67C3×#217—61no—
68C3×#217—62no—
69C3×C4⋊Dic3—31no—
70C3×C4⋊Dic3—31no—
71C3×C4⋊Dic3—31no—
72C3×C4⋊Dic3—31no—
73C3×C4⋊Dic3—61no—
74C2×C3×C3⋊C8—31no—
75C2×C3×C3⋊C8—31no—
76(C2×D3)⋊Dic3—61no—
77(C2×D3)⋊Dic3—61no—
78Dic32—61no—
79Dic32—61no—
80C22×H(3,𝔽3)—41no—
81C22×H(3,𝔽3)—41no—
82C22×H(3,𝔽3)—41no—
83C22×H(3,𝔽3)C2×D411yescharacteristic
84C2×#272—21no—
85C2×#272—21no—
86C2×#272—21no—
87C4×H(3,𝔽3)#4111yescharacteristic
88C4×H(3,𝔽3)#4111yescharacteristic
89#1084—21no—
90#1084—41no—
91(C3⋊Dic3)⋊C3—41no—
92(C3⋊Dic3)⋊C3—41no—
93C3×#121—91no—
94#938—31no—
95#938—31no—
96#938—61no—
97#926—91no—
98#892—91no—
99#892—91no—
100C3×#102—91no—
101C3×#102—91no—
102#832—31no—
103#832—31no—
104#832—91no—
105#832—91no—
106#830—91no—
107(C2×D3)⋊C8—91no—
108(C2×D3)⋊C8—91no—
109#818—91no—
110#817—91no—
111#808—91no—
112#808—91no—
113#807—91no—
114#807—91no—
115C23×C32—22no—
116C23×C32—21no—
117C23×C32—21no—
118C22×C3×D3—31no—
119C22×C3×D3—31no—
120C22×C3×D3—61no—
121C32×D4—21no—
122C32×D4—21no—
123C32×D4—21no—
124C32×D4—21no—
125C32×D4—41no—
126C32×D4—41no—
127C2×C32×C4—21no—
128C2×C32×C4C22×D311yescharacteristic
129C2×C32×C4C22×D311yescharacteristic
130C2×C3⋊Dic3—21no—
131C2×C3⋊Dic3—21no—
132C3×#67—124no—
133C3×#67—122no—
134C3×#67—122no—
135C2×C3×Dic3—31no—
136C2×C3×Dic3—31no—
137C2×C3×Dic3—61no—
138C2×C3×Dic3—62no—
139C2×C3×Dic3—61no—
140C2×C3×Dic3—61no—
141C2×C3×Dic3—61no—
142C2×C3×Dic3—61no—
143C3×C4×D3—61no—
144C3×C4×D3—61no—
145C3×C4×D3—61no—
146C3×C4×D3—61no—
147C3×C4×D3—121no—
148C3×C4×D3—121no—
149C3⋊(C3⋊C8)—21no—
150C3⋊(C3⋊C8)—21no—
151C3×C3⋊C8—61no—
152C3×C3⋊C8—61no—
153#482—271noSylow
154C2×H(3,𝔽3)—81no—
155C2×H(3,𝔽3)#12311yescharacteristic
156C2×H(3,𝔽3)#13511yescharacteristic
157C2×H(3,𝔽3)AGL(1,ℤ/8)11yescharacteristic
158#272—41no—
159C2×C3×D4—31no—
160C2×C3×D4—31no—
161C2×C3×D4—61no—
162C2×C3×D4—181no—
163C2×C3×D4D3211yescharacteristic
164C22×C3×C4—91no—
165C2×#67—61no—
166C2×#67—61no—
167C2×#67—62no—
168C2×C4×D3—31no—
169C2×C4×D3—31no—
170C2×C4×D3—61no—
171C2×C3×C8—91no—
172C2×C3×C8—91no—
173C3×C4⋊C4—91no—
174C3×C4⋊C4—91no—
175C3×C4⋊C4—91no—
176C3×C22⋊C4—181no—
177#217—61no—
178#217—61no—
179#217—62no—
180#217—181no—
181(C2×D3)⋊C4—181no—
182(C2×D3)⋊C4—181no—
183C4⋊Dic3—31no—
184C4⋊Dic3—31no—
185C4⋊Dic3—31no—
186C4⋊Dic3—31no—
187C4⋊Dic3—61no—
188C4⋊Dic3—91no—
189C4⋊Dic3—91no—
190C4×Dic3—181no—
191C4×Dic3—181no—
192C4×Dic3—181no—
193C2×C3⋊C8—31no—
194C2×C3⋊C8—31no—
195C2×C3⋊C8—91no—
196C2×C3⋊C8—91no—
197C22×C32—21no—
198C22×C32—42no—
199C22×C32—42no—
200C22×C32—42no—
201C22×C32—41no—
202C22×C32—41no—
203C22×C32—41no—
204C22×C32—41no—
205C22×C32—41no—
206C22×C32—41no—
207C22×C32D3×D411yescharacteristic
208C22×C32D3×D411yescharacteristic
209C2×C3×D3—62no—
210C2×C3×D3—61no—
211C2×C3×D3—61no—
212C2×C3×D3—61no—
213C2×C3×D3—61no—
214C2×C3×D3—61no—
215C2×C3×D3—61no—
216C2×C3×D3—121no—
217C2×C3×D3—121no—
218C32×C4—21no—
219C32×C4—21no—
220C32×C4#23511yescharacteristic
221C32×C4#23511yescharacteristic
222C32×C4#23711yescharacteristic
223C32×C4#23711yescharacteristic
224C3⋊Dic3—41no—
225C3⋊Dic3—41no—
226C3×Dic3—61no—
227C3×Dic3—61no—
228C3×Dic3—121no—
229C3×Dic3—122no—
230C3×Dic3—121no—
231C3×Dic3—121no—
232C3×Dic3—121no—
233C3×Dic3—121no—
234#121—91no—
235#117—271no—
236#107—271no—
237#106—271no—
238#102—91no—
239#102—91no—
240C22⋊C8—271no—
241H(3,𝔽3)#48211yescharacteristic, complemented, Sylow
242C23×C3—21no—
243C23×C3—61no—
244C23×C3—61no—
245C23×C3—62no—
246C23×C3—91no—
247C22×D3—31no—
248C22×D3—31no—
249C22×D3—61no—
250C3×D4—21no—
251C3×D4—21no—
252C3×D4—61no—
253C3×D4—61no—
254C3×D4—61no—
255C3×D4—61no—
256C3×D4—121no—
257C3×D4—121no—
258C3×D4—362no—
259C2×C3×C4—31no—
260C2×C3×C4—31no—
261C2×C3×C4—61no—
262C2×C3×C4—91no—
263C2×C3×C4—181no—
264C2×C3×C4—181no—
265C2×C3×C4—181no—
266C2×C3×C4—181no—
267C2×C3×C4—181no—
268C2×C3×C4C2×D3211yescharacteristic
269#67—122no—
270#67—122no—
271#67—124no—
272C2×Dic3—31no—
273C2×Dic3—31no—
274C2×Dic3—61no—
275C2×Dic3—61no—
276C2×Dic3—61no—
277C2×Dic3—62no—
278C2×Dic3—61no—
279C2×Dic3—61no—
280C2×Dic3—181no—
281C2×Dic3—181no—
282C4×D3—61no—
283C4×D3—61no—
284C4×D3—61no—
285C4×D3—61no—
286C4×D3—121no—
287C4×D3—121no—
288C3×C8—181no—
289C3×C8—181no—
290C3⋊C8—61no—
291C3⋊C8—61no—
292C3⋊C8—181no—
293C3⋊C8—181no—
294C2×C32—21no—
295C2×C32—21no—
296C2×C32—21no—
297C2×C32—82no—
298C2×C32—81no—
299C2×C32—81no—
300C2×C32#88311yescharacteristic
301C2×C32#88311yescharacteristic
302C2×C32#91111yescharacteristic
303C2×C32#91211yescharacteristic
304C2×C32#91411yescharacteristic
305C2×C32#91611yescharacteristic
306C3×D3—122no—
307C3×D3—121no—
308C3×D3—121no—
309C2×D4—181no—
310C2×D4((C3⋊D3)⋊C3)⋊C211yescharacteristic
311C22×C4—91no—
312C2×C8—91no—
313C2×C8—91no—
314C4⋊C4—91no—
315C4⋊C4—91no—
316C4⋊C4—91no—
317C22⋊C4—181no—
318C22⋊C4—541no—
319C42—541no—
320C22×C3—31no—
321C22×C3—31no—
322C22×C3—41no—
323C22×C3—41no—
324C22×C3—41no—
325C22×C3—61no—
326C22×C3—121no—
327C22×C3—121no—
328C22×C3—121no—
329C22×C3—121no—
330C22×C3—121no—
331C22×C3—121no—
332C22×C3—122no—
333C22×C3—122no—
334C22×C3—122no—
335C22×C3—181no—
336C22×C3—181no—
337C22×C3—181no—
338C22×C3#381111yescharacteristic
339C2×D3—61no—
340C2×D3—61no—
341C2×D3—61no—
342C2×D3—61no—
343C2×D3—61no—
344C2×D3—61no—
345C2×D3—62no—
346C2×D3—121no—
347C2×D3—121no—
348C3×C4—31no—
349C3×C4—31no—
350C3×C4—31no—
351C3×C4—31no—
352C3×C4—61no—
353C3×C4—61no—
354C3×C4—181no—
355C3×C4—361no—
356C3×C4—361no—
357C3×C4—361no—
358C3×C4#380411yescharacteristic
359C3×C4#380411yescharacteristic
360Dic3—61no—
361Dic3—61no—
362Dic3—121no—
363Dic3—121no—
364Dic3—121no—
365Dic3—122no—
366Dic3—121no—
367Dic3—121no—
368Dic3—361no—
369Dic3—361no—
370C32—21no—
371C32#477911yescharacteristic, complemented
372C32#478411yescharacteristic, complemented
373C23—21no—
374C23—91no—
375D4—21no—
376D4—21no—
377D4—362no—
378C2×C4—91no—
379C2×C4—181no—
380C2×C4—181no—
381C2×C4—181no—
382C2×C4—181no—
383C2×C4—181no—
384C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
385C8—181no—
386C8—181no—
387C2×C3—31no—
388C2×C3—31no—
389C2×C3—31no—
390C2×C3—31no—
391C2×C3—31no—
392C2×C3—31no—
393C2×C3—61no—
394C2×C3—61no—
395C2×C3—61no—
396C2×C3—81no—
397C2×C3—241no—
398C2×C3—241no—
399C2×C3—242no—
400C2×C3—361no—
401C2×C3#6393611yescharacteristic
402C2×C3#6393711yescharacteristic
403C2×C3#6397511yescharacteristic
404D3—121no—
405D3—121no—
406D3—122no—
407C22—41no—
408C22—41no—
409C22—41no—
410C22—181no—
411C22—181no—
412C22—181no—
413C22#8867711yescharacteristic
414C4—181no—
415C4—361no—
416C4—361no—
417C4—361no—
418C4#8867111yescharacteristic
419C4#8867111yescharacteristic
420C3—31no—
421C3—31no—
422C3—61no—
423C3#1059142511yescharacteristic
424C2—81no—
425C2—361no—
426C2#1172119211yescharacteristic
427C2#1172119311yescharacteristic
428C2#1172131911yescharacteristic
subgroupG/H
centerC22#88677
commutator, cosocleC2×C4×H(3,𝔽3)C23
FrattiniC2×C3×C4C2×D32
FittingC2×D4×H(3,𝔽3)C22
socleC22×C3#3811
prime radical#11721848C2
hypercenterC2×D4((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)#482
LCSG ▷ C2×C4×H(3,𝔽3) ▷ C2×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C22 ◁ C2×C4 ◁ C2×D4

Complements

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

NH ≅ G/Ntype
2. #11721089C2semidirect
3. #63659⋊C3C2semidirect
4. #63659⋊C3C2semidirect
5. #11720323C2semidirect
6. #11720307C2semidirect
7. #11720306C2semidirect
26. #63659D3semidirect
27. #63659D3semidirect
241. H(3,𝔽3)#482semidirect
371. C32#4779semidirect
372. C32#4784semidirect

Maximal subgroups

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

subgroupindex
#117218482
#117210892
#63659⋊C32
#117203232
#117203072
#117203062
#105914233
#105914223