Finite Group Databasebeta

← #13137980

Subgroups — #13137980

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#11721763C211yesmaximal, characteristic
2#11721089C211yesmaximal, characteristic, complemented
3#63635⋊C3C211yesmaximal, characteristic, complemented
4#63634⋊C3C211yesmaximal, characteristic, complemented
5#11720324C211yesmaximal, characteristic, complemented
6#11720324C211yesmaximal, characteristic, complemented
7#11720323C211yesmaximal, characteristic, complemented
8#10590449—31nomaximal
9#10590446—31nomaximal
10C2×C4⋊#272—21no—
11C2×C4⋊#272C2211yescharacteristic
12C2×C4×#272C2211yescharacteristic
13C4⋊C4×H(3,𝔽3)C2211yescharacteristic
14#88542—21no—
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×#895—31no—
22C3×#895—31no—
23C3×#895—61no—
24#63841—31no—
25#63841—31no—
26#63635D311yescharacteristic, complemented
27#63634D311yescharacteristic, complemented
28C3×#808—31no—
29C3×#807—31no—
30#63563—31no—
31#63563—31no—
32#63562—31no—
33#63562—31no—
34(C2×D3)⋊(C3⋊C8)—31no—
35(C2×D3)⋊(C3⋊C8)—31no—
36C22×#272—21no—
37C22×#272D411yescharacteristic
38C2×C4×H(3,𝔽3)—21no—
39C2×C4×H(3,𝔽3)C2311yescharacteristic
40C2×#1084D411yescharacteristic
41C4⋊#272—21no—
42C4⋊#272—21no—
43C4⋊#272—42no—
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#4885—91no—
52#4866—91no—
53C2×C3×D12—31no—
54C2×C3×D12—31no—
55C2×C3×D12—61no—
56C2×C3×D12—61no—
57C2×C3×D12—61no—
58C2×C3×D12—62no—
59C2×C3×C4×D3—31no—
60C2×C3×C4×D3—31no—
61C2×C3×C4×D3—61no—
62C32×C4⋊C4—21no—
63C32×C4⋊C4C2×D311yescharacteristic
64C32×C4⋊C4C2×D311yescharacteristic
65C4⋊(C3⋊Dic3)C2×D311yescharacteristic
66C4⋊(C3⋊Dic3)C2×D311yescharacteristic
67C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
68C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
69C3×(C2×D3)⋊C4—61no—
70C3×(C2×D3)⋊C4—61no—
71C3×(C2×D3)⋊C4—62no—
72C3×C4⋊Dic3—31no—
73C3×C4⋊Dic3—31no—
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)C2×D411yescharacteristic
81C2×#272—21no—
82C2×#272—21no—
83C2×#272—21no—
84C2×#272—41no—
85C2×#272—41no—
86C2×#272—41no—
87C4×H(3,𝔽3)—41no—
88C4×H(3,𝔽3)#4111yescharacteristic
89C4×H(3,𝔽3)#4111yescharacteristic
90#1084—21no—
91(C3⋊Dic3)⋊C3—41no—
92(C3⋊Dic3)⋊C3—41no—
93C3×#121—91no—
94#926—91no—
95#895—31no—
96#895—31no—
97#895—61no—
98#892—91no—
99#892—91no—
100C3×#107—91no—
101C3×#106—91no—
102#832—91no—
103#832—91no—
104#830—91no—
105#821—91no—
106#821—91no—
107(C2×D3)⋊C8—91no—
108(C2×D3)⋊C8—91no—
109#809—91no—
110#809—91no—
111#808—31no—
112#808—91no—
113#807—31no—
114#807—91no—
115C22×C3×D3—31no—
116C22×C3×D3—31no—
117C22×C3×D3—61no—
118C22×C3×D3—62no—
119C22×C3×D3—61no—
120C22×C3×D3—61no—
121C2×C32×C4—22no—
122C2×C32×C4—21no—
123C2×C32×C4—21no—
124C2×C32×C4—21no—
125C2×C32×C4C22×D311yescharacteristic
126C2×C32×C4C22×D311yescharacteristic
127C2×C3⋊Dic3—21no—
128C2×C3⋊Dic3—21no—
129C2×C3×Dic3—31no—
130C2×C3×Dic3—31no—
131C2×C3×Dic3—61no—
132C2×C3×Dic3—61no—
133C2×C3×Dic3—61no—
134C3×D12—61no—
135C3×D12—61no—
136C3×D12—61no—
137C3×D12—61no—
138C3×D12—121no—
139C3×D12—121no—
140C3×D12—124no—
141C3×D12—122no—
142C3×D12—122no—
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)#12311yescharacteristic
155C2×H(3,𝔽3)#13511yescharacteristic
156C2×H(3,𝔽3)AGL(1,ℤ/8)11yescharacteristic
157#272—41no—
158#272—81no—
159C2×C3×D4—91no—
160C2×C3×D4—181no—
161C22×C3×C4—91no—
162C2×D12—31no—
163C2×D12—31no—
164C2×D12—61no—
165C2×D12—61no—
166C2×D12—61no—
167C2×D12—62no—
168C2×C4×D3—31no—
169C2×C4×D3—31no—
170C2×C4×D3—61no—
171C2×C3×C8—91no—
172C2×C3×C8—91no—
173C3×C4⋊C4—31no—
174C3×C4⋊C4—31no—
175C3×C4⋊C4—61no—
176C3×C4⋊C4—91no—
177C3×C4⋊C4—91no—
178C3×C4⋊C4D3211yescharacteristic
179C3×C22⋊C4—181no—
180#217—181no—
181(C2×D3)⋊C4—61no—
182(C2×D3)⋊C4—61no—
183(C2×D3)⋊C4—62no—
184(C2×D3)⋊C4—181no—
185(C2×D3)⋊C4—181no—
186C4⋊Dic3—31no—
187C4⋊Dic3—31no—
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×C32D3×D411yescharacteristic
199C22×C32D3×D411yescharacteristic
200C2×C3×D3—62no—
201C2×C3×D3—61no—
202C2×C3×D3—61no—
203C2×C3×D3—61no—
204C2×C3×D3—61no—
205C2×C3×D3—61no—
206C2×C3×D3—61no—
207C2×C3×D3—121no—
208C2×C3×D3—121no—
209C2×C3×D3—122no—
210C2×C3×D3—122no—
211C2×C3×D3—122no—
212C2×C3×D3—121no—
213C2×C3×D3—121no—
214C2×C3×D3—121no—
215C2×C3×D3—121no—
216C2×C3×D3—121no—
217C2×C3×D3—121no—
218C32×C4—21no—
219C32×C4—21no—
220C32×C4—42no—
221C32×C4—41no—
222C32×C4—41no—
223C32×C4#23511yescharacteristic
224C32×C4#23511yescharacteristic
225C32×C4#23911yescharacteristic
226C32×C4#23911yescharacteristic
227C3⋊Dic3—41no—
228C3⋊Dic3—41no—
229C3×Dic3—61no—
230C3×Dic3—61no—
231C3×Dic3—121no—
232C3×Dic3—121no—
233C3×Dic3—121no—
234#121—91no—
235#117—271no—
236#107—91no—
237#106—91no—
238#102—271no—
239#102—271no—
240C22⋊C8—271no—
241H(3,𝔽3)#48211yescharacteristic, complemented, Sylow
242C23×C3—91no—
243C23×C3—181no—
244C22×D3—31no—
245C22×D3—31no—
246C22×D3—61no—
247C22×D3—61no—
248C22×D3—61no—
249C22×D3—62no—
250C3×D4—181no—
251C3×D4—181no—
252C3×D4—362no—
253C2×C3×C4—21no—
254C2×C3×C4—31no—
255C2×C3×C4—31no—
256C2×C3×C4—61no—
257C2×C3×C4—61no—
258C2×C3×C4—62no—
259C2×C3×C4—61no—
260C2×C3×C4—91no—
261C2×C3×C4—181no—
262C2×C3×C4—181no—
263C2×C3×C4—181no—
264C2×C3×C4—181no—
265C2×C3×C4C2×D3211yescharacteristic
266C2×Dic3—31no—
267C2×Dic3—31no—
268C2×Dic3—61no—
269C2×Dic3—61no—
270C2×Dic3—61no—
271C2×Dic3—181no—
272C2×Dic3—181no—
273D12—61no—
274D12—61no—
275D12—61no—
276D12—61no—
277D12—121no—
278D12—121no—
279D12—122no—
280D12—122no—
281D12—124no—
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#89411yescharacteristic
298C2×C32#89411yescharacteristic
299C2×C32#91111yescharacteristic
300C2×C32#91411yescharacteristic
301C2×C32#91611yescharacteristic
302C2×C32#91911yescharacteristic
303C3×D3—122no—
304C3×D3—121no—
305C3×D3—121no—
306C3×D3—242no—
307C3×D3—241no—
308C3×D3—241no—
309C2×D4—91no—
310C2×D4—181no—
311C22×C4—91no—
312C2×C8—91no—
313C2×C8—91no—
314C4⋊C4—91no—
315C4⋊C4—91no—
316C4⋊C4((C3⋊D3)⋊C3)⋊C211yescharacteristic
317C22⋊C4—181no—
318C22⋊C4—541no—
319C42—541no—
320C22×C3—31no—
321C22×C3—31no—
322C22×C3—61no—
323C22×C3—181no—
324C22×C3—181no—
325C22×C3—181no—
326C22×C3—361no—
327C22×C3—361no—
328C22×C3—361no—
329C22×C3#381111yescharacteristic
330C2×D3—61no—
331C2×D3—61no—
332C2×D3—61no—
333C2×D3—61no—
334C2×D3—61no—
335C2×D3—61no—
336C2×D3—62no—
337C2×D3—121no—
338C2×D3—121no—
339C2×D3—121no—
340C2×D3—121no—
341C2×D3—121no—
342C2×D3—121no—
343C2×D3—121no—
344C2×D3—121no—
345C2×D3—122no—
346C2×D3—122no—
347C2×D3—122no—
348C3×C4—31no—
349C3×C4—31no—
350C3×C4—31no—
351C3×C4—31no—
352C3×C4—41no—
353C3×C4—61no—
354C3×C4—61no—
355C3×C4—121no—
356C3×C4—121no—
357C3×C4—122no—
358C3×C4—181no—
359C3×C4—361no—
360C3×C4—361no—
361C3×C4#379911yescharacteristic
362C3×C4#379911yescharacteristic
363Dic3—61no—
364Dic3—61no—
365Dic3—121no—
366Dic3—121no—
367Dic3—121no—
368Dic3—361no—
369Dic3—361no—
370C32—21no—
371C32#486611yescharacteristic, complemented
372C32#488511yescharacteristic, complemented
373C23—91no—
374C23—181no—
375D4—181no—
376D4—181no—
377D4—362no—
378C2×C4—21no—
379C2×C4—91no—
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—361no—
397C2×C3—721no—
398C2×C3#6389711yescharacteristic
399C2×C3#6393711yescharacteristic
400C2×C3#6395011yescharacteristic
401D3—121no—
402D3—121no—
403D3—122no—
404D3—241no—
405D3—241no—
406D3—242no—
407C22—181no—
408C22—181no—
409C22—181no—
410C22—361no—
411C22—361no—
412C22—361no—
413C22#8867711yescharacteristic
414C4—41no—
415C4—181no—
416C4—361no—
417C4—361no—
418C4#8865211yescharacteristic
419C4#8865211yescharacteristic
420C3—31no—
421C3—31no—
422C3—61no—
423C3#1059045111yescharacteristic
424C2—361no—
425C2—721no—
426C2#1172115111yescharacteristic
427C2#1172119311yescharacteristic
428C2#1172123011yescharacteristic
subgroupG/H
centerC22#88677
commutator, cosocleC2×C4×H(3,𝔽3)C23
FrattiniC2×C3×C4C2×D32
FittingC4⋊C4×H(3,𝔽3)C22
socleC22×C3#3811
prime radical#11721763C2
hypercenterC4⋊C4((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 ◁ C4⋊C4

Complements

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

NH ≅ G/Ntype
2. #11721089C2semidirect
3. #63635⋊C3C2semidirect
4. #63634⋊C3C2semidirect
5. #11720324C2semidirect
6. #11720324C2semidirect
7. #11720323C2semidirect
26. #63635D3semidirect
27. #63634D3semidirect
241. H(3,𝔽3)#482semidirect
371. C32#4866semidirect
372. C32#4885semidirect

Maximal subgroups

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

subgroupindex
#117217632
#117210892
#63635⋊C32
#63634⋊C32
#117203242
#117203232
#105904493
#105904463