Finite Group Databasebeta

← #13140185

Subgroups — #13140185

Order 1728.

929 conjugacy classes of subgroups; only the 89 normal subgroups are shown.

normal subgroups only

This group has 929 conjugacy classes of subgroups — too many to draw. Only its 89 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
1C22×C4⋊#272C211yesmaximal, characteristic, complemented
2#64139⋊C3C211yesmaximal, characteristic, complemented
3C2×#88436C212yesmaximal, complemented
4#63657⋊C3C211yesmaximal, characteristic, complemented
5#11720324C212yesmaximal, complemented
6#10591162—31nomaximal
7#10591160—31nomaximal
8C23×#272—21no—
9C22×C4×H(3,𝔽3)C2211yescharacteristic
10C2×C4⋊#272—21no—
11C2×C4⋊#272—24no—
12C2×C4⋊#272C2212yes—
13C2×#638⋊C3—21no—
14C2×C4⋊((C3⋊D3)⋊C3)C2211yescharacteristic, complemented
15#3752⋊C3—21no—
16C4⋊((C3⋊Dic3)⋊C3)C2211yescharacteristic, complemented
17C2×(C3⋊(C3⋊C8))⋊C3C2212yescomplemented
18#88436—24no—
19C22×C3×D12—31no—
20C22×C3×D12—31no—
21C22×C3×D12—61no—
22#64139D311yescharacteristic, complemented
23C3×#930—31no—
24C2×#3714—32no—
25C2×#3713—32no—
26#63657D311yescharacteristic, complemented
27C3×#830—31no—
28#63563—32no—
29#63562—32no—
30C23×H(3,𝔽3)D411yescharacteristic
31C22×#272—24no—
32C22×#272—22no—
33C22×#272—42no—
34C22×#272—42no—
35C22×(C3⋊D3)⋊C3—21no—
36C2×C4×H(3,𝔽3)—22no—
37C2×C4×H(3,𝔽3)D411yescharacteristic
38C2×C4×H(3,𝔽3)C2311yescharacteristic
39C4⋊#272—24no—
40C4⋊#272—22no—
41C4⋊#272D414yescomplemented
42#638⋊C3—42no—
43C2×(C3⋊Dic3)⋊C3—21no—
44C4⋊((C3⋊D3)⋊C3)—22no—
45(C3⋊(C3⋊C8))⋊C3—22no—
46#5217—91no—
47#5036—91no—
48#4732—91no—
49C23×C3×D3—62no—
50C23×C3×D3—61no—
51C23×C3×D3—61no—
52C22×C32×C4—21no—
53C22×C32×C4C2×D311yescharacteristic
54C22×C32×C4C2×D311yescharacteristic
55C2×#638—21no—
56C2×C4⋊(C3⋊D3)C2×D311yescharacteristic, complemented
57C2×C3×#67—61no—
58C2×C3×D12—32no—
59C2×C3×D12—32no—
60C2×C3×D12—31no—
61C2×C3×D12—64no—
62C2×C3×D12—64no—
63C2×C3×D12—62no—
64C2×C3×D12—68no—
65C2×C3×D12—62no—
66C2×C3×D12—61no—
67C2×C3×D12—61no—
68#3752—21no—
69C4⋊(C3⋊Dic3)C2×D311yescharacteristic, complemented
70C2×C3⋊(C3⋊C8)C2×D312yescomplemented
71C3×(C2×D3)⋊C4—61no—
72C3×C4⋊Dic3—31no—
73C2×C3×C3⋊C8—32no—
74#3714—64no—
75#3713—64no—
76C22×H(3,𝔽3)—21no—
77C22×H(3,𝔽3)—21no—
78C22×H(3,𝔽3)D812yescomplemented
79C22×H(3,𝔽3)C2×D411yescharacteristic
80C2×#272—24no—
81C2×#272—48no—
82C2×#272—42no—
83C2×#272—42no—
84C2×(C3⋊D3)⋊C3—41no—
85C2×(C3⋊D3)⋊C3—41no—
86C2×(C3⋊D3)⋊C3—41no—
87C4×H(3,𝔽3)—21no—
88C4×H(3,𝔽3)C2×D412yes—
89(C3⋊Dic3)⋊C3—41no—
90C22×C3×D4—91no—
91C22×D12—31no—
92C22×D12—31no—
93C22×D12—61no—
94C3×#121—91no—
95C2×#213—92no—
96C2×#213—92no—
97#930—31no—
98#930—91no—
99C2×D24—92no—
100C3×C22⋊C8—91no—
101#832—92no—
102#830—31no—
103#830—91no—
104#821—92no—
105#809—92no—
106C23×C32—21no—
107C23×C32D1211yescharacteristic
108C23×C32#6711yescharacteristic
109C22×C3⋊D3—21no—
110C22×C3×D3—68no—
111C22×C3×D3—64no—
112C22×C3×D3—64no—
113C22×C3×D3—64no—
114C22×C3×D3—62no—
115C22×C3×D3—62no—
116C22×C3×D3—61no—
117C22×C3×D3—124no—
118C22×C3×D3—122no—
119C22×C3×D3—122no—
120C22×C3×D3—124no—
121C22×C3×D3—122no—
122C22×C3×D3—122no—
123C2×C32×C4—24no—
124C2×C32×C4—22no—
125C2×C32×C4—22no—
126C2×C32×C4—21no—
127C2×C32×C4—21no—
128C2×C32×C4D1211yescharacteristic
129C2×C32×C4#6711yescharacteristic
130C2×C32×C4C22×D311yescharacteristic
131C2×C32×C4C22×D311yescharacteristic
132#638—42no—
133C2×C3⋊Dic3—21no—
134C4⋊(C3⋊D3)—22no—
135C3×#67—122no—
136C2×C3×Dic3—61no—
137C3×D12—34no—
138C3×D12—34no—
139C3×D12—64no—
140C3×D12—68no—
141C3×D12—64no—
142C3×D12—64no—
143C3×D12—64no—
144C3×D12—62no—
145C3×D12—62no—
146C3×D12—62no—
147C3⋊(C3⋊C8)—22no—
148C3×C3⋊C8—62no—
149#447—271noSylow
150C2×H(3,𝔽3)—22no—
151C2×H(3,𝔽3)#12011yescharacteristic
152C2×H(3,𝔽3)C2×D811yescharacteristic
153C2×H(3,𝔽3)AGL(1,ℤ/8)11yescharacteristic
154#272—44no—
155(C3⋊D3)⋊C3—81no—
156C24×C3—181no—
157C23×D3—61no—
158C23×D3—61no—
159C23×D3—62no—
160C2×C3×D4—91no—
161C2×C3×D4—92no—
162C2×C3×D4—184no—
163C2×C3×D4—181no—
164C2×C3×D4—181no—
165C22×C3×C4—31no—
166C22×C3×C4—31no—
167C22×C3×C4—61no—
168C22×C3×C4D3211yescharacteristic
169C2×#67—61no—
170C2×#67—181no—
171C2×D12—32no—
172C2×D12—32no—
173C2×D12—31no—
174C2×D12—62no—
175C2×D12—64no—
176C2×D12—64no—
177C2×D12—61no—
178C2×D12—61no—
179C2×D12—68no—
180C2×D12—62no—
181C2×D12—91no—
182C2×C3×C8—92no—
183C3×C4⋊C4—91no—
184C3×C22⋊C4—181no—
185#213—184no—
186#213—184no—
187(C2×D3)⋊C4—61no—
188(C2×D3)⋊C4—181no—
189C4⋊Dic3—31no—
190C4⋊Dic3—91no—
191C2×C3⋊C8—32no—
192C2×C3⋊C8—92no—
193D24—184no—
194C22×C32—22no—
195C22×C32—22no—
196C22×C32—21no—
197C22×C32—21no—
198C22×C32—21no—
199C22×C32—21no—
200C22×C32—22no—
201C22×C32—21no—
202C22×C32D2412yescomplemented
203C22×C32#21312yescomplemented
204C22×C32C2×D1211yescharacteristic
205C22×C32C2×#6711yescharacteristic
206C2×C3⋊D3—41no—
207C2×C3⋊D3—41no—
208C2×C3⋊D3—41no—
209C2×C3×D3—68no—
210C2×C3×D3—64no—
211C2×C3×D3—64no—
212C2×C3×D3—1216no—
213C2×C3×D3—128no—
214C2×C3×D3—128no—
215C2×C3×D3—124no—
216C2×C3×D3—124no—
217C2×C3×D3—122no—
218C2×C3×D3—122no—
219C2×C3×D3—122no—
220C2×C3×D3—122no—
221C2×C3×D3—121no—
222C2×C3×D3—121no—
223C2×C3×D3—121no—
224C32×C4—22no—
225C32×C4—21no—
226C32×C4—21no—
227C32×C4—22no—
228C32×C4D3×D412yes—
229C32×C4D3×D412yes—
230C3⋊Dic3—41no—
231C3×Dic3—121no—
232C22×D4—91no—
233C2×D8—272no—
234#121—91no—
235#102—272no—
236C22⋊C8—91no—
237H(3,𝔽3)#44711yescharacteristic, complemented, Sylow
238C23×C3—31no—
239C23×C3—31no—
240C23×C3—61no—
241C23×C3—184no—
242C23×C3—181no—
243C23×C3—182no—
244C23×C3—362no—
245C23×C3—362no—
246C23×C3#62611yescharacteristic
247C22×D3—62no—
248C22×D3—62no—
249C22×D3—64no—
250C22×D3—64no—
251C22×D3—68no—
252C22×D3—64no—
253C22×D3—61no—
254C22×D3—122no—
255C22×D3—122no—
256C22×D3—122no—
257C22×D3—122no—
258C22×D3—124no—
259C22×D3—124no—
260C22×D3—181no—
261C3×D4—94no—
262C3×D4—184no—
263C3×D4—182no—
264C3×D4—182no—
265C3×D4—362no—
266C2×C3×C4—22no—
267C2×C3×C4—31no—
268C2×C3×C4—31no—
269C2×C3×C4—31no—
270C2×C3×C4—31no—
271C2×C3×C4—62no—
272C2×C3×C4—62no—
273C2×C3×C4—64no—
274C2×C3×C4—61no—
275C2×C3×C4—61no—
276C2×C3×C4—181no—
277C2×C3×C4#62611yescharacteristic
278C2×C3×C4C2×D3211yescharacteristic
279#67—122no—
280#67—362no—
281C2×Dic3—61no—
282C2×Dic3—181no—
283D12—34no—
284D12—34no—
285D12—64no—
286D12—62no—
287D12—62no—
288D12—64no—
289D12—64no—
290D12—68no—
291D12—64no—
292D12—62no—
293D12—182no—
294C3×C8—182no—
295C3⋊C8—62no—
296C3⋊C8—182no—
297C2×C32—24no—
298C2×C32—22no—
299C2×C32—22no—
300C2×C32—21no—
301C2×C32—21no—
302C2×C32—21no—
303C2×C32#88211yescharacteristic
304C2×C32C2×D2411yescharacteristic
305C2×C32#90811yescharacteristic
306C2×C32C2×#21311yescharacteristic
307C2×C32#93711yescharacteristic
308C2×C32#94911yescharacteristic
309C3⋊D3—81no—
310C3×D3—128no—
311C3×D3—124no—
312C3×D3—124no—
313C3×D3—241no—
314C24—181no—
315C2×D4—92no—
316C2×D4—91no—
317C2×D4—184no—
318C2×D4—181no—
319C2×D4—181no—
320C22×C4((C3⋊D3)⋊C3)⋊C211yescharacteristic
321D8—544no—
322C2×C8—92no—
323C4⋊C4—91no—
324C22⋊C4—181no—
325C22×C3—21no—
326C22×C3—21no—
327C22×C3—31no—
328C22×C3—31no—
329C22×C3—32no—
330C22×C3—32no—
331C22×C3—61no—
332C22×C3—61no—
333C22×C3—61no—
334C22×C3—61no—
335C22×C3—62no—
336C22×C3—62no—
337C22×C3—61no—
338C22×C3—62no—
339C22×C3—184no—
340C22×C3—368no—
341C22×C3—361no—
342C22×C3—361no—
343C22×C3—361no—
344C22×C3—362no—
345C22×C3—362no—
346C22×C3#371412yes—
347C22×C3C2×#62611yescharacteristic
348C2×D3—64no—
349C2×D3—64no—
350C2×D3—68no—
351C2×D3—122no—
352C2×D3—122no—
353C2×D3—122no—
354C2×D3—122no—
355C2×D3—128no—
356C2×D3—128no—
357C2×D3—1216no—
358C2×D3—124no—
359C2×D3—124no—
360C2×D3—121no—
361C2×D3—121no—
362C2×D3—121no—
363C2×D3—361no—
364C2×D3—361no—
365C2×D3—361no—
366C3×C4—21no—
367C3×C4—32no—
368C3×C4—32no—
369C3×C4—61no—
370C3×C4—61no—
371C3×C4—62no—
372C3×C4—62no—
373C3×C4—361no—
374C3×C4#381112yes—
375Dic3—121no—
376Dic3—361no—
377C32—21no—
378C32#473211yescharacteristic, complemented
379C32#503611yescharacteristic, complemented
380C23—184no—
381C23—182no—
382C23—181no—
383C23—362no—
384C23—362no—
385C23#620011yescharacteristic
386D4—94no—
387D4—184no—
388D4—182no—
389D4—182no—
390D4—362no—
391C2×C4—22no—
392C2×C4—181no—
393C2×C4#620011yescharacteristic
394C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
395C8—182no—
396C2×C3—22no—
397C2×C3—31no—
398C2×C3—31no—
399C2×C3—31no—
400C2×C3—31no—
401C2×C3—31no—
402C2×C3—31no—
403C2×C3—62no—
404C2×C3—62no—
405C2×C3—64no—
406C2×C3—61no—
407C2×C3—61no—
408C2×C3—61no—
409C2×C3—364no—
410C2×C3—721no—
411C2×C3C2×#371411yescharacteristic
412C2×C3#6382511yescharacteristic
413C2×C3#6398111yescharacteristic
414D3—124no—
415D3—124no—
416D3—128no—
417D3—241no—
418D3—721no—
419C22—21no—
420C22—21no—
421C22—184no—
422C22—368no—
423C22—362no—
424C22—362no—
425C22—361no—
426C22—361no—
427C22—361no—
428C22#8843612yescomplemented
429C22C2×#620011yescharacteristic
430C4—21no—
431C4—361no—
432C4#8867712yes—
433C3—31no—
434C3—31no—
435C3—61no—
436C3#1059116811yescharacteristic
437C2—22no—
438C2—364no—
439C2—721no—
440C2C2×#8843611yescharacteristic
441C2#1172106811yescharacteristic
442C2#1172132311yescharacteristic
subgroupG/H
centerC22C2×#6200
commutator, cosocleC2×C4×H(3,𝔽3)C23
FrattiniC2×C3×C4C2×D32
FittingC22×C4×H(3,𝔽3)C22
socleC22×C3C2×#626
hypercenterC22×C4((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)#447
LCSG ▷ C2×C4×H(3,𝔽3) ▷ C2×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C22 ◁ C22×C4

Complements

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

NH ≅ G/Ntype
1. C22×C4⋊#272C2semidirect
2. #64139⋊C3C2semidirect
3. C2×#88436C2semidirect
4. #63657⋊C3C2semidirect
5. #11720324C2semidirect
14. C2×C4⋊((C3⋊D3)⋊C3)C22semidirect
16. C4⋊((C3⋊Dic3)⋊C3)C22semidirect
17. C2×(C3⋊(C3⋊C8))⋊C3C22semidirect
22. #64139D3semidirect
26. #63657D3semidirect
41. C4⋊#272D4semidirect
56. C2×C4⋊(C3⋊D3)C2×D3semidirect
69. C4⋊(C3⋊Dic3)C2×D3semidirect
70. C2×C3⋊(C3⋊C8)C2×D3semidirect
78. C22×H(3,𝔽3)D8semidirect
202. C22×C32D24semidirect
203. C22×C32#213semidirect
237. H(3,𝔽3)#447semidirect
378. C32#4732semidirect
379. C32#5036semidirect
428. C22#88436semidirect

Maximal subgroups

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

subgroupindex
C22×C4⋊#2722
#64139⋊C32
C2×#884362
#63657⋊C32
#117203242
#105911623
#105911603