Finite Group Databasebeta

← #13140181

Subgroups — #13140181

Order 1728.

787 conjugacy classes of subgroups; only the 93 normal subgroups are shown.

normal subgroups only

This group has 787 conjugacy classes of subgroups — too many to draw. Only its 93 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⋊((C3⋊D3)⋊C3)C211yesmaximal, characteristic, complemented
2#11721830C211yesmaximal, characteristic, complemented
3C2×#88436C212yesmaximal, complemented
4#63657⋊C3C211yesmaximal, characteristic, complemented
5#11720308C212yesmaximal, complemented
6#10591168—31nomaximal
7#10591160—31nomaximal
8C23×(C3⋊D3)⋊C3—21no—
9C22×C4×H(3,𝔽3)C2211yescharacteristic
10C2×#6235—21no—
11C2×C4⋊#272C2211yescharacteristic, complemented
12C2×C4⋊((C3⋊D3)⋊C3)—21no—
13C2×C4⋊((C3⋊D3)⋊C3)—24no—
14C2×C4⋊((C3⋊D3)⋊C3)C2212yes—
15#88542—21no—
16C4⋊#1084C2211yescharacteristic, complemented
17C2×(C3⋊(C3⋊C8))⋊C3C2212yescomplemented
18#88436—24no—
19C22×C4⋊(C3⋊D3)D311yescharacteristic, complemented
20C22×C3×D12—31no—
21C3×#930—31no—
22C3×#930—31no—
23C3×#930—61no—
24C2×#3714—32no—
25C2×#3713—32no—
26#63657D311yescharacteristic, complemented
27C3×#830—31no—
28#63564—32no—
29#63562—32no—
30C23×H(3,𝔽3)D411yescharacteristic
31C22×#272—21no—
32C22×(C3⋊D3)⋊C3—24no—
33C22×(C3⋊D3)⋊C3—22no—
34C22×(C3⋊D3)⋊C3—42no—
35C22×(C3⋊D3)⋊C3—42no—
36C2×C4×H(3,𝔽3)—22no—
37C2×C4×H(3,𝔽3)D411yescharacteristic
38C2×C4×H(3,𝔽3)C2311yescharacteristic
39#6235—42no—
40C2×#1084—21no—
41C4⋊#272—22no—
42C4⋊((C3⋊D3)⋊C3)—24no—
43C4⋊((C3⋊D3)⋊C3)—22no—
44C4⋊((C3⋊D3)⋊C3)D414yescomplemented
45(C3⋊(C3⋊C8))⋊C3—22no—
46#5217—91no—
47#5036—91no—
48#4732—91no—
49C23×C3⋊D3—21no—
50C23×C3×D3—61no—
51C22×C32×C4—21no—
52C22×C32×C4C2×D311yescharacteristic
53C22×C32×C4C2×D311yescharacteristic
54C2×C4⋊(C3⋊D3)—21no—
55C2×C4⋊(C3⋊D3)—24no—
56C2×C4⋊(C3⋊D3)C2×D312yes—
57C2×C3×#67—62no—
58C2×C3×#67—61no—
59C2×C3×#67—61no—
60C2×C3×D12—32no—
61C2×C3×D12—31no—
62C2×C3×D12—31no—
63C2×C3×D12—64no—
64C2×C3×D12—61no—
65C2×C3×D12—61no—
66C2×C3⋊(C3⋊C8)C2×D312yescomplemented
67C3×(C2×D3)⋊C4—62no—
68C3×(C2×D3)⋊C4—61no—
69C3×(C2×D3)⋊C4—61no—
70C3×C4⋊Dic3—31no—
71C3×C4⋊Dic3—31no—
72C3×C4⋊Dic3—61no—
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—41no—
81C2×#272—41no—
82C2×#272—41no—
83C2×(C3⋊D3)⋊C3—24no—
84C2×(C3⋊D3)⋊C3—48no—
85C2×(C3⋊D3)⋊C3—42no—
86C2×(C3⋊D3)⋊C3—42no—
87C4×H(3,𝔽3)—21no—
88C4×H(3,𝔽3)C2×D412yes—
89#1084—41no—
90C22×C3×D4—91no—
91C22×D12—31no—
92C22×D12—91no—
93C3×#121—91no—
94C2×#213—92no—
95C2×#213—92no—
96#930—31no—
97#930—31no—
98#930—61no—
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—24no—
110C22×C3⋊D3—22no—
111C22×C3⋊D3—42no—
112C22×C3⋊D3—42no—
113C22×C3×D3—64no—
114C22×C3×D3—62no—
115C22×C3×D3—62no—
116C22×C3×D3—61no—
117C22×C3×D3—61no—
118C22×C3×D3—122no—
119C22×C3×D3—122no—
120C2×C32×C4—22no—
121C2×C32×C4—22no—
122C2×C32×C4—21no—
123C2×C32×C4—21no—
124C2×C32×C4—42no—
125C2×C32×C4D1211yescharacteristic
126C2×C32×C4#6711yescharacteristic
127C2×C32×C4C22×D311yescharacteristic
128C2×C32×C4C22×D311yescharacteristic
129C4⋊(C3⋊D3)—24no—
130C4⋊(C3⋊D3)—22no—
131C4⋊(C3⋊D3)#6714yescomplemented
132C3×#67—124no—
133C3×#67—122no—
134C3×#67—122no—
135C2×C3×Dic3—62no—
136C2×C3×Dic3—61no—
137C2×C3×Dic3—61no—
138C3×D12—34no—
139C3×D12—64no—
140C3×D12—62no—
141C3×D12—62no—
142C3×D12—62no—
143C3×D12—122no—
144C3⋊(C3⋊C8)—22no—
145C3×C3⋊C8—62no—
146#447—271noSylow
147C2×H(3,𝔽3)—22no—
148C2×H(3,𝔽3)#12011yescharacteristic
149C2×H(3,𝔽3)C2×D811yescharacteristic
150C2×H(3,𝔽3)AGL(1,ℤ/8)11yescharacteristic
151#272—81no—
152(C3⋊D3)⋊C3—44no—
153C24×C3—181no—
154C23×D3—61no—
155C23×D3—181no—
156C2×C3×D4—92no—
157C2×C3×D4—91no—
158C2×C3×D4—181no—
159C2×C3×D4—184no—
160C2×C3×D4—181no—
161C22×C3×C4—31no—
162C22×C3×C4—31no—
163C22×C3×C4—61no—
164C22×C3×C4D3211yescharacteristic
165C2×#67—61no—
166C2×#67—61no—
167C2×#67—62no—
168C2×D12—32no—
169C2×D12—31no—
170C2×D12—31no—
171C2×D12—61no—
172C2×D12—64no—
173C2×D12—61no—
174C2×D12—92no—
175C2×D12—184no—
176C2×D12—181no—
177C2×C3×C8—92no—
178C3×C4⋊C4—91no—
179C3×C22⋊C4—181no—
180#213—184no—
181#213—184no—
182(C2×D3)⋊C4—61no—
183(C2×D3)⋊C4—61no—
184(C2×D3)⋊C4—62no—
185C4⋊Dic3—31no—
186C4⋊Dic3—31no—
187C4⋊Dic3—61no—
188C2×C3⋊C8—32no—
189C2×C3⋊C8—92no—
190D24—184no—
191C22×C32—21no—
192C22×C32—21no—
193C22×C32—21no—
194C22×C32—21no—
195C22×C32—22no—
196C22×C32—21no—
197C22×C32—41no—
198C22×C32—41no—
199C22×C32D2412yescomplemented
200C22×C32#21312yescomplemented
201C22×C32C2×D1211yescharacteristic
202C22×C32C2×#6711yescharacteristic
203C2×C3⋊D3—24no—
204C2×C3⋊D3—48no—
205C2×C3⋊D3—42no—
206C2×C3⋊D3—42no—
207C2×C3×D3—64no—
208C2×C3×D3—128no—
209C2×C3×D3—122no—
210C2×C3×D3—122no—
211C2×C3×D3—122no—
212C2×C3×D3—122no—
213C2×C3×D3—122no—
214C2×C3×D3—121no—
215C2×C3×D3—121no—
216C2×C3×D3—121no—
217C2×C3×D3—121no—
218C2×C3×D3—121no—
219C2×C3×D3—121no—
220C32×C4—21no—
221C32×C4—21no—
222C32×C4—22no—
223C32×C4—41no—
224C32×C4D3×D412yes—
225C32×C4C2×#6712yes—
226C3×Dic3—122no—
227C3×Dic3—121no—
228C3×Dic3—121no—
229C22×D4—91no—
230C2×D8—272no—
231#121—91no—
232#102—272no—
233C22⋊C8—91no—
234H(3,𝔽3)#44711yescharacteristic, complemented, Sylow
235C23×C3—31no—
236C23×C3—31no—
237C23×C3—61no—
238C23×C3—182no—
239C23×C3—184no—
240C23×C3—181no—
241C23×C3—362no—
242C23×C3—362no—
243C23×C3#62611yescharacteristic
244C22×D3—64no—
245C22×D3—62no—
246C22×D3—61no—
247C22×D3—61no—
248C22×D3—62no—
249C22×D3—122no—
250C22×D3—122no—
251C22×D3—182no—
252C22×D3—184no—
253C22×D3—362no—
254C22×D3—362no—
255C3×D4—94no—
256C3×D4—182no—
257C3×D4—184no—
258C3×D4—182no—
259C3×D4—362no—
260C2×C3×C4—22no—
261C2×C3×C4—31no—
262C2×C3×C4—31no—
263C2×C3×C4—31no—
264C2×C3×C4—31no—
265C2×C3×C4—62no—
266C2×C3×C4—62no—
267C2×C3×C4—61no—
268C2×C3×C4—61no—
269C2×C3×C4—122no—
270C2×C3×C4—181no—
271C2×C3×C4#62611yescharacteristic
272C2×C3×C4C2×D3211yescharacteristic
273#67—122no—
274#67—122no—
275#67—124no—
276C2×Dic3—61no—
277C2×Dic3—61no—
278C2×Dic3—62no—
279D12—34no—
280D12—64no—
281D12—62no—
282D12—62no—
283D12—62no—
284D12—94no—
285D12—122no—
286D12—182no—
287D12—184no—
288C3×C8—182no—
289C3⋊C8—62no—
290C3⋊C8—182no—
291C2×C32—22no—
292C2×C32—22no—
293C2×C32—21no—
294C2×C32—21no—
295C2×C32—21no—
296C2×C32—42no—
297C2×C32#88211yescharacteristic
298C2×C32C2×D2411yescharacteristic
299C2×C32#90811yescharacteristic
300C2×C32C2×#21311yescharacteristic
301C2×C32#93211yescharacteristic
302C2×C32#95311yescharacteristic
303C3⋊D3—44no—
304C3×D3—124no—
305C3×D3—242no—
306C3×D3—241no—
307C3×D3—241no—
308C24—181no—
309C2×D4—92no—
310C2×D4—91no—
311C2×D4—184no—
312C2×D4—181no—
313C2×D4—181no—
314C22×C4((C3⋊D3)⋊C3)⋊C211yescharacteristic
315D8—544no—
316C2×C8—92no—
317C4⋊C4—91no—
318C22⋊C4—181no—
319C22×C3—21no—
320C22×C3—21no—
321C22×C3—31no—
322C22×C3—31no—
323C22×C3—32no—
324C22×C3—32no—
325C22×C3—61no—
326C22×C3—61no—
327C22×C3—61no—
328C22×C3—61no—
329C22×C3—61no—
330C22×C3—62no—
331C22×C3—121no—
332C22×C3—121no—
333C22×C3—184no—
334C22×C3—362no—
335C22×C3—362no—
336C22×C3—368no—
337C22×C3—361no—
338C22×C3—361no—
339C22×C3—361no—
340C22×C3#371412yes—
341C22×C3C2×#62611yescharacteristic
342C2×D3—64no—
343C2×D3—128no—
344C2×D3—122no—
345C2×D3—122no—
346C2×D3—121no—
347C2×D3—121no—
348C2×D3—121no—
349C2×D3—121no—
350C2×D3—121no—
351C2×D3—121no—
352C2×D3—122no—
353C2×D3—122no—
354C2×D3—122no—
355C2×D3—184no—
356C2×D3—362no—
357C2×D3—362no—
358C2×D3—368no—
359C3×C4—21no—
360C3×C4—32no—
361C3×C4—32no—
362C3×C4—61no—
363C3×C4—61no—
364C3×C4—62no—
365C3×C4—121no—
366C3×C4—361no—
367C3×C4D3×#6712yes—
368Dic3—121no—
369Dic3—121no—
370Dic3—122no—
371C32—21no—
372C32#473211yescharacteristic, complemented
373C32#521711yescharacteristic, complemented
374C23—184no—
375C23—182no—
376C23—181no—
377C23—362no—
378C23—362no—
379C23#620011yescharacteristic
380D4—94no—
381D4—184no—
382D4—182no—
383D4—182no—
384D4—362no—
385C2×C4—22no—
386C2×C4—181no—
387C2×C4#620011yescharacteristic
388C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
389C8—182no—
390C2×C3—22no—
391C2×C3—31no—
392C2×C3—31no—
393C2×C3—31no—
394C2×C3—31no—
395C2×C3—31no—
396C2×C3—31no—
397C2×C3—62no—
398C2×C3—62no—
399C2×C3—61no—
400C2×C3—61no—
401C2×C3—61no—
402C2×C3—122no—
403C2×C3—364no—
404C2×C3—721no—
405C2×C3C2×#371411yescharacteristic
406C2×C3#6383011yescharacteristic
407C2×C3#6397711yescharacteristic
408D3—124no—
409D3—241no—
410D3—241no—
411D3—242no—
412D3—364no—
413C22—21no—
414C22—21no—
415C22—184no—
416C22—368no—
417C22—362no—
418C22—362no—
419C22—361no—
420C22—361no—
421C22—361no—
422C22#8843612yescomplemented
423C22C2×#620011yescharacteristic
424C4—21no—
425C4—361no—
426C4#8867612yes—
427C3—31no—
428C3—31no—
429C3—61no—
430C3#1059116211yescharacteristic
431C2—22no—
432C2—364no—
433C2—721no—
434C2C2×#8843611yescharacteristic
435C2#1172107111yescharacteristic
436C2#1172132111yescharacteristic
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⋊((C3⋊D3)⋊C3)C2semidirect
2. #11721830C2semidirect
3. C2×#88436C2semidirect
4. #63657⋊C3C2semidirect
5. #11720308C2semidirect
11. C2×C4⋊#272C22semidirect
16. C4⋊#1084C22semidirect
17. C2×(C3⋊(C3⋊C8))⋊C3C22semidirect
19. C22×C4⋊(C3⋊D3)D3semidirect
26. #63657D3semidirect
44. C4⋊((C3⋊D3)⋊C3)D4semidirect
66. C2×C3⋊(C3⋊C8)C2×D3semidirect
78. C22×H(3,𝔽3)D8semidirect
131. C4⋊(C3⋊D3)#67semidirect
199. C22×C32D24semidirect
200. C22×C32#213semidirect
234. H(3,𝔽3)#447semidirect
372. C32#4732semidirect
373. C32#5217semidirect
422. C22#88436semidirect

Maximal subgroups

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

subgroupindex
C22×C4⋊((C3⋊D3)⋊C3)2
#117218302
C2×#884362
#63657⋊C32
#117203082
#105911683
#105911603