Finite Group Databasebeta

← C2×#11721191 #13151390

Subgroups — C2×#11721191 #13151390

Order 1728.

1207 conjugacy classes of subgroups; only the 147 normal subgroups are shown.

normal subgroups only

This group has 1207 conjugacy classes of subgroups — too many to draw. Only its 147 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
1C2×#88744C211yesmaximal, characteristic, complemented
2C2×#88655C211yesmaximal, characteristic, complemented
3C2×#3753⋊C3C212yesmaximal, complemented
4#11721191C218yesmaximal, direct product factor
5C2×#88438C212yesmaximal, complemented
6C2×#88435C211yesmaximal, characteristic, complemented
7C2×#63925—32nomaximal
8C22×((C3⋊D3)⋊C3)⋊C2—21no—
9C2×D4×H(3,𝔽3)C2211yescharacteristic, complemented
10C2×#6235—21no—
11C22×#1084—21no—
12#88744—22no—
13#88744C2214yescomplemented
14C2×C4×#272C2211yescharacteristic
15C2×#6229C2211yescharacteristic, complemented
16C2×C4⋊((C3⋊D3)⋊C3)C2212yes—
17C2×#6198—21no—
18#88655—22no—
19#88655C2214yescomplemented
20#3753⋊C3C2218yescomplemented
21C2×(C3⋊(C3⋊C8))⋊C3C2212yescomplemented
22#88438C2218yescomplemented
23#88435C2214yescomplemented
24C2×C3×#237—32no—
25C2×C3×#237—61no—
26C2×D3×D12—32no—
27C2×#3753D312yescomplemented
28C2×C3×#213—32no—
29#63925—316no—
30C2×#3717—32no—
31C2×#3715—32no—
32C2×#3711—32no—
33C23×H(3,𝔽3)—21no—
34C22×#272D411yescharacteristic
35C22×(C3⋊D3)⋊C3—22no—
36C2×((C3⋊D3)⋊C3)⋊C2—24no—
37C2×((C3⋊D3)⋊C3)⋊C2—44no—
38D4×H(3,𝔽3)—21no—
39D4×H(3,𝔽3)C2312yescomplemented
40C2×C4×H(3,𝔽3)C2311yescharacteristic
41#6235—24no—
42C2×#1084—21no—
43C2×#1084—24no—
44C2×#1084D411yescharacteristic
45C4×#272D412yes—
46C4×#272C2312yes—
47#6229—21no—
48#6229C2312yescomplemented
49C4⋊((C3⋊D3)⋊C3)—22no—
50C4⋊((C3⋊D3)⋊C3)C2314yes—
51#6198—24no—
52(C3⋊(C3⋊C8))⋊C3C2314yescomplemented
53C2×#949—91no—
54C2×#911—92no—
55C22×D32—62no—
56C2×C32×D4—21no—
57C2×C32×D4C2×D312yescomplemented
58C2×C4⋊(C3⋊D3)C2×D312yes—
59C2×C3×#67—62no—
60C2×C3×#67—62no—
61C22×C3×Dic3—62no—
62C22×C3×Dic3—62no—
63C3×#237—38no—
64C3×#237—64no—
65C3×#237—64no—
66C3×#237—64no—
67C2×C3×D12—32no—
68C2×C3×C4×D3—32no—
69C2×C3×C4×D3—61no—
70C2×C3×Dic6—32no—
71C2×C3×Dic6—61no—
72C2×#626—62no—
73D3×D12—38no—
74D3×D12—64no—
75#3753C2×D318yescomplemented
76C2×C3⋊(C3⋊C8)C2×D312yescomplemented
77C3×#213—38no—
78C2×C3×C3⋊C8—32no—
79#3717—38no—
80#3715—38no—
81#3711—38no—
82C22×H(3,𝔽3)—22no—
83C22×H(3,𝔽3)—42no—
84C22×H(3,𝔽3)C2×D411yescharacteristic
85C2×#272—22no—
86C2×#272C2×D412yes—
87C2×(C3⋊D3)⋊C3—24no—
88C2×(C3⋊D3)⋊C3—44no—
89((C3⋊D3)⋊C3)⋊C2—48no—
90C4×H(3,𝔽3)C2×D411yescharacteristic
91C4×H(3,𝔽3)C2411yescharacteristic
92#1084—22no—
93#1084C2×D412yes—
94C2×C3×#41—91no—
95C2×#237—32no—
96C2×#237—61no—
97C2×D3×D4—92no—
98C22×D12—91no—
99C2×C3×D8—92no—
100#949—98no—
101C2×#215—92no—
102C2×#214—92no—
103C2×#213—32no—
104C2×#213—92no—
105C2×#208—91no—
106#911—916no—
107C2×#204—92no—
108C2×ΓL(1,𝔽25)—92no—
109C23×C32—22no—
110C23×C32—22no—
111C22×C3⋊D3—22no—
112C22×C3×D3—32no—
113C22×C3×D3—61no—
114C22×C3×D3—62no—
115C2×D32—68no—
116C2×D32—128no—
117C32×D4—22no—
118C32×D4—22no—
119C32×D4—22no—
120C32×D4C22×D314yescomplemented
121C2×C32×C4—21no—
122C2×C32×C4C22×D312yes—
123C4⋊(C3⋊D3)—22no—
124C4⋊(C3⋊D3)C22×D314yes—
125C3×#67—68no—
126C3×#67—68no—
127C2×C3×Dic3—32no—
128C2×C3×Dic3—61no—
129C2×C3×Dic3—68no—
130C2×C3×Dic3—62no—
131C2×C3×Dic3—68no—
132C2×C3×Dic3—62no—
133C3×D12—34no—
134C3×D12—62no—
135C3×C4×D3—34no—
136C3×C4×D3—34no—
137C3×C4×D3—62no—
138C3×C4×D3—62no—
139C3×Dic6—34no—
140C3×Dic6—62no—
141C3×Dic6—62no—
142C3×Dic6—62no—
143#626—68no—
144C3⋊(C3⋊C8)C22×D314yescomplemented
145C3×C3⋊C8—34no—
146C2×AGL(1,ℤ/8)—271noSylow
147C2×H(3,𝔽3)—42no—
148C2×H(3,𝔽3)AGL(1,ℤ/8)12yescomplemented
149C2×H(3,𝔽3)C22×D411yescharacteristic
150#272—22no—
151(C3⋊D3)⋊C3—44no—
152C23×D3—182no—
153C23×D3—181no—
154C3×#41—94no—
155C3×#41—182no—
156C2×C3×Q8—91no—
157C2×C3×D4—32no—
158C2×C3×D4—61no—
159C2×C3×D4—92no—
160C2×C3×D4—181no—
161C2×C3×D4D3211yescharacteristic
162C22×C3×C4—91no—
163C22×C3×C4—181no—
164C2×#67—62no—
165C2×#67—62no—
166C2×#67—182no—
167C22×Dic3—62no—
168C22×Dic3—62no—
169#237—38no—
170#237—64no—
171#237—64no—
172#237—64no—
173D3×D4—98no—
174D3×D4—184no—
175C2×D12—32no—
176C2×D12—92no—
177C2×D12—94no—
178C2×D12—182no—
179C2×D12—181no—
180C2×C4×D3—32no—
181C2×C4×D3—61no—
182C2×Dic6—32no—
183C2×Dic6—61no—
184C3×D8—98no—
185C2×C3×C8—92no—
186#215—98no—
187#214—98no—
188#213—38no—
189#213—98no—
190#208—94no—
191C2×C3⋊C8—32no—
192C2×C3⋊C8—92no—
193#204—98no—
194ΓL(1,𝔽25)—98no—
195C22×C32—24no—
196C22×C32—24no—
197C22×C32—21no—
198C22×C32—44no—
199C22×C32—44no—
200C22×C32D3×D412yes—
201C2×C3⋊D3—24no—
202C2×C3⋊D3—44no—
203C2×C3×D3—34no—
204C2×C3×D3—62no—
205C2×C3×D3—64no—
206C2×C3×D3—64no—
207C2×C3×D3—122no—
208C2×C3×D3—124no—
209D32—1216no—
210C32×C4—21no—
211C32×C4—21no—
212C32×C4D3×D412yes—
213C32×C4C23×D312yes—
214C3×Dic3—34no—
215C3×Dic3—62no—
216C3×Dic3—64no—
217C3×Dic3—64no—
218C2×#41—91no—
219C22×D4—271no—
220AGL(1,ℤ/8)—278no—
221C2×SD16—272no—
222C2×D8—92no—
223C2×M16—271no—
224H(3,𝔽3)C2×AGL(1,ℤ/8)11yescharacteristic, complemented, Sylow
225C23×C3—21no—
226C23×C3—62no—
227C23×C3—62no—
228C23×C3—91no—
229C23×C3—182no—
230C22×D3—32no—
231C22×D3—61no—
232C22×D3—62no—
233C22×D3—182no—
234C22×D3—188no—
235C22×D3—184no—
236C22×D3—368no—
237C22×D3—364no—
238C3×Q8—92no—
239C3×Q8—181no—
240C3×D4—21no—
241C3×D4—34no—
242C3×D4—62no—
243C3×D4—62no—
244C3×D4—62no—
245C3×D4—94no—
246C3×D4—182no—
247C3×D4—184no—
248C3×D4C2×D3212yes—
249C2×C3×C4—32no—
250C2×C3×C4—61no—
251C2×C3×C4—91no—
252C2×C3×C4—92no—
253C2×C3×C4—92no—
254C2×C3×C4—184no—
255C2×C3×C4—181no—
256C2×C3×C4C2×D3211yescharacteristic
257#67—68no—
258#67—68no—
259#67—188no—
260C2×Dic3—32no—
261C2×Dic3—61no—
262C2×Dic3—68no—
263C2×Dic3—62no—
264C2×Dic3—62no—
265C2×Dic3—68no—
266D12—34no—
267D12—62no—
268D12—94no—
269D12—182no—
270D12—184no—
271C4×D3—34no—
272C4×D3—34no—
273C4×D3—62no—
274C4×D3—62no—
275Dic6—34no—
276Dic6—62no—
277Dic6—62no—
278Dic6—62no—
279C3×C8—94no—
280C3⋊C8—34no—
281C3⋊C8—94no—
282C2×C32—21no—
283C2×C32—22no—
284C2×C32—44no—
285C2×C32—44no—
286C2×C32#91114yescomplemented
287C2×C32C2×D3×D412yes—
288C3⋊D3—44no—
289C3×D3—64no—
290C3×D3—122no—
291C3×D3—124no—
292C24—541no—
293#41—94no—
294#41—182no—
295C2×Q8—91no—
296C2×D4—92no—
297C2×D4—181no—
298C2×D4—274no—
299C2×D4—542no—
300C2×D4—541no—
301C2×D4((C3⋊D3)⋊C3)⋊C211yescharacteristic, complemented
302C22×C4—91no—
303C22×C4—181no—
304SD16—278no—
305D8—98no—
306M16—274no—
307C2×C8—92no—
308C22×C3—22no—
309C22×C3—32no—
310C22×C3—42no—
311C22×C3—64no—
312C22×C3—64no—
313C22×C3—61no—
314C22×C3—92no—
315C22×C3—124no—
316C22×C3—124no—
317C22×C3—184no—
318C22×C3—182no—
319C22×C3—364no—
320C22×C3#381111yescharacteristic
321C2×D3—34no—
322C2×D3—62no—
323C2×D3—64no—
324C2×D3—64no—
325C2×D3—122no—
326C2×D3—124no—
327C2×D3—184no—
328C2×D3—364no—
329C2×D3—3616no—
330C2×D3—368no—
331C3×C4—32no—
332C3×C4—32no—
333C3×C4—61no—
334C3×C4—61no—
335C3×C4—92no—
336C3×C4—182no—
337C3×C4#381111yescharacteristic
338C3×C4C22×D3211yescharacteristic
339Dic3—34no—
340Dic3—62no—
341Dic3—64no—
342Dic3—64no—
343C32—21no—
344C32C2×#91112yescomplemented
345C23—21no—
346C23—91no—
347C23—182no—
348C23—544no—
349C23—1084no—
350Q8—92no—
351Q8—181no—
352D4—21no—
353D4—94no—
354D4—184no—
355D4—182no—
356D4—544no—
357D4C2×((C3⋊D3)⋊C3)⋊C212yescomplemented
358C2×C4—91no—
359C2×C4—92no—
360C2×C4—92no—
361C2×C4—184no—
362C2×C4—181no—
363C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
364C8—94no—
365C2×C3—32no—
366C2×C3—34no—
367C2×C3—42no—
368C2×C3—61no—
369C2×C3—62no—
370C2×C3—124no—
371C2×C3—124no—
372C2×C3—182no—
373C2×C3—364no—
374C2×C3#6392712yes—
375C2×C3C2×#381111yescharacteristic
376D3—64no—
377D3—122no—
378D3—124no—
379D3—364no—
380C22—22no—
381C22—42no—
382C22—92no—
383C22—182no—
384C22—184no—
385C22—364no—
386C22—1088no—
387C22#8867711yescharacteristic
388C4—92no—
389C4—182no—
390C4#8867711yescharacteristic
391C4C22×((C3⋊D3)⋊C3)⋊C211yescharacteristic
392C3—32no—
393C3—61no—
394C3C2×#6392711yescharacteristic
395C2—42no—
396C2—182no—
397C2—364no—
398C2#1172119112yesdirect product factor
399C2C2×#8867711yescharacteristic
subgroupG/H
centerC22#88677
commutator, cosocleC4×H(3,𝔽3)C24
FrattiniC3×C4C22×D32
FittingC2×D4×H(3,𝔽3)C22
socleC22×C3#3811
hypercenterC2×D4((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)C2×AGL(1,ℤ/8)
LCSG ▷ 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
1. C2×#88744C2semidirect
2. C2×#88655C2semidirect
3. C2×#3753⋊C3C2semidirect
4. #11721191C2direct (N × H)
5. C2×#88438C2semidirect
6. C2×#88435C2semidirect
9. C2×D4×H(3,𝔽3)C22semidirect
13. #88744C22semidirect
15. C2×#6229C22semidirect
19. #88655C22semidirect
20. #3753⋊C3C22semidirect
21. C2×(C3⋊(C3⋊C8))⋊C3C22semidirect
22. #88438C22semidirect
23. #88435C22semidirect
27. C2×#3753D3semidirect
39. D4×H(3,𝔽3)C23semidirect
48. #6229C23semidirect
52. (C3⋊(C3⋊C8))⋊C3C23semidirect
57. C2×C32×D4C2×D3semidirect
75. #3753C2×D3semidirect
76. C2×C3⋊(C3⋊C8)C2×D3semidirect
120. C32×D4C22×D3semidirect
144. C3⋊(C3⋊C8)C22×D3semidirect
148. C2×H(3,𝔽3)AGL(1,ℤ/8)semidirect
224. H(3,𝔽3)C2×AGL(1,ℤ/8)semidirect
286. C2×C32#911semidirect
301. C2×D4((C3⋊D3)⋊C3)⋊C2semidirect
344. C32C2×#911semidirect
357. D4C2×((C3⋊D3)⋊C3)⋊C2semidirect
398. C2#11721191direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#887442
C2×#886552
C2×#3753⋊C32
#117211912
C2×#884382
C2×#884352
C2×#639253