Finite Group Databasebeta

← C2×#11721188 #13151307

Subgroups — C2×#11721188 #13151307

Order 1728.

1279 conjugacy classes of subgroups; only the 155 normal subgroups are shown.

normal subgroups only

This group has 1279 conjugacy classes of subgroups — too many to draw. Only its 155 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×D4×#272C211yesmaximal, characteristic
2C2×#88651C211yesmaximal, characteristic, complemented
3C2×#3754⋊C3C212yesmaximal, complemented
4#11721188C218yesmaximal, direct product factor
5C2×#88437C212yesmaximal, complemented
6C2×#88434C211yesmaximal, characteristic, complemented
7C2×D3×#214—32nomaximal
8C23×#272—21no—
9C2×D4×H(3,𝔽3)C2211yescharacteristic
10C2×#6235—21no—
11D4×#272—22no—
12D4×#272C2214yes—
13C2×C4⋊#272C2211yescharacteristic
14C2×C4×#272C2211yescharacteristic
15C2×#634⋊C3C2212yescomplemented
16C2×#6199—21no—
17C2×#6196—21no—
18#88651—22no—
19#88651C2214yescomplemented
20#3754⋊C3C2218yescomplemented
21C2×(C3⋊(C3⋊C8))⋊C3C2212yescomplemented
22#88437C2218yescomplemented
23#88434C2214yescomplemented
24C2×C3×D3×D4—32no—
25C2×C3×D3×D4—61no—
26C2×D3×Dic6—32no—
27C2×#3754D312yescomplemented
28C2×C3×#214—32no—
29D3×#214—316no—
30C2×#3716—32no—
31C2×#3715—32no—
32C2×D3×C3⋊C8—32no—
33C23×H(3,𝔽3)—21no—
34C22×#272—21no—
35C22×#272—24no—
36C22×#272—44no—
37C22×#272D411yescharacteristic
38D4×H(3,𝔽3)—21no—
39D4×H(3,𝔽3)C2312yes—
40C2×C4×H(3,𝔽3)C2311yescharacteristic
41#6235—24no—
42C2×#1084D411yescharacteristic
43C4⋊#272—21no—
44C4⋊#272C2312yes—
45C4×#272D412yes—
46C4×#272C2312yes—
47C2×(C3⋊Dic3)⋊C3—22no—
48#634⋊C3—22no—
49#634⋊C3C2314yescomplemented
50#6199—24no—
51#6196—24no—
52(C3⋊(C3⋊C8))⋊C3C2314yescomplemented
53C22×#214—91no—
54C2×D3×SD16—92no—
55C23×C3×D3—62no—
56C23×C3×D3—62no—
57C2×C32×D4—21no—
58C2×C32×D4C2×D312yes—
59C2×#634C2×D312yescomplemented
60C2×C3×#67—62no—
61C2×C3×#67—62no—
62C3×D3×D4—38no—
63C3×D3×D4—64no—
64C3×D3×D4—64no—
65C3×D3×D4—64no—
66C2×C3×D12—32no—
67C2×C3×D12—61no—
68C2×C3×C4×D3—32no—
69C2×C3×C4×D3—61no—
70C2×C3×Dic6—32no—
71C2×#627—62no—
72C2×D3×Dic3—62no—
73D3×Dic6—38no—
74D3×Dic6—64no—
75#3754C2×D318yescomplemented
76C2×C3⋊(C3⋊C8)C2×D312yescomplemented
77C3×#214—38no—
78C2×C3×C3⋊C8—32no—
79#3716—38no—
80#3715—38no—
81D3×C3⋊C8—38no—
82C22×H(3,𝔽3)—22no—
83C22×H(3,𝔽3)—42no—
84C22×H(3,𝔽3)C2×D411yescharacteristic
85C2×#272—22no—
86C2×#272—42no—
87C2×#272—48no—
88C2×#272SD1614yescomplemented
89C2×#272C2×D412yes—
90C4×H(3,𝔽3)C2×D411yescharacteristic
91C4×H(3,𝔽3)C2411yescharacteristic
92#1084C2×D412yes—
93(C3⋊Dic3)⋊C3—24no—
94C22×C3×D4—91no—
95C2×D3×Q8—92no—
96C2×D3×D4—32no—
97C2×D3×D4—61no—
98C22×Dic6—91no—
99C2×C3×SD16—92no—
100C2×#215—92no—
101C2×#214—32no—
102C2×#214—92no—
103C2×#214—92no—
104C2×#214—98no—
105C22×C3⋊C8—91no—
106D3×SD16—916no—
107C2×#204—92no—
108C2×D3×C8—92no—
109C23×C32—22no—
110C23×C32—22no—
111C22×C3×D3—32no—
112C22×C3×D3—61no—
113C22×C3×D3—68no—
114C22×C3×D3—62no—
115C22×C3×D3—68no—
116C22×C3×D3—62no—
117C22×C3×D3—128no—
118C22×C3×D3—128no—
119C32×D4—22no—
120C32×D4—22no—
121C32×D4—22no—
122C32×D4C22×D314yes—
123C2×C32×C4—21no—
124C2×C32×C4C22×D312yes—
125C2×C3⋊Dic3—22no—
126#634—22no—
127#634C22×D314yescomplemented
128C3×#67—68no—
129C3×#67—68no—
130C2×C3×Dic3—32no—
131C2×C3×Dic3—61no—
132C2×C3×Dic3—62no—
133C3×D12—34no—
134C3×D12—62no—
135C3×D12—62no—
136C3×D12—62no—
137C3×C4×D3—34no—
138C3×C4×D3—34no—
139C3×C4×D3—62no—
140C3×C4×D3—62no—
141C3×Dic6—34no—
142C3×Dic6—62no—
143#627—68no—
144D3×Dic3—68no—
145C3⋊(C3⋊C8)C22×D314yescomplemented
146C3×C3⋊C8—34no—
147C22×SD16—271noSylow
148C2×H(3,𝔽3)—42no—
149C2×H(3,𝔽3)C2×SD1612yescomplemented
150C2×H(3,𝔽3)C22×D411yescharacteristic
151#272—42no—
152#272C2×SD1614yescomplemented
153C24×C3—181no—
154C23×D3—62no—
155C23×D3—62no—
156C2×C3×Q8—92no—
157C2×C3×D4—32no—
158C2×C3×D4—61no—
159C2×C3×D4—94no—
160C2×C3×D4—91no—
161C2×C3×D4—182no—
162C2×C3×D4—181no—
163C2×C3×D4D3211yescharacteristic
164C22×C3×C4—91no—
165C2×#67—62no—
166C2×#67—62no—
167C22×Dic3—181no—
168D3×Q8—98no—
169D3×Q8—184no—
170D3×D4—38no—
171D3×D4—64no—
172D3×D4—64no—
173D3×D4—64no—
174C2×D12—32no—
175C2×D12—61no—
176C2×C4×D3—32no—
177C2×C4×D3—61no—
178C2×C4×D3—182no—
179C2×Dic6—32no—
180C2×Dic6—92no—
181C2×Dic6—94no—
182C2×Dic6—182no—
183C2×Dic6—182no—
184C2×Dic6—181no—
185C3×SD16—98no—
186C2×C3×C8—92no—
187#215—98no—
188#214—38no—
189#214—98no—
190#214—98no—
191C2×C3⋊C8—32no—
192C2×C3⋊C8—92no—
193C2×C3⋊C8—94no—
194#204—98no—
195D3×C8—98no—
196C22×C32—24no—
197C22×C32—24no—
198C22×C32—21no—
199C22×C32—44no—
200C22×C32—44no—
201C22×C32D3×D412yes—
202C2×C3×D3—38no—
203C2×C3×D3—34no—
204C2×C3×D3—64no—
205C2×C3×D3—62no—
206C2×C3×D3—64no—
207C2×C3×D3—64no—
208C2×C3×D3—1216no—
209C2×C3×D3—1216no—
210C2×C3×D3—124no—
211C2×C3×D3—124no—
212C32×C4—21no—
213C32×C4—21no—
214C32×C4D3×D412yes—
215C32×C4C23×D312yes—
216C3⋊Dic3—24no—
217C3×Dic3—34no—
218C3×Dic3—62no—
219C3×Dic3—64no—
220C22×Q8—271no—
221C22×D4—91no—
222C2×SD16—92no—
223C2×SD16—272no—
224C2×SD16—278no—
225C22×C8—271no—
226H(3,𝔽3)C22×SD1611yescharacteristic, complemented, Sylow
227C23×C3—21no—
228C23×C3—62no—
229C23×C3—62no—
230C23×C3—91no—
231C23×C3—184no—
232C23×C3—181no—
233C23×C3—364no—
234C22×D3—32no—
235C22×D3—61no—
236C22×D3—68no—
237C22×D3—62no—
238C22×D3—62no—
239C22×D3—68no—
240C22×D3—128no—
241C22×D3—128no—
242C3×Q8—94no—
243C3×Q8—182no—
244C3×D4—21no—
245C3×D4—34no—
246C3×D4—62no—
247C3×D4—62no—
248C3×D4—62no—
249C3×D4—92no—
250C3×D4—184no—
251C3×D4—181no—
252C3×D4C2×D3212yes—
253C2×C3×C4—32no—
254C2×C3×C4—61no—
255C2×C3×C4—91no—
256C2×C3×C4—92no—
257C2×C3×C4—92no—
258C2×C3×C4—182no—
259C2×C3×C4C2×D3211yescharacteristic
260#67—68no—
261#67—68no—
262C2×Dic3—32no—
263C2×Dic3—61no—
264C2×Dic3—62no—
265C2×Dic3—182no—
266C2×Dic3—184no—
267D12—34no—
268D12—62no—
269D12—62no—
270D12—62no—
271C4×D3—34no—
272C4×D3—34no—
273C4×D3—62no—
274C4×D3—62no—
275C4×D3—188no—
276Dic6—34no—
277Dic6—62no—
278Dic6—94no—
279Dic6—182no—
280Dic6—188no—
281Dic6—184no—
282C3×C8—94no—
283C3⋊C8—34no—
284C3⋊C8—94no—
285C2×C32—21no—
286C2×C32—22no—
287C2×C32—44no—
288C2×C32—44no—
289C2×C32D3×SD1614yescomplemented
290C2×C32C2×D3×D412yes—
291C3×D3—38no—
292C3×D3—64no—
293C3×D3—124no—
294C3×D3—124no—
295C24—181no—
296C2×Q8—92no—
297C2×Q8—274no—
298C2×Q8—541no—
299C2×Q8—542no—
300C2×D4—94no—
301C2×D4—91no—
302C2×D4—182no—
303C2×D4—181no—
304C2×D4((C3⋊D3)⋊C3)⋊C211yescharacteristic
305C22×C4—91no—
306C22×C4—541no—
307SD16—98no—
308SD16—278no—
309C2×C8—92no—
310C2×C8—274no—
311C22×C3—22no—
312C22×C3—32no—
313C22×C3—42no—
314C22×C3—64no—
315C22×C3—64no—
316C22×C3—61no—
317C22×C3—92no—
318C22×C3—94no—
319C22×C3—124no—
320C22×C3—124no—
321C22×C3—182no—
322C22×C3—368no—
323C22×C3—362no—
324C22×C3#381111yescharacteristic
325C2×D3—34no—
326C2×D3—38no—
327C2×D3—62no—
328C2×D3—64no—
329C2×D3—64no—
330C2×D3—64no—
331C2×D3—1216no—
332C2×D3—1216no—
333C2×D3—124no—
334C2×D3—124no—
335C3×C4—32no—
336C3×C4—32no—
337C3×C4—61no—
338C3×C4—61no—
339C3×C4—92no—
340C3×C4—184no—
341C3×C4#381111yescharacteristic
342C3×C4C22×D3211yescharacteristic
343Dic3—34no—
344Dic3—62no—
345Dic3—64no—
346Dic3—184no—
347C32—21no—
348C32C2×D3×SD1612yescomplemented
349C23—21no—
350C23—91no—
351C23—184no—
352C23—181no—
353C23—364no—
354Q8—94no—
355Q8—182no—
356Q8—544no—
357D4—21no—
358D4—92no—
359D4—184no—
360D4—181no—
361D4C2×((C3⋊D3)⋊C3)⋊C212yes—
362C2×C4—91no—
363C2×C4—92no—
364C2×C4—92no—
365C2×C4—182no—
366C2×C4—544no—
367C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
368C8—94no—
369C2×C3—32no—
370C2×C3—34no—
371C2×C3—42no—
372C2×C3—61no—
373C2×C3—62no—
374C2×C3—94no—
375C2×C3—124no—
376C2×C3—124no—
377C2×C3—362no—
378C2×C3#6393012yes—
379C2×C3C2×#381111yescharacteristic
380D3—38no—
381D3—64no—
382D3—124no—
383D3—124no—
384C22—22no—
385C22—42no—
386C22—92no—
387C22—94no—
388C22—182no—
389C22—368no—
390C22—362no—
391C22#8867711yescharacteristic
392C4—92no—
393C4—184no—
394C4#8867711yescharacteristic
395C4C22×((C3⋊D3)⋊C3)⋊C211yescharacteristic
396C3—32no—
397C3—61no—
398C3C2×#6393011yescharacteristic
399C2—42no—
400C2—94no—
401C2—362no—
402C2#1172118812yesdirect product factor
403C2C2×#8867711yescharacteristic
subgroupG/H
centerC22#88677
commutator, cosocleC4×H(3,𝔽3)C24
FrattiniC3×C4C22×D32
FittingC2×D4×H(3,𝔽3)C22
socleC22×C3#3811
prime radicalC2×D4×#272C2
hypercenterC2×D4((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)C22×SD16
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
2. C2×#88651C2semidirect
3. C2×#3754⋊C3C2semidirect
4. #11721188C2direct (N × H)
5. C2×#88437C2semidirect
6. C2×#88434C2semidirect
15. C2×#634⋊C3C22semidirect
19. #88651C22semidirect
20. #3754⋊C3C22semidirect
21. C2×(C3⋊(C3⋊C8))⋊C3C22semidirect
22. #88437C22semidirect
23. #88434C22semidirect
27. C2×#3754D3semidirect
49. #634⋊C3C23semidirect
52. (C3⋊(C3⋊C8))⋊C3C23semidirect
59. C2×#634C2×D3semidirect
75. #3754C2×D3semidirect
76. C2×C3⋊(C3⋊C8)C2×D3semidirect
88. C2×#272SD16semidirect
127. #634C22×D3semidirect
145. C3⋊(C3⋊C8)C22×D3semidirect
149. C2×H(3,𝔽3)C2×SD16semidirect
152. #272C2×SD16semidirect
226. H(3,𝔽3)C22×SD16semidirect
289. C2×C32D3×SD16semidirect
348. C32C2×D3×SD16semidirect
402. C2#11721188direct (N × H)

Maximal subgroups

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

subgroupindex
C2×D4×#2722
C2×#886512
C2×#3754⋊C32
#117211882
C2×#884372
C2×#884342
C2×D3×#2143