Finite Group Databasebeta

← C2×#11722840 #13170578

Subgroups — C2×#11722840 #13170578

Order 1728.

1426 conjugacy classes of subgroups; only the 212 normal subgroups are shown.

normal subgroups only

This group has 1426 conjugacy classes of subgroups — too many to draw. Only its 212 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×C4⋊#1113C211yesmaximal, characteristic
2C2×C3×#3797C211yesmaximal, characteristic, complemented
3C2×#88793C212yesmaximal, complemented
4C2×C3⋊D3×C3⋊C8C211yesmaximal, characteristic, complemented
5C2×C3×#3717C212yesmaximal, complemented
6#11722840C218yesmaximal, direct product factor
7C2×D3×#215—32nomaximal
8C2×#63796—31nomaximal
9C22×#1113—21no—
10C2×C3×C4⋊(C3⋊D3)C2212yes—
11C2×C3×C4×C3⋊D3C2211yescharacteristic
12C2×C32×Dic6C2212yescomplemented
13C2×#6295—21no—
14C4⋊#1113—22no—
15C4⋊#1113C2214yes—
16C2×C3×#627—21no—
17C2×C3×#624—21no—
18C3×#3797—22no—
19C3×#3797C2214yescomplemented
20C2×C3⋊(C3⋊(C3⋊C8))C2211yescharacteristic, complemented
21C2×C32×C3⋊C8C2211yescharacteristic, complemented
22#88793C2218yescomplemented
23C3⋊D3×C3⋊C8C2214yescomplemented
24C3×#3717C2218yescomplemented
25C2×C3×D3×Q8—32no—
26C2×#3802—31no—
27C2×D3×D12—32no—
28C2×#3797D311yescharacteristic, complemented
29C2×#3755—32no—
30C2×C8×C3⋊D3—31no—
31C2×C3×#215—32no—
32C2×C3×#204—32no—
33D3×#215—316no—
34C2×#3717D312yescomplemented
35C2×#3715—32no—
36C2×#3715—32no—
37C2×#3715—62no—
38C2×D3×C3⋊C8—32no—
39C2×D3×C3⋊C8—61no—
40#63796—38no—
41C22×C3×C3⋊D3—22no—
42C22×C3×C3⋊D3D411yescharacteristic
43C2×#1113—24no—
44C2×#1113—44no—
45C2×C33×C4C2311yescharacteristic
46C2×C3×C3⋊Dic3D411yescharacteristic
47C3×C4⋊(C3⋊D3)—22no—
48C3×C4⋊(C3⋊D3)C2314yes—
49C3×C4×C3⋊D3D412yes—
50C3×C4×C3⋊D3C2312yes—
51C2×C32×Dic3—22no—
52C32×Dic6—22no—
53C32×Dic6C2314yescomplemented
54#6295—24no—
55C3×#627—24no—
56C3×#624—24no—
57C3⋊(C3⋊(C3⋊C8))C2312yescomplemented
58C32×C3⋊C8C2312yescomplemented
59C22×#215—91no—
60C2×D3×SD16—92no—
61C22×D32—62no—
62C22×D32—61no—
63C2×C32×Q8—32no—
64C2×C4⋊(C3⋊D3)C2×D312yes—
65C2×C4×C3⋊D3C2×D311yescharacteristic
66C3×D3×Q8—38no—
67C3×D3×Q8—64no—
68C2×C3×D12—32no—
69C2×C3×D12—32no—
70C2×C3×D12—62no—
71C2×C3×C4×D3—32no—
72C2×C3×C4×D3—62no—
73C2×C3×C4×D3—61no—
74C2×C3×Dic6—22no—
75C2×C3×Dic6—62no—
76C2×C3×Dic6C2×D312yescomplemented
77C2×C3×Dic6C2×D312yescomplemented
78C2×#627—21no—
79C2×#626—62no—
80C2×#625—61no—
81C2×#624—21no—
82#3802—34no—
83#3802—62no—
84D3×D12—38no—
85D3×D12—64no—
86#3797—22no—
87#3797C2×D314yescomplemented
88C2×C32×C8—31no—
89#3755—38no—
90C2×C3⋊(C3⋊C8)—32no—
91C2×C3⋊(C3⋊C8)—31no—
92C2×C3⋊(C3⋊C8)—61no—
93C2×C3⋊(C3⋊C8)—62no—
94C2×C3⋊(C3⋊C8)—121no—
95C8×C3⋊D3—34no—
96C3×#215—38no—
97C2×C3×C3⋊C8—21no—
98C2×C3×C3⋊C8C2×D312yescomplemented
99C3×#204—38no—
100#3717C2×D318yescomplemented
101#3715—38no—
102#3715—38no—
103#3715—68no—
104D3×C3⋊C8—38no—
105D3×C3⋊C8—64no—
106C22×C33C2×D411yescharacteristic
107C2×C3×C3⋊D3—24no—
108C2×C3×C3⋊D3—44no—
109C2×C3×C3⋊D3SD1614yescomplemented
110C2×C3×C3⋊D3C2×D412yes—
111#1113—48no—
112C33×C4C2×D411yescharacteristic
113C33×C4C2411yescharacteristic
114C3×C3⋊Dic3C2×D412yes—
115C32×Dic3—24no—
116C22×C3×Q8—91no—
117C2×D3×Q8—32no—
118C2×D3×D4—92no—
119C22×D12—91no—
120C2×C3×SD16—92no—
121C2×#215—32no—
122C2×#215—98no—
123C2×#215—92no—
124C2×#215—92no—
125C2×#215—182no—
126C2×#214—92no—
127C22×C3⋊C8—91no—
128D3×SD16—916no—
129C2×#204—32no—
130C2×D3×C8—92no—
131C2×D3×C8—181no—
132C22×C3⋊D3—22no—
133C22×C3⋊D3#6711yescharacteristic
134C22×C3×D3—32no—
135C22×C3×D3—61no—
136C22×C3×D3—62no—
137C22×C3×D3—62no—
138C22×C3×D3—122no—
139C2×D32—68no—
140C2×D32—64no—
141C2×D32—128no—
142C2×D32—124no—
143C32×Q8—34no—
144C32×Q8—62no—
145C2×C32×C4—21no—
146C2×C32×C4—22no—
147C2×C32×C4—41no—
148C2×C32×C4—62no—
149C2×C32×C4C22×D312yes—
150C2×C32×C4C22×D311yescharacteristic
151C2×C3⋊Dic3#6711yescharacteristic
152C4⋊(C3⋊D3)—22no—
153C4⋊(C3⋊D3)C22×D314yes—
154C4×C3⋊D3#6712yes—
155C4×C3⋊D3C22×D312yes—
156C2×C3×Dic3—22no—
157C2×C3×Dic3—22no—
158C2×C3×Dic3—32no—
159C2×C3×Dic3—42no—
160C2×C3×Dic3—61no—
161C3×D12—34no—
162C3×D12—34no—
163C3×D12—64no—
164C3×D12—62no—
165C3×D12—62no—
166C3×D12—122no—
167C3×C4×D3—34no—
168C3×C4×D3—34no—
169C3×C4×D3—62no—
170C3×C4×D3—62no—
171C3×C4×D3—68no—
172C3×Dic6—22no—
173C3×Dic6—22no—
174C3×Dic6—24no—
175C3×Dic6—42no—
176C3×Dic6—68no—
177C3×Dic6C22×D314yescomplemented
178C3×Dic6C22×D314yescomplemented
179#627—24no—
180#626—68no—
181#625—64no—
182#624—24no—
183C32×C8—32no—
184C3⋊(C3⋊C8)—34no—
185C3⋊(C3⋊C8)—32no—
186C3⋊(C3⋊C8)—62no—
187C3⋊(C3⋊C8)—64no—
188C3⋊(C3⋊C8)—122no—
189C3×C3⋊C8—22no—
190C3×C3⋊C8C22×D314yescomplemented
191C22×SD16—271noSylow
192C2×C33C2×SD1612yescomplemented
193C2×C33C22×D411yescharacteristic
194C3×C3⋊D3—44no—
195C3×C3⋊D3C2×SD1614yescomplemented
196C23×D3—182no—
197C23×D3—181no—
198C2×C3×Q8—32no—
199C2×C3×Q8—32no—
200C2×C3×Q8—62no—
201C2×C3×Q8—94no—
202C2×C3×Q8—181no—
203C2×C3×Q8—182no—
204C2×C3×D4—92no—
205C22×C3×C4—91no—
206C22×C3×C4—181no—
207C2×#67—182no—
208D3×Q8—38no—
209D3×Q8—64no—
210D3×D4—98no—
211D3×D4—184no—
212C2×D12—32no—
213C2×D12—32no—
214C2×D12—62no—
215C2×D12—94no—
216C2×D12—182no—
217C2×D12—181no—
218C2×C4×D3—32no—
219C2×C4×D3—61no—
220C2×C4×D3—62no—
221C2×Dic6—62no—
222C2×Dic6D3212yescomplemented
223C3×SD16—98no—
224C2×C3×C8—32no—
225C2×C3×C8—61no—
226#215—38no—
227#215—98no—
228#215—98no—
229#215—188no—
230#214—98no—
231C2×C3⋊C8—92no—
232C2×C3⋊C8—91no—
233C2×C3⋊C8—94no—
234C2×C3⋊C8—181no—
235C2×C3⋊C8—182no—
236C2×C3⋊C8—361no—
237C2×C3⋊C8D3211yescharacteristic, complemented
238#204—38no—
239D3×C8—98no—
240D3×C8—184no—
241C22×C32—21no—
242C22×C32—22no—
243C22×C32—41no—
244C22×C32D3×D412yes—
245C22×C32C2×#6711yescharacteristic
246C2×C3⋊D3—24no—
247C2×C3⋊D3—44no—
248C2×C3⋊D3#21514yescomplemented
249C2×C3⋊D3C2×#6712yes—
250C2×C3×D3—38no—
251C2×C3×D3—34no—
252C2×C3×D3—64no—
253C2×C3×D3—62no—
254C2×C3×D3—64no—
255C2×C3×D3—64no—
256C2×C3×D3—124no—
257C2×C3×D3—124no—
258C2×C3×D3—124no—
259C2×C3×D3—244no—
260D32—1216no—
261D32—128no—
262C32×C4—21no—
263C32×C4—22no—
264C32×C4—21no—
265C32×C4—22no—
266C32×C4—41no—
267C32×C4—41no—
268C32×C4—64no—
269C32×C4D3×D412yes—
270C32×C4C2×#6711yescharacteristic
271C32×C4C23×D312yes—
272C32×C4C23×D311yescharacteristic
273C3⋊Dic3C2×#6712yes—
274C3×Dic3—24no—
275C3×Dic3—24no—
276C3×Dic3—34no—
277C3×Dic3—44no—
278C3×Dic3—62no—
279C22×Q8—91no—
280C22×D4—271no—
281C2×SD16—92no—
282C2×SD16—278no—
283C2×SD16—272no—
284C22×C8—271no—
285C33C22×SD1611yescharacteristic, complemented, Sylow
286C23×C3—91no—
287C23×C3—182no—
288C22×D3—32no—
289C22×D3—61no—
290C22×D3—62no—
291C22×D3—62no—
292C22×D3—122no—
293C22×D3—188no—
294C22×D3—184no—
295C22×D3—368no—
296C22×D3—364no—
297C3×Q8—34no—
298C3×Q8—34no—
299C3×Q8—64no—
300C3×Q8—62no—
301C3×Q8—62no—
302C3×Q8—122no—
303C3×Q8—184no—
304C3×D4—94no—
305C3×D4—182no—
306C2×C3×C4—21no—
307C2×C3×C4—22no—
308C2×C3×C4—41no—
309C2×C3×C4—62no—
310C2×C3×C4—62no—
311C2×C3×C4—91no—
312C2×C3×C4—92no—
313C2×C3×C4—92no—
314C2×C3×C4—122no—
315C2×C3×C4—184no—
316C2×C3×C4C2×D3211yescharacteristic
317C2×C3×C4C2×D3212yes—
318#67—188no—
319C2×Dic3—22no—
320C2×Dic3—32no—
321C2×Dic3—61no—
322D12—34no—
323D12—34no—
324D12—64no—
325D12—62no—
326D12—62no—
327D12—122no—
328D12—184no—
329C4×D3—34no—
330C4×D3—34no—
331C4×D3—62no—
332C4×D3—62no—
333C4×D3—68no—
334Dic6—22no—
335Dic6—68no—
336Dic6C2×D3214yescomplemented
337C3×C8—34no—
338C3×C8—62no—
339C3⋊C8—94no—
340C3⋊C8—92no—
341C3⋊C8—182no—
342C3⋊C8—184no—
343C3⋊C8—362no—
344C3⋊C8C2×D3212yescomplemented
345C2×C32—21no—
346C2×C32—22no—
347C2×C32—22no—
348C2×C32—24no—
349C2×C32—42no—
350C2×C32—41no—
351C2×C32D3×SD1614yescomplemented
352C2×C32C2×#21512yescomplemented
353C2×C32C2×D3×D412yes—
354C2×C32C22×#6711yescharacteristic
355C3⋊D3—44no—
356C3⋊D3C2×#21514yescomplemented
357C3×D3—38no—
358C3×D3—64no—
359C3×D3—124no—
360C3×D3—124no—
361C3×D3—244no—
362C24—541no—
363C2×Q8—32no—
364C2×Q8—94no—
365C2×Q8—181no—
366C2×Q8—182no—
367C2×D4—92no—
368C2×D4—274no—
369C2×D4—542no—
370C2×D4—541no—
371C22×C4—91no—
372C22×C4—181no—
373SD16—98no—
374SD16—278no—
375C2×C8—31no—
376C2×C8—274no—
377C2×C8—271no—
378C22×C3—21no—
379C22×C3—22no—
380C22×C3—41no—
381C22×C3—92no—
382C22×C3—94no—
383C22×C3—184no—
384C22×C3—364no—
385C22×C3#380211yescharacteristic
386C22×C3D3×#6712yes—
387C2×D3—34no—
388C2×D3—38no—
389C2×D3—62no—
390C2×D3—64no—
391C2×D3—64no—
392C2×D3—64no—
393C2×D3—124no—
394C2×D3—124no—
395C2×D3—124no—
396C2×D3—244no—
397C2×D3—3616no—
398C2×D3—368no—
399C3×C4—21no—
400C3×C4—22no—
401C3×C4—21no—
402C3×C4—22no—
403C3×C4—41no—
404C3×C4—41no—
405C3×C4—64no—
406C3×C4—64no—
407C3×C4—92no—
408C3×C4—124no—
409C3×C4#380211yescharacteristic
410C3×C4D3×#6712yes—
411C3×C4C22×D3211yescharacteristic
412C3×C4C22×D3212yes—
413Dic3—24no—
414Dic3—34no—
415Dic3—62no—
416C32—21no—
417C32—22no—
418C32—41no—
419C32C2×D3×SD1612yescomplemented
420C32C22×#21511yescharacteristic, complemented
421C23—91no—
422C23—182no—
423C23—544no—
424C23—1084no—
425Q8—34no—
426Q8—62no—
427Q8—184no—
428D4—94no—
429D4—182no—
430D4—544no—
431C2×C4—62no—
432C2×C4—91no—
433C2×C4—92no—
434C2×C4—92no—
435C2×C4—184no—
436C2×C4D3311yescharacteristic
437C8—32no—
438C8—272no—
439C2×C3—22no—
440C2×C3—24no—
441C2×C3—21no—
442C2×C3—22no—
443C2×C3—42no—
444C2×C3—41no—
445C2×C3—94no—
446C2×C3—364no—
447C2×C3#6379612yescomplemented
448C2×C3D3×#21514yescomplemented
449C2×C3C2×#380211yescharacteristic
450C2×C3C2×D3×#6712yes—
451D3—38no—
452D3—64no—
453D3—124no—
454D3—124no—
455D3—244no—
456C22—94no—
457C22—92no—
458C22—184no—
459C22—364no—
460C22—1088no—
461C22#8895311yescharacteristic
462C4—64no—
463C4—92no—
464C4#8895311yescharacteristic
465C4C2×D3311yescharacteristic
466C3—21no—
467C3—22no—
468C3—41no—
469C3C2×#6379611yescharacteristic, complemented
470C3C2×D3×#21512yescomplemented
471C2—94no—
472C2—364no—
473C2#1172284012yesdirect product factor
474C2C2×#8895311yescharacteristic
subgroupG/H
centerC22#88953
commutator, cosocleC33×C4C24
FrattiniC4C2×D33
FittingC2×C33×C4C23
socleC22×C33C2×D4
prime radicalC2×C4⋊#1113C2
hypercenterC2×C4D33
hypocenterC33C22×SD16
LCSG ▷ C33×C4 ▷ C2×C33 ▷ C33
UCSC1 ◁ C22 ◁ C2×C4

Complements

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

NH ≅ G/Ntype
2. C2×C3×#3797C2semidirect
3. C2×#88793C2semidirect
4. C2×C3⋊D3×C3⋊C8C2semidirect
5. C2×C3×#3717C2semidirect
6. #11722840C2direct (N × H)
12. C2×C32×Dic6C22semidirect
19. C3×#3797C22semidirect
20. C2×C3⋊(C3⋊(C3⋊C8))C22semidirect
21. C2×C32×C3⋊C8C22semidirect
22. #88793C22semidirect
23. C3⋊D3×C3⋊C8C22semidirect
24. C3×#3717C22semidirect
28. C2×#3797D3semidirect
34. C2×#3717D3semidirect
53. C32×Dic6C23semidirect
57. C3⋊(C3⋊(C3⋊C8))C23semidirect
58. C32×C3⋊C8C23semidirect
76. C2×C3×Dic6C2×D3semidirect
77. C2×C3×Dic6C2×D3semidirect
87. #3797C2×D3semidirect
98. C2×C3×C3⋊C8C2×D3semidirect
100. #3717C2×D3semidirect
109. C2×C3×C3⋊D3SD16semidirect
177. C3×Dic6C22×D3semidirect
178. C3×Dic6C22×D3semidirect
190. C3×C3⋊C8C22×D3semidirect
192. C2×C33C2×SD16semidirect
195. C3×C3⋊D3C2×SD16semidirect
222. C2×Dic6D32semidirect
237. C2×C3⋊C8D32semidirect
248. C2×C3⋊D3#215semidirect
285. C33C22×SD16semidirect
336. Dic6C2×D32semidirect
344. C3⋊C8C2×D32semidirect
351. C2×C32D3×SD16semidirect
352. C2×C32C2×#215semidirect
356. C3⋊D3C2×#215semidirect
419. C32C2×D3×SD16semidirect
420. C32C22×#215semidirect
447. C2×C3#63796semidirect
448. C2×C3D3×#215semidirect
469. C3C2×#63796semidirect
470. C3C2×D3×#215semidirect
473. C2#11722840direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C4⋊#11132
C2×C3×#37972
C2×#887932
C2×C3⋊D3×C3⋊C82
C2×C3×#37172
#117228402
C2×D3×#2153
C2×#637963