Finite Group Databasebeta

← C4×#88655 #13147897

Subgroups — C4×#88655 #13147897

Order 1728.

1599 conjugacy classes of subgroups; only the 227 normal subgroups are shown.

normal subgroups only

This group has 1599 conjugacy classes of subgroups — too many to draw. Only its 227 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×#88655C211yesmaximal, characteristic
2C2×C4×((C3⋊D3)⋊C3)⋊C2C212yesmaximal, complemented
3C42×#272C211yesmaximal, characteristic, complemented
4C4×C4⋊((C3⋊D3)⋊C3)C212yesmaximal, complemented
5#11721161C212yesmaximal, complemented
6#11721152C211yesmaximal, characteristic, complemented
7#11721138C212yesmaximal, complemented
8C4×#6198C212yesmaximal, complemented
9#11721136C212yesmaximal, complemented
10C4×D3×D12—32nomaximal
11C22×((C3⋊D3)⋊C3)⋊C2C2212yes—
12C2×C4×#272C2211yescharacteristic
13C2×C4×#272C2211yescharacteristic
14C2×C4×#272C2211yescharacteristic
15C2×C4⋊((C3⋊D3)⋊C3)C2212yes—
16C2×C4×(C3⋊D3)⋊C3C2214yescomplemented
17C2×#6199C2212yes—
18C2×#6198C2212yes—
19C2×((C3⋊D3)⋊C3)⋊C4C2214yescomplemented
20#88655C418yesdirect product factor
21C4×((C3⋊D3)⋊C3)⋊C2—28no—
22C42×H(3,𝔽3)C2211yescharacteristic, complemented
23C4×#1084C2211yescharacteristic, complemented
24C4×#1084C2211yescharacteristic, complemented
25C4×#1084C2211yescharacteristic, complemented
26#3752⋊C3C2214yescomplemented
27C4⋊((C3⋊Dic3)⋊C3)C2212yescomplemented
28#88450C2214yescomplemented
29#88449C2212yescomplemented
30C2×D3×D12—32no—
31C2×C4×D32—34no—
32C4×C4⋊(C3⋊D3)D312yescomplemented
33C3×C4×D12—32no—
34C3×C42×D3—32no—
35C3×C42×D3—61no—
36D3×(C2×D3)⋊C4—34no—
37C4×#626—34no—
38#63894—32no—
39Dic3×D12—32no—
40D3×C4⋊Dic3—32no—
41#63880—34no—
42C22×#272C2311yescharacteristic
43C22×(C3⋊D3)⋊C3C2314yes—
44C2×((C3⋊D3)⋊C3)⋊C2—24no—
45C2×((C3⋊D3)⋊C3)⋊C2—24no—
46C2×((C3⋊D3)⋊C3)⋊C2C2×C418yescomplemented
47C2×C4×H(3,𝔽3)C2311yescharacteristic
48C2×C4×H(3,𝔽3)C2311yescharacteristic
49C2×C4×H(3,𝔽3)C2311yescharacteristic
50C2×#1084C2311yescharacteristic
51C2×#1084C2311yescharacteristic
52C2×#1084C2311yescharacteristic
53C4×#272—22no—
54C4×#272C2×C414yescomplemented
55C4×#272D414yescomplemented
56C2×(C3⋊Dic3)⋊C3C2314yes—
57C4⋊((C3⋊D3)⋊C3)C2×C418yescomplemented
58C4×(C3⋊D3)⋊C3—28no—
59#6199—24no—
60#6198C2×C418yescomplemented
61((C3⋊D3)⋊C3)⋊C4—28no—
62C4×D3×D4—92no—
63C2×C4×D12—91no—
64C22×D32—34no—
65C2×C4⋊(C3⋊D3)C2×D312yes—
66C2×C4×C3⋊D3C2×D314yescomplemented
67C2×C3×D12—32no—
68C2×C3×C4×D3—32no—
69C2×C3×C4×D3—32no—
70C2×C3×C4×D3—34no—
71C2×C3×C4×D3—32no—
72C2×C3×C4×D3—61no—
73C2×C3×C4×D3—61no—
74C2×C3×C4×D3—61no—
75C2×#626—34no—
76C2×#624—34no—
77C2×D3×Dic3—34no—
78C2×D3×Dic3—34no—
79D3×D12—316no—
80C4×D32—616no—
81C32×C42—21no—
82C32×C42C2×D312yescomplemented
83#3752C2×D314yescomplemented
84C4⋊(C3⋊Dic3)C2×D312yescomplemented
85C3×(C2×D3)⋊C4—34no—
86C3×C4⋊Dic3—32no—
87C3×C4×Dic3—32no—
88C3×C4×Dic3—32no—
89C3×C4×Dic3—32no—
90C3×C4×Dic3—61no—
91C3×C4×Dic3—61no—
92C3×C4×Dic3—61no—
93Dic3⋊Dic3—34no—
94#3722—34no—
95(C2×D3)⋊Dic3—34no—
96C22×H(3,𝔽3)C2411yescharacteristic
97C2×#272C22×C412yes—
98C2×#272C2×D412yes—
99C2×#272#4112yes—
100C2×(C3⋊D3)⋊C3—24no—
101C2×(C3⋊D3)⋊C3—24no—
102C2×(C3⋊D3)⋊C3C22×C418yescomplemented
103((C3⋊D3)⋊C3)⋊C2—216no—
104C4×H(3,𝔽3)—21no—
105C4×H(3,𝔽3)C22×C412yescomplemented
106C4×H(3,𝔽3)C2×D412yescomplemented
107#1084—21no—
108#1084C22×C412yescomplemented
109#1084C2×D412yescomplemented
110(C3⋊Dic3)⋊C3—24no—
111C2×D3×D4—92no—
112C22×D12—91no—
113C22×C4×D3—92no—
114C22×C4×D3—94no—
115C3×C4×D4—92no—
116C2×C3×C42—91no—
117D4×Dic3—92no—
118C4×#67—94no—
119C2×(C2×D3)⋊C4—92no—
120C2×C4⋊Dic3—91no—
121#893—92no—
122D3×C4⋊C4—92no—
123#67⋊C4—94no—
124D3×C22⋊C4—94no—
125C4×D12—32no—
126C4×D12—92no—
127C4×D12—92no—
128C4×D12—92no—
129C4×D12—92no—
130C42×D3—32no—
131C42×D3—61no—
132C22×C3⋊D3C22×D314yes—
133C22×C3×D3—32no—
134C22×C3×D3—34no—
135C22×C3×D3—61no—
136C2×D32—316no—
137C2×D32—68no—
138C2×D32—68no—
139C2×C32×C4—21no—
140C2×C32×C4—21no—
141C2×C32×C4—21no—
142C2×C32×C4C22×D312yes—
143C2×C32×C4C22×D312yes—
144C2×C32×C4C22×D312yes—
145C2×C3⋊Dic3C22×D314yes—
146C4⋊(C3⋊D3)C4×D318yescomplemented
147C4×C3⋊D3—28no—
148C2×C3×Dic3—32no—
149C2×C3×Dic3—32no—
150C2×C3×Dic3—34no—
151C2×C3×Dic3—32no—
152C2×C3×Dic3—61no—
153C2×C3×Dic3—61no—
154C2×C3×Dic3—61no—
155C3×D12—38no—
156C3×C4×D3—38no—
157C3×C4×D3—38no—
158C3×C4×D3—64no—
159C3×C4×D3—64no—
160C3×C4×D3—64no—
161C3×C4×D3—64no—
162C3×C4×D3—68no—
163#626—316no—
164#624—68no—
165D3×Dic3—68no—
166D3×Dic3—68no—
167C2×C4×D4—271noSylow
168C2×H(3,𝔽3)C23×C411yescharacteristic
169C2×H(3,𝔽3)C22×D411yescharacteristic
170C2×H(3,𝔽3)C2×#4111yescharacteristic
171#272C4×D414yescomplemented
172(C3⋊D3)⋊C3—28no—
173C23×D3—92no—
174C23×D3—94no—
175C2×C3×D4—92no—
176C22×C3×C4—94no—
177C22×C3×C4—91no—
178C22×C3×C4—91no—
179C22×C3×C4—91no—
180C2×#67—94no—
181C22×Dic3—92no—
182C22×Dic3—94no—
183D3×D4—916no—
184C2×D12—32no—
185C2×D12—92no—
186C2×D12—98no—
187C2×D12—92no—
188C2×C4×D3—32no—
189C2×C4×D3—32no—
190C2×C4×D3—32no—
191C2×C4×D3—34no—
192C2×C4×D3—61no—
193C2×C4×D3—61no—
194C2×C4×D3—61no—
195C2×C4×D3—94no—
196C2×C4×D3—94no—
197C2×C4×D3—94no—
198C2×C4×D3—94no—
199C2×C4×D3—1816no—
200C2×C4×D3—188no—
201C3×C4⋊C4—92no—
202C3×C22⋊C4—94no—
203C3×C42—32no—
204C3×C42—61no—
205C3×C42—91no—
206C3×C42—91no—
207C3×C42—91no—
208C3×C42D3211yescharacteristic
209#217—94no—
210(C2×D3)⋊C4—34no—
211(C2×D3)⋊C4—94no—
212(C2×D3)⋊C4—94no—
213(C2×D3)⋊C4—94no—
214C4⋊Dic3—32no—
215C4⋊Dic3—92no—
216C4⋊Dic3—92no—
217Dic3⋊C4—94no—
218C4×Dic3—32no—
219C4×Dic3—32no—
220C4×Dic3—32no—
221C4×Dic3—61no—
222C4×Dic3—61no—
223C4×Dic3—61no—
224C22×C32—21no—
225C22×C32C23×D312yes—
226C2×C3⋊D3—24no—
227C2×C3⋊D3—24no—
228C2×C3⋊D3C2×C4×D318yescomplemented
229C2×C3×D3—34no—
230C2×C3×D3—34no—
231C2×C3×D3—34no—
232C2×C3×D3—38no—
233C2×C3×D3—62no—
234C2×C3×D3—62no—
235C2×C3×D3—62no—
236C2×C3×D3—64no—
237C2×C3×D3—64no—
238D32—632no—
239C32×C4—22no—
240C32×C4—22no—
241C32×C4—22no—
242C32×C4—22no—
243C32×C4C2×C4×D314yescomplemented
244C32×C4D3×D414yescomplemented
245C3⋊Dic3—24no—
246C3×Dic3—34no—
247C3×Dic3—34no—
248C3×Dic3—62no—
249C3×Dic3—62no—
250C3×Dic3—62no—
251C3×Dic3—64no—
252C3×Dic3—62no—
253C22×D4—271no—
254C23×C4—272no—
255C4×D4—92no—
256C4×D4—272no—
257C4×D4—272no—
258C4×D4—272no—
259C2×C4⋊C4—271no—
260C2×C22⋊C4—272no—
261C2×C42—91no—
262H(3,𝔽3)C2×C4×D411yescharacteristic, complemented, Sylow
263C23×C3—94no—
264C23×C3—91no—
265C22×D3—32no—
266C22×D3—34no—
267C22×D3—61no—
268C22×D3—94no—
269C22×D3—98no—
270C22×D3—916no—
271C22×D3—188no—
272C22×D3—188no—
273C22×D3—184no—
274C22×D3—184no—
275C3×D4—98no—
276C2×C3×C4—32no—
277C2×C3×C4—32no—
278C2×C3×C4—32no—
279C2×C3×C4—61no—
280C2×C3×C4—61no—
281C2×C3×C4—61no—
282C2×C3×C4—91no—
283C2×C3×C4—91no—
284C2×C3×C4—94no—
285C2×C3×C4—94no—
286C2×C3×C4—94no—
287C2×C3×C4—91no—
288C2×C3×C4—182no—
289C2×C3×C4—188no—
290C2×C3×C4C2×D3211yescharacteristic
291C2×C3×C4C2×D3211yescharacteristic
292C2×C3×C4C2×D3211yescharacteristic
293#67—916no—
294C2×Dic3—32no—
295C2×Dic3—32no—
296C2×Dic3—32no—
297C2×Dic3—34no—
298C2×Dic3—61no—
299C2×Dic3—61no—
300C2×Dic3—61no—
301C2×Dic3—94no—
302C2×Dic3—188no—
303C2×Dic3—184no—
304D12—38no—
305D12—98no—
306D12—98no—
307C4×D3—38no—
308C4×D3—38no—
309C4×D3—64no—
310C4×D3—64no—
311C4×D3—64no—
312C4×D3—64no—
313C4×D3—68no—
314C4×D3—188no—
315C4×D3—188no—
316C4×D3—188no—
317C4×D3—188no—
318C2×C32—21no—
319C2×C32—21no—
320C2×C32—21no—
321C2×C32C22×C4×D312yes—
322C2×C32C2×D3×D412yes—
323C2×C32D3×#4112yes—
324C3⋊D3—28no—
325C3×D3—38no—
326C3×D3—64no—
327C3×D3—68no—
328C24—272no—
329C2×D4—92no—
330C2×D4—278no—
331C2×D4—272no—
332C22×C4—91no—
333C22×C4—91no—
334C22×C4—91no—
335C22×C4—94no—
336C22×C4—272no—
337C22×C4—274no—
338C22×C4—548no—
339C4⋊C4—92no—
340C4⋊C4—272no—
341C22⋊C4—94no—
342C22⋊C4—274no—
343C42—91no—
344C42—91no—
345C42—91no—
346C42((C3⋊D3)⋊C3)⋊C211yescharacteristic, complemented
347C22×C3—32no—
348C22×C3—61no—
349C22×C3—98no—
350C22×C3—92no—
351C22×C3—92no—
352C22×C3—92no—
353C22×C3—184no—
354C22×C3—184no—
355C22×C3C22×D3211yescharacteristic
356C2×D3—34no—
357C2×D3—34no—
358C2×D3—34no—
359C2×D3—38no—
360C2×D3—62no—
361C2×D3—62no—
362C2×D3—62no—
363C2×D3—64no—
364C2×D3—64no—
365C2×D3—98no—
366C2×D3—184no—
367C2×D3—184no—
368C2×D3—1832no—
369C2×D3—1816no—
370C3×C4—21no—
371C3×C4—34no—
372C3×C4—34no—
373C3×C4—62no—
374C3×C4—62no—
375C3×C4—62no—
376C3×C4—62no—
377C3×C4—92no—
378C3×C4—92no—
379C3×C4—184no—
380C3×C4—181no—
381C3×C4C4×D3212yes—
382C3×C4#380212yes—
383Dic3—34no—
384Dic3—34no—
385Dic3—62no—
386Dic3—62no—
387Dic3—62no—
388Dic3—62no—
389Dic3—64no—
390Dic3—184no—
391C32—21no—
392C32C4×D3×D412yescomplemented
393C23—91no—
394C23—94no—
395C23—278no—
396C23—544no—
397C23—544no—
398D4—98no—
399D4—278no—
400C2×C4—91no—
401C2×C4—91no—
402C2×C4—94no—
403C2×C4—94no—
404C2×C4—91no—
405C2×C4—94no—
406C2×C4—182no—
407C2×C4—188no—
408C2×C4—548no—
409C2×C4—544no—
410C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
411C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
412C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
413C2×C3—32no—
414C2×C3—32no—
415C2×C3—32no—
416C2×C3—61no—
417C2×C3—61no—
418C2×C3—61no—
419C2×C3—94no—
420C2×C3—188no—
421C2×C3C2×C4×D3211yescharacteristic
422C2×C3C2×#380211yescharacteristic
423C2×C3#6430611yescharacteristic
424D3—38no—
425D3—64no—
426D3—68no—
427D3—188no—
428C22—92no—
429C22—92no—
430C22—92no—
431C22—98no—
432C22—184no—
433C22—184no—
434C22—5416no—
435C22C22×((C3⋊D3)⋊C3)⋊C211yescharacteristic
436C4—21no—
437C4—92no—
438C4—92no—
439C4—184no—
440C4—181no—
441C4C4×((C3⋊D3)⋊C3)⋊C212yescomplemented
442C4#8865512yesdirect product factor
443C3—32no—
444C3—61no—
445C3C4×#380211yescharacteristic
446C2—94no—
447C2—188no—
448C2C2×C4×((C3⋊D3)⋊C3)⋊C211yescharacteristic
449C2C2×#8865511yescharacteristic
450C2#1172249111yescharacteristic
subgroupG/H
centerC2×C4C2×((C3⋊D3)⋊C3)⋊C2
commutatorC2×H(3,𝔽3)C23×C4
Frattini, socleC22×C3C22×D32
FittingC42×H(3,𝔽3)C22
prime radicalC2×#88655C2
cosocleC22×H(3,𝔽3)C24
hypercenterC42((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)C2×C4×D4
LCSG ▷ C2×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C2×C4 ◁ C42

Complements

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

NH ≅ G/Ntype
2. C2×C4×((C3⋊D3)⋊C3)⋊C2C2semidirect
3. C42×#272C2semidirect
4. C4×C4⋊((C3⋊D3)⋊C3)C2semidirect
5. #11721161C2semidirect
6. #11721152C2semidirect
7. #11721138C2semidirect
8. C4×#6198C2semidirect
9. #11721136C2semidirect
16. C2×C4×(C3⋊D3)⋊C3C22semidirect
19. C2×((C3⋊D3)⋊C3)⋊C4C22semidirect
20. #88655C4direct (N × H)
22. C42×H(3,𝔽3)C22semidirect
23. C4×#1084C22semidirect
24. C4×#1084C22semidirect
25. C4×#1084C22semidirect
26. #3752⋊C3C22semidirect
27. C4⋊((C3⋊Dic3)⋊C3)C22semidirect
28. #88450C22semidirect
29. #88449C22semidirect
32. C4×C4⋊(C3⋊D3)D3semidirect
46. C2×((C3⋊D3)⋊C3)⋊C2C2×C4semidirect
54. C4×#272C2×C4semidirect
55. C4×#272D4semidirect
57. C4⋊((C3⋊D3)⋊C3)C2×C4semidirect
60. #6198C2×C4semidirect
66. C2×C4×C3⋊D3C2×D3semidirect
82. C32×C42C2×D3semidirect
83. #3752C2×D3semidirect
84. C4⋊(C3⋊Dic3)C2×D3semidirect
102. C2×(C3⋊D3)⋊C3C22×C4semidirect
105. C4×H(3,𝔽3)C22×C4semidirect
106. C4×H(3,𝔽3)C2×D4semidirect
108. #1084C22×C4semidirect
109. #1084C2×D4semidirect
146. C4⋊(C3⋊D3)C4×D3semidirect
171. #272C4×D4semidirect
228. C2×C3⋊D3C2×C4×D3semidirect
243. C32×C4C2×C4×D3semidirect
244. C32×C4D3×D4semidirect
262. H(3,𝔽3)C2×C4×D4semidirect
346. C42((C3⋊D3)⋊C3)⋊C2semidirect
392. C32C4×D3×D4semidirect
441. C4C4×((C3⋊D3)⋊C3)⋊C2semidirect
442. C4#88655direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#886552
C2×C4×((C3⋊D3)⋊C3)⋊C22
C42×#2722
C4×C4⋊((C3⋊D3)⋊C3)2
#117211612
#117211522
#117211382
C4×#61982
#117211362
C4×D3×D123