Finite Group Databasebeta

← D4×#6199 #13151436

Subgroups — D4×#6199 #13151436

Order 1728.

1727 conjugacy classes of subgroups; only the 239 normal subgroups are shown.

normal subgroups only

This group has 1727 conjugacy classes of subgroups — too many to draw. Only its 239 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
2C22×#6199C212yesmaximal, complemented
3D4×(C3⋊Dic3)⋊C3C212yesmaximal, complemented
4#11721320C212yesmaximal, complemented
5#11721293C214yesmaximal, complemented
6#11721152C211yesmaximal, characteristic, complemented
7#11721149C212yesmaximal, complemented
8C4×#6199C211yesmaximal, characteristic, complemented
9D3×D4×Dic3—32nomaximal
10C23×#272C2212yes—
11C2×D4×H(3,𝔽3)C2211yescharacteristic
12C2×#6235C2212yes—
13D4×#272C418yescomplemented
14C2×C4⋊#272C2211yescharacteristic
15C2×C4×#272C2211yescharacteristic
16C22×(C3⋊Dic3)⋊C3C2214yescomplemented
17C2×#6199—28no—
18C2×#6199C2212yes—
19C2×#6199C2211yescharacteristic
20C2×#6199C2211yescharacteristic
21C2×#6199C2214yescomplemented
22#3757⋊C3C2214yescomplemented
23C4⋊((C3⋊Dic3)⋊C3)C2212yescomplemented
24C4×(C3⋊Dic3)⋊C3C2212yescomplemented
25#88451C2214yescomplemented
26#88449C2212yescomplemented
27#88447C2212yescomplemented
28C2×C3×D3×D4—32no—
29C2×C3×D3×D4—61no—
30C22×D3×Dic3—34no—
31D4×C3⋊Dic3D312yescomplemented
32C3×D4×Dic3—32no—
33#67⋊Dic3—34no—
34Dic3×#67—34no—
35D3×#217—34no—
36#63898—32no—
37Dic3×D12—32no—
38D3×C4⋊Dic3—32no—
39C4×D3×Dic3—32no—
40C23×H(3,𝔽3)C2312yes—
41C22×#272—24no—
42C22×#272—24no—
43C22×#272C2×C418yescomplemented
44C22×#272C2312yes—
45C22×#272C2311yescharacteristic
46D4×H(3,𝔽3)C2×C414yescomplemented
47C2×C4×H(3,𝔽3)C2311yescharacteristic
48#6235C2×C418yescomplemented
49C2×#1084C2311yescharacteristic
50C4⋊#272C2×C414yescomplemented
51C4×#272C2×C414yescomplemented
52C2×(C3⋊Dic3)⋊C3—28no—
53C2×(C3⋊Dic3)⋊C3C2314yes—
54C2×(C3⋊Dic3)⋊C3C2312yes—
55C2×(C3⋊Dic3)⋊C3C2312yes—
56#6199—24no—
57#6199—22no—
58#6199—28no—
59#6199D414yesdirect product factor
60C2×D4×Dic3—91no—
61C4×D3×D4—92no—
62C23×C3×D3—34no—
63C23×C3×D3—62no—
64C2×C32×D4—21no—
65C2×C32×D4C2×D312yes—
66C22×C3⋊Dic3C2×D314yescomplemented
67C2×C3×#67—34no—
68C2×C3×#67—62no—
69C22×C3×Dic3—34no—
70C3×D3×D4—316no—
71C3×D3×D4—68no—
72C2×C3×D12—32no—
73C2×C3×D12—61no—
74C2×C3×C4×D3—32no—
75C2×C3×C4×D3—61no—
76C2×D3×Dic3—32no—
77C2×D3×Dic3—34no—
78C2×D3×Dic3—34no—
79C2×D3×Dic3—34no—
80C2×D3×Dic3—32no—
81C2×D3×Dic3—616no—
82#3757C2×D314yescomplemented
83C4⋊(C3⋊Dic3)C2×D312yescomplemented
84C4×C3⋊Dic3C2×D312yescomplemented
85C3×#217—34no—
86C3×C4⋊Dic3—32no—
87C3×C4×Dic3—32no—
88Dic3⋊Dic3—34no—
89(C2×D3)⋊Dic3—34no—
90(C2×D3)⋊Dic3—34no—
91Dic32—32no—
92Dic32—32no—
93C22×H(3,𝔽3)—22no—
94C22×H(3,𝔽3)—22no—
95C22×H(3,𝔽3)C22×C414yescomplemented
96C22×H(3,𝔽3)C2411yescharacteristic
97C2×#272—22no—
98C2×#272—22no—
99C2×#272—216no—
100C2×#272C22×C412yes—
101C2×#272C22×C414yescomplemented
102C2×#272C2×D412yes—
103C2×#272#4112yes—
104C4×H(3,𝔽3)C22×C412yescomplemented
105#1084C22×C412yescomplemented
106(C3⋊Dic3)⋊C3—24no—
107(C3⋊Dic3)⋊C3—22no—
108(C3⋊Dic3)⋊C3C2×D414yescomplemented
109C22×C3×D4—91no—
110C23×Dic3—92no—
111C2×D3×D4—32no—
112C2×D3×D4—61no—
113C22×C4×D3—94no—
114C3×C4×D4—92no—
115C2×#217—92no—
116D4×Dic3—32no—
117D4×Dic3—92no—
118D4×Dic3—92no—
119D4×Dic3—94no—
120C4×#67—94no—
121C2×C4⋊Dic3—91no—
122C2×C4×Dic3—91no—
123#893—92no—
124D3×C4⋊C4—92no—
125#67⋊C4—94no—
126D3×C22⋊C4—94no—
127C4×D12—92no—
128C42×D3—92no—
129C23×C32—22no—
130C23×C32C22×D314yes—
131C22×C3×D3—32no—
132C22×C3×D3—316no—
133C22×C3×D3—34no—
134C22×C3×D3—61no—
135C22×C3×D3—68no—
136C22×C3×D3—62no—
137C22×C3×D3—68no—
138C22×C3×D3—68no—
139C22×C3×D3—124no—
140C22×C3×D3—124no—
141C32×D4—24no—
142C32×D4C4×D318yescomplemented
143C2×C32×C4—21no—
144C2×C32×C4C22×D312yes—
145C2×C3⋊Dic3—28no—
146C2×C3⋊Dic3C22×D312yes—
147C2×C3⋊Dic3C22×D314yes—
148C2×C3⋊Dic3C22×D312yes—
149C3×#67—316no—
150C3×#67—68no—
151C2×C3×Dic3—32no—
152C2×C3×Dic3—32no—
153C2×C3×Dic3—32no—
154C2×C3×Dic3—34no—
155C2×C3×Dic3—61no—
156C2×C3×Dic3—68no—
157C3×D12—38no—
158C3×D12—64no—
159C3×C4×D3—38no—
160C3×C4×D3—64no—
161D3×Dic3—38no—
162D3×Dic3—68no—
163D3×Dic3—64no—
164D3×Dic3—68no—
165D3×Dic3—68no—
166C2×C4×D4—271noSylow
167C2×H(3,𝔽3)—24no—
168C2×H(3,𝔽3)C23×C411yescharacteristic
169C2×H(3,𝔽3)C22×D411yescharacteristic
170C2×H(3,𝔽3)C2×#4111yescharacteristic
171#272—24no—
172#272C4×D414yescomplemented
173C24×C3—92no—
174C23×D3—34no—
175C23×D3—62no—
176C2×C3×D4—32no—
177C2×C3×D4—61no—
178C2×C3×D4—92no—
179C2×C3×D4—98no—
180C2×C3×D4—91no—
181C2×C3×D4D3211yescharacteristic
182C22×C3×C4—91no—
183C22×C3×C4—94no—
184C2×#67—34no—
185C2×#67—62no—
186C22×Dic3—34no—
187C22×Dic3—94no—
188C22×Dic3—91no—
189C22×Dic3—91no—
190C22×Dic3—92no—
191C22×Dic3—94no—
192C22×Dic3—188no—
193D3×D4—316no—
194D3×D4—68no—
195C2×D12—32no—
196C2×D12—61no—
197C2×C4×D3—32no—
198C2×C4×D3—61no—
199C2×C4×D3—92no—
200C2×C4×D3—92no—
201C2×C4×D3—94no—
202C2×C4×D3—94no—
203C2×C4×D3—94no—
204C2×C4×D3—1816no—
205C3×C4⋊C4—92no—
206C3×C22⋊C4—94no—
207C3×C42—92no—
208#217—34no—
209#217—94no—
210#217—94no—
211(C2×D3)⋊C4—94no—
212(C2×D3)⋊C4—94no—
213C4⋊Dic3—32no—
214C4⋊Dic3—92no—
215C4⋊Dic3—92no—
216Dic3⋊C4—94no—
217C4×Dic3—32no—
218C4×Dic3—92no—
219C4×Dic3—92no—
220C4×Dic3—92no—
221C4×Dic3—92no—
222C22×C32—24no—
223C22×C32—24no—
224C22×C32—21no—
225C22×C32—24no—
226C22×C32—42no—
227C22×C32—42no—
228C22×C32C2×C4×D318yescomplemented
229C22×C32C23×D312yes—
230C2×C3×D3—34no—
231C2×C3×D3—34no—
232C2×C3×D3—34no—
233C2×C3×D3—38no—
234C2×C3×D3—62no—
235C2×C3×D3—62no—
236C2×C3×D3—62no—
237C2×C3×D3—64no—
238C2×C3×D3—632no—
239C2×C3×D3—64no—
240C2×C3×D3—64no—
241C2×C3×D3—1216no—
242C2×C3×D3—122no—
243C2×C3×D3—122no—
244C32×C4—22no—
245C32×C4C2×C4×D314yescomplemented
246C3⋊Dic3—24no—
247C3⋊Dic3—22no—
248C3⋊Dic3D3×D414yescomplemented
249C3×Dic3—34no—
250C3×Dic3—34no—
251C3×Dic3—62no—
252C3×Dic3—64no—
253C3×Dic3—62no—
254C22×D4—91no—
255C23×C4—272no—
256C4×D4—92no—
257C4×D4—272no—
258C4×D4—274no—
259C2×C4⋊C4—271no—
260C2×C22⋊C4—272no—
261C2×C42—271no—
262H(3,𝔽3)C2×C4×D411yescharacteristic, complemented, Sylow
263C23×C3—34no—
264C23×C3—62no—
265C23×C3—91no—
266C23×C3—92no—
267C23×C3—98no—
268C23×C3—184no—
269C23×C3—184no—
270C23×C3C2×D3212yes—
271C22×D3—32no—
272C22×D3—34no—
273C22×D3—316no—
274C22×D3—61no—
275C22×D3—62no—
276C22×D3—68no—
277C22×D3—68no—
278C22×D3—68no—
279C22×D3—124no—
280C22×D3—124no—
281C3×D4—38no—
282C3×D4—64no—
283C3×D4—98no—
284C3×D4—94no—
285C3×D4#62414yes—
286C2×C3×C4—32no—
287C2×C3×C4—61no—
288C2×C3×C4—92no—
289C2×C3×C4—91no—
290C2×C3×C4—94no—
291C2×C3×C4—92no—
292C2×C3×C4—94no—
293C2×C3×C4—188no—
294C2×C3×C4C2×D3211yescharacteristic
295#67—316no—
296#67—68no—
297C2×Dic3—32no—
298C2×Dic3—32no—
299C2×Dic3—32no—
300C2×Dic3—34no—
301C2×Dic3—61no—
302C2×Dic3—68no—
303C2×Dic3—92no—
304C2×Dic3—92no—
305C2×Dic3—94no—
306C2×Dic3—94no—
307C2×Dic3—188no—
308C2×Dic3—188no—
309C2×Dic3—182no—
310C2×Dic3—184no—
311D12—38no—
312D12—64no—
313C4×D3—38no—
314C4×D3—64no—
315C4×D3—98no—
316C4×D3—188no—
317C4×D3—188no—
318C4×D3—184no—
319C4×D3—188no—
320C2×C32—28no—
321C2×C32—21no—
322C2×C32—21no—
323C2×C32—21no—
324C2×C32—44no—
325C2×C32C22×C4×D312yes—
326C2×C32C2×D3×D412yes—
327C2×C32D3×#4112yes—
328C3×D3—38no—
329C3×D3—64no—
330C3×D3—68no—
331C3×D3—124no—
332C24—92no—
333C2×D4—92no—
334C2×D4—98no—
335C2×D4—91no—
336C2×D4((C3⋊D3)⋊C3)⋊C211yescharacteristic
337C22×C4—91no—
338C22×C4—94no—
339C22×C4—271no—
340C22×C4—271no—
341C22×C4—272no—
342C22×C4—274no—
343C22×C4—548no—
344C4⋊C4—92no—
345C4⋊C4—272no—
346C22⋊C4—94no—
347C22⋊C4—274no—
348C42—92no—
349C42—272no—
350C22×C3—22no—
351C22×C3—22no—
352C22×C3—32no—
353C22×C3—38no—
354C22×C3—64no—
355C22×C3—64no—
356C22×C3—61no—
357C22×C3—64no—
358C22×C3—92no—
359C22×C3—92no—
360C22×C3—92no—
361C22×C3—94no—
362C22×C3—122no—
363C22×C3—122no—
364C22×C3—1816no—
365C22×C3—182no—
366C22×C3—182no—
367C22×C3C2×#62414yes—
368C22×C3C22×D3211yescharacteristic
369C2×D3—34no—
370C2×D3—34no—
371C2×D3—34no—
372C2×D3—38no—
373C2×D3—62no—
374C2×D3—62no—
375C2×D3—62no—
376C2×D3—64no—
377C2×D3—632no—
378C2×D3—64no—
379C2×D3—64no—
380C2×D3—1216no—
381C2×D3—122no—
382C2×D3—122no—
383C3×C4—34no—
384C3×C4—62no—
385C3×C4—94no—
386C3×C4—92no—
387C3×C4—184no—
388C3×C4—182no—
389C3×C4C2×#62412yes—
390Dic3—34no—
391Dic3—34no—
392Dic3—62no—
393Dic3—64no—
394Dic3—62no—
395Dic3—94no—
396Dic3—184no—
397Dic3—182no—
398C32—21no—
399C32C4×D3×D412yescomplemented
400C23—91no—
401C23—92no—
402C23—98no—
403C23—184no—
404C23—184no—
405C23C2×((C3⋊D3)⋊C3)⋊C212yes—
406D4—98no—
407D4—94no—
408D4#619914yesdirect product factor
409C2×C4—92no—
410C2×C4—91no—
411C2×C4—94no—
412C2×C4—92no—
413C2×C4—94no—
414C2×C4—188no—
415C2×C4—274no—
416C2×C4—548no—
417C2×C4—544no—
418C2×C4—542no—
419C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
420C2×C3—24no—
421C2×C3—32no—
422C2×C3—32no—
423C2×C3—32no—
424C2×C3—68no—
425C2×C3—61no—
426C2×C3—61no—
427C2×C3—61no—
428C2×C3—94no—
429C2×C3—124no—
430C2×C3—184no—
431C2×C3D32×D411yescharacteristic
432C2×C3#6431211yescharacteristic
433C2×C3C22×#62411yescharacteristic
434D3—38no—
435D3—64no—
436D3—68no—
437D3—124no—
438C22—22no—
439C22—22no—
440C22—92no—
441C22—92no—
442C22—92no—
443C22—94no—
444C22—1816no—
445C22—182no—
446C22—182no—
447C22C2×#619914yescomplemented
448C22C22×((C3⋊D3)⋊C3)⋊C211yescharacteristic
449C4—94no—
450C4—92no—
451C4—184no—
452C4—182no—
453C4C2×#619912yescomplemented
454C3—32no—
455C3—61no—
456C3D4×#62411yescharacteristic
457C2—24no—
458C2—94no—
459C2—184no—
460C2D4×((C3⋊D3)⋊C3)⋊C211yescharacteristic
461C2#1172250511yescharacteristic
462C2C22×#619911yescharacteristic
subgroupG/H
centerC22C22×((C3⋊D3)⋊C3)⋊C2
commutatorC2×H(3,𝔽3)C23×C4
Frattini, socleC22×C3C22×D32
FittingC2×D4×H(3,𝔽3)C22
prime radicalC2×D4×#272C2
cosocleC22×H(3,𝔽3)C24
hypercenterC2×D4((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)C2×C4×D4
LCSG ▷ C2×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C22 ◁ C2×D4

Complements

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

NH ≅ G/Ntype
2. C22×#6199C2semidirect
3. D4×(C3⋊Dic3)⋊C3C2semidirect
4. #11721320C2semidirect
5. #11721293C2semidirect
6. #11721152C2semidirect
7. #11721149C2semidirect
8. C4×#6199C2semidirect
13. D4×#272C4semidirect
16. C22×(C3⋊Dic3)⋊C3C22semidirect
21. C2×#6199C22semidirect
22. #3757⋊C3C22semidirect
23. C4⋊((C3⋊Dic3)⋊C3)C22semidirect
24. C4×(C3⋊Dic3)⋊C3C22semidirect
25. #88451C22semidirect
26. #88449C22semidirect
27. #88447C22semidirect
31. D4×C3⋊Dic3D3semidirect
43. C22×#272C2×C4semidirect
46. D4×H(3,𝔽3)C2×C4semidirect
48. #6235C2×C4semidirect
50. C4⋊#272C2×C4semidirect
51. C4×#272C2×C4semidirect
59. #6199D4direct (N × H)
66. C22×C3⋊Dic3C2×D3semidirect
82. #3757C2×D3semidirect
83. C4⋊(C3⋊Dic3)C2×D3semidirect
84. C4×C3⋊Dic3C2×D3semidirect
95. C22×H(3,𝔽3)C22×C4semidirect
101. C2×#272C22×C4semidirect
104. C4×H(3,𝔽3)C22×C4semidirect
105. #1084C22×C4semidirect
108. (C3⋊Dic3)⋊C3C2×D4semidirect
142. C32×D4C4×D3semidirect
172. #272C4×D4semidirect
228. C22×C32C2×C4×D3semidirect
245. C32×C4C2×C4×D3semidirect
248. C3⋊Dic3D3×D4semidirect
262. H(3,𝔽3)C2×C4×D4semidirect
399. C32C4×D3×D4semidirect
408. D4#6199direct (N × H)
447. C22C2×#6199semidirect
453. C4C2×#6199semidirect

Maximal subgroups

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

subgroupindex
C2×D4×#2722
C22×#61992
D4×(C3⋊Dic3)⋊C32
#117213202
#117212932
#117211522
#117211492
C4×#61992
D3×D4×Dic33