Finite Group Databasebeta

← D4×#6196 #13151299

Subgroups — D4×#6196 #13151299

Order 1728.

1074 conjugacy classes of subgroups; only the 193 normal subgroups are shown.

normal subgroups only

This group has 1074 conjugacy classes of subgroups — too many to draw. Only its 193 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
1C22×#6196C212yesmaximal, complemented
2D4×#1084C211yesmaximal, characteristic
3D4×(C3⋊Dic3)⋊C3C212yesmaximal
4#11721309C212yesmaximal, complemented
5#11721299C214yesmaximal, complemented
6#11721117C211yesmaximal, characteristic, complemented
7#11721116C212yesmaximal, complemented
8C4×#6196C211yesmaximal, characteristic, complemented
9D4×#627—32nomaximal
10C2×D4×H(3,𝔽3)C2211yescharacteristic
11C22×#1084C2212yes—
12C22×(C3⋊Dic3)⋊C3C2214yes—
13C2×#6196—28no—
14C2×#6196C2214yes—
15C2×#6196C2212yes—
16C2×#6196C2211yescharacteristic
17#88552C2212yes—
18C4⋊#1084C2211yescharacteristic
19C4×#1084C2211yescharacteristic
20#3757⋊C3C2214yes—
21C4⋊((C3⋊Dic3)⋊C3)C2212yes—
22C4×(C3⋊Dic3)⋊C3C2212yes—
23#88449C2212yes—
24#88449C2211yescharacteristic
25#88448C2212yes—
26#88448C2214yes—
27C22×#627—34no—
28D4×C3⋊Dic3D312yes—
29C3×D4×Dic3—32no—
30C3×D4×Dic3—32no—
31C3×D4×Dic3—61no—
32#63982—34no—
33#63964—34no—
34#63964—34no—
35#63919—32no—
36#63918—32no—
37#63918—32no—
38C4×#627—32no—
39C23×H(3,𝔽3)C2312yes—
40D4×H(3,𝔽3)Q814yescomplemented
41C2×C4×H(3,𝔽3)C2311yescharacteristic
42C2×#1084—24no—
43C2×#1084C2312yes—
44C2×#1084C2311yescharacteristic
45C2×#1084C2311yescharacteristic
46C2×(C3⋊Dic3)⋊C3—28no—
47C2×(C3⋊Dic3)⋊C3C2314yes—
48C2×(C3⋊Dic3)⋊C3C2312yes—
49C2×(C3⋊Dic3)⋊C3C2312yes—
50#6196—24no—
51#6196—28no—
52#6196D414yesdirect product factor
53D4×Dic6—91no—
54D4×Dic6—92no—
55C2×C32×D4—21no—
56C2×C32×D4C2×D312yes—
57C22×C3⋊Dic3C2×D314yes—
58C22×C3×Dic3—34no—
59C22×C3×Dic3—34no—
60C22×C3×Dic3—62no—
61C2×#627—32no—
62C2×#627—34no—
63C2×#627—34no—
64C2×#627—34no—
65C2×#627—616no—
66#3757C2×D314yes—
67C4⋊(C3⋊Dic3)C2×D312yes—
68C4×C3⋊Dic3C2×D312yes—
69C3×#217—34no—
70C3×#217—34no—
71C3×#217—62no—
72C3×C4⋊Dic3—32no—
73C3×C4⋊Dic3—32no—
74C3×C4⋊Dic3—61no—
75C3×C4×Dic3—32no—
76C3×C4×Dic3—32no—
77C3×C4×Dic3—61no—
78#3724—32no—
79#3724—34no—
80Dic3⋊Dic3—32no—
81Dic3⋊Dic3—34no—
82Dic3⋊Dic3—32no—
83Dic3⋊Dic3—34no—
84C22×H(3,𝔽3)—22no—
85C22×H(3,𝔽3)—22no—
86C22×H(3,𝔽3)C2×Q814yescomplemented
87C22×H(3,𝔽3)C2411yescharacteristic
88C4×H(3,𝔽3)C2×Q812yescomplemented
89#1084—22no—
90#1084—21no—
91#1084C2×D412yes—
92(C3⋊Dic3)⋊C3—24no—
93(C3⋊Dic3)⋊C3—22no—
94(C3⋊Dic3)⋊C3C2×D414yes—
95C22×Dic6—92no—
96C22×Dic6—94no—
97C3×C4×D4—92no—
98C3×C4×D4—91no—
99D4×Dic3—32no—
100D4×Dic3—32no—
101D4×Dic3—61no—
102D4×Dic3—92no—
103#924—92no—
104#924—94no—
105#888—92no—
106#888—92no—
107#888—92no—
108#878—94no—
109#878—94no—
110#878—94no—
111#869—91no—
112#869—92no—
113C4×Dic6—91no—
114C4×Dic6—92no—
115C23×C32—22no—
116C23×C32C22×D314yes—
117C32×D4—24no—
118C32×D4Dic618yescomplemented
119C2×C32×C4—21no—
120C2×C32×C4C22×D312yes—
121C2×C3⋊Dic3—28no—
122C2×C3⋊Dic3C22×D312yes—
123C2×C3⋊Dic3C22×D314yes—
124C2×C3⋊Dic3C22×D312yes—
125C2×C3×Dic3—32no—
126C2×C3×Dic3—32no—
127C2×C3×Dic3—32no—
128C2×C3×Dic3—34no—
129C2×C3×Dic3—32no—
130C2×C3×Dic3—34no—
131C2×C3×Dic3—61no—
132C2×C3×Dic3—61no—
133C2×C3×Dic3—62no—
134C2×C3×Dic3—68no—
135C2×C3×Dic3—68no—
136C2×C3×Dic3—124no—
137#627—38no—
138#627—68no—
139#627—68no—
140#627—68no—
141D4×Q8—271noSylow
142C2×H(3,𝔽3)—24no—
143C2×H(3,𝔽3)C22×D411yescharacteristic
144C2×H(3,𝔽3)C22×Q811yescharacteristic
145C2×H(3,𝔽3)2−1+411yescharacteristic
146C2×C3×D4—32no—
147C2×C3×D4—61no—
148C2×C3×D4D3211yescharacteristic
149C22×C3×C4—94no—
150C22×C3×C4—92no—
151C22×Dic3—34no—
152C22×Dic3—34no—
153C22×Dic3—62no—
154C22×Dic3—94no—
155C2×Dic6—91no—
156C2×Dic6—92no—
157C2×Dic6—92no—
158C2×Dic6—94no—
159C2×Dic6—94no—
160C2×Dic6—94no—
161C2×Dic6—94no—
162C2×Dic6—188no—
163C2×Dic6—1816no—
164C3×C4⋊C4—92no—
165C3×C4⋊C4—91no—
166C3×C22⋊C4—94no—
167C3×C22⋊C4—92no—
168C3×C42—92no—
169C3×C42—91no—
170#217—34no—
171#217—34no—
172#217—62no—
173#217—94no—
174C4⋊Dic3—32no—
175C4⋊Dic3—32no—
176C4⋊Dic3—61no—
177C4⋊Dic3—92no—
178C4⋊Dic3—91no—
179C4⋊Dic3—92no—
180C4⋊Dic3—92no—
181C4⋊Dic3—94no—
182Dic3⋊C4—92no—
183Dic3⋊C4—94no—
184Dic3⋊C4—92no—
185Dic3⋊C4—92no—
186Dic3⋊C4—94no—
187Dic3⋊C4—94no—
188C4×Dic3—32no—
189C4×Dic3—32no—
190C4×Dic3—61no—
191C4×Dic3—92no—
192C22×C32—24no—
193C22×C32—24no—
194C22×C32—21no—
195C22×C32—24no—
196C22×C32—42no—
197C22×C32—42no—
198C22×C32C2×Dic618yescomplemented
199C22×C32C23×D312yes—
200C32×C4—22no—
201C32×C4C2×Dic614yescomplemented
202C3⋊Dic3—24no—
203C3⋊Dic3—22no—
204C3⋊Dic3D3×D414yes—
205C3×Dic3—34no—
206C3×Dic3—34no—
207C3×Dic3—62no—
208C3×Dic3—64no—
209C3×Dic3—64no—
210C3×Dic3—62no—
211C3×Dic3—62no—
212C3×Dic3—121no—
213C3×Dic3—122no—
214C22×Q8—272no—
215C4⋊Q8—271no—
216C4⋊Q8—272no—
217C22⋊Q8—274no—
218C22⋊Q8—272no—
219C4×Q8—271no—
220C4×D4—91no—
221C4×D4—92no—
222H(3,𝔽3)D4×Q811yescharacteristic, complemented, Sylow
223C23×C3—34no—
224C23×C3—62no—
225C23×C3C2×D3212yes—
226C3×D4—38no—
227C3×D4—64no—
228C3×D4#62714yes—
229C2×C3×C4—32no—
230C2×C3×C4—61no—
231C2×C3×C4—92no—
232C2×C3×C4—91no—
233C2×C3×C4—92no—
234C2×C3×C4—94no—
235C2×C3×C4—91no—
236C2×C3×C4—92no—
237C2×C3×C4—184no—
238C2×C3×C4—188no—
239C2×C3×C4C2×D3211yescharacteristic
240C2×Dic3—32no—
241C2×Dic3—32no—
242C2×Dic3—32no—
243C2×Dic3—34no—
244C2×Dic3—32no—
245C2×Dic3—34no—
246C2×Dic3—61no—
247C2×Dic3—61no—
248C2×Dic3—62no—
249C2×Dic3—68no—
250C2×Dic3—68no—
251C2×Dic3—92no—
252C2×Dic3—92no—
253C2×Dic3—94no—
254C2×Dic3—124no—
255C2×Dic3—188no—
256Dic6—94no—
257Dic6—98no—
258Dic6—188no—
259Dic6—188no—
260Dic6—184no—
261Dic6—188no—
262Dic6—188no—
263C2×C32—28no—
264C2×C32—21no—
265C2×C32—21no—
266C2×C32—21no—
267C2×C32—44no—
268C2×C32C22×Dic612yes—
269C2×C32C2×D3×D412yes—
270C2×C32#101012yes—
271C2×Q8—271no—
272C2×Q8—272no—
273C2×Q8—274no—
274C2×Q8—548no—
275C2×D4((C3⋊D3)⋊C3)⋊C211yescharacteristic
276C22×C4—92no—
277C22×C4—94no—
278C4⋊C4—91no—
279C4⋊C4—92no—
280C4⋊C4—271no—
281C4⋊C4—272no—
282C4⋊C4—272no—
283C4⋊C4—274no—
284C22⋊C4—92no—
285C22⋊C4—94no—
286C42—91no—
287C42—92no—
288C22×C3—22no—
289C22×C3—22no—
290C22×C3—32no—
291C22×C3—38no—
292C22×C3—64no—
293C22×C3—64no—
294C22×C3—61no—
295C22×C3—64no—
296C22×C3—122no—
297C22×C3—122no—
298C22×C3C2×#62714yes—
299C22×C3C22×D3211yescharacteristic
300C3×C4—34no—
301C3×C4—62no—
302C3×C4—94no—
303C3×C4—92no—
304C3×C4—184no—
305C3×C4—182no—
306C3×C4—181no—
307C3×C4—182no—
308C3×C4C2×#62712yes—
309Dic3—34no—
310Dic3—34no—
311Dic3—62no—
312Dic3—64no—
313Dic3—62no—
314Dic3—62no—
315Dic3—64no—
316Dic3—94no—
317Dic3—122no—
318Dic3—121no—
319Dic3—184no—
320Dic3—182no—
321C32—21no—
322C32D4×Dic612yescomplemented
323C23C2×((C3⋊D3)⋊C3)⋊C212yes—
324Q8—274no—
325Q8—544no—
326Q8—548no—
327D4#619614yesdirect product factor
328C2×C4—91no—
329C2×C4—92no—
330C2×C4—91no—
331C2×C4—92no—
332C2×C4—92no—
333C2×C4—94no—
334C2×C4—184no—
335C2×C4—188no—
336C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
337C2×C3—24no—
338C2×C3—32no—
339C2×C3—32no—
340C2×C3—32no—
341C2×C3—68no—
342C2×C3—61no—
343C2×C3—61no—
344C2×C3—61no—
345C2×C3—124no—
346C2×C3#6430911yescharacteristic
347C2×C3D32×D411yescharacteristic
348C2×C3C22×#62711yescharacteristic
349C22—22no—
350C22—22no—
351C22C2×#619614yescomplemented
352C22C22×((C3⋊D3)⋊C3)⋊C211yescharacteristic
353C4—92no—
354C4—94no—
355C4—181no—
356C4—182no—
357C4—182no—
358C4—184no—
359C4C2×#619612yescomplemented
360C3—32no—
361C3—61no—
362C3D4×#62711yescharacteristic
363C2—24no—
364C2#1172250211yescharacteristic
365C2D4×((C3⋊D3)⋊C3)⋊C211yescharacteristic
366C2C22×#619611yescharacteristic
subgroupG/H
centerC22C22×((C3⋊D3)⋊C3)⋊C2
commutator, cosocleC22×H(3,𝔽3)C24
Frattini, socleC22×C3C22×D32
Fitting, prime radicalC2×D4×H(3,𝔽3)C22
hypercenterC2×D4((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)D4×Q8
LCSG ▷ C22×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C22 ◁ C2×D4

Complements

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

NH ≅ G/Ntype
1. C22×#6196C2semidirect
4. #11721309C2semidirect
5. #11721299C2semidirect
6. #11721117C2semidirect
7. #11721116C2semidirect
8. C4×#6196C2semidirect
40. D4×H(3,𝔽3)Q8semidirect
52. #6196D4direct (N × H)
86. C22×H(3,𝔽3)C2×Q8semidirect
88. C4×H(3,𝔽3)C2×Q8semidirect
118. C32×D4Dic6semidirect
198. C22×C32C2×Dic6semidirect
201. C32×C4C2×Dic6semidirect
222. H(3,𝔽3)D4×Q8semidirect
322. C32D4×Dic6semidirect
327. D4#6196direct (N × H)
351. C22C2×#6196semidirect
359. C4C2×#6196semidirect

Maximal subgroups

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

subgroupindex
C22×#61962
D4×#10842
D4×(C3⋊Dic3)⋊C32
#117213092
#117212992
#117211172
#117211162
C4×#61962
D4×#6273