Finite Group Databasebeta

← C4×#88439 #13137436

Subgroups — C4×#88439 #13137436

Order 1728.

455 conjugacy classes of subgroups; only the 99 normal subgroups are shown.

normal subgroups only

This group has 455 conjugacy classes of subgroups — too many to draw. Only its 99 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
1C4×#6229C211yesmaximal, characteristic
2C4×#634⋊C3C211yesmaximal, characteristic
3C2×#88439C211yesmaximal, characteristic
4C4×(C3⋊(C3⋊C8))⋊C3C211yesmaximal, characteristic
5#11720326C211yesmaximal, characteristic
6#11720309C211yesmaximal, characteristic
7#11720306C211yesmaximal, characteristic
8C4×#3719—31nomaximal
9C4×#3718—31nomaximal
10C2×#6229C2211yescharacteristic
11C2×#634⋊C3C2211yescharacteristic
12C42×H(3,𝔽3)C2211yescharacteristic
13C4⋊#1084C2211yescharacteristic
14#88540—21no—
15C4×#1084—21no—
16C4⋊((C3⋊Dic3)⋊C3)C2211yescharacteristic
17#3750⋊C3—21no—
18C4×(C3⋊Dic3)⋊C3—21no—
19C2×(C3⋊(C3⋊C8))⋊C3C2211yescharacteristic
20C2×(C3⋊(C3⋊C8))⋊C3C2211yescharacteristic
21#88439C414yesdirect product factor
22C4×#634D311yescharacteristic
23C3×C4×Dic6—31no—
24C3×C4×Dic6—31no—
25C3×C4×Dic6—31no—
26C3×C4×Dic6—61no—
27C2×#3719—31no—
28C2×#3718—31no—
29C4×C3⋊(C3⋊C8)D311yescharacteristic
30C3×C4×C3⋊C8—31no—
31#63576—31no—
32#63575—31no—
33#63567—31no—
34#63566—31no—
35#63565—31no—
36#63565—31no—
37C2×C4×H(3,𝔽3)D411yescharacteristic
38C2×C4×H(3,𝔽3)D411yescharacteristic
39C2×C4×H(3,𝔽3)C2311yescharacteristic
40C2×#1084—21no—
41C2×#1084—21no—
42#6229—21no—
43#6229C2×C412yes—
44C2×(C3⋊Dic3)⋊C3—21no—
45C2×(C3⋊Dic3)⋊C3—21no—
46#634⋊C3—21no—
47#634⋊C3C2×C412yes—
48(C3⋊(C3⋊C8))⋊C3—21no—
49(C3⋊(C3⋊C8))⋊C3C2×C412yes—
50C4×#216—91no—
51C4×#216—91no—
52C4×Dic12—91no—
53C2×#634C2×D311yescharacteristic
54C2×C3×Dic6—31no—
55C2×C3×Dic6—31no—
56C2×C3×Dic6—31no—
57C2×C3×Dic6—61no—
58C32×C42—21no—
59C32×C42C2×D311yescharacteristic
60C32×C42C2×D311yescharacteristic
61C4⋊(C3⋊Dic3)C2×D311yescharacteristic
62#3750—21no—
63C4×C3⋊Dic3—21no—
64C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
65C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
66C3×C4⋊Dic3—31no—
67C3×C4⋊Dic3—31no—
68C3×C4⋊Dic3—31no—
69C3×C4⋊Dic3—61no—
70C3×Dic3⋊C4—62no—
71C3×Dic3⋊C4—61no—
72C3×Dic3⋊C4—61no—
73C3×Dic3⋊C4—61no—
74C3×C4×Dic3—62no—
75C3×C4×Dic3—61no—
76C3×C4×Dic3—61no—
77C3×C4×Dic3—61no—
78C2×C3×C3⋊C8—31no—
79C2×C3×C3⋊C8—31no—
80#3719—34no—
81#3718—34no—
82C22×H(3,𝔽3)C2×D411yescharacteristic
83C4×H(3,𝔽3)—21no—
84C4×H(3,𝔽3)Q1612yescomplemented
85C4×H(3,𝔽3)C22×C411yescharacteristic
86C4×H(3,𝔽3)#4111yescharacteristic
87#1084—22no—
88#1084—41no—
89(C3⋊Dic3)⋊C3—22no—
90(C3⋊Dic3)⋊C3—41no—
91C3×C4×Q8—91no—
92C3×C4×Q8—91no—
93C2×#216—91no—
94C2×#216—91no—
95C2×Dic12—91no—
96C4×Dic6—31no—
97C4×Dic6—31no—
98C4×Dic6—31no—
99C4×Dic6—61no—
100C4×Dic6—91no—
101C3×C4×C8—91no—
102#835—91no—
103#835—91no—
104#818—91no—
105#816—91no—
106#816—91no—
107#810—91no—
108#810—91no—
109#807—91no—
110#807—91no—
111C4×C3⋊C8—31no—
112C4×C3⋊C8—91no—
113C2×C32×C4—21no—
114C2×C32×C4—21no—
115C2×C32×C4—21no—
116C2×C32×C4D1211yescharacteristic
117C2×C32×C4D1211yescharacteristic
118C2×C32×C4#6711yescharacteristic
119C2×C32×C4#6711yescharacteristic
120C2×C32×C4C22×D311yescharacteristic
121C2×C32×C4C22×D311yescharacteristic
122C2×C3⋊Dic3—21no—
123C2×C3⋊Dic3—21no—
124#634—21no—
125#634C4×D312yes—
126C2×C3×Dic3—62no—
127C2×C3×Dic3—61no—
128C2×C3×Dic3—61no—
129C2×C3×Dic3—61no—
130C2×C3×Dic3—62no—
131C2×C3×Dic3—61no—
132C2×C3×Dic3—61no—
133C2×C3×Dic3—61no—
134C3×Dic6—32no—
135C3×Dic6—32no—
136C3×Dic6—32no—
137C3×Dic6—62no—
138C3×Dic6—62no—
139C3×Dic6—61no—
140C3×Dic6—61no—
141C3×Dic6—61no—
142C3⋊(C3⋊C8)—21no—
143C3⋊(C3⋊C8)C4×D312yes—
144C3×C3⋊C8—32no—
145C3×C3⋊C8—61no—
146C4×Q16—271noSylow
147C2×H(3,𝔽3)C4×D411yescharacteristic
148C2×H(3,𝔽3)C2×Q1611yescharacteristic
149C2×H(3,𝔽3)#13511yescharacteristic
150C2×C3×Q8—91no—
151C2×C3×Q8—91no—
152C2×Dic6—31no—
153C2×Dic6—31no—
154C2×Dic6—31no—
155C2×Dic6—61no—
156C2×Dic6—91no—
157C2×C3×C8—91no—
158C2×C3×C8—91no—
159C3×C4⋊C4—91no—
160C3×C4⋊C4—91no—
161C3×C4⋊C4—181no—
162C3×C4⋊C4—181no—
163C3×C42—31no—
164C3×C42—31no—
165C3×C42—61no—
166C3×C42—181no—
167C3×C42—181no—
168C3×C42D3211yescharacteristic
169#216—94no—
170#216—94no—
171C4⋊Dic3—31no—
172C4⋊Dic3—31no—
173C4⋊Dic3—31no—
174C4⋊Dic3—61no—
175C4⋊Dic3—91no—
176Dic3⋊C4—61no—
177Dic3⋊C4—61no—
178Dic3⋊C4—62no—
179Dic3⋊C4—61no—
180Dic3⋊C4—181no—
181C4×Dic3—61no—
182C4×Dic3—61no—
183C4×Dic3—62no—
184C4×Dic3—61no—
185C4×Dic3—181no—
186C2×C3⋊C8—31no—
187C2×C3⋊C8—31no—
188C2×C3⋊C8—91no—
189C2×C3⋊C8—91no—
190Dic12—94no—
191C22×C32—21no—
192C22×C32C2×D1211yescharacteristic
193C22×C32C2×#6711yescharacteristic
194C32×C4—21no—
195C32×C4—21no—
196C32×C4—22no—
197C32×C4—21no—
198C32×C4—21no—
199C32×C4—41no—
200C32×C4Dic1212yescomplemented
201C32×C4#21612yescomplemented
202C32×C4C2×C4×D311yescharacteristic
203C32×C4C2×C4×D311yescharacteristic
204C32×C4#23511yescharacteristic
205C32×C4#23511yescharacteristic
206C3⋊Dic3—22no—
207C3⋊Dic3—41no—
208C3×Dic3—64no—
209C3×Dic3—62no—
210C3×Dic3—62no—
211C3×Dic3—62no—
212C3×Dic3—122no—
213C3×Dic3—121no—
214C3×Dic3—121no—
215C3×Dic3—121no—
216C2×Q16—271no—
217C4×Q8—91no—
218C4×Q8—91no—
219#107—271no—
220#103—271no—
221#103—271no—
222C4×C8—91no—
223H(3,𝔽3)C4×Q1611yescharacteristic, complemented, Sylow
224C3×Q8—92no—
225C3×Q8—92no—
226C3×Q8—181no—
227C3×Q8—181no—
228C2×C3×C4—31no—
229C2×C3×C4—31no—
230C2×C3×C4—31no—
231C2×C3×C4—31no—
232C2×C3×C4—31no—
233C2×C3×C4—31no—
234C2×C3×C4—61no—
235C2×C3×C4—61no—
236C2×C3×C4—61no—
237C2×C3×C4—181no—
238C2×C3×C4—181no—
239C2×C3×C4—181no—
240C2×C3×C4—181no—
241C2×C3×C4#62611yescharacteristic
242C2×C3×C4#62611yescharacteristic
243C2×C3×C4C2×D3211yescharacteristic
244C2×Dic3—61no—
245C2×Dic3—61no—
246C2×Dic3—62no—
247C2×Dic3—61no—
248C2×Dic3—61no—
249C2×Dic3—61no—
250C2×Dic3—62no—
251C2×Dic3—61no—
252C2×Dic3—181no—
253C2×Dic3—181no—
254Dic6—32no—
255Dic6—32no—
256Dic6—32no—
257Dic6—62no—
258Dic6—61no—
259Dic6—61no—
260Dic6—62no—
261Dic6—61no—
262Dic6—92no—
263Dic6—181no—
264C3×C8—92no—
265C3×C8—181no—
266C3⋊C8—32no—
267C3⋊C8—61no—
268C3⋊C8—92no—
269C3⋊C8—181no—
270C2×C32—21no—
271C2×C32—21no—
272C2×C32—21no—
273C2×C32C4×D1211yescharacteristic
274C2×C32#90411yescharacteristic
275C2×C32C2×Dic1211yescharacteristic
276C2×C32C4×#6711yescharacteristic
277C2×C32C2×#21611yescharacteristic
278C2×C32#95011yescharacteristic
279C2×Q8—91no—
280C2×Q8—91no—
281Q16—274no—
282C2×C8—91no—
283C2×C8—91no—
284C4⋊C4—91no—
285C4⋊C4—91no—
286C4⋊C4—181no—
287C4⋊C4—181no—
288C42—181no—
289C42—181no—
290C42((C3⋊D3)⋊C3)⋊C211yescharacteristic
291C22×C3—31no—
292C22×C3—31no—
293C22×C3—61no—
294C22×C3C2×#62611yescharacteristic
295C3×C4—21no—
296C3×C4—31no—
297C3×C4—31no—
298C3×C4—31no—
299C3×C4—31no—
300C3×C4—32no—
301C3×C4—32no—
302C3×C4—61no—
303C3×C4—61no—
304C3×C4—61no—
305C3×C4—61no—
306C3×C4—62no—
307C3×C4—121no—
308C3×C4—182no—
309C3×C4—182no—
310C3×C4—361no—
311C3×C4—361no—
312C3×C4#371912yes—
313C3×C4#379811yescharacteristic
314C3×C4C4×D3211yescharacteristic
315Dic3—62no—
316Dic3—62no—
317Dic3—64no—
318Dic3—62no—
319Dic3—121no—
320Dic3—121no—
321Dic3—122no—
322Dic3—121no—
323Dic3—182no—
324Dic3—361no—
325C32—21no—
326C32C4×Dic1211yescharacteristic, complemented
327C32C4×#21611yescharacteristic, complemented
328Q8—92no—
329Q8—92no—
330Q8—181no—
331Q8—181no—
332C2×C4—181no—
333C2×C4—181no—
334C2×C4—181no—
335C2×C4—181no—
336C2×C4#620011yescharacteristic
337C2×C4#620011yescharacteristic
338C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
339C8—92no—
340C8—181no—
341C2×C3—31no—
342C2×C3—31no—
343C2×C3—31no—
344C2×C3—31no—
345C2×C3—31no—
346C2×C3—31no—
347C2×C3—61no—
348C2×C3—61no—
349C2×C3—61no—
350C2×C3#6382911yescharacteristic
351C2×C3C2×#371911yescharacteristic
352C2×C3C4×#62611yescharacteristic
353C22C2×#620011yescharacteristic
354C4—21no—
355C4—182no—
356C4—182no—
357C4—361no—
358C4—361no—
359C4#8843912yesdirect product factor
360C4#8865011yescharacteristic
361C4C4×((C3⋊D3)⋊C3)⋊C211yescharacteristic
362C3—31no—
363C3—31no—
364C3—61no—
365C3C4×#371911yescharacteristic
366C2#1172107311yescharacteristic
367C2C2×#8843911yescharacteristic
368C2C4×#620011yescharacteristic
subgroupG/H
centerC2×C4#6200
commutatorC4×H(3,𝔽3)C22×C4
FrattiniC2×C3×C4C2×D32
FittingC42×H(3,𝔽3)C22
socleC22×C3C2×#626
prime radicalC22×H(3,𝔽3)C2×D4
cosocleC2×C4×H(3,𝔽3)C23
hypercenterC42((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)C4×Q16
LCSG ▷ C4×H(3,𝔽3) ▷ 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
21. #88439C4direct (N × H)
84. C4×H(3,𝔽3)Q16semidirect
200. C32×C4Dic12semidirect
201. C32×C4#216semidirect
223. H(3,𝔽3)C4×Q16semidirect
326. C32C4×Dic12semidirect
327. C32C4×#216semidirect
359. C4#88439direct (N × H)

Maximal subgroups

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

subgroupindex
C4×#62292
C4×#634⋊C32
C2×#884392
C4×(C3⋊(C3⋊C8))⋊C32
#117203262
#117203092
#117203062
C4×#37193
C4×#37183