Finite Group Databasebeta

← C4×#88651 #13147851

Subgroups — C4×#88651 #13147851

Order 1728.

1151 conjugacy classes of subgroups; only the 227 normal subgroups are shown.

normal subgroups only

This group has 1151 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×#88651C211yesmaximal, characteristic
2C42×#272C211yesmaximal, characteristic
3C4×#634⋊C3C212yesmaximal, complemented
4#11721152C211yesmaximal, characteristic
5#11721147C212yesmaximal
6C4×#6199C212yesmaximal
7C4×#6196C212yesmaximal, complemented
8#11721084C212yesmaximal, complemented
9#11721078C212yesmaximal, complemented
10C4×D3×Dic6—32nomaximal
11C2×C4×#272C2211yescharacteristic
12C2×C4×#272C2211yescharacteristic
13C2×C4×#272C2211yescharacteristic
14C2×#634⋊C3C2212yes—
15C2×#6199C2212yes—
16C2×#6199C2212yes—
17C2×#6196C2212yes—
18#88651C418yesdirect product factor
19C42×H(3,𝔽3)C2211yescharacteristic
20C4×#1084C2211yescharacteristic
21C4×#1084C2211yescharacteristic
22C4×#1084C2211yescharacteristic
23C4⋊((C3⋊Dic3)⋊C3)C2212yes—
24#3750⋊C3C2214yes—
25C4×(C3⋊Dic3)⋊C3C2214yes—
26#88449C2212yes—
27#88448C2214yes—
28#88447C2214yes—
29C2×D3×Dic6—32no—
30C4×#634D312yescomplemented
31C3×C42×D3—32no—
32C3×C42×D3—61no—
33C3×C4×Dic6—32no—
34C4×#627—34no—
35D3×C4⋊Dic3—32no—
36D3×Dic3⋊C4—34no—
37C4×D3×Dic3—34no—
38#63844—32no—
39Dic3×Dic6—32no—
40#63837—34no—
41C22×#272C2311yescharacteristic
42C2×C4×H(3,𝔽3)C2311yescharacteristic
43C2×C4×H(3,𝔽3)C2311yescharacteristic
44C2×C4×H(3,𝔽3)C2311yescharacteristic
45C2×#1084C2311yescharacteristic
46C2×#1084C2311yescharacteristic
47C2×#1084C2311yescharacteristic
48C4×#272—22no—
49C4×#272C2×C414yes—
50C4×#272Q814yescomplemented
51C2×(C3⋊Dic3)⋊C3C2314yes—
52C2×(C3⋊Dic3)⋊C3C2314yes—
53#634⋊C3C2×C418yescomplemented
54#6199—24no—
55#6199C2×C418yes—
56#6196C2×C418yescomplemented
57C4×D3×Q8—92no—
58C2×C4×Dic6—91no—
59C2×#634C2×D312yes—
60C2×C3×C4×D3—32no—
61C2×C3×C4×D3—32no—
62C2×C3×C4×D3—32no—
63C2×C3×C4×D3—61no—
64C2×C3×C4×D3—61no—
65C2×C3×C4×D3—61no—
66C2×C3×Dic6—32no—
67C2×#627—34no—
68C2×D3×Dic3—34no—
69C2×D3×Dic3—34no—
70D3×Dic6—316no—
71C32×C42—21no—
72C32×C42C2×D312yes—
73C4⋊(C3⋊Dic3)C2×D312yes—
74#3750C2×D314yes—
75C4×C3⋊Dic3C2×D314yes—
76C3×C4⋊Dic3—32no—
77C3×Dic3⋊C4—34no—
78C3×C4×Dic3—32no—
79C3×C4×Dic3—32no—
80C3×C4×Dic3—34no—
81C3×C4×Dic3—32no—
82C3×C4×Dic3—61no—
83C3×C4×Dic3—61no—
84C3×C4×Dic3—61no—
85#3724—34no—
86Dic3⋊Dic3—34no—
87Dic3⋊Dic3—34no—
88Dic32—34no—
89Dic32—34no—
90C22×H(3,𝔽3)C2411yescharacteristic
91C2×#272C22×C412yes—
92C2×#272C2×Q812yes—
93C2×#272#4112yes—
94C4×H(3,𝔽3)—21no—
95C4×H(3,𝔽3)C22×C412yes—
96C4×H(3,𝔽3)C2×Q812yescomplemented
97#1084—21no—
98#1084C22×C412yes—
99#1084C2×Q812yescomplemented
100(C3⋊Dic3)⋊C3—24no—
101(C3⋊Dic3)⋊C3C22×C418yes—
102C2×D3×Q8—92no—
103C22×Dic6—91no—
104C3×C4×Q8—92no—
105C2×C3×C42—91no—
106Q8×Dic3—92no—
107C2×C4⋊Dic3—91no—
108C2×Dic3⋊C4—92no—
109C2×C4×Dic3—92no—
110D3×C4⋊C4—94no—
111D3×C4⋊C4—92no—
112#887—94no—
113#887—92no—
114C42×D3—32no—
115C42×D3—61no—
116C42×D3—94no—
117C4×Dic6—32no—
118C4×Dic6—92no—
119C4×Dic6—92no—
120C4×Dic6—94no—
121C4×Dic6—92no—
122C4×Dic6—92no—
123C22×C3×D3—32no—
124C22×C3×D3—61no—
125C2×C32×C4—21no—
126C2×C32×C4—21no—
127C2×C32×C4—21no—
128C2×C32×C4C22×D312yes—
129C2×C32×C4C22×D312yes—
130C2×C32×C4C22×D312yes—
131C2×C3⋊Dic3C22×D314yes—
132C2×C3⋊Dic3C22×D314yes—
133#634C4×D318yescomplemented
134C2×C3×Dic3—34no—
135C2×C3×Dic3—32no—
136C2×C3×Dic3—32no—
137C2×C3×Dic3—34no—
138C2×C3×Dic3—32no—
139C2×C3×Dic3—61no—
140C2×C3×Dic3—61no—
141C2×C3×Dic3—61no—
142C3×C4×D3—38no—
143C3×C4×D3—38no—
144C3×C4×D3—64no—
145C3×C4×D3—64no—
146C3×C4×D3—64no—
147C3×C4×D3—64no—
148C3×Dic6—38no—
149#627—316no—
150D3×Dic3—316no—
151D3×Dic3—68no—
152C2×C4×Q8—271noSylow
153C2×H(3,𝔽3)C23×C411yescharacteristic
154C2×H(3,𝔽3)C22×Q811yescharacteristic
155C2×H(3,𝔽3)C2×#4111yescharacteristic
156#272C4×Q814yescomplemented
157C2×C3×Q8—92no—
158C22×C3×C4—91no—
159C22×C3×C4—91no—
160C22×C3×C4—91no—
161C22×Dic3—92no—
162C22×Dic3—92no—
163D3×Q8—916no—
164C2×C4×D3—32no—
165C2×C4×D3—32no—
166C2×C4×D3—32no—
167C2×C4×D3—61no—
168C2×C4×D3—61no—
169C2×C4×D3—61no—
170C2×C4×D3—94no—
171C2×C4×D3—94no—
172C2×Dic6—32no—
173C2×Dic6—92no—
174C2×Dic6—98no—
175C2×Dic6—92no—
176C2×Dic6—94no—
177C3×C4⋊C4—94no—
178C3×C4⋊C4—92no—
179C3×C42—32no—
180C3×C42—61no—
181C3×C42—94no—
182C3×C42—91no—
183C3×C42—91no—
184C3×C42—91no—
185C3×C42D3211yescharacteristic
186C4⋊Dic3—32no—
187C4⋊Dic3—92no—
188C4⋊Dic3—92no—
189C4⋊Dic3—94no—
190Dic3⋊C4—34no—
191Dic3⋊C4—94no—
192Dic3⋊C4—94no—
193Dic3⋊C4—94no—
194Dic3⋊C4—94no—
195C4×Dic3—32no—
196C4×Dic3—32no—
197C4×Dic3—32no—
198C4×Dic3—34no—
199C4×Dic3—61no—
200C4×Dic3—61no—
201C4×Dic3—61no—
202C4×Dic3—94no—
203C4×Dic3—94no—
204C4×Dic3—94no—
205C4×Dic3—94no—
206C22×C32—21no—
207C22×C32C23×D312yes—
208C2×C3×D3—34no—
209C2×C3×D3—34no—
210C2×C3×D3—34no—
211C2×C3×D3—62no—
212C2×C3×D3—62no—
213C2×C3×D3—62no—
214C32×C4—22no—
215C32×C4—22no—
216C32×C4—22no—
217C32×C4—22no—
218C32×C4C2×C4×D314yes—
219C32×C4D3×Q814yescomplemented
220C3⋊Dic3—24no—
221C3⋊Dic3C2×C4×D318yes—
222C3×Dic3—38no—
223C3×Dic3—34no—
224C3×Dic3—34no—
225C3×Dic3—62no—
226C3×Dic3—62no—
227C3×Dic3—62no—
228C3×Dic3—64no—
229C3×Dic3—62no—
230C22×Q8—271no—
231C4×Q8—92no—
232C4×Q8—272no—
233C4×Q8—272no—
234C4×Q8—272no—
235C2×C4⋊C4—272no—
236C2×C4⋊C4—271no—
237C2×C42—91no—
238C2×C42—272no—
239H(3,𝔽3)C2×C4×Q811yescharacteristic, complemented, Sylow
240C23×C3—91no—
241C22×D3—32no—
242C22×D3—61no—
243C3×Q8—98no—
244C2×C3×C4—32no—
245C2×C3×C4—32no—
246C2×C3×C4—32no—
247C2×C3×C4—61no—
248C2×C3×C4—61no—
249C2×C3×C4—61no—
250C2×C3×C4—94no—
251C2×C3×C4—91no—
252C2×C3×C4—91no—
253C2×C3×C4—94no—
254C2×C3×C4—94no—
255C2×C3×C4—94no—
256C2×C3×C4—91no—
257C2×C3×C4—182no—
258C2×C3×C4C2×D3211yescharacteristic
259C2×C3×C4C2×D3211yescharacteristic
260C2×C3×C4C2×D3211yescharacteristic
261C2×Dic3—32no—
262C2×Dic3—32no—
263C2×Dic3—34no—
264C2×Dic3—32no—
265C2×Dic3—34no—
266C2×Dic3—61no—
267C2×Dic3—61no—
268C2×Dic3—61no—
269C2×Dic3—94no—
270C2×Dic3—94no—
271C2×Dic3—98no—
272C2×Dic3—184no—
273C4×D3—38no—
274C4×D3—38no—
275C4×D3—64no—
276C4×D3—64no—
277C4×D3—64no—
278C4×D3—64no—
279C4×D3—916no—
280C4×D3—188no—
281Dic6—38no—
282Dic6—98no—
283Dic6—98no—
284Dic6—916no—
285C2×C32—21no—
286C2×C32—21no—
287C2×C32—21no—
288C2×C32C22×C4×D312yes—
289C2×C32C2×D3×Q812yes—
290C2×C32D3×#4112yes—
291C3×D3—38no—
292C3×D3—64no—
293C2×Q8—92no—
294C2×Q8—272no—
295C2×Q8—278no—
296C22×C4—91no—
297C22×C4—91no—
298C22×C4—91no—
299C22×C4—272no—
300C22×C4—272no—
301C4⋊C4—94no—
302C4⋊C4—92no—
303C4⋊C4—272no—
304C4⋊C4—274no—
305C42—91no—
306C42—91no—
307C42—91no—
308C42—94no—
309C42—274no—
310C42((C3⋊D3)⋊C3)⋊C211yescharacteristic
311C22×C3—32no—
312C22×C3—61no—
313C22×C3—92no—
314C22×C3—92no—
315C22×C3—92no—
316C22×C3C22×D3211yescharacteristic
317C2×D3—34no—
318C2×D3—34no—
319C2×D3—34no—
320C2×D3—62no—
321C2×D3—62no—
322C2×D3—62no—
323C3×C4—21no—
324C3×C4—34no—
325C3×C4—34no—
326C3×C4—62no—
327C3×C4—62no—
328C3×C4—62no—
329C3×C4—62no—
330C3×C4—98no—
331C3×C4—92no—
332C3×C4—92no—
333C3×C4—184no—
334C3×C4—181no—
335C3×C4#379712yes—
336C3×C4C4×D3212yes—
337Dic3—34no—
338Dic3—34no—
339Dic3—38no—
340Dic3—62no—
341Dic3—62no—
342Dic3—62no—
343Dic3—62no—
344Dic3—64no—
345Dic3—98no—
346Dic3—184no—
347C32—21no—
348C32C4×D3×Q812yescomplemented
349C23—91no—
350Q8—98no—
351Q8—278no—
352C2×C4—91no—
353C2×C4—91no—
354C2×C4—94no—
355C2×C4—94no—
356C2×C4—94no—
357C2×C4—91no—
358C2×C4—94no—
359C2×C4—182no—
360C2×C4—278no—
361C2×C4—544no—
362C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
363C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
364C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
365C2×C3—32no—
366C2×C3—32no—
367C2×C3—32no—
368C2×C3—61no—
369C2×C3—61no—
370C2×C3—61no—
371C2×C3—94no—
372C2×C3C2×#379711yescharacteristic
373C2×C3C2×C4×D3211yescharacteristic
374C2×C3#6430611yescharacteristic
375D3—38no—
376D3—64no—
377C22—92no—
378C22—92no—
379C22—92no—
380C22C22×((C3⋊D3)⋊C3)⋊C211yescharacteristic
381C4—21no—
382C4—92no—
383C4—92no—
384C4—98no—
385C4—181no—
386C4—184no—
387C4#8865112yesdirect product factor
388C4C4×((C3⋊D3)⋊C3)⋊C212yes—
389C3—32no—
390C3—61no—
391C3C4×#379711yescharacteristic
392C2—94no—
393C2C2×#8865111yescharacteristic
394C2C2×C4×((C3⋊D3)⋊C3)⋊C211yescharacteristic
395C2#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 radicalC22×#272C23
cosocleC22×H(3,𝔽3)C24
hypercenterC42((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)C2×C4×Q8
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
3. C4×#634⋊C3C2semidirect
7. C4×#6196C2semidirect
8. #11721084C2semidirect
9. #11721078C2semidirect
18. #88651C4direct (N × H)
30. C4×#634D3semidirect
50. C4×#272Q8semidirect
53. #634⋊C3C2×C4semidirect
56. #6196C2×C4semidirect
96. C4×H(3,𝔽3)C2×Q8semidirect
99. #1084C2×Q8semidirect
133. #634C4×D3semidirect
156. #272C4×Q8semidirect
219. C32×C4D3×Q8semidirect
239. H(3,𝔽3)C2×C4×Q8semidirect
348. C32C4×D3×Q8semidirect
387. C4#88651direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#886512
C42×#2722
C4×#634⋊C32
#117211522
#117211472
C4×#61992
C4×#61962
#117210842
#117210782
C4×D3×Dic63