Finite Group Databasebeta

← C2×#11721064 #13150599

Subgroups — C2×#11721064 #13150599

Order 1728.

1095 conjugacy classes of subgroups.

The subgroup lattice has 418 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#88740C211yesmaximal, characteristic
2C2×#3748⋊C3C211yesmaximal, characteristic, complemented
3C22×(C3⋊(C3⋊C8))⋊C3C211yesmaximal, characteristic, complemented
4C2×#88435C212yesmaximal, complemented
5#11721064C218yesmaximal, direct product factor
6C2×#88434C212yesmaximal, complemented
7C2×#63815—31nomaximal
8C2×#63814—31nomaximal
9C22×C4×H(3,𝔽3)C2211yescharacteristic
10C2×#6235C412yes—
11#88740C414yes—
12#88740C2214yes—
13C2×C4⋊#272C411yescharacteristic
14C2×C4×#272C2212yes—
15C2×#6229C411yescharacteristic
16#3748⋊C3C2214yescomplemented
17C2×(C3⋊(C3⋊C8))⋊C3C2211yescharacteristic
18C2×(C3⋊(C3⋊C8))⋊C3C2212yes—
19C2×(C3⋊(C3⋊C8))⋊C3C2211yescharacteristic
20C2×(C3⋊(C3⋊C8))⋊C3C2214yescomplemented
21#88435C2218yescomplemented
22#88434C2218yescomplemented
23C2×C3×#235—31no—
24C2×C3×#235—31no—
25C2×C3×#235—61no—
26C2×#3748D311yescharacteristic, complemented
27C22×C3⋊(C3⋊C8)D311yescharacteristic, complemented
28C2×C3×#208—31no—
29C22×C3×C3⋊C8—31no—
30C2×#3711—32no—
31C2×#3711—32no—
32#63815—38no—
33#63814—38no—
34C2×D3×C3⋊C8—32no—
35C2×D3×C3⋊C8—32no—
36C23×H(3,𝔽3)C2×C411yescharacteristic
37C22×#272C2×C412yes—
38C2×C4×H(3,𝔽3)C2×C411yescharacteristic
39C2×C4×H(3,𝔽3)C2×C412yes—
40C2×C4×H(3,𝔽3)C2311yescharacteristic
41C2×C4×H(3,𝔽3)C2312yes—
42#6235C2×C418yes—
43C2×#1084C2×C412yes—
44C4⋊#272C2×C414yes—
45C4×#272C2×C414yes—
46C4×#272C2314yes—
47#6229C2×C414yes—
48(C3⋊(C3⋊C8))⋊C3C2312yes—
49(C3⋊(C3⋊C8))⋊C3C2314yes—
50(C3⋊(C3⋊C8))⋊C3C2312yes—
51C2×#948—91no—
52C2×#907—91no—
53C2×#901—91no—
54C22×C32×C4—21no—
55C22×C32×C4C2×D311yescharacteristic
56C22×C32×C4C2×D311yescharacteristic
57C2×C3×#67—32no—
58C2×C3×#67—32no—
59C2×C3×#67—62no—
60C3×#235—34no—
61C3×#235—34no—
62C3×#235—34no—
63C3×#235—34no—
64C3×#235—64no—
65C3×#235—64no—
66C2×C3×D12—31no—
67C2×C3×D12—31no—
68C2×C3×D12—61no—
69C2×C3×C4×D3—32no—
70C2×C3×C4×D3—32no—
71C2×C3×C4×D3—62no—
72C2×C3×Dic6—31no—
73C2×C3×Dic6—31no—
74C2×C3×Dic6—61no—
75#3748C2×D314yescomplemented
76C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
77C2×C3⋊(C3⋊C8)C2×D312yes—
78C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
79C2×C3⋊(C3⋊C8)C2×D314yescomplemented
80C3×#208—34no—
81C2×C3×C3⋊C8—31no—
82C2×C3×C3⋊C8—34no—
83C2×C3×C3⋊C8—31no—
84C2×C3×C3⋊C8—32no—
85#3711—38no—
86#3711—38no—
87D3×C3⋊C8—38no—
88D3×C3⋊C8—38no—
89C22×H(3,𝔽3)—22no—
90C22×H(3,𝔽3)C22×C412yes—
91C22×H(3,𝔽3)C22×C411yescharacteristic
92C2×#272—24no—
93C2×#272C22×C414yes—
94C4×H(3,𝔽3)C22×C412yes—
95C4×H(3,𝔽3)C22×C411yescharacteristic
96C4×H(3,𝔽3)C2411yescharacteristic
97#1084C22×C414yes—
98C2×C3×#41—91no—
99C2×#235—31no—
100C2×#235—31no—
101C2×#235—61no—
102C2×C3×M16—91no—
103C22×C3×C8—91no—
104#948—98no—
105C2×#208—31no—
106C2×#208—91no—
107C2×#208—92no—
108C22×C3⋊C8—31no—
109C22×C3⋊C8—91no—
110C22×C3⋊C8—92no—
111#907—98no—
112#901—98no—
113C2×ΓL(1,𝔽25)—92no—
114C2×ΓL(1,𝔽25)—92no—
115C2×D3×C8—92no—
116C2×D3×C8—92no—
117C23×C32—21no—
118C23×C32C4×D311yescharacteristic
119C23×C32C4×D311yescharacteristic
120C22×C3×D3—32no—
121C22×C3×D3—32no—
122C22×C3×D3—62no—
123C2×C32×C4—21no—
124C2×C32×C4—22no—
125C2×C32×C4—21no—
126C2×C32×C4—22no—
127C2×C32×C4C4×D311yescharacteristic
128C2×C32×C4C4×D311yescharacteristic
129C2×C32×C4C4×D312yes—
130C2×C32×C4C4×D312yes—
131C2×C32×C4C22×D311yescharacteristic
132C2×C32×C4C22×D311yescharacteristic
133C2×C32×C4C22×D312yes—
134C2×C32×C4C22×D312yes—
135C3×#67—38no—
136C3×#67—38no—
137C3×#67—68no—
138C2×C3×Dic3—32no—
139C2×C3×Dic3—32no—
140C2×C3×Dic3—62no—
141C3×D12—34no—
142C3×D12—34no—
143C3×D12—64no—
144C3×C4×D3—34no—
145C3×C4×D3—34no—
146C3×C4×D3—34no—
147C3×C4×D3—34no—
148C3×C4×D3—64no—
149C3×C4×D3—64no—
150C3×Dic6—34no—
151C3×Dic6—34no—
152C3×Dic6—64no—
153C3⋊(C3⋊C8)C22×D312yes—
154C3⋊(C3⋊C8)C22×D314yes—
155C3⋊(C3⋊C8)C22×D312yes—
156C3×C3⋊C8—32no—
157C3×C3⋊C8—34no—
158C3×C3⋊C8—32no—
159C2×#131—271noSylow
160C2×H(3,𝔽3)—22no—
161C2×H(3,𝔽3)#13112yescomplemented
162C2×H(3,𝔽3)C23×C411yescharacteristic
163#272—24no—
164C3×#41—94no—
165C3×#41—94no—
166C2×C3×Q8—91no—
167C2×C3×D4—92no—
168C2×C3×D4—91no—
169C22×C3×C4—31no—
170C22×C3×C4—31no—
171C22×C3×C4—61no—
172C22×C3×C4—92no—
173C22×C3×C4D3211yescharacteristic
174C2×#67—32no—
175C2×#67—32no—
176C2×#67—62no—
177#235—34no—
178#235—34no—
179#235—34no—
180#235—34no—
181#235—64no—
182#235—64no—
183C2×D12—31no—
184C2×D12—31no—
185C2×D12—61no—
186C2×C4×D3—32no—
187C2×C4×D3—32no—
188C2×C4×D3—62no—
189C2×Dic6—31no—
190C2×Dic6—31no—
191C2×Dic6—61no—
192C3×M16—94no—
193C2×C3×C8—91no—
194C2×C3×C8—92no—
195C2×C3×C8—91no—
196C2×C3×C8—94no—
197#208—34no—
198#208—94no—
199#208—98no—
200C2×C3⋊C8—31no—
201C2×C3⋊C8—31no—
202C2×C3⋊C8—34no—
203C2×C3⋊C8—32no—
204C2×C3⋊C8—91no—
205C2×C3⋊C8—92no—
206C2×C3⋊C8—91no—
207C2×C3⋊C8—94no—
208C2×C3⋊C8—98no—
209ΓL(1,𝔽25)—98no—
210ΓL(1,𝔽25)—98no—
211D3×C8—98no—
212D3×C8—98no—
213C22×C32—22no—
214C22×C32—22no—
215C22×C32—21no—
216C22×C32—22no—
217C22×C32—42no—
218C22×C32C2×C4×D312yes—
219C22×C32C2×C4×D312yes—
220C22×C32C2×C4×D311yescharacteristic
221C22×C32C2×C4×D311yescharacteristic
222C2×C3×D3—34no—
223C2×C3×D3—34no—
224C2×C3×D3—64no—
225C2×C3×D3—64no—
226C2×C3×D3—64no—
227C2×C3×D3—124no—
228C32×C4—21no—
229C32×C4—22no—
230C32×C4—21no—
231C32×C4C2×C4×D312yes—
232C32×C4C2×C4×D312yes—
233C32×C4C2×C4×D311yescharacteristic
234C32×C4C2×C4×D311yescharacteristic
235C32×C4C23×D311yescharacteristic
236C32×C4C23×D311yescharacteristic
237C3×Dic3—34no—
238C3×Dic3—34no—
239C3×Dic3—64no—
240C2×#41—91no—
241#131—278no—
242C2×M16—91no—
243C2×M16—272no—
244C22×C8—91no—
245C22×C8—272no—
246H(3,𝔽3)C2×#13111yescharacteristic, complemented, Sylow
247C23×C3—31no—
248C23×C3—31no—
249C23×C3—61no—
250C23×C3—92no—
251C23×C3#62411yescharacteristic
252C22×D3—32no—
253C22×D3—32no—
254C22×D3—62no—
255C3×Q8—94no—
256C3×D4—98no—
257C3×D4—94no—
258C2×C3×C4—31no—
259C2×C3×C4—31no—
260C2×C3×C4—32no—
261C2×C3×C4—32no—
262C2×C3×C4—32no—
263C2×C3×C4—32no—
264C2×C3×C4—31no—
265C2×C3×C4—31no—
266C2×C3×C4—61no—
267C2×C3×C4—62no—
268C2×C3×C4—62no—
269C2×C3×C4—61no—
270C2×C3×C4—92no—
271C2×C3×C4—94no—
272C2×C3×C4—94no—
273C2×C3×C4#62411yescharacteristic
274C2×C3×C4#62412yes—
275C2×C3×C4C2×D3212yes—
276C2×C3×C4C2×D3211yescharacteristic
277#67—38no—
278#67—38no—
279#67—68no—
280C2×Dic3—32no—
281C2×Dic3—32no—
282C2×Dic3—62no—
283D12—34no—
284D12—34no—
285D12—64no—
286C4×D3—34no—
287C4×D3—34no—
288C4×D3—34no—
289C4×D3—34no—
290C4×D3—64no—
291C4×D3—64no—
292Dic6—34no—
293Dic6—34no—
294Dic6—64no—
295C3×C8—92no—
296C3×C8—94no—
297C3×C8—92no—
298C3⋊C8—32no—
299C3⋊C8—32no—
300C3⋊C8—34no—
301C3⋊C8—92no—
302C3⋊C8—94no—
303C3⋊C8—92no—
304C2×C32—22no—
305C2×C32—22no—
306C2×C32—21no—
307C2×C32—22no—
308C2×C32—42no—
309C2×C32#90112yescomplemented
310C2×C32#90712yescomplemented
311C2×C32C22×C4×D311yescharacteristic
312C2×C32C22×C4×D311yescharacteristic
313C3×D3—64no—
314C3×D3—64no—
315C3×D3—124no—
316#41—94no—
317#41—94no—
318C2×Q8—91no—
319C2×D4—92no—
320C2×D4—91no—
321C22×C4—92no—
322C22×C4((C3⋊D3)⋊C3)⋊C211yescharacteristic
323M16—94no—
324M16—278no—
325C2×C8—94no—
326C2×C8—91no—
327C2×C8—91no—
328C2×C8—92no—
329C2×C8—278no—
330C22×C3—22no—
331C22×C3—31no—
332C22×C3—31no—
333C22×C3—32no—
334C22×C3—32no—
335C22×C3—62no—
336C22×C3—62no—
337C22×C3—61no—
338C22×C3—62no—
339C22×C3—94no—
340C22×C3—122no—
341C22×C3—184no—
342C22×C3C2×#62412yes—
343C22×C3C2×#62411yescharacteristic
344C2×D3—34no—
345C2×D3—34no—
346C2×D3—64no—
347C2×D3—64no—
348C2×D3—64no—
349C2×D3—124no—
350C3×C4—31no—
351C3×C4—31no—
352C3×C4—32no—
353C3×C4—32no—
354C3×C4—31no—
355C3×C4—31no—
356C3×C4—61no—
357C3×C4—62no—
358C3×C4—61no—
359C3×C4—94no—
360C3×C4C2×#62411yescharacteristic
361C3×C4C2×#62412yes—
362C3×C4C22×D3211yescharacteristic
363Dic3—34no—
364Dic3—34no—
365Dic3—64no—
366C32—21no—
367C32C2×#90111yescharacteristic, complemented
368C32C2×#90711yescharacteristic, complemented
369C23—92no—
370C23#619911yescharacteristic
371Q8—94no—
372D4—98no—
373D4—94no—
374C2×C4—92no—
375C2×C4—94no—
376C2×C4—94no—
377C2×C4#619911yescharacteristic
378C2×C4#619912yes—
379C2×C4C2×((C3⋊D3)⋊C3)⋊C212yes—
380C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
381C8—92no—
382C8—92no—
383C8—94no—
384C2×C3—22no—
385C2×C3—31no—
386C2×C3—31no—
387C2×C3—32no—
388C2×C3—32no—
389C2×C3—62no—
390C2×C3—62no—
391C2×C3—61no—
392C2×C3—62no—
393C2×C3—122no—
394C2×C3—184no—
395C2×C3#6381712yes—
396C2×C3C22×#62411yescharacteristic
397D3—64no—
398D3—64no—
399D3—124no—
400C22—22no—
401C22—94no—
402C22—184no—
403C22C2×#619912yes—
404C22C2×#619911yescharacteristic
405C4—94no—
406C4C2×#619911yescharacteristic
407C4C2×#619912yes—
408C4C22×((C3⋊D3)⋊C3)⋊C211yescharacteristic
409C3—31no—
410C3—31no—
411C3—61no—
412C3C2×#6381711yescharacteristic
413C2—22no—
414C2—184no—
415C2#1172106412yesdirect product factor
416C2C22×#619911yescharacteristic
subgroupG/H
centerC2×C4C2×((C3⋊D3)⋊C3)⋊C2
commutatorC2×H(3,𝔽3)C23×C4
FrattiniC3×C4C22×D32
FittingC22×C4×H(3,𝔽3)C22
socleC22×C3C2×#624
prime radicalC2×#88740C2
cosocleC4×H(3,𝔽3)C24
hypercenterC22×C4((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)C2×#131
LCSG ▷ C2×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C2×C4 ◁ C22×C4

Complements

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

NH ≅ G/Ntype
2. C2×#3748⋊C3C2semidirect
3. C22×(C3⋊(C3⋊C8))⋊C3C2semidirect
4. C2×#88435C2semidirect
5. #11721064C2direct (N × H)
6. C2×#88434C2semidirect
16. #3748⋊C3C22semidirect
20. C2×(C3⋊(C3⋊C8))⋊C3C22semidirect
21. #88435C22semidirect
22. #88434C22semidirect
26. C2×#3748D3semidirect
27. C22×C3⋊(C3⋊C8)D3semidirect
75. #3748C2×D3semidirect
79. C2×C3⋊(C3⋊C8)C2×D3semidirect
161. C2×H(3,𝔽3)#131semidirect
246. H(3,𝔽3)C2×#131semidirect
309. C2×C32#901semidirect
310. C2×C32#907semidirect
367. C32C2×#901semidirect
368. C32C2×#907semidirect
415. C2#11721064direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#887402
C2×#3748⋊C32
C22×(C3⋊(C3⋊C8))⋊C32
C2×#884352
#117210642
C2×#884342
C2×#638153
C2×#638143