Finite Group Databasebeta

← C2×#11721094 #13150621

Subgroups — C2×#11721094 #13150621

Order 1728.

1363 conjugacy classes of subgroups.

The subgroup lattice has 428 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C4×#272C211yesmaximal, characteristic
2C22×#634⋊C3C211yesmaximal, characteristic, complemented
3C2×C4⋊((C3⋊Dic3)⋊C3)C211yesmaximal, characteristic, complemented
4C2×#88451C212yesmaximal
5C2×#88448C212yesmaximal, complemented
6#11721094C218yesmaximal, direct product factor
7C2×#63857—31nomaximal
8C2×#63856—31nomaximal
9C23×#272C2211yescharacteristic
10C22×C4×H(3,𝔽3)C2211yescharacteristic
11C22×#1084C2211yescharacteristic
12C2×C4×#272—24no—
13C2×C4×#272C2214yes—
14C22×(C3⋊Dic3)⋊C3C2212yes—
15C22×(C3⋊Dic3)⋊C3C2212yes—
16C2×#634⋊C3—24no—
17C2×#634⋊C3C2214yescomplemented
18C4⋊((C3⋊Dic3)⋊C3)C2214yescomplemented
19#88451C2218yes—
20#88448C2218yescomplemented
21C22×#634D311yescharacteristic, complemented
22C22×C3×C4×D3—31no—
23C22×C3×C4×D3—31no—
24C22×C3×C4×D3—61no—
25C22×C3×Dic6—31no—
26C2×C4⋊(C3⋊Dic3)D311yescharacteristic, complemented
27C2×C3×C4⋊Dic3—31no—
28C2×#3724—32no—
29C2×Dic3⋊Dic3—32no—
30C2×(C2×D3)⋊Dic3—32no—
31C2×(C2×D3)⋊Dic3—32no—
32#63857—38no—
33#63856—38no—
34C23×H(3,𝔽3)C2311yescharacteristic
35C22×#272—22no—
36C22×#272—22no—
37C22×#272Q814yescomplemented
38C22×#272C2312yes—
39C2×C4×H(3,𝔽3)D414yes—
40C2×C4×H(3,𝔽3)C2312yes—
41C2×#1084—22no—
42C2×#1084C2312yes—
43C4×#272—28no—
44C2×(C3⋊Dic3)⋊C3—24no—
45C2×(C3⋊Dic3)⋊C3—24no—
46C2×(C3⋊Dic3)⋊C3C2314yes—
47C2×(C3⋊Dic3)⋊C3C2314yes—
48#634⋊C3—28no—
49C2×#946—91no—
50C2×#924—91no—
51C2×#897—91no—
52C23×C3×D3—31no—
53C23×C3×D3—31no—
54C23×C3×D3—61no—
55C22×C32×C4—21no—
56C22×C32×C4C2×D311yescharacteristic
57C22×C32×C4C2×D311yescharacteristic
58C22×C3⋊Dic3C2×D312yes—
59C22×C3⋊Dic3C2×D312yes—
60C2×#634—24no—
61C2×#634C2×D314yescomplemented
62C22×C3×Dic3—32no—
63C22×C3×Dic3—32no—
64C22×C3×Dic3—31no—
65C22×C3×Dic3—31no—
66C22×C3×Dic3—61no—
67C2×C3×C4×D3—34no—
68C2×C3×C4×D3—34no—
69C2×C3×C4×D3—64no—
70C2×C3×C4×D3—68no—
71C2×C3×C4×D3—64no—
72C2×C3×C4×D3—64no—
73C2×C3×Dic6—34no—
74C2×C3×Dic6—64no—
75C4⋊(C3⋊Dic3)C2×D314yescomplemented
76C3×C4⋊Dic3—34no—
77#3724—38no—
78Dic3⋊Dic3—38no—
79(C2×D3)⋊Dic3—38no—
80(C2×D3)⋊Dic3—38no—
81C22×H(3,𝔽3)C2×D412yes—
82C22×H(3,𝔽3)C2×Q812yes—
83C22×H(3,𝔽3)#4112yes—
84C22×H(3,𝔽3)C2411yescharacteristic
85C2×#272—28no—
86C2×#272—22no—
87C2×#272—22no—
88C2×#272C2×Q814yescomplemented
89C4×H(3,𝔽3)C2×D414yes—
90#1084—22no—
91(C3⋊Dic3)⋊C3—24no—
92(C3⋊Dic3)⋊C3—24no—
93C22×C3×Q8—91no—
94C23×C3×C4—91no—
95C22×C4×D3—31no—
96C22×C4×D3—31no—
97C22×C4×D3—61no—
98C22×Dic6—31no—
99C22×Dic6—91no—
100C2×C3×C4⋊C4—91no—
101C2×#217—92no—
102#946—98no—
103C2×(C2×D3)⋊C4—92no—
104C2×(C2×D3)⋊C4—92no—
105C2×C4⋊Dic3—31no—
106C2×C4⋊Dic3—91no—
107C2×C4⋊Dic3—92no—
108#924—98no—
109C2×Dic3⋊C4—92no—
110C2×Dic3⋊C4—92no—
111#897—98no—
112C23×C32—21no—
113C23×C32C22×D311yescharacteristic
114C23×C32C22×D311yescharacteristic
115C22×C3×D3—34no—
116C22×C3×D3—34no—
117C22×C3×D3—32no—
118C22×C3×D3—32no—
119C22×C3×D3—64no—
120C22×C3×D3—62no—
121C22×C3×D3—64no—
122C22×C3×D3—64no—
123C22×C3×D3—62no—
124C22×C3×D3—62no—
125C22×C3×D3—62no—
126C22×C3×D3—62no—
127C2×C32×C4—24no—
128C2×C32×C4—22no—
129C2×C32×C4D1214yes—
130C2×C32×C4#6714yes—
131C2×C32×C4C22×D312yes—
132C2×C32×C4C22×D312yes—
133C2×C3⋊Dic3—24no—
134C2×C3⋊Dic3—24no—
135C2×C3⋊Dic3C22×D314yes—
136C2×C3⋊Dic3C22×D314yes—
137#634—28no—
138C2×C3×Dic3—34no—
139C2×C3×Dic3—34no—
140C2×C3×Dic3—32no—
141C2×C3×Dic3—32no—
142C2×C3×Dic3—62no—
143C2×C3×Dic3—64no—
144C2×C3×Dic3—64no—
145C2×C3×Dic3—62no—
146C2×C3×Dic3—64no—
147C2×C3×Dic3—62no—
148C3×C4×D3—616no—
149C3×C4×D3—68no—
150C3×C4×D3—68no—
151C3×Dic6—68no—
152C2×C22⋊Q8—271noSylow
153C2×H(3,𝔽3)C22⋊Q814yescomplemented
154C2×H(3,𝔽3)C22×D411yescharacteristic
155C2×H(3,𝔽3)C22×Q811yescharacteristic
156C2×H(3,𝔽3)C2×#4111yescharacteristic
157#272—24no—
158C24×C3—91no—
159C23×D3—31no—
160C23×D3—31no—
161C23×D3—61no—
162C2×C3×Q8—94no—
163C2×C3×Q8—184no—
164C22×C3×C4—31no—
165C22×C3×C4—31no—
166C22×C3×C4—61no—
167C22×C3×C4—92no—
168C22×C3×C4—92no—
169C22×C3×C4—91no—
170C22×C3×C4—94no—
171C22×C3×C4—184no—
172C22×C3×C4D3211yescharacteristic
173C22×Dic3—31no—
174C22×Dic3—31no—
175C22×Dic3—32no—
176C22×Dic3—32no—
177C22×Dic3—61no—
178C22×Dic3—92no—
179C22×Dic3—92no—
180C2×C4×D3—34no—
181C2×C4×D3—34no—
182C2×C4×D3—64no—
183C2×C4×D3—64no—
184C2×C4×D3—64no—
185C2×C4×D3—68no—
186C2×Dic6—34no—
187C2×Dic6—64no—
188C2×Dic6—94no—
189C2×Dic6—184no—
190C3×C4⋊C4—94no—
191#217—98no—
192(C2×D3)⋊C4—98no—
193(C2×D3)⋊C4—98no—
194C4⋊Dic3—34no—
195C4⋊Dic3—94no—
196C4⋊Dic3—98no—
197Dic3⋊C4—98no—
198Dic3⋊C4—98no—
199C22×C32—21no—
200C22×C32—22no—
201C22×C32—22no—
202C22×C32—22no—
203C22×C32C2×D1212yes—
204C22×C32#23712yes—
205C22×C32D3×Q812yes—
206C22×C32D3×Q812yes—
207C22×C32#23912yes—
208C22×C32C2×#6712yes—
209C22×C32C23×D311yescharacteristic
210C22×C32C23×D311yescharacteristic
211C2×C3×D3—34no—
212C2×C3×D3—34no—
213C2×C3×D3—64no—
214C2×C3×D3—64no—
215C2×C3×D3—64no—
216C2×C3×D3—616no—
217C2×C3×D3—68no—
218C2×C3×D3—68no—
219C2×C3×D3—62no—
220C2×C3×D3—62no—
221C2×C3×D3—62no—
222C2×C3×D3—62no—
223C32×C4—24no—
224C32×C4C2×D1214yes—
225C32×C4C2×#6714yes—
226C3⋊Dic3—24no—
227C3⋊Dic3—24no—
228C3×Dic3—64no—
229C3×Dic3—64no—
230C3×Dic3—62no—
231C3×Dic3—64no—
232C3×Dic3—62no—
233C22×Q8—91no—
234C23×C4—91no—
235C22⋊Q8—278no—
236C2×C4⋊C4—91no—
237C2×C4⋊C4—272no—
238C2×C22⋊C4—272no—
239H(3,𝔽3)C2×C22⋊Q811yescharacteristic, complemented, Sylow
240C23×C3—31no—
241C23×C3—31no—
242C23×C3—61no—
243C23×C3—92no—
244C23×C3—94no—
245C23×C3—182no—
246C23×C3—182no—
247C23×C3C2×D3211yescharacteristic
248C22×D3—32no—
249C22×D3—32no—
250C22×D3—34no—
251C22×D3—34no—
252C22×D3—62no—
253C22×D3—64no—
254C22×D3—62no—
255C22×D3—62no—
256C22×D3—62no—
257C22×D3—62no—
258C22×D3—64no—
259C22×D3—64no—
260C3×Q8—188no—
261C2×C3×C4—34no—
262C2×C3×C4—34no—
263C2×C3×C4—32no—
264C2×C3×C4—32no—
265C2×C3×C4—64no—
266C2×C3×C4—62no—
267C2×C3×C4—94no—
268C2×C3×C4—94no—
269C2×C3×C4—92no—
270C2×C3×C4—184no—
271C2×C3×C4—184no—
272C2×C3×C4—182no—
273C2×C3×C4—188no—
274C2×C3×C4#62614yes—
275C2×C3×C4C2×D3212yes—
276C2×Dic3—32no—
277C2×Dic3—32no—
278C2×Dic3—34no—
279C2×Dic3—34no—
280C2×Dic3—62no—
281C2×Dic3—62no—
282C2×Dic3—62no—
283C2×Dic3—64no—
284C2×Dic3—64no—
285C2×Dic3—64no—
286C2×Dic3—94no—
287C2×Dic3—94no—
288C2×Dic3—184no—
289C2×Dic3—184no—
290C4×D3—68no—
291C4×D3—68no—
292C4×D3—616no—
293Dic6—68no—
294Dic6—188no—
295C2×C32—21no—
296C2×C32—21no—
297C2×C32—21no—
298C2×C32—24no—
299C2×C32#89714yescomplemented
300C2×C32#94614yescomplemented
301C2×C32C22×D1211yescharacteristic
302C2×C32C2×#23711yescharacteristic
303C2×C32C2×D3×Q811yescharacteristic
304C2×C32C2×D3×Q811yescharacteristic
305C2×C32C2×#23911yescharacteristic
306C2×C32C22×#6711yescharacteristic
307C3×D3—68no—
308C3×D3—64no—
309C3×D3—64no—
310C24—91no—
311C2×Q8—94no—
312C2×Q8—184no—
313C22×C4—92no—
314C22×C4—92no—
315C22×C4—91no—
316C22×C4—94no—
317C22×C4—184no—
318C22×C4((C3⋊D3)⋊C3)⋊C211yescharacteristic
319C4⋊C4—94no—
320C4⋊C4—278no—
321C22⋊C4—278no—
322C22×C3—31no—
323C22×C3—31no—
324C22×C3—32no—
325C22×C3—32no—
326C22×C3—32no—
327C22×C3—32no—
328C22×C3—32no—
329C22×C3—32no—
330C22×C3—61no—
331C22×C3—62no—
332C22×C3—62no—
333C22×C3—62no—
334C22×C3—94no—
335C22×C3—182no—
336C22×C3—182no—
337C22×C3—188no—
338C22×C3#379612yes—
339C22×C3#379712yes—
340C22×C3C2×#62612yes—
341C22×C3C22×D3211yescharacteristic
342C2×D3—34no—
343C2×D3—34no—
344C2×D3—64no—
345C2×D3—62no—
346C2×D3—62no—
347C2×D3—68no—
348C2×D3—68no—
349C2×D3—62no—
350C2×D3—62no—
351C2×D3—64no—
352C2×D3—616no—
353C2×D3—64no—
354C3×C4—34no—
355C3×C4—34no—
356C3×C4—64no—
357C3×C4—184no—
358C3×C4—184no—
359C3×C4—182no—
360C3×C4C2×#62614yes—
361Dic3—62no—
362Dic3—62no—
363Dic3—64no—
364Dic3—64no—
365Dic3—64no—
366Dic3—184no—
367Dic3—184no—
368C32—21no—
369C32C2×#89711yescharacteristic, complemented
370C32C2×#94611yescharacteristic, complemented
371C23—92no—
372C23—94no—
373C23—182no—
374C23—182no—
375C23C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
376Q8—188no—
377C2×C4—94no—
378C2×C4—94no—
379C2×C4—92no—
380C2×C4—184no—
381C2×C4—182no—
382C2×C4—184no—
383C2×C4—188no—
384C2×C4#620014yes—
385C2×C4C2×((C3⋊D3)⋊C3)⋊C212yes—
386C2×C3—31no—
387C2×C3—31no—
388C2×C3—34no—
389C2×C3—34no—
390C2×C3—31no—
391C2×C3—31no—
392C2×C3—31no—
393C2×C3—31no—
394C2×C3—61no—
395C2×C3—64no—
396C2×C3—61no—
397C2×C3—61no—
398C2×C3—184no—
399C2×C3#6387114yes—
400C2×C3C2×#379611yescharacteristic
401C2×C3C2×#379711yescharacteristic
402C2×C3C22×#62611yescharacteristic
403D3—64no—
404D3—64no—
405D3—68no—
406C22—94no—
407C22—182no—
408C22—188no—
409C22—182no—
410C22#8864912yes—
411C22#8865112yes—
412C22C2×#620012yes—
413C22C22×((C3⋊D3)⋊C3)⋊C211yescharacteristic
414C4—182no—
415C4—184no—
416C4—184no—
417C4C2×#620014yes—
418C3—31no—
419C3—31no—
420C3—61no—
421C3C2×#6387111yescharacteristic
422C2—184no—
423C2#1172109414yesdirect product factor
424C2C2×#8864911yescharacteristic
425C2C2×#8865111yescharacteristic
426C2C22×#620011yescharacteristic
subgroupG/H
centerC23C2×((C3⋊D3)⋊C3)⋊C2
commutator, cosocleC22×H(3,𝔽3)C24
FrattiniC22×C3C22×D32
FittingC22×C4×H(3,𝔽3)C22
socleC23×C3C2×D32
prime radicalC23×#272C22
hypercenterC22×C4((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)C2×C22⋊Q8
LCSG ▷ C22×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C23 ◁ C22×C4

Complements

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

NH ≅ G/Ntype
2. C22×#634⋊C3C2semidirect
3. C2×C4⋊((C3⋊Dic3)⋊C3)C2semidirect
5. C2×#88448C2semidirect
6. #11721094C2direct (N × H)
17. C2×#634⋊C3C22semidirect
18. C4⋊((C3⋊Dic3)⋊C3)C22semidirect
20. #88448C22semidirect
21. C22×#634D3semidirect
26. C2×C4⋊(C3⋊Dic3)D3semidirect
37. C22×#272Q8semidirect
61. C2×#634C2×D3semidirect
75. C4⋊(C3⋊Dic3)C2×D3semidirect
88. C2×#272C2×Q8semidirect
153. C2×H(3,𝔽3)C22⋊Q8semidirect
239. H(3,𝔽3)C2×C22⋊Q8semidirect
299. C2×C32#897semidirect
300. C2×C32#946semidirect
369. C32C2×#897semidirect
370. C32C2×#946semidirect
423. C2#11721094direct (N × H)

Maximal subgroups

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

subgroupindex
C22×C4×#2722
C22×#634⋊C32
C2×C4⋊((C3⋊Dic3)⋊C3)2
C2×#884512
C2×#884482
#117210942
C2×#638573
C2×#638563