Finite Group Databasebeta

← C2×#11720335 #13141149

Subgroups — C2×#11720335 #13141149

Order 1728.

971 conjugacy classes of subgroups; only the 123 normal subgroups are shown.

normal subgroups only

This group has 971 conjugacy classes of subgroups — too many to draw. Only its 123 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×#6235C211yesmaximal, characteristic
2C2×#3757⋊C3C212yesmaximal, complemented
3#11720335C214yesmaximal, direct product factor
4C2×#63580—32nomaximal
5C24×H(3,𝔽3)C2211yescharacteristic
6C23×#272C411yescharacteristic, complemented
7C2×#6235—22no—
8C2×#6235—22no—
9C2×#6235C412yescomplemented
10C2×#6235C2212yes—
11C22×#1084C411yescharacteristic, complemented
12C22×(C3⋊Dic3)⋊C3—22no—
13#3757⋊C3—22no—
14#3757⋊C3C2214yescomplemented
15C22×C3×#67—32no—
16C22×C3×#67—61no—
17C2×#3757D312yescomplemented
18C2×C3×#217—32no—
19#63580—38no—
20C23×H(3,𝔽3)—42no—
21C23×H(3,𝔽3)C2×C411yescharacteristic
22C23×H(3,𝔽3)C2×C411yescharacteristic
23C23×H(3,𝔽3)D412yes—
24C23×H(3,𝔽3)D412yes—
25C23×H(3,𝔽3)C2311yescharacteristic
26C22×#272—21no—
27C22×#272—21no—
28C22×#272—42no—
29C22×#272C2×C412yescomplemented
30#6235—24no—
31#6235—24no—
32C2×#1084—21no—
33C2×#1084—21no—
34C2×#1084C2×C412yescomplemented
35C2×(C3⋊Dic3)⋊C3—24no—
36C2×(C3⋊Dic3)⋊C3—44no—
37C2×#834—91no—
38C2×#806—92no—
39C24×C32—21no—
40C24×C32C2×D312yes—
41C23×C3×D3—32no—
42C23×C3×D3—61no—
43C22×C3⋊Dic3—22no—
44C2×C3×#67—34no—
45C2×C3×#67—34no—
46C2×C3×#67—62no—
47C2×C3×#67—62no—
48C2×C3×#67—64no—
49C2×C3×#67—64no—
50C2×C3×#67—62no—
51C2×C3×#67—62no—
52C2×C3×#67—62no—
53C2×C3×#67—62no—
54C22×C3×Dic3—32no—
55C22×C3×Dic3—61no—
56C22×C3×Dic3—62no—
57#3757—22no—
58#3757C2×D314yescomplemented
59C3×#217—34no—
60C3×#217—62no—
61C22×H(3,𝔽3)—22no—
62C22×H(3,𝔽3)—24no—
63C22×H(3,𝔽3)—41no—
64C22×H(3,𝔽3)—42no—
65C22×H(3,𝔽3)—41no—
66C22×H(3,𝔽3)C22⋊C411yescharacteristic
67C22×H(3,𝔽3)C22⋊C412yes—
68C22×H(3,𝔽3)C22⋊C411yescharacteristic
69C22×H(3,𝔽3)C22×C411yescharacteristic
70C22×H(3,𝔽3)C2×D412yes—
71C2×#272—22no—
72C2×#272—42no—
73C2×#272—42no—
74C2×#272—42no—
75#1084—22no—
76(C3⋊Dic3)⋊C3—44no—
77C22×C3×D4—91no—
78C22×#67—32no—
79C22×#67—61no—
80C2×C3×C22⋊C4—92no—
81C2×#217—32no—
82C2×#217—92no—
83#834—94no—
84#806—98no—
85C23×C32—21no—
86C23×C32—21no—
87C23×C32—22no—
88C23×C32—22no—
89C23×C32—21no—
90C23×C32—44no—
91C23×C32—44no—
92C23×C32C4×D312yes—
93C23×C32C4×D312yes—
94C23×C32D1212yes—
95C23×C32D1212yes—
96C23×C32#6712yes—
97C23×C32#6712yes—
98C23×C32C22×D312yes—
99C22×C3×D3—34no—
100C22×C3×D3—62no—
101C22×C3×D3—62no—
102C22×C3×D3—62no—
103C22×C3×D3—62no—
104C22×C3×D3—62no—
105C22×C3×D3—124no—
106C22×C3×D3—124no—
107C2×C3⋊Dic3—24no—
108C2×C3⋊Dic3—44no—
109C3×#67—68no—
110C3×#67—68no—
111C3×#67—64no—
112C3×#67—64no—
113C3×#67—64no—
114C3×#67—64no—
115C2×C3×Dic3—34no—
116C2×C3×Dic3—62no—
117C2×C3×Dic3—62no—
118C2×C3×Dic3—62no—
119C2×C3×Dic3—62no—
120C2×C3×Dic3—62no—
121C2×C3×Dic3—64no—
122C2×C3×Dic3—124no—
123C2×#99—271noSylow
124C2×H(3,𝔽3)—21no—
125C2×H(3,𝔽3)—21no—
126C2×H(3,𝔽3)—22no—
127C2×H(3,𝔽3)—22no—
128C2×H(3,𝔽3)#9912yescomplemented
129C2×H(3,𝔽3)C2×C22⋊C411yescharacteristic
130#272—42no—
131C24×C3—32no—
132C24×C3—61no—
133C24×C3—91no—
134C24×C3D3211yescharacteristic
135C23×D3—32no—
136C23×D3—61no—
137C2×C3×D4—92no—
138C2×C3×D4—92no—
139C2×C3×D4—182no—
140C2×C3×D4—182no—
141C22×C3×C4—91no—
142C22×C3×C4—182no—
143C2×#67—34no—
144C2×#67—34no—
145C2×#67—62no—
146C2×#67—62no—
147C2×#67—62no—
148C2×#67—62no—
149C2×#67—62no—
150C2×#67—62no—
151C2×#67—64no—
152C2×#67—64no—
153C22×Dic3—32no—
154C22×Dic3—61no—
155C22×Dic3—62no—
156C22×Dic3—182no—
157C3×C22⋊C4—94no—
158C3×C22⋊C4—182no—
159#217—34no—
160#217—62no—
161#217—94no—
162#217—182no—
163C22×C32—24no—
164C22×C32—24no—
165C22×C32—24no—
166C22×C32—24no—
167C22×C32—21no—
168C22×C32—21no—
169C22×C32—22no—
170C22×C32—21no—
171C22×C32—22no—
172C22×C32—44no—
173C22×C32—42no—
174C22×C32—42no—
175C22×C32—42no—
176C22×C32—42no—
177C22×C32—42no—
178C22×C32—44no—
179C22×C32—42no—
180C22×C32(C2×D3)⋊C412yes—
181C22×C32(C2×D3)⋊C412yes—
182C22×C32(C2×D3)⋊C412yes—
183C22×C32(C2×D3)⋊C412yes—
184C22×C32C2×C4×D312yes—
185C22×C32C2×D1212yes—
186C22×C32C2×#6712yes—
187C2×C3×D3—64no—
188C2×C3×D3—64no—
189C2×C3×D3—124no—
190C2×C3×D3—124no—
191C2×C3×D3—124no—
192C2×C3×D3—124no—
193C2×C3×D3—124no—
194C2×C3×D3—124no—
195C3⋊Dic3—44no—
196C3×Dic3—64no—
197C3×Dic3—64no—
198C3×Dic3—124no—
199C22×D4—91no—
200C2×C22⋊C4—92no—
201#99—274no—
202H(3,𝔽3)C2×#9911yescharacteristic, complemented, Sylow
203C23×C3—32no—
204C23×C3—32no—
205C23×C3—32no—
206C23×C3—32no—
207C23×C3—32no—
208C23×C3—32no—
209C23×C3—32no—
210C23×C3—42no—
211C23×C3—61no—
212C23×C3—61no—
213C23×C3—62no—
214C23×C3—62no—
215C23×C3—61no—
216C23×C3—92no—
217C23×C3—124no—
218C23×C3—124no—
219C23×C3—181no—
220C23×C3—181no—
221C23×C3—362no—
222C23×C3#62411yescharacteristic
223C23×C3#62411yescharacteristic
224C23×C3#62612yes—
225C23×C3#62612yes—
226C23×C3C2×D3211yescharacteristic
227C22×D3—34no—
228C22×D3—62no—
229C22×D3—62no—
230C22×D3—62no—
231C22×D3—62no—
232C22×D3—62no—
233C22×D3—124no—
234C22×D3—124no—
235C3×D4—184no—
236C3×D4—184no—
237C2×C3×C4—92no—
238C2×C3×C4—181no—
239C2×C3×C4—181no—
240C2×C3×C4—184no—
241C2×C3×C4—364no—
242#67—64no—
243#67—64no—
244#67—64no—
245#67—64no—
246#67—68no—
247#67—68no—
248C2×Dic3—34no—
249C2×Dic3—62no—
250C2×Dic3—62no—
251C2×Dic3—62no—
252C2×Dic3—62no—
253C2×Dic3—62no—
254C2×Dic3—64no—
255C2×Dic3—124no—
256C2×Dic3—184no—
257C2×Dic3—364no—
258C2×C32—22no—
259C2×C32—22no—
260C2×C32—22no—
261C2×C32—22no—
262C2×C32—22no—
263C2×C32—22no—
264C2×C32—22no—
265C2×C32—22no—
266C2×C32—22no—
267C2×C32—21no—
268C2×C32—42no—
269C2×C32—42no—
270C2×C32#80614yescomplemented
271C2×C32C2×(C2×D3)⋊C412yes—
272C3×D3—124no—
273C3×D3—124no—
274C24—91no—
275C24((C3⋊D3)⋊C3)⋊C211yescharacteristic
276C2×D4—92no—
277C2×D4—92no—
278C2×D4—182no—
279C2×D4—182no—
280C22×C4—91no—
281C22×C4—182no—
282C22⋊C4—94no—
283C22⋊C4—182no—
284C22×C3—24no—
285C22×C3—22no—
286C22×C3—32no—
287C22×C3—32no—
288C22×C3—32no—
289C22×C3—32no—
290C22×C3—32no—
291C22×C3—32no—
292C22×C3—32no—
293C22×C3—41no—
294C22×C3—42no—
295C22×C3—41no—
296C22×C3—64no—
297C22×C3—64no—
298C22×C3—64no—
299C22×C3—64no—
300C22×C3—61no—
301C22×C3—62no—
302C22×C3—61no—
303C22×C3—62no—
304C22×C3—61no—
305C22×C3—124no—
306C22×C3—122no—
307C22×C3—122no—
308C22×C3—122no—
309C22×C3—122no—
310C22×C3—122no—
311C22×C3—124no—
312C22×C3—122no—
313C22×C3—182no—
314C22×C3—362no—
315C22×C3—362no—
316C22×C3—362no—
317C22×C3#372211yescharacteristic
318C22×C3#372212yes—
319C22×C3#372211yescharacteristic
320C22×C3C2×#62411yescharacteristic
321C22×C3C2×#62612yes—
322C2×D3—64no—
323C2×D3—64no—
324C2×D3—124no—
325C2×D3—124no—
326C2×D3—124no—
327C2×D3—124no—
328C2×D3—124no—
329C2×D3—124no—
330C3×C4—182no—
331C3×C4—364no—
332Dic3—64no—
333Dic3—64no—
334Dic3—124no—
335Dic3—364no—
336C32—21no—
337C32C2×#80612yescomplemented
338C23—42no—
339C23—92no—
340C23—181no—
341C23—181no—
342C23—362no—
343C23#619911yescharacteristic
344C23#619911yescharacteristic
345C23#620012yes—
346C23#620012yes—
347C23C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
348D4—184no—
349D4—184no—
350C2×C4—92no—
351C2×C4—181no—
352C2×C4—181no—
353C2×C4—184no—
354C2×C4—364no—
355C2×C3—22no—
356C2×C3—21no—
357C2×C3—21no—
358C2×C3—22no—
359C2×C3—32no—
360C2×C3—34no—
361C2×C3—62no—
362C2×C3—62no—
363C2×C3—62no—
364C2×C3—62no—
365C2×C3—62no—
366C2×C3—62no—
367C2×C3—62no—
368C2×C3—62no—
369C2×C3—61no—
370C2×C3—62no—
371C2×C3—122no—
372C2×C3—122no—
373C2×C3—362no—
374C2×C3#6358112yes—
375C2×C3C2×#372211yescharacteristic
376D3—124no—
377D3—124no—
378C22—24no—
379C22—22no—
380C22—41no—
381C22—42no—
382C22—41no—
383C22—182no—
384C22—362no—
385C22—362no—
386C22—362no—
387C22#8845111yescharacteristic
388C22#8845112yes—
389C22#8845111yescharacteristic
390C22C2×#619911yescharacteristic
391C22C2×#620012yes—
392C4—182no—
393C4—364no—
394C3—32no—
395C3—61no—
396C3C2×#6358111yescharacteristic
397C2—22no—
398C2—21no—
399C2—21no—
400C2—22no—
401C2—362no—
402C2#1172033512yesdirect product factor
403C2C2×#8845111yescharacteristic
subgroupG/H
centerC22#88451
commutatorC22×H(3,𝔽3)C22×C4
FrattiniC23×C3C2×D32
FittingC24×H(3,𝔽3)C22
socleC22×C3#3722
prime radicalC22×#6235C2
cosocleC23×H(3,𝔽3)C23
hypercenterC24((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)C2×#99
LCSG ▷ C22×H(3,𝔽3) ▷ C2×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C22 ◁ C24

Complements

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

NH ≅ G/Ntype
2. C2×#3757⋊C3C2semidirect
3. #11720335C2direct (N × H)
6. C23×#272C4semidirect
9. C2×#6235C4semidirect
11. C22×#1084C4semidirect
14. #3757⋊C3C22semidirect
17. C2×#3757D3semidirect
29. C22×#272C2×C4semidirect
34. C2×#1084C2×C4semidirect
58. #3757C2×D3semidirect
128. C2×H(3,𝔽3)#99semidirect
202. H(3,𝔽3)C2×#99semidirect
270. C2×C32#806semidirect
337. C32C2×#806semidirect
402. C2#11720335direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#62352
C2×#3757⋊C32
#117203352
C2×#635803