Finite Group Databasebeta

← C2×#11722781 #13167392

Subgroups — C2×#11722781 #13167392

Order 1728.

752 conjugacy classes of subgroups; only the 79 normal subgroups are shown.

normal subgroups only

This group has 752 conjugacy classes of subgroups — too many to draw. Only its 79 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×C4⋊#1113C211yesmaximal, characteristic
2C2×C3×#3788C211yesmaximal, characteristic, complemented
3C2×#88809C211yesmaximal, characteristic, complemented
4#11722781C214yesmaximal, direct product factor
5C2×#63727—31nomaximal
6C22×#1113C412yes—
7C2×C3×C4⋊(C3⋊D3)—21no—
8C2×C3×C4×C3⋊D3C2211yescharacteristic
9C2×#6295—21no—
10C4⋊#1113—21no—
11C4⋊#1113—21no—
12C4⋊#1113C412yes—
13C4⋊#1113C2212yes—
14C2×C3×#622—21no—
15C3×#3788—21no—
16C3×#3788C2212yescomplemented
17C2×C3×C3⋊(C3⋊C8)—21no—
18#88809—21no—
19#88809C2212yescomplemented
20C2×#3802—31no—
21C2×D3×D12—61no—
22C2×#3788D311yescharacteristic, complemented
23C2×#3712—31no—
24C2×#3711—61no—
25#63727—34no—
26C22×C3×C3⋊D3—22no—
27C22×C3×C3⋊D3C2×C411yescharacteristic
28C2×#1113—22no—
29C2×#1113—44no—
30C2×#1113C2×C414yes—
31C2×C33×C4D411yescharacteristic
32C2×C3×C3⋊Dic3D411yescharacteristic
33C3×C4⋊(C3⋊D3)—22no—
34C3×C4⋊(C3⋊D3)—22no—
35C3×C4×C3⋊D3C2×C411yescharacteristic
36C3×C4×C3⋊D3D411yescharacteristic
37C3×C4×C3⋊D3D411yescharacteristic
38C3×C4×C3⋊D3C2311yescharacteristic
39#6295—22no—
40#6295—22no—
41C3×#622—22no—
42C3×C3⋊(C3⋊C8)—22no—
43C2×#823—91nomaximal
44C22×D32—32no—
45C22×D32—62no—
46C2×C4⋊(C3⋊D3)—21no—
47C2×C4×C3⋊D3C2×D311yescharacteristic
48C2×C3×D12—61no—
49C2×C3×D12—61no—
50C2×C3×D12—121no—
51C2×C3×C4×D3—61no—
52C2×C3×C4×D3—61no—
53C2×#626—62no—
54C2×#625—61no—
55#3802—32no—
56#3802—32no—
57#3802—61no—
58#3802—61no—
59D3×D12—62no—
60D3×D12—62no—
61D3×D12—62no—
62D3×D12—62no—
63C2×#622—21no—
64#3788—21no—
65#3788C2×D312yescomplemented
66C2×C3⋊(C3⋊C8)—21no—
67C2×C3×C3⋊C8—61no—
68C2×C3×C3⋊C8—61no—
69C2×C3×C3⋊C8—121no—
70#3712—32no—
71#3712—61no—
72#3711—62no—
73#3711—62no—
74C22×C33C22⋊C411yescharacteristic
75C2×C3×C3⋊D3—22no—
76C2×C3×C3⋊D3—24no—
77C2×C3×C3⋊D3—44no—
78C2×C3×C3⋊D3C22⋊C411yescharacteristic
79C2×C3×C3⋊D3C22×C411yescharacteristic
80#1113—44no—
81#1113—44no—
82C33×C4C22⋊C411yescharacteristic
83C33×C4C2×D411yescharacteristic
84C3×C3⋊Dic3C22⋊C411yescharacteristic
85C3×C3⋊Dic3C2×D411yescharacteristic
86C2×D3×D4—181no—
87C22×D12—91no—
88C2×C3×M16—91no—
89C2×#208—91no—
90C2×ΓL(1,𝔽25)—181no—
91#823—94no—
92C22×C3⋊D3—22no—
93C22×C3⋊D3C4×D311yescharacteristic
94C22×C3×D3—61no—
95C22×C3×D3—61no—
96C22×C3×D3—62no—
97C22×C3×D3—62no—
98C22×C3×D3—122no—
99C2×D32—34no—
100C2×D32—64no—
101C2×D32—64no—
102C2×D32—62no—
103C2×D32—128no—
104C2×D32—124no—
105C2×C32×C4—21no—
106C2×C32×C4—21no—
107C2×C32×C4—41no—
108C2×C32×C4—41no—
109C2×C32×C4D1211yescharacteristic
110C2×C3⋊Dic3#6711yescharacteristic
111C4⋊(C3⋊D3)—22no—
112C4⋊(C3⋊D3)—22no—
113C4×C3⋊D3C4×D311yescharacteristic
114C4×C3⋊D3D1211yescharacteristic
115C4×C3⋊D3#6711yescharacteristic
116C4×C3⋊D3C22×D311yescharacteristic
117C2×C3×Dic3—61no—
118C2×C3×Dic3—61no—
119C3×D12—62no—
120C3×D12—62no—
121C3×D12—62no—
122C3×D12—62no—
123C3×D12—122no—
124C3×D12—122no—
125C3×C4×D3—61no—
126C3×C4×D3—61no—
127C3×C4×D3—61no—
128C3×C4×D3—61no—
129C3×C4×D3—61no—
130C3×C4×D3—61no—
131C3×C4×D3—61no—
132C3×C4×D3—61no—
133#626—64no—
134#626—64no—
135#625—62no—
136#625—62no—
137#622—22no—
138C3⋊(C3⋊C8)—22no—
139C3×C3⋊C8—62no—
140C3×C3⋊C8—62no—
141C3×C3⋊C8—122no—
142C2×#100—271noSylow
143C2×C33#10012yescomplemented
144C2×C33C2×C22⋊C411yescharacteristic
145C3×C3⋊D3—21no—
146C3×C3⋊D3—21no—
147C3×C3⋊D3—44no—
148C23×D3—92no—
149C23×D3—182no—
150C2×C3×D4—181no—
151C22×C3×C4—91no—
152C2×#67—182no—
153D3×D4—182no—
154D3×D4—182no—
155D3×D4—182no—
156D3×D4—182no—
157C2×D12—61no—
158C2×D12—61no—
159C2×D12—92no—
160C2×D12—92no—
161C2×D12—121no—
162C2×D12—181no—
163C2×D12—181no—
164C2×D12—181no—
165C2×C4×D3—61no—
166C2×C4×D3—61no—
167C3×M16—92no—
168C3×M16—181no—
169C2×C3×C8—181no—
170C2×C3×C8—181no—
171#208—92no—
172#208—181no—
173C2×C3⋊C8—61no—
174C2×C3⋊C8—61no—
175C2×C3⋊C8—121no—
176ΓL(1,𝔽25)—182no—
177ΓL(1,𝔽25)—182no—
178C22×C32—21no—
179C22×C32—21no—
180C22×C32—41no—
181C22×C32—41no—
182C22×C32(C2×D3)⋊C411yescharacteristic
183C2×C3⋊D3—22no—
184C2×C3⋊D3—24no—
185C2×C3⋊D3—44no—
186C2×C3⋊D3(C2×D3)⋊C411yescharacteristic
187C2×C3⋊D3C2×C4×D311yescharacteristic
188C2×C3×D3—61no—
189C2×C3×D3—61no—
190C2×C3×D3—61no—
191C2×C3×D3—61no—
192C2×C3×D3—64no—
193C2×C3×D3—64no—
194C2×C3×D3—64no—
195C2×C3×D3—122no—
196C2×C3×D3—124no—
197C2×C3×D3—124no—
198C2×C3×D3—124no—
199C2×C3×D3—244no—
200D32—128no—
201D32—128no—
202D32—124no—
203D32—124no—
204C32×C4—21no—
205C32×C4—21no—
206C32×C4—21no—
207C32×C4—21no—
208C32×C4—41no—
209C32×C4—41no—
210C32×C4—41no—
211C32×C4—41no—
212C32×C4(C2×D3)⋊C411yescharacteristic
213C32×C4C2×D1211yescharacteristic
214C3⋊Dic3(C2×D3)⋊C411yescharacteristic
215C3⋊Dic3C2×#6711yescharacteristic
216C3×Dic3—61no—
217C3×Dic3—61no—
218C3×Dic3—61no—
219C3×Dic3—61no—
220C22×D4—271no—
221C2×M16—91no—
222C2×M16—271no—
223#100—274no—
224C33C2×#10011yescharacteristic, complemented, Sylow
225C23×C3—91no—
226C23×C3—182no—
227C22×D3—61no—
228C22×D3—61no—
229C22×D3—62no—
230C22×D3—62no—
231C22×D3—94no—
232C22×D3—122no—
233C22×D3—184no—
234C22×D3—182no—
235C22×D3—184no—
236C22×D3—368no—
237C22×D3—364no—
238C3×D4—182no—
239C3×D4—182no—
240C2×C3×C4—21no—
241C2×C3×C4—21no—
242C2×C3×C4—41no—
243C2×C3×C4—41no—
244C2×C3×C4—91no—
245C2×C3×C4—91no—
246C2×C3×C4—91no—
247C2×C3×C4—91no—
248C2×C3×C4—91no—
249C2×C3×C4GO+(4,𝔽2)11yescharacteristic
250#67—184no—
251#67—184no—
252C2×Dic3—61no—
253C2×Dic3—61no—
254D12—62no—
255D12—62no—
256D12—62no—
257D12—62no—
258D12—122no—
259D12—122no—
260D12—182no—
261D12—182no—
262C4×D3—61no—
263C4×D3—61no—
264C4×D3—61no—
265C4×D3—61no—
266C4×D3—61no—
267C4×D3—61no—
268C4×D3—61no—
269C4×D3—61no—
270C3×C8—182no—
271C3×C8—182no—
272C3⋊C8—62no—
273C3⋊C8—62no—
274C3⋊C8—122no—
275C2×C32—21no—
276C2×C32—21no—
277C2×C32—22no—
278C2×C32—22no—
279C2×C32—42no—
280C2×C32—42no—
281C2×C32—41no—
282C2×C32—41no—
283C2×C32#82312yescomplemented
284C2×C32C2×(C2×D3)⋊C411yescharacteristic
285C3⋊D3—21no—
286C3⋊D3—21no—
287C3⋊D3—44no—
288C3×D3—62no—
289C3×D3—62no—
290C3×D3—121no—
291C3×D3—121no—
292C3×D3—124no—
293C3×D3—124no—
294C3×D3—244no—
295C24—272no—
296C2×D4—181no—
297C2×D4—272no—
298C2×D4—272no—
299C2×D4—541no—
300C2×D4—541no—
301C2×D4—541no—
302C22×C4—91no—
303M16—92no—
304M16—181no—
305M16—272no—
306M16—541no—
307C2×C8—181no—
308C2×C8—181no—
309C22×C3—21no—
310C22×C3—21no—
311C22×C3—41no—
312C22×C3—41no—
313C22×C3—91no—
314C22×C3—91no—
315C22×C3—184no—
316C22×C3—182no—
317C22×C3—364no—
318C22×C3D32⋊C411yescharacteristic
319C2×D3—61no—
320C2×D3—61no—
321C2×D3—61no—
322C2×D3—61no—
323C2×D3—64no—
324C2×D3—64no—
325C2×D3—64no—
326C2×D3—122no—
327C2×D3—124no—
328C2×D3—124no—
329C2×D3—124no—
330C2×D3—244no—
331C2×D3—368no—
332C2×D3—368no—
333C2×D3—364no—
334C2×D3—364no—
335C3×C4—21no—
336C3×C4—21no—
337C3×C4—21no—
338C3×C4—21no—
339C3×C4—41no—
340C3×C4—41no—
341C3×C4—41no—
342C3×C4—41no—
343C3×C4—91no—
344C3×C4—91no—
345C3×C4D32⋊C411yescharacteristic
346C3×C4C2×GO+(4,𝔽2)11yescharacteristic
347Dic3—61no—
348Dic3—61no—
349Dic3—61no—
350Dic3—61no—
351C32—21no—
352C32—21no—
353C32—41no—
354C32—41no—
355C32C2×#82311yescharacteristic, complemented
356C23—91no—
357C23—182no—
358C23—274no—
359C23—542no—
360C23—1084no—
361D4—182no—
362D4—182no—
363D4—542no—
364D4—542no—
365C2×C4—91no—
366C2×C4—91no—
367C2×C4—91no—
368C2×C4—91no—
369C2×C4—91no—
370C2×C4#157⋊D311yescharacteristic
371C8—182no—
372C8—182no—
373C2×C3—22no—
374C2×C3—22no—
375C2×C3—21no—
376C2×C3—21no—
377C2×C3—42no—
378C2×C3—42no—
379C2×C3—41no—
380C2×C3—41no—
381C2×C3—181no—
382C2×C3—181no—
383C2×C3—364no—
384C2×C3#6372712yescomplemented
385C2×C3C2×D32⋊C411yescharacteristic
386D3—62no—
387D3—62no—
388D3—121no—
389D3—121no—
390D3—124no—
391D3—124no—
392D3—244no—
393C22—91no—
394C22—91no—
395C22—182no—
396C22—184no—
397C22—364no—
398C22—1084no—
399C22—1084no—
400C22#8893911yescharacteristic
401C4—91no—
402C4—91no—
403C4#8893911yescharacteristic
404C4C2×#157⋊D311yescharacteristic
405C3—21no—
406C3—21no—
407C3—41no—
408C3—41no—
409C3C2×#6372711yescharacteristic, complemented
410C2—181no—
411C2—181no—
412C2—364no—
413C2#1172278112yesdirect product factor
414C2C2×#8893911yescharacteristic
subgroupG/H
centerC22#88939
commutatorC2×C3×C3⋊D3C22×C4
FrattiniC4C2×#157⋊D3
FittingC2×C33×C4D4
socleC22×C33C22⋊C4
prime radicalC2×C4⋊#1113C2
cosocleC3×C4×C3⋊D3C23
hypercenterC2×C4#157⋊D3
hypocenterC33C2×#100
LCSG ▷ C2×C3×C3⋊D3 ▷ C2×C33 ▷ C33
UCSC1 ◁ C22 ◁ C2×C4

Complements

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

NH ≅ G/Ntype
2. C2×C3×#3788C2semidirect
3. C2×#88809C2semidirect
4. #11722781C2direct (N × H)
16. C3×#3788C22semidirect
19. #88809C22semidirect
22. C2×#3788D3semidirect
65. #3788C2×D3semidirect
143. C2×C33#100semidirect
224. C33C2×#100semidirect
283. C2×C32#823semidirect
355. C32C2×#823semidirect
384. C2×C3#63727semidirect
409. C3C2×#63727semidirect
413. C2#11722781direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C4⋊#11132
C2×C3×#37882
C2×#888092
#117227812
C2×#637273
C2×#8239