Finite Group Databasebeta

← C2×#11722757 #13167020

Subgroups — C2×#11722757 #13167020

Order 1728.

792 conjugacy classes of subgroups; only the 95 normal subgroups are shown.

normal subgroups only

This group has 792 conjugacy classes of subgroups — too many to draw. Only its 95 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×C4⋊(C3⋊#157)C211yesmaximal, characteristic, complemented
3C2×C3×#3710C211yesmaximal, characteristic, complemented
4#11722757C214yesmaximal, direct product factor
5C2×#63729—31nomaximal
6C22×#1113—21no—
7C22×C3⋊#157—21no—
8C2×C3×C4⋊(C3⋊D3)—21no—
9C2×C3×C4×C3⋊D3C2211yescharacteristic
10C2×#6295C412yescomplemented
11C4⋊#1113—21no—
12C4⋊#1113—21no—
13C4⋊#1113C412yescomplemented
14C4⋊#1113C2212yes—
15C4⋊(C3⋊#157)—21no—
16C4⋊(C3⋊#157)C2212yescomplemented
17C2×C32×C3⋊C8—21no—
18C3×#3710—21no—
19C3×#3710C2212yescomplemented
20C2×#3802—31no—
21C2×D3×D12—61no—
22C2×C4⋊#157—31no—
23C2×C3×D3×C8—61no—
24C2×#3710D311yescharacteristic, complemented
25#63729—34no—
26C22×C3×C3⋊D3—22no—
27C22×C3×C3⋊D3D411yescharacteristic
28C2×#1113—22no—
29C2×#1113—22no—
30C2×#1113—42no—
31C2×#1113—42no—
32C2×C3⋊#157—22no—
33C2×C3⋊#157—42no—
34C2×C33×C4D411yescharacteristic
35C2×C3×C3⋊Dic3C2×C411yescharacteristic
36C3×C4⋊(C3⋊D3)—22no—
37C3×C4⋊(C3⋊D3)—22no—
38C3×C4×C3⋊D3C2×C411yescharacteristic
39C3×C4×C3⋊D3D411yescharacteristic
40C3×C4×C3⋊D3D411yescharacteristic
41C3×C4×C3⋊D3C2311yescharacteristic
42#6295—22no—
43#6295C2×C414yescomplemented
44C32×C3⋊C8—22no—
45C2×#821—91nomaximal
46C22×D32—62no—
47C22×D32—61no—
48C22×#157—61no—
49C2×C4⋊(C3⋊D3)—21no—
50C2×C4×C3⋊D3C2×D311yescharacteristic
51C2×C3×D12—61no—
52C2×C3×D12—61no—
53C2×C3×D12—121no—
54C2×C3×C4×D3—61no—
55C2×C3×C4×D3—61no—
56C2×#626—62no—
57C2×#625—32no—
58#3802—32no—
59#3802—32no—
60#3802—61no—
61#3802—61no—
62D3×D12—62no—
63D3×D12—62no—
64D3×D12—62no—
65D3×D12—62no—
66C4⋊#157—32no—
67C4⋊#157—61no—
68C2×C32×C8—61no—
69C2×C3×C3⋊C8—21no—
70C2×C3×C3⋊C8—21no—
71C2×C3×C3⋊C8—41no—
72C3×D3×C8—62no—
73C3×D3×C8—62no—
74#3710—21no—
75#3710C2×D312yescomplemented
76C22×C33C22⋊C411yescharacteristic
77C2×C3×C3⋊D3—24no—
78C2×C3×C3⋊D3—44no—
79C2×C3×C3⋊D3C22⋊C411yescharacteristic
80C2×C3×C3⋊D3D812yes—
81C2×C3×C3⋊D3SD1612yes—
82C2×C3×C3⋊D3C2×D411yescharacteristic
83#1113—44no—
84#1113—42no—
85#1113—42no—
86C3⋊#157—42no—
87C33×C4C22⋊C411yescharacteristic
88C33×C4C2×D411yescharacteristic
89C3×C3⋊Dic3C22⋊C411yescharacteristic
90C3×C3⋊Dic3C22×C411yescharacteristic
91C2×D3×D4—181no—
92C22×D12—91no—
93C22×C3×C8—91no—
94C2×C4⋊Dic3—91no—
95C2×D3×C8—61no—
96#821—94no—
97C22×C3⋊D3—22no—
98C22×C3⋊D3D1211yescharacteristic
99C22×C3×D3—61no—
100C22×C3×D3—61no—
101C22×C3×D3—62no—
102C22×C3×D3—62no—
103C22×C3×D3—122no—
104C2×D32—64no—
105C2×D32—64no—
106C2×D32—62no—
107C2×D32—62no—
108C2×D32—124no—
109C2×D32—124no—
110C2×D32—122no—
111C2×D32—122no—
112C2×#157—62no—
113C2×#157—122no—
114C2×C32×C4—21no—
115C2×C32×C4—21no—
116C2×C32×C4—41no—
117C2×C32×C4—41no—
118C2×C32×C4#6711yescharacteristic
119C2×C3⋊Dic3C4×D311yescharacteristic
120C4⋊(C3⋊D3)—22no—
121C4⋊(C3⋊D3)—22no—
122C4×C3⋊D3C4×D311yescharacteristic
123C4×C3⋊D3D1211yescharacteristic
124C4×C3⋊D3#6711yescharacteristic
125C4×C3⋊D3C22×D311yescharacteristic
126C2×C3×Dic3—61no—
127C2×C3×Dic3—61no—
128C3×D12—62no—
129C3×D12—62no—
130C3×D12—62no—
131C3×D12—62no—
132C3×D12—122no—
133C3×D12—122no—
134C3×C4×D3—61no—
135C3×C4×D3—61no—
136C3×C4×D3—61no—
137C3×C4×D3—61no—
138C3×C4×D3—61no—
139C3×C4×D3—61no—
140C3×C4×D3—61no—
141C3×C4×D3—61no—
142#626—64no—
143#626—64no—
144#625—34no—
145#625—62no—
146C32×C8—62no—
147C3×C3⋊C8—22no—
148C3×C3⋊C8—22no—
149C3×C3⋊C8—42no—
150C2×#102—271noSylow
151C2×C33#10212yescomplemented
152C2×C33C2×C22⋊C411yescharacteristic
153C3×C3⋊D3—44no—
154C3×C3⋊D3#10212yescomplemented
155C3×C3⋊D3C2×D811yescharacteristic
156C3×C3⋊D3C2×SD1611yescharacteristic
157C23×D3—181no—
158C23×D3—182no—
159C2×C3×D4—181no—
160C22×C3×C4—91no—
161C2×#67—182no—
162C22×Dic3—181no—
163D3×D4—182no—
164D3×D4—182no—
165D3×D4—182no—
166D3×D4—182no—
167C2×D12—61no—
168C2×D12—61no—
169C2×D12—92no—
170C2×D12—92no—
171C2×D12—92no—
172C2×D12—121no—
173C2×D12—181no—
174C2×D12—181no—
175C2×C4×D3—61no—
176C2×C4×D3—61no—
177C2×C3×C8—61no—
178C2×C3×C8—61no—
179C2×C3×C8—92no—
180C2×C3×C8—121no—
181C2×C3×C8—181no—
182C4⋊Dic3—92no—
183C4⋊Dic3—181no—
184C2×C3⋊C8—21no—
185D3×C8—62no—
186D3×C8—62no—
187C22×C32—21no—
188C22×C32—21no—
189C22×C32—41no—
190C22×C32—41no—
191C22×C32(C2×D3)⋊C411yescharacteristic
192C2×C3⋊D3—24no—
193C2×C3⋊D3—44no—
194C2×C3⋊D3#20412yes—
195C2×C3⋊D3D2412yes—
196C2×C3⋊D3(C2×D3)⋊C411yescharacteristic
197C2×C3⋊D3C2×D1211yescharacteristic
198C2×C3×D3—61no—
199C2×C3×D3—61no—
200C2×C3×D3—62no—
201C2×C3×D3—62no—
202C2×C3×D3—62no—
203C2×C3×D3—62no—
204C2×C3×D3—61no—
205C2×C3×D3—61no—
206C2×C3×D3—64no—
207C2×C3×D3—64no—
208C2×C3×D3—124no—
209C2×C3×D3—124no—
210C2×C3×D3—124no—
211C2×C3×D3—244no—
212D32—128no—
213D32—124no—
214D32—124no—
215D32—124no—
216D32—122no—
217D32—122no—
218#157—122no—
219C32×C4—21no—
220C32×C4—21no—
221C32×C4—21no—
222C32×C4—21no—
223C32×C4—41no—
224C32×C4—41no—
225C32×C4—41no—
226C32×C4—41no—
227C32×C4(C2×D3)⋊C411yescharacteristic
228C32×C4C2×#6711yescharacteristic
229C3⋊Dic3(C2×D3)⋊C411yescharacteristic
230C3⋊Dic3C2×C4×D311yescharacteristic
231C3×Dic3—61no—
232C3×Dic3—61no—
233C3×Dic3—61no—
234C3×Dic3—61no—
235C22×D4—271no—
236C22×C8—91no—
237C2×C4⋊C4—271no—
238#102—274no—
239C33C2×#10211yescharacteristic, complemented, Sylow
240C23×C3—91no—
241C23×C3—182no—
242C22×D3—61no—
243C22×D3—61no—
244C22×D3—62no—
245C22×D3—62no—
246C22×D3—122no—
247C22×D3—182no—
248C22×D3—182no—
249C22×D3—184no—
250C22×D3—184no—
251C22×D3—364no—
252C22×D3—364no—
253C22×D3—362no—
254C22×D3—362no—
255C3×D4—182no—
256C3×D4—182no—
257C2×C3×C4—21no—
258C2×C3×C4—21no—
259C2×C3×C4—41no—
260C2×C3×C4—41no—
261C2×C3×C4—91no—
262C2×C3×C4—91no—
263C2×C3×C4—91no—
264C2×C3×C4—91no—
265C2×C3×C4—91no—
266C2×C3×C4GO+(4,𝔽2)11yescharacteristic
267#67—184no—
268#67—184no—
269C2×Dic3—61no—
270C2×Dic3—61no—
271C2×Dic3—182no—
272C2×Dic3—362no—
273D12—62no—
274D12—62no—
275D12—62no—
276D12—62no—
277D12—94no—
278D12—122no—
279D12—122no—
280D12—182no—
281C4×D3—61no—
282C4×D3—61no—
283C4×D3—61no—
284C4×D3—61no—
285C4×D3—61no—
286C4×D3—61no—
287C4×D3—61no—
288C4×D3—61no—
289C3×C8—62no—
290C3×C8—62no—
291C3×C8—122no—
292C3⋊C8—22no—
293C2×C32—21no—
294C2×C32—21no—
295C2×C32—22no—
296C2×C32—22no—
297C2×C32—42no—
298C2×C32—42no—
299C2×C32—41no—
300C2×C32—41no—
301C2×C32#82112yescomplemented
302C2×C32C2×(C2×D3)⋊C411yescharacteristic
303C3⋊D3—44no—
304C3⋊D3#82112yescomplemented
305C3⋊D3C2×#20411yescharacteristic
306C3⋊D3C2×D2411yescharacteristic
307C3×D3—62no—
308C3×D3—62no—
309C3×D3—61no—
310C3×D3—61no—
311C3×D3—61no—
312C3×D3—61no—
313C3×D3—124no—
314C3×D3—124no—
315C3×D3—244no—
316C24—541no—
317C2×D4—181no—
318C2×D4—272no—
319C2×D4—272no—
320C2×D4—272no—
321C2×D4—541no—
322C2×D4—541no—
323C22×C4—91no—
324C22×C4—541no—
325C2×C8—61no—
326C2×C8—92no—
327C2×C8—181no—
328C4⋊C4—272no—
329C4⋊C4—541no—
330C22×C3—21no—
331C22×C3—21no—
332C22×C3—41no—
333C22×C3—41no—
334C22×C3—91no—
335C22×C3—91no—
336C22×C3—92no—
337C22×C3—92no—
338C22×C3—184no—
339C22×C3—364no—
340C22×C3D32⋊C411yescharacteristic
341C2×D3—61no—
342C2×D3—61no—
343C2×D3—61no—
344C2×D3—61no—
345C2×D3—62no—
346C2×D3—62no—
347C2×D3—62no—
348C2×D3—62no—
349C2×D3—64no—
350C2×D3—64no—
351C2×D3—124no—
352C2×D3—124no—
353C2×D3—124no—
354C2×D3—244no—
355C2×D3—368no—
356C2×D3—364no—
357C2×D3—364no—
358C2×D3—362no—
359C2×D3—362no—
360C2×D3—364no—
361C3×C4—21no—
362C3×C4—21no—
363C3×C4—21no—
364C3×C4—21no—
365C3×C4—41no—
366C3×C4—41no—
367C3×C4—41no—
368C3×C4—41no—
369C3×C4—91no—
370C3×C4—91no—
371C3×C4D32⋊C411yescharacteristic
372C3×C4C2×GO+(4,𝔽2)11yescharacteristic
373Dic3—61no—
374Dic3—61no—
375Dic3—61no—
376Dic3—61no—
377Dic3—362no—
378C32—21no—
379C32—21no—
380C32—41no—
381C32—41no—
382C32C2×#82111yescharacteristic, complemented
383C23—91no—
384C23—182no—
385C23—542no—
386C23—542no—
387C23—1082no—
388C23—1082no—
389D4—182no—
390D4—182no—
391D4—274no—
392D4—542no—
393C2×C4—91no—
394C2×C4—91no—
395C2×C4—91no—
396C2×C4—91no—
397C2×C4—91no—
398C2×C4—542no—
399C2×C4—1082no—
400C2×C4D32⋊D311yescharacteristic
401C8—62no—
402C2×C3—22no—
403C2×C3—22no—
404C2×C3—21no—
405C2×C3—21no—
406C2×C3—42no—
407C2×C3—42no—
408C2×C3—41no—
409C2×C3—41no—
410C2×C3—91no—
411C2×C3—91no—
412C2×C3—92no—
413C2×C3—364no—
414C2×C3#6372912yescomplemented
415C2×C3C2×D32⋊C411yescharacteristic
416D3—62no—
417D3—61no—
418D3—61no—
419D3—62no—
420D3—61no—
421D3—61no—
422D3—124no—
423D3—124no—
424D3—244no—
425C22—92no—
426C22—92no—
427C22—91no—
428C22—91no—
429C22—184no—
430C22—364no—
431C22—1082no—
432C22—1084no—
433C22—1082no—
434C22#8893211yescharacteristic
435C4—91no—
436C4—91no—
437C4—1082no—
438C4#8893211yescharacteristic
439C4C2×D32⋊D311yescharacteristic
440C3—21no—
441C3—21no—
442C3—41no—
443C3—41no—
444C3C2×#6372911yescharacteristic, complemented
445C2—91no—
446C2—92no—
447C2—91no—
448C2—364no—
449C2#1172275712yesdirect product factor
450C2C2×#8893211yescharacteristic
subgroupG/H
centerC22#88932
commutatorC3×C3⋊Dic3C22×C4
FrattiniC4C2×D32⋊D3
FittingC2×C33×C4D4
socleC22×C33C22⋊C4
prime radicalC2×C4⋊#1113C2
cosocleC3×C4×C3⋊D3C23
hypercenterC2×C4D32⋊D3
hypocenterC33C2×#102
LCSG ▷ C3×C3⋊Dic3 ▷ C2×C33 ▷ C33
UCSC1 ◁ C22 ◁ C2×C4

Complements

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

NH ≅ G/Ntype
2. C2×C4⋊(C3⋊#157)C2semidirect
3. C2×C3×#3710C2semidirect
4. #11722757C2direct (N × H)
10. C2×#6295C4semidirect
13. C4⋊#1113C4semidirect
16. C4⋊(C3⋊#157)C22semidirect
19. C3×#3710C22semidirect
24. C2×#3710D3semidirect
43. #6295C2×C4semidirect
75. #3710C2×D3semidirect
151. C2×C33#102semidirect
154. C3×C3⋊D3#102semidirect
239. C33C2×#102semidirect
301. C2×C32#821semidirect
304. C3⋊D3#821semidirect
382. C32C2×#821semidirect
414. C2×C3#63729semidirect
444. C3C2×#63729semidirect
449. C2#11722757direct (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×C4⋊(C3⋊#157)2
C2×C3×#37102
#117227572
C2×#637293
C2×#8219