Finite Group Databasebeta

← C2×#11722706 #13166361

Subgroups — C2×#11722706 #13166361

Order 1728.

984 conjugacy classes of subgroups; only the 98 normal subgroups are shown.

normal subgroups only

This group has 984 conjugacy classes of subgroups — too many to draw. Only its 98 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×C3⋊D3×D12C211yesmaximal, characteristic
2C2×#88988C211yesmaximal, characteristic, complemented
3C2×C3×C4⋊#157C211yesmaximal, characteristic, complemented
4#11722706C214yesmaximal, direct product factor
5C2×#63782—31nomaximal
6C22×D3×C3⋊D3—21no—
7C22×C3×#157—21no—
8C2×C4⋊(C3⋊(C3⋊D3))C411yescharacteristic, complemented
9C2×C3×C4×C3⋊D3C2211yescharacteristic
10C2×C32×D12C411yescharacteristic, complemented
11C2×#6291—21no—
12C3⋊D3×D12—21no—
13C3⋊D3×D12—21no—
14C3⋊D3×D12C412yescomplemented
15C3⋊D3×D12C2212yes—
16C2×C3⋊#622—21no—
17#88988—21no—
18#88988C2212yescomplemented
19C3×C4⋊#157—21no—
20C3×C4⋊#157C2212yescomplemented
21C2×D4×C3⋊D3—31no—
22C2×D3×D12—62no—
23C2×C4⋊#157D311yescharacteristic, complemented
24C2×#3787—31no—
25#63782—34no—
26C22×C3⋊(C3⋊D3)—21no—
27C22×C3×C3⋊D3D411yescharacteristic
28C22×C32×D3—21no—
29C2×D3×C3⋊D3—22no—
30C2×D3×C3⋊D3—22no—
31C2×D3×C3⋊D3—42no—
32C2×D3×C3⋊D3—42no—
33C2×C3×#157—22no—
34C2×C3×#157—42no—
35C2×C33×C4C2×C411yescharacteristic
36C4⋊(C3⋊(C3⋊D3))—21no—
37C4⋊(C3⋊(C3⋊D3))C2×C412yescomplemented
38C2×C3×C3⋊Dic3D411yescharacteristic
39C3×C4×C3⋊D3C2×C411yescharacteristic
40C3×C4×C3⋊D3D411yescharacteristic
41C3×C4×C3⋊D3D411yescharacteristic
42C3×C4×C3⋊D3C2311yescharacteristic
43C32×D12—21no—
44C32×D12C2×C412yescomplemented
45#6291—22no—
46#6291—22no—
47C3⋊#622—22no—
48C2×#809—91nomaximal
49C23×C3⋊D3—61no—
50C22×D32—64no—
51C22×#157—21no—
52C2×C32×D4—31no—
53C2×#638—61no—
54D4×C3⋊D3—32no—
55D4×C3⋊D3—32no—
56D4×C3⋊D3—61no—
57D4×C3⋊D3—61no—
58C2×C4⋊(C3⋊D3)—31no—
59C2×C4⋊(C3⋊D3)—62no—
60C2×C4⋊(C3⋊D3)—122no—
61C2×C4×C3⋊D3C2×D311yescharacteristic
62C2×C3×D12—22no—
63C2×C3×C4×D3—62no—
64C2×#626—64no—
65D3×D12—64no—
66D3×D12—64no—
67D3×D12—64no—
68D3×D12—64no—
69C2×#622—61no—
70C4⋊#157—21no—
71C4⋊#157C2×D312yescomplemented
72#3787—32no—
73#3787—61no—
74C22×C33C22⋊C411yescharacteristic
75C2×C3⋊(C3⋊D3)—22no—
76C2×C3⋊(C3⋊D3)—42no—
77C2×C3×C3⋊D3C22⋊C411yescharacteristic
78C2×C3×C3⋊D3D812yes—
79C2×C3×C3⋊D3SD1612yes—
80C2×C3×C3⋊D3C2×D411yescharacteristic
81C2×C32×D3—22no—
82C2×C32×D3—42no—
83D3×C3⋊D3—44no—
84D3×C3⋊D3—42no—
85D3×C3⋊D3—42no—
86C3×#157—42no—
87C33×C4C22⋊C411yescharacteristic
88C33×C4C22×C411yescharacteristic
89C3×C3⋊Dic3C22⋊C411yescharacteristic
90C3×C3⋊Dic3C2×D411yescharacteristic
91C2×D3×D4—182no—
92C22×D12—91no—
93C2×C3×C4⋊C4—91no—
94C22×C3⋊C8—91no—
95#809—94no—
96C23×C32—61no—
97C22×C3⋊D3—61no—
98C22×C3⋊D3—62no—
99C22×C3⋊D3—62no—
100C22×C3⋊D3—64no—
101C22×C3⋊D3—124no—
102C22×C3⋊D3—122no—
103C22×C3⋊D3—122no—
104C22×C3⋊D3#6711yescharacteristic
105C22×C3×D3—24no—
106C22×C3×D3—62no—
107C2×D32—68no—
108C2×D32—68no—
109C2×D32—128no—
110C2×D32—128no—
111C2×#157—22no—
112C2×#157—42no—
113C32×D4—32no—
114C32×D4—61no—
115C2×C32×C4—22no—
116C2×C32×C4—42no—
117C2×C32×C4C4×D311yescharacteristic
118#638—62no—
119#638—62no—
120C2×C3⋊Dic3D1211yescharacteristic
121C4⋊(C3⋊D3)—32no—
122C4⋊(C3⋊D3)—64no—
123C4⋊(C3⋊D3)—61no—
124C4⋊(C3⋊D3)—64no—
125C4⋊(C3⋊D3)—124no—
126C4⋊(C3⋊D3)—124no—
127C4×C3⋊D3C4×D311yescharacteristic
128C4×C3⋊D3D1211yescharacteristic
129C4×C3⋊D3#6711yescharacteristic
130C4×C3⋊D3C22×D311yescharacteristic
131C2×C3×Dic3—62no—
132C3×D12—24no—
133C3×D12—24no—
134C3×C4×D3—62no—
135C3×C4×D3—62no—
136C3×C4×D3—62no—
137C3×C4×D3—62no—
138#626—68no—
139#626—68no—
140#622—62no—
141C2×#102—271noSylow
142C2×C33#10212yescomplemented
143C2×C33C2×C22⋊C411yescharacteristic
144C3⋊(C3⋊D3)—42no—
145C3×C3⋊D3#10212yescomplemented
146C3×C3⋊D3C2×D811yescharacteristic
147C3×C3⋊D3C2×SD1611yescharacteristic
148C32×D3—42no—
149C23×D3—181no—
150C23×D3—184no—
151C2×C3×D4—62no—
152C22×C3×C4—91no—
153C22×C3×C4—181no—
154C2×#67—184no—
155D3×D4—184no—
156D3×D4—184no—
157D3×D4—184no—
158D3×D4—184no—
159C2×D12—91no—
160C2×D12—92no—
161C2×D12—92no—
162C2×D12—182no—
163C2×D12—181no—
164C2×D12—181no—
165C2×D12—181no—
166C2×D12—362no—
167C2×D12#15711yescharacteristic, complemented
168C2×C4×D3—62no—
169C3×C4⋊C4—92no—
170C3×C4⋊C4—181no—
171C2×C3⋊C8—92no—
172C2×C3⋊C8—181no—
173C2×C3⋊C8—181no—
174C22×C32—22no—
175C22×C32—42no—
176C22×C32—62no—
177C22×C32—122no—
178C22×C32(C2×D3)⋊C411yescharacteristic
179C2×C3⋊D3—62no—
180C2×C3⋊D3—68no—
181C2×C3⋊D3—128no—
182C2×C3⋊D3—122no—
183C2×C3⋊D3—124no—
184C2×C3⋊D3—122no—
185C2×C3⋊D3—122no—
186C2×C3⋊D3—128no—
187C2×C3⋊D3—248no—
188C2×C3⋊D3(C2×D3)⋊C411yescharacteristic
189C2×C3⋊D3#21312yes—
190C2×C3⋊D3#21512yes—
191C2×C3⋊D3C2×#6711yescharacteristic
192C2×C3×D3—28no—
193C2×C3×D3—48no—
194C2×C3×D3—62no—
195C2×C3×D3—64no—
196C2×C3×D3—64no—
197C2×C3×D3—62no—
198D32—1216no—
199D32—128no—
200D32—128no—
201#157—42no—
202C32×C4—22no—
203C32×C4—22no—
204C32×C4—42no—
205C32×C4—42no—
206C32×C4(C2×D3)⋊C411yescharacteristic
207C32×C4C2×C4×D311yescharacteristic
208C3⋊Dic3(C2×D3)⋊C411yescharacteristic
209C3⋊Dic3C2×D1211yescharacteristic
210C3×Dic3—62no—
211C3×Dic3—62no—
212C22×D4—271no—
213C22×C8—271no—
214C2×C4⋊C4—91no—
215#102—274no—
216C33C2×#10211yescharacteristic, complemented, Sylow
217C23×C3—64no—
218C23×C3—91no—
219C22×D3—21no—
220C22×D3—62no—
221C22×D3—182no—
222C22×D3—182no—
223C22×D3—188no—
224C22×D3—188no—
225C22×D3—184no—
226C22×D3—181no—
227C22×D3—364no—
228C22×D3—362no—
229C22×D3—362no—
230C22×D3—368no—
231C22×D3—368no—
232C3×D4—64no—
233C3×D4—64no—
234C2×C3×C4—22no—
235C2×C3×C4—42no—
236C2×C3×C4—91no—
237C2×C3×C4—91no—
238C2×C3×C4—91no—
239C2×C3×C4—91no—
240C2×C3×C4—91no—
241C2×C3×C4—182no—
242C2×C3×C4—362no—
243C2×C3×C4C2×#15711yescharacteristic
244#67—188no—
245#67—188no—
246C2×Dic3—62no—
247D12—21no—
248D12—92no—
249D12—184no—
250D12—184no—
251D12—182no—
252D12—182no—
253D12—181no—
254D12—364no—
255D12—364no—
256D12C2×#15712yescomplemented
257C4×D3—62no—
258C4×D3—62no—
259C4×D3—62no—
260C4×D3—62no—
261C3⋊C8—182no—
262C2×C32—22no—
263C2×C32—24no—
264C2×C32—44no—
265C2×C32—42no—
266C2×C32—122no—
267C2×C32#80912yescomplemented
268C2×C32C2×(C2×D3)⋊C411yescharacteristic
269C3⋊D3—122no—
270C3⋊D3—128no—
271C3⋊D3—248no—
272C3⋊D3#80912yescomplemented
273C3⋊D3C2×#21311yescharacteristic
274C3⋊D3C2×#21511yescharacteristic
275C3×D3—48no—
276C3×D3—64no—
277C3×D3—62no—
278C3×D3—62no—
279C24—541no—
280C2×D4—31no—
281C2×D4—272no—
282C2×D4—272no—
283C2×D4—271no—
284C2×D4—541no—
285C2×D4—541no—
286C2×D4—541no—
287C22×C4—91no—
288C22×C4—181no—
289C2×C8—272no—
290C2×C8—541no—
291C2×C8—541no—
292C4⋊C4—92no—
293C4⋊C4—181no—
294C22×C3—22no—
295C22×C3—42no—
296C22×C3—68no—
297C22×C3—91no—
298C22×C3—91no—
299C22×C3—92no—
300C22×C3—92no—
301C22×C3—128no—
302C22×C3C22⋊#15711yescharacteristic
303C2×D3—22no—
304C2×D3—42no—
305C2×D3—62no—
306C2×D3—62no—
307C2×D3—64no—
308C2×D3—64no—
309C2×D3—188no—
310C2×D3—182no—
311C2×D3—368no—
312C2×D3—362no—
313C2×D3—362no—
314C2×D3—364no—
315C2×D3—3616no—
316C2×D3—368no—
317C2×D3—368no—
318C2×D3—368no—
319C2×D3—362no—
320C2×D3—728no—
321C3×C4—22no—
322C3×C4—22no—
323C3×C4—42no—
324C3×C4—42no—
325C3×C4—91no—
326C3×C4—91no—
327C3×C4—362no—
328C3×C4C22⋊#15711yescharacteristic
329C3×C4C22×#15711yescharacteristic
330Dic3—62no—
331Dic3—62no—
332C32—22no—
333C32—42no—
334C32C2×#80911yescharacteristic, complemented
335C23—61no—
336C23—91no—
337C23—542no—
338C23—542no—
339C23—541no—
340C23—1082no—
341C23—1082no—
342D4—32no—
343D4—61no—
344D4—272no—
345D4—542no—
346D4—542no—
347D4—541no—
348C2×C4—91no—
349C2×C4—91no—
350C2×C4—91no—
351C2×C4—91no—
352C2×C4—91no—
353C2×C4—182no—
354C2×C4—362no—
355C2×C4D3×#15711yescharacteristic
356C8—542no—
357C2×C3—24no—
358C2×C3—22no—
359C2×C3—44no—
360C2×C3—42no—
361C2×C3—91no—
362C2×C3—91no—
363C2×C3—92no—
364C2×C3—128no—
365C2×C3#6378212yescomplemented
366C2×C3C2×C22⋊#15711yescharacteristic
367D3—42no—
368D3—64no—
369D3—62no—
370D3—62no—
371D3—368no—
372D3—362no—
373D3—728no—
374C22—62no—
375C22—92no—
376C22—92no—
377C22—91no—
378C22—91no—
379C22—122no—
380C22—542no—
381C22—1082no—
382C22—1084no—
383C22—1082no—
384C22—1082no—
385C22(C2×D3)⋊#15711yescharacteristic
386C4—91no—
387C4—91no—
388C4—362no—
389C4(C2×D3)⋊#15711yescharacteristic
390C4C2×D3×#15711yescharacteristic
391C3—22no—
392C3—42no—
393C3C2×#6378211yescharacteristic, complemented
394C2—91no—
395C2—92no—
396C2—91no—
397C2—122no—
398C2—1082no—
399C2#1172270612yesdirect product factor
400C2C2×(C2×D3)⋊#15711yescharacteristic
subgroupG/H
centerC22(C2×D3)⋊#157
commutatorC33×C4C22×C4
FrattiniC4C2×D3×#157
FittingC2×C33×C4C2×C4
socleC22×C33C22⋊C4
prime radicalC2×C3⋊D3×D12C2
cosocleC3×C4×C3⋊D3C23
hypercenterC2×C4D3×#157
hypocenterC33C2×#102
LCSG ▷ C33×C4 ▷ C2×C33 ▷ C33
UCSC1 ◁ C22 ◁ C2×C4

Complements

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

NH ≅ G/Ntype
2. C2×#88988C2semidirect
3. C2×C3×C4⋊#157C2semidirect
4. #11722706C2direct (N × H)
8. C2×C4⋊(C3⋊(C3⋊D3))C4semidirect
10. C2×C32×D12C4semidirect
14. C3⋊D3×D12C4semidirect
18. #88988C22semidirect
20. C3×C4⋊#157C22semidirect
23. C2×C4⋊#157D3semidirect
37. C4⋊(C3⋊(C3⋊D3))C2×C4semidirect
44. C32×D12C2×C4semidirect
71. C4⋊#157C2×D3semidirect
142. C2×C33#102semidirect
145. C3×C3⋊D3#102semidirect
167. C2×D12#157semidirect
216. C33C2×#102semidirect
256. D12C2×#157semidirect
267. C2×C32#809semidirect
272. C3⋊D3#809semidirect
334. C32C2×#809semidirect
365. C2×C3#63782semidirect
393. C3C2×#63782semidirect
399. C2#11722706direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3⋊D3×D122
C2×#889882
C2×C3×C4⋊#1572
#117227062
C2×#637823
C2×#8099