Finite Group Databasebeta

← C2×#11723321 #13176641

Subgroups — C2×#11723321 #13176641

Order 1728.

2092 conjugacy classes of subgroups.

The subgroup lattice has 476 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#89026C211yesmaximal, characteristic
2C2×#89019C211yesmaximal, characteristic, complemented
3C2×C8×C3⋊(C3⋊D3)C211yesmaximal, characteristic, complemented
4C2×C3×#3744C211yesmaximal, characteristic, complemented
5C2×C32×#204C211yesmaximal, characteristic, complemented
6C2×#88793C211yesmaximal, characteristic, complemented
7C2×#88792C211yesmaximal, characteristic, complemented
8#11723321C218yesmaximal, direct product factor
9C2×SD16×C3⋊D3—31nomaximal
10C2×#63796—34nomaximal
11C22×D3×C3⋊D3—21no—
12C2×C4×C3⋊(C3⋊D3)C2211yescharacteristic
13C2×C3×C4⋊(C3⋊D3)C2211yescharacteristic
14C2×C3×#634C2211yescharacteristic, complemented
15C2×C32×D12C2211yescharacteristic
16C2×C32×Dic6C2211yescharacteristic, complemented
17C2×#6293—21no—
18C2×#6290—21no—
19C2×#6289—21no—
20#89026—22no—
21#89026C2214yes—
22#89019—22no—
23#89019C2214yescomplemented
24C2×C33×C8C2211yescharacteristic, complemented
25C2×C3⋊(C3⋊(C3⋊C8))C2211yescharacteristic, complemented
26C8×C3⋊(C3⋊D3)C2214yescomplemented
27C3×#3744C2214yescomplemented
28C32×#204C2214yescomplemented
29#88793C2214yescomplemented
30#88792C2214yescomplemented
31C2×Q8×C3⋊D3—31no—
32C2×D4×C3⋊D3—31no—
33C2×#3802—34no—
34C2×#3797—34no—
35C2×C32×SD16—31no—
36C2×#3755—31no—
37C2×#3754—31no—
38SD16×C3⋊D3—38no—
39C2×#3744D311yescharacteristic, complemented
40C2×C8×C3⋊D3—31no—
41C2×C8×C3⋊D3—34no—
42C2×C8×C3⋊D3—64no—
43C2×C3×#204—34no—
44C2×C3×#204D314yescomplemented
45C2×#3715—34no—
46C2×#3715—34no—
47#63796—332no—
48C22×C3⋊(C3⋊D3)D411yescharacteristic
49C22×C3×C3⋊D3—21no—
50C22×C32×D3—21no—
51C2×D3×C3⋊D3—24no—
52C2×D3×C3⋊D3—44no—
53C2×C33×C4C2311yescharacteristic
54C2×C3⋊(C3⋊Dic3)D411yescharacteristic
55C4×C3⋊(C3⋊D3)D412yes—
56C4×C3⋊(C3⋊D3)C2312yes—
57C2×C3×C3⋊Dic3—21no—
58C3×C4⋊(C3⋊D3)—21no—
59C3×C4⋊(C3⋊D3)C2312yes—
60C3×#634—21no—
61C3×#634C2312yescomplemented
62C2×C32×Dic3—21no—
63C32×D12—21no—
64C32×D12C2312yes—
65C32×Dic6—21no—
66C32×Dic6C2312yescomplemented
67#6293—24no—
68#6290—24no—
69#6289—24no—
70C33×C8C2312yescomplemented
71C3⋊(C3⋊(C3⋊C8))C2312yescomplemented
72C2×D3×SD16—94no—
73C2×D3×SD16—91no—
74C23×C3⋊D3—61no—
75C22×D32—64no—
76C2×C32×Q8—31no—
77C2×C32×D4—31no—
78C2×#638—61no—
79Q8×C3⋊D3—34no—
80Q8×C3⋊D3—62no—
81D4×C3⋊D3—34no—
82D4×C3⋊D3—62no—
83C2×C4⋊(C3⋊D3)C2×D311yescharacteristic
84C2×C4×C3⋊D3—31no—
85C2×C4×C3⋊D3—34no—
86C2×C4×C3⋊D3—61no—
87C2×C4×C3⋊D3—64no—
88C2×#634—61no—
89C2×#634C2×D311yescharacteristic, complemented
90C2×C3×D12—34no—
91C2×C3×D12C2×D314yes—
92C2×C3×Dic6—34no—
93C2×C3×Dic6C2×D314yescomplemented
94C2×#627—64no—
95C2×#625—64no—
96C2×#624—64no—
97#3802—316no—
98#3802—68no—
99#3797—316no—
100#3797—68no—
101C32×SD16—34no—
102C2×C32×C8—24no—
103C2×C32×C8C2×D314yescomplemented
104C2×C32×C8C2×D311yescharacteristic, complemented
105#3755—34no—
106#3754—34no—
107C2×C3⋊(C3⋊C8)—31no—
108C2×C3⋊(C3⋊C8)—34no—
109C2×C3⋊(C3⋊C8)—64no—
110#3744C2×D314yescomplemented
111C8×C3⋊D3—34no—
112C8×C3⋊D3—316no—
113C8×C3⋊D3—616no—
114C3×#204—316no—
115C3×#204C2×D3116yescomplemented
116#3715—316no—
117#3715—316no—
118C22×C33C2×D411yescharacteristic
119C2×C3⋊(C3⋊D3)SD1614yescomplemented
120C2×C3⋊(C3⋊D3)C2×D412yes—
121C2×C3×C3⋊D3—22no—
122C2×C3×C3⋊D3—42no—
123C2×C32×D3—22no—
124C2×C32×D3—42no—
125D3×C3⋊D3—48no—
126C33×C4C2×D411yescharacteristic
127C33×C4C2411yescharacteristic
128C3⋊(C3⋊Dic3)C2×D412yes—
129C3×C3⋊Dic3—22no—
130C32×Dic3—22no—
131C2×D3×Q8—94no—
132C2×D3×Q8—91no—
133C2×D3×D4—94no—
134C2×D3×D4—91no—
135C2×C3×SD16—34no—
136C2×C3×SD16—91no—
137C2×#215—94no—
138C2×#215—91no—
139C2×#214—94no—
140C2×#214—91no—
141D3×SD16—932no—
142D3×SD16—98no—
143C2×#204—34no—
144C2×#204C3⋊D311yescharacteristic, complemented
145C2×D3×C8—94no—
146C2×D3×C8—91no—
147C2×D3×C8—184no—
148C23×C32—61no—
149C22×C3⋊D3—21no—
150C22×C3⋊D3—31no—
151C22×C3⋊D3—34no—
152C22×C3⋊D3—64no—
153C22×C3⋊D3—64no—
154C22×C3⋊D3—124no—
155C22×C3×D3—24no—
156C22×C3×D3—64no—
157C2×D32—616no—
158C2×D32—1216no—
159C32×Q8—32no—
160C32×Q8—61no—
161C32×D4—32no—
162C32×D4—61no—
163C2×C32×C4—24no—
164C2×C32×C4—61no—
165C2×C32×C4C22×D314yes—
166C2×C32×C4C22×D311yescharacteristic
167#638—64no—
168C2×C3⋊Dic3—21no—
169C2×C3⋊Dic3—31no—
170C2×C3⋊Dic3—34no—
171C2×C3⋊Dic3—64no—
172C4⋊(C3⋊D3)—21no—
173C4⋊(C3⋊D3)C22×D312yes—
174C4×C3⋊D3—32no—
175C4×C3⋊D3—32no—
176C4×C3⋊D3—38no—
177C4×C3⋊D3—38no—
178C4×C3⋊D3—68no—
179C4×C3⋊D3—68no—
180C4×C3⋊D3—64no—
181#634—21no—
182#634—64no—
183#634C22×D312yescomplemented
184C2×C3×Dic3—24no—
185C2×C3×Dic3—64no—
186C3×D12—24no—
187C3×D12—38no—
188C3×D12—64no—
189C3×D12C22×D318yes—
190C3×Dic6—24no—
191C3×Dic6—38no—
192C3×Dic6—64no—
193C3×Dic6C22×D318yescomplemented
194#627—616no—
195#625—616no—
196#624—616no—
197C32×C8—28no—
198C32×C8C22×D318yescomplemented
199C32×C8C22×D312yescomplemented
200C3⋊(C3⋊C8)—32no—
201C3⋊(C3⋊C8)—38no—
202C3⋊(C3⋊C8)—68no—
203C22×SD16—271noSylow
204C2×C33C2×SD1612yescomplemented
205C2×C33C22×D411yescharacteristic
206C3⋊(C3⋊D3)C2×SD1614yescomplemented
207C3×C3⋊D3—42no—
208C32×D3—42no—
209C23×D3—181no—
210C23×D3—184no—
211C2×C3×Q8—34no—
212C2×C3×Q8—91no—
213C2×C3×D4—34no—
214C2×C3×D4—91no—
215C2×#67—181no—
216C2×#67—184no—
217D3×Q8—916no—
218D3×Q8—94no—
219D3×Q8—182no—
220D3×Q8—188no—
221D3×D4—916no—
222D3×D4—94no—
223D3×D4—182no—
224D3×D4—188no—
225C2×D12—34no—
226C2×D12C2×C3⋊D311yescharacteristic
227C2×C4×D3—94no—
228C2×C4×D3—91no—
229C2×C4×D3—184no—
230C2×C4×D3—181no—
231C2×C4×D3—184no—
232C2×Dic6—34no—
233C2×Dic6—181no—
234C2×Dic6—184no—
235C2×Dic6C2×C3⋊D311yescharacteristic, complemented
236C3×SD16—316no—
237C3×SD16—94no—
238C2×C3×C8—24no—
239C2×C3×C8D3214yescomplemented
240C2×C3×C8C2×C3⋊D311yescharacteristic, complemented
241#215—916no—
242#215—94no—
243#214—916no—
244#214—94no—
245C2×C3⋊C8—94no—
246C2×C3⋊C8—91no—
247C2×C3⋊C8—184no—
248#204—316no—
249#204C2×C3⋊D314yescomplemented
250D3×C8—916no—
251D3×C8—94no—
252D3×C8—1816no—
253C22×C32—24no—
254C22×C32—62no—
255C22×C32—122no—
256C22×C32D3×D414yes—
257C22×C32D3×D411yescharacteristic
258C2×C3⋊D3—22no—
259C2×C3⋊D3—34no—
260C2×C3⋊D3—32no—
261C2×C3⋊D3—316no—
262C2×C3⋊D3—38no—
263C2×C3⋊D3—42no—
264C2×C3⋊D3—616no—
265C2×C3⋊D3—68no—
266C2×C3⋊D3—128no—
267C2×C3×D3—28no—
268C2×C3×D3—48no—
269C2×C3×D3—68no—
270C2×C3×D3—128no—
271D32—1232no—
272C32×C4—24no—
273C32×C4—24no—
274C32×C4—62no—
275C32×C4D3×D414yes—
276C32×C4D3×D411yescharacteristic
277C32×C4C23×D314yes—
278C32×C4C23×D311yescharacteristic
279C3⋊Dic3—22no—
280C3⋊Dic3—32no—
281C3⋊Dic3—38no—
282C3⋊Dic3—68no—
283C3×Dic3—28no—
284C3×Dic3—68no—
285C22×Q8—271no—
286C22×D4—271no—
287C2×SD16—31no—
288C2×SD16—91no—
289C2×SD16—278no—
290C2×SD16—271no—
291C2×SD16—271no—
292C22×C8—271no—
293C33C22×SD1611yescharacteristic, complemented, Sylow
294C23×C3—64no—
295C23×C3—181no—
296C22×D3—21no—
297C22×D3—64no—
298C22×D3—94no—
299C22×D3—91no—
300C22×D3—184no—
301C22×D3—184no—
302C22×D3—1816no—
303C22×D3—364no—
304C22×D3—3616no—
305C3×Q8—38no—
306C3×Q8—64no—
307C3×Q8—92no—
308C3×Q8—181no—
309C3×D4—38no—
310C3×D4—64no—
311C3×D4—92no—
312C3×D4—181no—
313C2×C3×C4—24no—
314C2×C3×C4—64no—
315C2×C3×C4—181no—
316C2×C3×C4C2×D3214yes—
317C2×C3×C4C22×C3⋊D311yescharacteristic
318#67—184no—
319#67—1816no—
320C2×Dic3—21no—
321C2×Dic3—64no—
322C2×Dic3—94no—
323C2×Dic3—91no—
324C2×Dic3—184no—
325D12—21no—
326D12—38no—
327D12—64no—
328D12C22×C3⋊D312yes—
329C4×D3—98no—
330C4×D3—92no—
331C4×D3—98no—
332C4×D3—92no—
333C4×D3—188no—
334C4×D3—188no—
335C4×D3—184no—
336C4×D3—1816no—
337Dic6—21no—
338Dic6—38no—
339Dic6—64no—
340Dic6—184no—
341Dic6—1816no—
342Dic6C22×C3⋊D312yescomplemented
343C3×C8—28no—
344C3×C8C2×D3218yescomplemented
345C3×C8C22×C3⋊D312yescomplemented
346C3⋊C8—98no—
347C3⋊C8—92no—
348C3⋊C8—188no—
349C2×C32—24no—
350C2×C32—28no—
351C2×C32—122no—
352C2×C32D3×SD1618yescomplemented
353C2×C32D3×SD1612yescomplemented
354C2×C32C2×D3×D414yes—
355C2×C32C2×D3×D411yescharacteristic
356C3⋊D3—34no—
357C3⋊D3—316no—
358C3⋊D3—42no—
359C3⋊D3—616no—
360C3×D3—48no—
361C3×D3—128no—
362C24—541no—
363C2×Q8—31no—
364C2×Q8—91no—
365C2×Q8—274no—
366C2×Q8—541no—
367C2×Q8—542no—
368C2×D4—31no—
369C2×D4—91no—
370C2×D4—274no—
371C2×D4—542no—
372C2×D4—541no—
373C22×C4—271no—
374C22×C4—541no—
375SD16—34no—
376SD16—94no—
377SD16—274no—
378SD16—274no—
379C2×C8—271no—
380C2×C8—274no—
381C2×C8D3×C3⋊D311yescharacteristic, complemented
382C22×C3—24no—
383C22×C3—68no—
384C22×C3—128no—
385C22×C3—182no—
386C22×C3—362no—
387C22×C3#380214yes—
388C22×C3D4×C3⋊D311yescharacteristic
389C2×D3—22no—
390C2×D3—42no—
391C2×D3—68no—
392C2×D3—98no—
393C2×D3—916no—
394C2×D3—92no—
395C2×D3—94no—
396C2×D3—128no—
397C2×D3—188no—
398C2×D3—1816no—
399C2×D3—368no—
400C2×D3—3632no—
401C3×C4—24no—
402C3×C4—24no—
403C3×C4—68no—
404C3×C4—182no—
405C3×C4#380214yes—
406C3×C4D4×C3⋊D311yescharacteristic
407C3×C4C22×D3214yes—
408C3×C4C23×C3⋊D311yescharacteristic
409Dic3—22no—
410Dic3—68no—
411Dic3—98no—
412Dic3—92no—
413Dic3—188no—
414C32—24no—
415C32C2×D3×SD1614yescomplemented
416C32C2×D3×SD1611yescharacteristic, complemented
417C23—61no—
418C23—181no—
419C23—271no—
420C23—544no—
421C23—1084no—
422Q8—32no—
423Q8—61no—
424Q8—92no—
425Q8—181no—
426Q8—544no—
427D4—32no—
428D4—61no—
429D4—92no—
430D4—181no—
431D4—544no—
432C2×C4—61no—
433C2×C4—181no—
434C2×C4—271no—
435C2×C4—272no—
436C2×C4—272no—
437C2×C4—544no—
438C2×C4C2×D3×C3⋊D311yescharacteristic
439C8—272no—
440C8C2×D3×C3⋊D312yescomplemented
441C2×C3—24no—
442C2×C3—28no—
443C2×C3—128no—
444C2×C3—362no—
445C2×C3#6379618yescomplemented
446C2×C3SD16×C3⋊D312yescomplemented
447C2×C3C2×#380214yes—
448C2×C3C2×D4×C3⋊D311yescharacteristic
449D3—42no—
450D3—916no—
451D3—94no—
452D3—128no—
453D3—1816no—
454C22—62no—
455C22—122no—
456C22—182no—
457C22—272no—
458C22—274no—
459C22—362no—
460C22—1088no—
461C22#8902611yescharacteristic
462C4—62no—
463C4—182no—
464C4—272no—
465C4#8902611yescharacteristic
466C4C22×D3×C3⋊D311yescharacteristic
467C3—24no—
468C3C2×#6379614yescomplemented
469C3C2×SD16×C3⋊D311yescharacteristic, complemented
470C2—122no—
471C2—274no—
472C2—362no—
473C2#1172332112yesdirect product factor
474C2C2×#8902611yescharacteristic
subgroupG/H
centerC22#89026
commutator, cosocleC33×C4C24
FrattiniC4C22×D3×C3⋊D3
FittingC2×C33×C8C22
socleC22×C33C2×D4
prime radicalC2×#89026C2
hypercenterC2×C8D3×C3⋊D3
hypocenterC33C22×SD16
LCSG ▷ C33×C4 ▷ C2×C33 ▷ C33
UCSC1 ◁ C22 ◁ C2×C4 ◁ C2×C8

Complements

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

NH ≅ G/Ntype
2. C2×#89019C2semidirect
3. C2×C8×C3⋊(C3⋊D3)C2semidirect
4. C2×C3×#3744C2semidirect
5. C2×C32×#204C2semidirect
6. C2×#88793C2semidirect
7. C2×#88792C2semidirect
8. #11723321C2direct (N × H)
14. C2×C3×#634C22semidirect
16. C2×C32×Dic6C22semidirect
23. #89019C22semidirect
24. C2×C33×C8C22semidirect
25. C2×C3⋊(C3⋊(C3⋊C8))C22semidirect
26. C8×C3⋊(C3⋊D3)C22semidirect
27. C3×#3744C22semidirect
28. C32×#204C22semidirect
29. #88793C22semidirect
30. #88792C22semidirect
39. C2×#3744D3semidirect
44. C2×C3×#204D3semidirect
61. C3×#634C23semidirect
66. C32×Dic6C23semidirect
70. C33×C8C23semidirect
71. C3⋊(C3⋊(C3⋊C8))C23semidirect
89. C2×#634C2×D3semidirect
93. C2×C3×Dic6C2×D3semidirect
103. C2×C32×C8C2×D3semidirect
104. C2×C32×C8C2×D3semidirect
110. #3744C2×D3semidirect
115. C3×#204C2×D3semidirect
119. C2×C3⋊(C3⋊D3)SD16semidirect
144. C2×#204C3⋊D3semidirect
183. #634C22×D3semidirect
193. C3×Dic6C22×D3semidirect
198. C32×C8C22×D3semidirect
199. C32×C8C22×D3semidirect
204. C2×C33C2×SD16semidirect
206. C3⋊(C3⋊D3)C2×SD16semidirect
235. C2×Dic6C2×C3⋊D3semidirect
239. C2×C3×C8D32semidirect
240. C2×C3×C8C2×C3⋊D3semidirect
249. #204C2×C3⋊D3semidirect
293. C33C22×SD16semidirect
342. Dic6C22×C3⋊D3semidirect
344. C3×C8C2×D32semidirect
345. C3×C8C22×C3⋊D3semidirect
352. C2×C32D3×SD16semidirect
353. C2×C32D3×SD16semidirect
381. C2×C8D3×C3⋊D3semidirect
415. C32C2×D3×SD16semidirect
416. C32C2×D3×SD16semidirect
440. C8C2×D3×C3⋊D3semidirect
445. C2×C3#63796semidirect
446. C2×C3SD16×C3⋊D3semidirect
468. C3C2×#63796semidirect
469. C3C2×SD16×C3⋊D3semidirect
473. C2#11723321direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#890262
C2×#890192
C2×C8×C3⋊(C3⋊D3)2
C2×C3×#37442
C2×C32×#2042
C2×#887932
C2×#887922
#117233212
C2×SD16×C3⋊D33
C2×#637963