Finite Group Databasebeta

← #13159415

Subgroups — #13159415

Order 1728.

1580 conjugacy classes of subgroups; only the 178 normal subgroups are shown.

normal subgroups only

This group has 1580 conjugacy classes of subgroups — too many to draw. Only its 178 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×#89026C211yesmaximal, characteristic
2C4⋊C4×C3⋊(C3⋊D3)C211yesmaximal, characteristic, complemented
3C2×#88785C211yesmaximal, characteristic, complemented
4C3×#63636C211yesmaximal, characteristic, complemented
5C32×#809C211yesmaximal, characteristic, complemented
6#11722034C211yesmaximal, characteristic, complemented
7#11722032C211yesmaximal, characteristic, complemented
8C3⋊D3×#102—31nomaximal
9#10590593—34nomaximal
10C22×D3×C3⋊D3—21no—
11C2×C4×C3⋊(C3⋊D3)—21no—
12C2×C4×C3⋊(C3⋊D3)C2211yescharacteristic
13C2×C3×C4⋊(C3⋊D3)C2211yescharacteristic
14C2×C32×D12C2211yescharacteristic
15C2×#6290—21no—
16#89026—22no—
17#89026C414yescomplemented
18C33×C4⋊C4C2211yescharacteristic, complemented
19C4⋊(C3⋊(C3⋊Dic3))C2211yescharacteristic, complemented
20#88857—21no—
21C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic, complemented
22C2×C32×C3⋊C8C2211yescharacteristic, complemented
23#88785—22no—
24C2×D4×C3⋊D3—31no—
25C2×#3802—34no—
26C2×C8×C3⋊D3—31no—
27C4⋊C4×C3⋊D3—31no—
28C4⋊C4×C3⋊D3—34no—
29C4⋊C4×C3⋊D3—64no—
30C2×#3710—34no—
31C32×#102—31no—
32#63659—31no—
33#63648—31no—
34#63636D311yescharacteristic, complemented
35C3×#809—34no—
36C3×#809D314yescomplemented
37#63563—34no—
38#63563—34no—
39C22×C3⋊(C3⋊D3)D411yescharacteristic
40C22×C3×C3⋊D3—21no—
41C22×C32×D3—21no—
42C2×D3×C3⋊D3—24no—
43C2×D3×C3⋊D3—42no—
44C2×D3×C3⋊D3—42no—
45C2×C33×C4—21no—
46C2×C33×C4C2311yescharacteristic
47C2×C3⋊(C3⋊Dic3)—21no—
48C2×C3⋊(C3⋊Dic3)D411yescharacteristic
49C4×C3⋊(C3⋊D3)—42no—
50C4×C3⋊(C3⋊D3)C2×C412yes—
51C4×C3⋊(C3⋊D3)D412yes—
52C3×C4⋊(C3⋊D3)—21no—
53C3×C4⋊(C3⋊D3)C2×C412yescomplemented
54C32×D12—21no—
55C32×D12C2×C412yescomplemented
56#6290—24no—
57C3×C3⋊(C3⋊C8)—21no—
58C32×C3⋊C8—21no—
59D3×#102—94no—
60D3×#102—91no—
61C23×C3⋊D3—61no—
62C22×D32—64no—
63C2×C32×D4—31no—
64C2×#638—61no—
65D4×C3⋊D3—34no—
66D4×C3⋊D3—62no—
67C2×C4⋊(C3⋊D3)C2×D311yescharacteristic
68C2×C4×C3⋊D3—31no—
69C2×C4×C3⋊D3—34no—
70C2×C4×C3⋊D3—64no—
71C2×C4×C3⋊D3—61no—
72C2×C4×C3⋊D3—64no—
73C2×C4×C3⋊D3—68no—
74C2×C3×D12—34no—
75C2×C3×D12C2×D314yes—
76C2×#625—64no—
77#3802—316no—
78#3802—68no—
79C2×C32×C8—31no—
80C32×C4⋊C4—24no—
81C32×C4⋊C4C2×D314yescomplemented
82C32×C4⋊C4C2×D311yescharacteristic, complemented
83C4⋊(C3⋊Dic3)—31no—
84C4⋊(C3⋊Dic3)—34no—
85C4⋊(C3⋊Dic3)—64no—
86#3750—61no—
87#3750—64no—
88#3750—68no—
89C2×C3⋊(C3⋊C8)C2×D311yescharacteristic, complemented
90C8×C3⋊D3—62no—
91C2×C3×C3⋊C8—34no—
92C2×C3×C3⋊C8C2×D314yescomplemented
93#3710—68no—
94C22×C33C2×D411yescharacteristic
95C2×C3⋊(C3⋊D3)C22⋊C412yes—
96C2×C3⋊(C3⋊D3)D812yes—
97C2×C3⋊(C3⋊D3)SD1612yes—
98C2×C3×C3⋊D3—22no—
99C2×C3×C3⋊D3—41no—
100C2×C3×C3⋊D3—41no—
101C2×C32×D3—22no—
102C2×C32×D3—41no—
103C2×C32×D3—41no—
104D3×C3⋊D3—48no—
105C33×C4—41no—
106C33×C4C22×C411yescharacteristic
107C33×C4C2×D411yescharacteristic
108C3⋊(C3⋊Dic3)—41no—
109C3⋊(C3⋊Dic3)C22⋊C412yes—
110C2×D3×D4—94no—
111C2×D3×D4—91no—
112C2×D3×C8—94no—
113C2×D3×C8—91no—
114D3×C4⋊C4—94no—
115D3×C4⋊C4—91no—
116D3×C4⋊C4—184no—
117C3×#102—34no—
118C3×#102—91no—
119#832—94no—
120#832—91no—
121#821—94no—
122#821—91no—
123#809—34no—
124#809C3⋊D311yescharacteristic, complemented
125C23×C32—61no—
126C22×C3⋊D3—21no—
127C22×C3⋊D3—31no—
128C22×C3⋊D3—34no—
129C22×C3⋊D3—64no—
130C22×C3⋊D3—64no—
131C22×C3⋊D3—122no—
132C22×C3⋊D3—122no—
133C22×C3×D3—24no—
134C22×C3×D3—64no—
135C2×D32—616no—
136C2×D32—128no—
137C2×D32—128no—
138C32×D4—32no—
139C32×D4—61no—
140C2×C32×C4—28no—
141C2×C32×C4—24no—
142C2×C32×C4—21no—
143C2×C32×C4—24no—
144C2×C32×C4C22×D314yes—
145C2×C32×C4C22×D311yescharacteristic
146#638—64no—
147C2×C3⋊Dic3—31no—
148C2×C3⋊Dic3—34no—
149C2×C3⋊Dic3—64no—
150C2×C3⋊Dic3—68no—
151C2×C3⋊Dic3—64no—
152C2×C3⋊Dic3—61no—
153C4⋊(C3⋊D3)—21no—
154C4⋊(C3⋊D3)C4×D312yescomplemented
155C4×C3⋊D3—32no—
156C4×C3⋊D3—32no—
157C4×C3⋊D3—38no—
158C4×C3⋊D3—38no—
159C4×C3⋊D3—68no—
160C4×C3⋊D3—68no—
161C4×C3⋊D3—122no—
162C4×C3⋊D3—128no—
163C4×C3⋊D3—1216no—
164C3×D12—24no—
165C3×D12—38no—
166C3×D12—64no—
167C3×D12C4×D318yescomplemented
168#625—616no—
169C32×C8—61no—
170C3⋊(C3⋊C8)—21no—
171C3×C3⋊C8—24no—
172C3×C3⋊C8—64no—
173C2×#102—271noSylow
174C2×C33C2×C22⋊C411yescharacteristic
175C2×C33C2×D811yescharacteristic
176C2×C33C2×SD1611yescharacteristic
177C3⋊(C3⋊D3)#10214yescomplemented
178C3×C3⋊D3—42no—
179C32×D3—42no—
180C23×D3—181no—
181C23×D3—184no—
182C2×C3×D4—34no—
183C2×C3×D4—91no—
184C2×#67—181no—
185C2×#67—184no—
186D3×D4—916no—
187D3×D4—94no—
188D3×D4—182no—
189D3×D4—188no—
190C2×D12—34no—
191C2×D12C2×C3⋊D311yescharacteristic
192C2×C4×D3—94no—
193C2×C4×D3—91no—
194C2×C4×D3—184no—
195C2×C4×D3—184no—
196C2×C4×D3—181no—
197C2×C4×D3—188no—
198C2×C3×C8—34no—
199C2×C3×C8—91no—
200C3×C4⋊C4—24no—
201C3×C4⋊C4D3214yescomplemented
202C3×C4⋊C4C2×C3⋊D311yescharacteristic, complemented
203C4⋊Dic3—94no—
204C4⋊Dic3—91no—
205C4⋊Dic3—184no—
206Dic3⋊C4—184no—
207Dic3⋊C4—181no—
208Dic3⋊C4—188no—
209C2×C3⋊C8—34no—
210C2×C3⋊C8C2×C3⋊D311yescharacteristic, complemented
211D3×C8—182no—
212D3×C8—188no—
213C22×C32—24no—
214C22×C32—62no—
215C22×C32—121no—
216C22×C32—121no—
217C22×C32D3×D414yes—
218C22×C32D3×D411yescharacteristic
219C2×C3⋊D3—22no—
220C2×C3⋊D3—32no—
221C2×C3⋊D3—32no—
222C2×C3⋊D3—32no—
223C2×C3⋊D3—38no—
224C2×C3⋊D3—38no—
225C2×C3⋊D3—38no—
226C2×C3⋊D3—41no—
227C2×C3⋊D3—41no—
228C2×C3⋊D3—68no—
229C2×C3⋊D3—68no—
230C2×C3⋊D3—68no—
231C2×C3⋊D3—128no—
232C2×C3×D3—28no—
233C2×C3×D3—44no—
234C2×C3×D3—44no—
235C2×C3×D3—68no—
236C2×C3×D3—124no—
237C2×C3×D3—124no—
238D32—1232no—
239C32×C4—24no—
240C32×C4—24no—
241C32×C4—48no—
242C32×C4—44no—
243C32×C4—41no—
244C32×C4C2×C4×D314yes—
245C32×C4C2×C4×D311yescharacteristic
246C32×C4D3×D414yes—
247C32×C4D3×D411yescharacteristic
248C3⋊Dic3—32no—
249C3⋊Dic3—38no—
250C3⋊Dic3—68no—
251C3⋊Dic3—128no—
252C3⋊Dic3—124no—
253C3⋊Dic3—121no—
254C22×D4—271no—
255C22×C8—271no—
256C2×C4⋊C4—271no—
257#102—31no—
258#102—91no—
259#102—271no—
260#102—271no—
261C33C2×#10211yescharacteristic, complemented, Sylow
262C23×C3—64no—
263C23×C3—181no—
264C22×D3—21no—
265C22×D3—64no—
266C22×D3—94no—
267C22×D3—91no—
268C22×D3—184no—
269C22×D3—184no—
270C22×D3—1816no—
271C22×D3—362no—
272C22×D3—362no—
273C22×D3—368no—
274C22×D3—368no—
275C3×D4—38no—
276C3×D4—64no—
277C3×D4—92no—
278C3×D4—181no—
279C2×C3×C4—28no—
280C2×C3×C4—24no—
281C2×C3×C4—21no—
282C2×C3×C4—24no—
283C2×C3×C4C2×D3214yes—
284C2×C3×C4C22×C3⋊D311yescharacteristic
285#67—184no—
286#67—1816no—
287C2×Dic3—94no—
288C2×Dic3—91no—
289C2×Dic3—184no—
290C2×Dic3—188no—
291C2×Dic3—184no—
292C2×Dic3—181no—
293D12—21no—
294D12—38no—
295D12—64no—
296D12C4×C3⋊D312yescomplemented
297C4×D3—98no—
298C4×D3—98no—
299C4×D3—92no—
300C4×D3—92no—
301C4×D3—188no—
302C4×D3—188no—
303C4×D3—362no—
304C4×D3—368no—
305C4×D3—3616no—
306C3×C8—64no—
307C3×C8—181no—
308C3⋊C8—21no—
309C3⋊C8—64no—
310C2×C32—24no—
311C2×C32—24no—
312C2×C32—24no—
313C2×C32—122no—
314C2×C32D3×C22⋊C414yes—
315C2×C32D3×C22⋊C411yescharacteristic
316C2×C32D3×D814yes—
317C2×C32D3×D811yescharacteristic
318C2×C32D3×SD1614yes—
319C2×C32D3×SD1611yescharacteristic
320C3⋊D3—34no—
321C3⋊D3—316no—
322C3⋊D3—42no—
323C3⋊D3—616no—
324C3×D3—48no—
325C3×D3—128no—
326C24—541no—
327C2×D4—31no—
328C2×D4—91no—
329C2×D4—274no—
330C2×D4—542no—
331C2×D4—541no—
332C22×C4—271no—
333C22×C4—541no—
334C2×C8—31no—
335C2×C8—91no—
336C2×C8—542no—
337C4⋊C4—271no—
338C4⋊C4—541no—
339C4⋊C4D3×C3⋊D311yescharacteristic, complemented
340C22×C3—24no—
341C22×C3—68no—
342C22×C3—124no—
343C22×C3—124no—
344C22×C3—182no—
345C22×C3—361no—
346C22×C3—361no—
347C22×C3#381114yes—
348C22×C3D4×C3⋊D311yescharacteristic
349C2×D3—22no—
350C2×D3—41no—
351C2×D3—41no—
352C2×D3—68no—
353C2×D3—92no—
354C2×D3—92no—
355C2×D3—92no—
356C2×D3—98no—
357C2×D3—98no—
358C2×D3—98no—
359C2×D3—124no—
360C2×D3—124no—
361C2×D3—188no—
362C2×D3—188no—
363C2×D3—188no—
364C2×D3—368no—
365C2×D3—3632no—
366C3×C4—24no—
367C3×C4—24no—
368C3×C4—48no—
369C3×C4—41no—
370C3×C4—44no—
371C3×C4C4×D3214yes—
372C3×C4#380214yes—
373C3×C4C2×C4×C3⋊D311yescharacteristic
374C3×C4D4×C3⋊D311yescharacteristic
375Dic3—92no—
376Dic3—98no—
377Dic3—188no—
378Dic3—368no—
379Dic3—361no—
380Dic3—364no—
381C32—24no—
382C32D3×#10214yescomplemented
383C32D3×#10211yescharacteristic, complemented
384C23—61no—
385C23—181no—
386C23—271no—
387C23—544no—
388C23—1082no—
389C23—1082no—
390D4—32no—
391D4—61no—
392D4—92no—
393D4—181no—
394D4—544no—
395C2×C4—21no—
396C2×C4—271no—
397C2×C4—272no—
398C2×C4—272no—
399C2×C4—541no—
400C2×C4—1082no—
401C2×C4C2×D3×C3⋊D311yescharacteristic
402C8—61no—
403C8—181no—
404C2×C3—24no—
405C2×C3—24no—
406C2×C3—24no—
407C2×C3—128no—
408C2×C3—362no—
409C2×C3#6392114yes—
410C2×C3#6392614yes—
411C2×C3#6394014yes—
412C2×C3C22⋊C4×C3⋊D311yescharacteristic
413C2×C3D8×C3⋊D311yescharacteristic
414C2×C3SD16×C3⋊D311yescharacteristic
415D3—42no—
416D3—916no—
417D3—94no—
418D3—128no—
419D3—1816no—
420C22—62no—
421C22—121no—
422C22—121no—
423C22—182no—
424C22—272no—
425C22—272no—
426C22—272no—
427C22—361no—
428C22—361no—
429C22—1088no—
430C22#8903911yescharacteristic
431C4—41no—
432C4—272no—
433C4—1081no—
434C4C4×D3×C3⋊D311yescharacteristic
435C4#8902611yescharacteristic
436C3—24no—
437C3#1059059314yescomplemented
438C3C3⋊D3×#10211yescharacteristic, complemented
439C2—122no—
440C2—274no—
441C2—362no—
442C2#1172353911yescharacteristic
443C2#1172354711yescharacteristic
444C2#1172357111yescharacteristic
subgroupG/H
centerC22#89039
commutatorC33×C4C22×C4
FrattiniC2×C4C2×D3×C3⋊D3
FittingC33×C4⋊C4C22
socleC22×C33C2×D4
prime radicalC2×#89026C2
cosocleC2×C33×C4C23
hypercenterC4⋊C4D3×C3⋊D3
hypocenterC33C2×#102
LCSG ▷ C33×C4 ▷ C2×C33 ▷ C33
UCSC1 ◁ C22 ◁ C2×C4 ◁ C4⋊C4

Complements

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

NH ≅ G/Ntype
2. C4⋊C4×C3⋊(C3⋊D3)C2semidirect
3. C2×#88785C2semidirect
4. C3×#63636C2semidirect
5. C32×#809C2semidirect
6. #11722034C2semidirect
7. #11722032C2semidirect
17. #89026C4semidirect
18. C33×C4⋊C4C22semidirect
19. C4⋊(C3⋊(C3⋊Dic3))C22semidirect
21. C2×C3×C3⋊(C3⋊C8)C22semidirect
22. C2×C32×C3⋊C8C22semidirect
34. #63636D3semidirect
36. C3×#809D3semidirect
53. C3×C4⋊(C3⋊D3)C2×C4semidirect
55. C32×D12C2×C4semidirect
81. C32×C4⋊C4C2×D3semidirect
82. C32×C4⋊C4C2×D3semidirect
89. C2×C3⋊(C3⋊C8)C2×D3semidirect
92. C2×C3×C3⋊C8C2×D3semidirect
124. #809C3⋊D3semidirect
154. C4⋊(C3⋊D3)C4×D3semidirect
167. C3×D12C4×D3semidirect
177. C3⋊(C3⋊D3)#102semidirect
201. C3×C4⋊C4D32semidirect
202. C3×C4⋊C4C2×C3⋊D3semidirect
210. C2×C3⋊C8C2×C3⋊D3semidirect
261. C33C2×#102semidirect
296. D12C4×C3⋊D3semidirect
339. C4⋊C4D3×C3⋊D3semidirect
382. C32D3×#102semidirect
383. C32D3×#102semidirect
437. C3#10590593semidirect
438. C3C3⋊D3×#102semidirect

Maximal subgroups

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

subgroupindex
C2×#890262
C4⋊C4×C3⋊(C3⋊D3)2
C2×#887852
C3×#636362
C32×#8092
#117220342
#117220322
C3⋊D3×#1023
#105905933