Finite Group Databasebeta

← #13160034

Subgroups — #13160034

Order 1728.

1516 conjugacy classes of subgroups; only the 178 normal subgroups are shown.

normal subgroups only

This group has 1516 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
2C2×C8×C3⋊(C3⋊D3)C211yesmaximal, characteristic, complemented
3#11723503C211yesmaximal, characteristic, complemented
4C3×#63648C211yesmaximal, characteristic, complemented
5C32×#821C211yesmaximal, characteristic, complemented
6#11722033C211yesmaximal, characteristic, complemented
7#11722031C211yesmaximal, characteristic, complemented
8C3⋊D3×#102—31nomaximal
9#10590940—34nomaximal
10C22×D3×C3⋊D3—21no—
11C2×C4×C3⋊(C3⋊D3)C2211yescharacteristic
12C2×C3×C4⋊(C3⋊D3)C2211yescharacteristic
13C2×C32×D12C2211yescharacteristic
14C2×#6290—21no—
15C2×#6289—21no—
16#89026—22no—
17#89026C414yescomplemented
18C2×C33×C8C2211yescharacteristic, complemented
19C2×C3⋊(C3⋊(C3⋊C8))C2211yescharacteristic, complemented
20C8×C3⋊(C3⋊D3)—22no—
21C3×C4⋊(C3⋊Dic3)C2211yescharacteristic, complemented
22C32×C4⋊Dic3C2211yescharacteristic, complemented
23#88806—21no—
24C2×D4×C3⋊D3—31no—
25C2×#3802—34no—
26C2×C8×C3⋊D3—31no—
27C2×C8×C3⋊D3—34no—
28C2×C8×C3⋊D3—64no—
29C4⋊C4×C3⋊D3—31no—
30#63900—34no—
31C32×#102—31no—
32#63659—31no—
33#63648D311yescharacteristic, complemented
34#63636—31no—
35C3×#821—34no—
36C3×#821D314yescomplemented
37#63562—34no—
38#63562—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×C4C2311yescharacteristic
46C2×C3⋊(C3⋊Dic3)D411yescharacteristic
47C4×C3⋊(C3⋊D3)C2×C412yes—
48C4×C3⋊(C3⋊D3)D412yes—
49C2×C3×C3⋊Dic3—21no—
50C3×C4⋊(C3⋊D3)—21no—
51C3×C4⋊(C3⋊D3)C2×C412yescomplemented
52C2×C32×Dic3—21no—
53C32×D12—21no—
54C32×D12C2×C412yescomplemented
55#6290—24no—
56#6289—42no—
57C33×C8—21no—
58C3⋊(C3⋊(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×C3×D12—34no—
73C2×C3×D12C2×D314yes—
74C2×#625—64no—
75C2×#624—64no—
76#3802—316no—
77#3802—68no—
78C2×C32×C8—24no—
79C2×C32×C8C2×D314yescomplemented
80C2×C32×C8C2×D311yescharacteristic, complemented
81C32×C4⋊C4—31no—
82C4⋊(C3⋊Dic3)C2×D311yescharacteristic, complemented
83#3750—61no—
84C2×C3⋊(C3⋊C8)—31no—
85C2×C3⋊(C3⋊C8)—34no—
86C2×C3⋊(C3⋊C8)—64no—
87C8×C3⋊D3—62no—
88C8×C3⋊D3—68no—
89C8×C3⋊D3—616no—
90C3×C4⋊Dic3—34no—
91C3×C4⋊Dic3C2×D314yescomplemented
92#3724—64no—
93C22×C33C2×D411yescharacteristic
94C2×C3⋊(C3⋊D3)C22⋊C412yes—
95C2×C3⋊(C3⋊D3)D812yes—
96C2×C3⋊(C3⋊D3)SD1612yes—
97C2×C3×C3⋊D3—22no—
98C2×C3×C3⋊D3—41no—
99C2×C3×C3⋊D3—41no—
100C2×C32×D3—22no—
101C2×C32×D3—41no—
102C2×C32×D3—41no—
103D3×C3⋊D3—48no—
104C33×C4C22×C411yescharacteristic
105C33×C4C2×D411yescharacteristic
106C3⋊(C3⋊Dic3)C22⋊C412yes—
107C3×C3⋊Dic3—41no—
108C32×Dic3—41no—
109C2×D3×D4—94no—
110C2×D3×D4—91no—
111C2×D3×C8—94no—
112C2×D3×C8—91no—
113C2×D3×C8—184no—
114D3×C4⋊C4—94no—
115D3×C4⋊C4—91no—
116C3×#102—34no—
117C3×#102—91no—
118#832—94no—
119#832—91no—
120#821—34no—
121#821C3⋊D311yescharacteristic, complemented
122#809—94no—
123#809—91no—
124C23×C32—61no—
125C22×C3⋊D3—21no—
126C22×C3⋊D3—31no—
127C22×C3⋊D3—34no—
128C22×C3⋊D3—64no—
129C22×C3⋊D3—64no—
130C22×C3⋊D3—122no—
131C22×C3⋊D3—122no—
132C22×C3×D3—24no—
133C22×C3×D3—64no—
134C2×D32—616no—
135C2×D32—128no—
136C2×D32—128no—
137C32×D4—32no—
138C32×D4—61no—
139C2×C32×C4—24no—
140C2×C32×C4—61no—
141C2×C32×C4C22×D314yes—
142C2×C32×C4C22×D311yescharacteristic
143#638—64no—
144C2×C3⋊Dic3—21no—
145C2×C3⋊Dic3—31no—
146C2×C3⋊Dic3—34no—
147C2×C3⋊Dic3—64no—
148C4⋊(C3⋊D3)—21no—
149C4⋊(C3⋊D3)C4×D312yescomplemented
150C4×C3⋊D3—32no—
151C4×C3⋊D3—32no—
152C4×C3⋊D3—38no—
153C4×C3⋊D3—38no—
154C4×C3⋊D3—68no—
155C4×C3⋊D3—68no—
156C4×C3⋊D3—122no—
157C2×C3×Dic3—24no—
158C2×C3×Dic3—64no—
159C3×D12—24no—
160C3×D12—38no—
161C3×D12—64no—
162C3×D12C4×D318yescomplemented
163#625—616no—
164#624—128no—
165C32×C8—28no—
166C32×C8—24no—
167C32×C8—21no—
168C3⋊(C3⋊C8)—68no—
169C3⋊(C3⋊C8)—64no—
170C3⋊(C3⋊C8)—61no—
171C2×#102—271noSylow
172C2×C33C2×C22⋊C411yescharacteristic
173C2×C33C2×D811yescharacteristic
174C2×C33C2×SD1611yescharacteristic
175C3⋊(C3⋊D3)#10214yescomplemented
176C3×C3⋊D3—42no—
177C32×D3—42no—
178C23×D3—181no—
179C23×D3—184no—
180C2×C3×D4—34no—
181C2×C3×D4—91no—
182C2×#67—181no—
183C2×#67—184no—
184D3×D4—916no—
185D3×D4—94no—
186D3×D4—182no—
187D3×D4—188no—
188C2×D12—34no—
189C2×D12C2×C3⋊D311yescharacteristic
190C2×C4×D3—94no—
191C2×C4×D3—91no—
192C2×C4×D3—184no—
193C2×C4×D3—181no—
194C2×C4×D3—184no—
195C2×C3×C8—24no—
196C2×C3×C8D3214yescomplemented
197C2×C3×C8C2×C3⋊D311yescharacteristic, complemented
198C3×C4⋊C4—34no—
199C3×C4⋊C4—91no—
200C4⋊Dic3—34no—
201C4⋊Dic3C2×C3⋊D311yescharacteristic, complemented
202Dic3⋊C4—181no—
203Dic3⋊C4—184no—
204C2×C3⋊C8—94no—
205C2×C3⋊C8—91no—
206C2×C3⋊C8—184no—
207D3×C8—188no—
208D3×C8—182no—
209D3×C8—1816no—
210C22×C32—24no—
211C22×C32—62no—
212C22×C32—121no—
213C22×C32—121no—
214C22×C32D3×D414yes—
215C22×C32D3×D411yescharacteristic
216C2×C3⋊D3—22no—
217C2×C3⋊D3—32no—
218C2×C3⋊D3—32no—
219C2×C3⋊D3—32no—
220C2×C3⋊D3—38no—
221C2×C3⋊D3—38no—
222C2×C3⋊D3—38no—
223C2×C3⋊D3—41no—
224C2×C3⋊D3—41no—
225C2×C3⋊D3—68no—
226C2×C3⋊D3—68no—
227C2×C3⋊D3—68no—
228C2×C3⋊D3—128no—
229C2×C3×D3—28no—
230C2×C3×D3—44no—
231C2×C3×D3—44no—
232C2×C3×D3—68no—
233C2×C3×D3—124no—
234C2×C3×D3—124no—
235D32—1232no—
236C32×C4—24no—
237C32×C4—24no—
238C32×C4—121no—
239C32×C4C2×C4×D314yes—
240C32×C4C2×C4×D311yescharacteristic
241C32×C4D3×D414yes—
242C32×C4D3×D411yescharacteristic
243C3⋊Dic3—32no—
244C3⋊Dic3—38no—
245C3⋊Dic3—41no—
246C3⋊Dic3—68no—
247C3×Dic3—44no—
248C3×Dic3—124no—
249C22×D4—271no—
250C22×C8—271no—
251C2×C4⋊C4—271no—
252#102—31no—
253#102—91no—
254#102—271no—
255#102—271no—
256C33C2×#10211yescharacteristic, complemented, Sylow
257C23×C3—64no—
258C23×C3—181no—
259C22×D3—21no—
260C22×D3—64no—
261C22×D3—94no—
262C22×D3—91no—
263C22×D3—184no—
264C22×D3—184no—
265C22×D3—1816no—
266C22×D3—362no—
267C22×D3—362no—
268C22×D3—368no—
269C22×D3—368no—
270C3×D4—38no—
271C3×D4—64no—
272C3×D4—92no—
273C3×D4—181no—
274C2×C3×C4—24no—
275C2×C3×C4—64no—
276C2×C3×C4—181no—
277C2×C3×C4C2×D3214yes—
278C2×C3×C4C22×C3⋊D311yescharacteristic
279#67—184no—
280#67—1816no—
281C2×Dic3—21no—
282C2×Dic3—64no—
283C2×Dic3—94no—
284C2×Dic3—91no—
285C2×Dic3—184no—
286D12—21no—
287D12—38no—
288D12—64no—
289D12C4×C3⋊D312yescomplemented
290C4×D3—98no—
291C4×D3—98no—
292C4×D3—92no—
293C4×D3—92no—
294C4×D3—188no—
295C4×D3—188no—
296C4×D3—362no—
297C4×D3—368no—
298C3×C8—28no—
299C3×C8—24no—
300C3×C8—21no—
301C3⋊C8—188no—
302C3⋊C8—184no—
303C3⋊C8—181no—
304C2×C32—24no—
305C2×C32—24no—
306C2×C32—24no—
307C2×C32—122no—
308C2×C32D3×C22⋊C414yes—
309C2×C32D3×C22⋊C411yescharacteristic
310C2×C32D3×D814yes—
311C2×C32D3×D811yescharacteristic
312C2×C32D3×SD1614yes—
313C2×C32D3×SD1611yescharacteristic
314C3⋊D3—34no—
315C3⋊D3—316no—
316C3⋊D3—42no—
317C3⋊D3—616no—
318C3×D3—48no—
319C3×D3—128no—
320C24—541no—
321C2×D4—31no—
322C2×D4—91no—
323C2×D4—274no—
324C2×D4—542no—
325C2×D4—541no—
326C22×C4—271no—
327C22×C4—541no—
328C2×C8—271no—
329C2×C8—542no—
330C2×C8D3×C3⋊D311yescharacteristic, complemented
331C4⋊C4—31no—
332C4⋊C4—91no—
333C4⋊C4—541no—
334C22×C3—24no—
335C22×C3—68no—
336C22×C3—124no—
337C22×C3—124no—
338C22×C3—182no—
339C22×C3—361no—
340C22×C3—361no—
341C22×C3#380214yes—
342C22×C3D4×C3⋊D311yescharacteristic
343C2×D3—22no—
344C2×D3—41no—
345C2×D3—41no—
346C2×D3—68no—
347C2×D3—98no—
348C2×D3—98no—
349C2×D3—92no—
350C2×D3—92no—
351C2×D3—92no—
352C2×D3—98no—
353C2×D3—124no—
354C2×D3—124no—
355C2×D3—188no—
356C2×D3—188no—
357C2×D3—188no—
358C2×D3—368no—
359C2×D3—3632no—
360C3×C4—24no—
361C3×C4—24no—
362C3×C4—124no—
363C3×C4—361no—
364C3×C4C4×D3214yes—
365C3×C4#381114yes—
366C3×C4C2×C4×C3⋊D311yescharacteristic
367C3×C4D4×C3⋊D311yescharacteristic
368Dic3—41no—
369Dic3—92no—
370Dic3—98no—
371Dic3—124no—
372Dic3—188no—
373C32—24no—
374C32D3×#10214yescomplemented
375C32D3×#10211yescharacteristic, complemented
376C23—61no—
377C23—181no—
378C23—271no—
379C23—544no—
380C23—1082no—
381C23—1082no—
382D4—32no—
383D4—61no—
384D4—92no—
385D4—181no—
386D4—544no—
387C2×C4—61no—
388C2×C4—181no—
389C2×C4—271no—
390C2×C4—272no—
391C2×C4—272no—
392C2×C4—1082no—
393C2×C4C2×D3×C3⋊D311yescharacteristic
394C8—21no—
395C8—541no—
396C2×C3—24no—
397C2×C3—24no—
398C2×C3—24no—
399C2×C3—128no—
400C2×C3—362no—
401C2×C3#6379614yes—
402C2×C3#6379714yes—
403C2×C3#6392114yes—
404C2×C3C22⋊C4×C3⋊D311yescharacteristic
405C2×C3D8×C3⋊D311yescharacteristic
406C2×C3SD16×C3⋊D311yescharacteristic
407D3—42no—
408D3—94no—
409D3—916no—
410D3—128no—
411D3—1816no—
412C22—62no—
413C22—121no—
414C22—121no—
415C22—182no—
416C22—272no—
417C22—272no—
418C22—272no—
419C22—361no—
420C22—361no—
421C22—1088no—
422C22#8902611yescharacteristic
423C4—121no—
424C4—272no—
425C4—361no—
426C4C4×D3×C3⋊D311yescharacteristic
427C4#8903911yescharacteristic
428C3—24no—
429C3#1059094014yescomplemented
430C3C3⋊D3×#10211yescharacteristic, complemented
431C2—122no—
432C2—274no—
433C2—362no—
434C2#1172332111yescharacteristic
435C2#1172332211yescharacteristic
436C2#1172353911yescharacteristic
subgroupG/H
centerC22#89026
commutatorC33×C4C22×C4
FrattiniC2×C4C2×D3×C3⋊D3
FittingC2×C33×C8C22
socleC22×C33C2×D4
prime radicalC2×#89026C2
cosocleC2×C33×C4C23
hypercenterC2×C8D3×C3⋊D3
hypocenterC33C2×#102
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×C8×C3⋊(C3⋊D3)C2semidirect
3. #11723503C2semidirect
4. C3×#63648C2semidirect
5. C32×#821C2semidirect
6. #11722033C2semidirect
7. #11722031C2semidirect
17. #89026C4semidirect
18. C2×C33×C8C22semidirect
19. C2×C3⋊(C3⋊(C3⋊C8))C22semidirect
21. C3×C4⋊(C3⋊Dic3)C22semidirect
22. C32×C4⋊Dic3C22semidirect
33. #63648D3semidirect
36. C3×#821D3semidirect
51. C3×C4⋊(C3⋊D3)C2×C4semidirect
54. C32×D12C2×C4semidirect
79. C2×C32×C8C2×D3semidirect
80. C2×C32×C8C2×D3semidirect
82. C4⋊(C3⋊Dic3)C2×D3semidirect
91. C3×C4⋊Dic3C2×D3semidirect
121. #821C3⋊D3semidirect
149. C4⋊(C3⋊D3)C4×D3semidirect
162. C3×D12C4×D3semidirect
175. C3⋊(C3⋊D3)#102semidirect
196. C2×C3×C8D32semidirect
197. C2×C3×C8C2×C3⋊D3semidirect
201. C4⋊Dic3C2×C3⋊D3semidirect
256. C33C2×#102semidirect
289. D12C4×C3⋊D3semidirect
330. C2×C8D3×C3⋊D3semidirect
374. C32D3×#102semidirect
375. C32D3×#102semidirect
429. C3#10590940semidirect
430. 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
C2×C8×C3⋊(C3⋊D3)2
#117235032
C3×#636482
C32×#8212
#117220332
#117220312
C3⋊D3×#1023
#105909403