Finite Group Databasebeta

← #13161963

Subgroups — #13161963

Order 1728.

860 conjugacy classes of subgroups; only the 119 normal subgroups are shown.

normal subgroups only

This group has 860 conjugacy classes of subgroups — too many to draw. Only its 119 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
1C3×D4×C3⋊Dic3C211yesmaximal, characteristic, complemented
2C2×C3×#3753C212yesmaximal, complemented
3#11723697C211yesmaximal, characteristic, complemented
4#11722078C211yesmaximal, characteristic, complemented
5#11722075C212yesmaximal, complemented
6#10591486—31nomaximal
7#10591481—32nomaximal
8C2×C33×D4C2211yescharacteristic
9C22×C3×C3⋊Dic3—21no—
10C2×C3×C4⋊(C3⋊D3)C2212yes—
11C2×#6295—21no—
12C2×#6295—21no—
13C3×#3757—21no—
14C3×#3753—24no—
15C3×C4⋊(C3⋊Dic3)C2211yescharacteristic
16C3×C4×C3⋊Dic3C2211yescharacteristic
17C2×C3×C3⋊(C3⋊C8)C2212yes—
18D4×C3⋊Dic3D311yescharacteristic, complemented
19C2×#3753D312yescomplemented
20C3×D4×Dic3—32no—
21C3×D4×Dic3—61no—
22C2×C3×#213—32no—
23C2×C3×#213—32no—
24C2×C3×#213—62no—
25#63914—31no—
26#63911—32no—
27#63572—31no—
28Dic3⋊(C3⋊C8)—32no—
29#63564—32no—
30#63563—32no—
31#63562—32no—
32C23×C33—21no—
33C22×C3×C3⋊D3—22no—
34C33×D4—21no—
35C33×D4D412yescomplemented
36C2×C33×C4C2311yescharacteristic
37C2×C3×C3⋊Dic3—21no—
38C2×C3×C3⋊Dic3—42no—
39C2×C3×C3⋊Dic3D411yescharacteristic
40C2×C3×C3⋊Dic3D411yescharacteristic
41C3×C4⋊(C3⋊D3)—22no—
42C3×C4⋊(C3⋊D3)—22no—
43#6295—42no—
44#6295—42no—
45C3×C3⋊(C3⋊C8)—22no—
46#5150—92no—
47#5015—91no—
48C2×C32×D4—21no—
49C2×C32×D4—22no—
50C2×C32×D4—41no—
51C2×C32×D4C2×D311yescharacteristic
52C2×C32×D4C2×D312yes—
53C22×C3⋊Dic3—21no—
54C2×C4⋊(C3⋊D3)C2×D312yes—
55C22×C3×Dic3—62no—
56C22×C3×Dic3—62no—
57C2×C3×D12—32no—
58C2×C3×D12—32no—
59C2×C3×D12—62no—
60C2×#626—62no—
61C2×#626—62no—
62C2×#625—61no—
63C2×#625—61no—
64#3757—21no—
65#3753—24no—
66C4⋊(C3⋊Dic3)C2×D311yescharacteristic
67C4×C3⋊Dic3C2×D311yescharacteristic
68C2×C3⋊(C3⋊C8)C2×D312yes—
69C3×#217—62no—
70C3×#217—62no—
71C3×#213—68no—
72C3×#213—64no—
73C3×#213—64no—
74C3×C4⋊Dic3—32no—
75C3×C4⋊Dic3—61no—
76C3×C4×Dic3—32no—
77C3×C4×Dic3—61no—
78C2×C3×C3⋊C8—32no—
79C2×C3×C3⋊C8—32no—
80C2×C3×C3⋊C8—62no—
81C22×C33—22no—
82C22×C33—41no—
83C22×C33—41no—
84C22×C33C2×D411yescharacteristic
85C2×C3×C3⋊D3—42no—
86C2×C3×C3⋊D3—42no—
87C2×C3×C3⋊D3—42no—
88C33×C4C2×D411yescharacteristic
89C33×C4#4111yescharacteristic
90C3×C3⋊Dic3—21no—
91C3×C3⋊Dic3—41no—
92C3×C3⋊Dic3D812yescomplemented
93C2×C3×D8—92no—
94C3×C4×D4—91no—
95#940—92no—
96D4×Dic3—32no—
97D4×Dic3—61no—
98C2×#213—32no—
99C2×#213—32no—
100C2×#213—62no—
101#874—91no—
102#832—92no—
103#821—92no—
104Dic3⋊C8—92no—
105#809—92no—
106C4⋊(C3⋊C8)—91no—
107C23×C32—21no—
108C23×C32—22no—
109C23×C32—22no—
110C23×C32—42no—
111C23×C32—42no—
112C22×C3⋊D3—22no—
113C22×C3×D3—62no—
114C22×C3×D3—62no—
115C22×C3×D3—122no—
116C32×D4—21no—
117C32×D4—22no—
118C32×D4—22no—
119C32×D4—22no—
120C32×D4—24no—
121C32×D4—42no—
122C32×D4—42no—
123C32×D4—42no—
124C32×D4D1212yescomplemented
125C32×D4#6714yescomplemented
126C2×C32×C4—22no—
127C2×C32×C4—21no—
128C2×C32×C4—41no—
129C2×C32×C4C22×D311yescharacteristic
130C2×C32×C4C22×D312yes—
131C2×C3⋊Dic3—21no—
132C2×C3⋊Dic3—42no—
133C2×C3⋊Dic3#6711yescharacteristic
134C2×C3⋊Dic3#6711yescharacteristic
135C4⋊(C3⋊D3)—22no—
136C4⋊(C3⋊D3)—22no—
137C2×C3×Dic3—32no—
138C2×C3×Dic3—32no—
139C2×C3×Dic3—61no—
140C2×C3×Dic3—61no—
141C2×C3×Dic3—62no—
142C2×C3×Dic3—62no—
143C2×C3×Dic3—124no—
144C2×C3×Dic3—124no—
145C3×D12—64no—
146C3×D12—62no—
147C3×D12—62no—
148C3×D12—62no—
149C3×D12—62no—
150C3×D12—122no—
151#626—124no—
152#626—124no—
153#625—122no—
154#625—122no—
155C3⋊(C3⋊C8)—22no—
156C3×C3⋊C8—64no—
157C3×C3⋊C8—62no—
158C3×C3⋊C8—62no—
159#459—271noSylow
160C2×C33—42no—
161C2×C33#12111yescharacteristic
162C2×C33C2×D811yescharacteristic
163C2×C33AGL(1,ℤ/8)11yescharacteristic
164C3×C3⋊D3—82no—
165C2×C3×D4—21no—
166C2×C3×D4—22no—
167C2×C3×D4—41no—
168C2×C3×D4—92no—
169C2×C3×D4D3211yescharacteristic
170C2×C3×D4D3212yes—
171C22×C3×C4—181no—
172C2×#67—182no—
173C2×#67—182no—
174C22×Dic3—62no—
175C22×Dic3—62no—
176C2×D12—32no—
177C2×D12—32no—
178C2×D12—62no—
179C2×D12—181no—
180C2×D12—181no—
181C3×D8—184no—
182C2×C3×C8—92no—
183C3×C4⋊C4—91no—
184C3×C22⋊C4—181no—
185C3×C42—91no—
186#217—62no—
187#217—62no—
188#213—68no—
189#213—64no—
190#213—64no—
191C4⋊Dic3—32no—
192C4⋊Dic3—61no—
193C4×Dic3—32no—
194C4×Dic3—61no—
195C2×C3⋊C8—32no—
196C2×C3⋊C8—32no—
197C2×C3⋊C8—62no—
198C22×C32—22no—
199C22×C32—24no—
200C22×C32—24no—
201C22×C32—22no—
202C22×C32—21no—
203C22×C32—41no—
204C22×C32—44no—
205C22×C32—44no—
206C22×C32—41no—
207C22×C32—41no—
208C22×C32—42no—
209C22×C32—42no—
210C22×C32—42no—
211C22×C32—42no—
212C22×C32—82no—
213C22×C32—82no—
214C22×C32—82no—
215C22×C32—82no—
216C22×C32D3×D412yes—
217C22×C32C2×#6711yescharacteristic
218C2×C3⋊D3—42no—
219C2×C3⋊D3—42no—
220C2×C3⋊D3—42no—
221C2×C3×D3—124no—
222C2×C3×D3—122no—
223C2×C3×D3—122no—
224C2×C3×D3—122no—
225C2×C3×D3—122no—
226C2×C3×D3—122no—
227C2×C3×D3—122no—
228C2×C3×D3—242no—
229C2×C3×D3—242no—
230C32×C4—22no—
231C32×C4—21no—
232C32×C4—21no—
233C32×C4—22no—
234C32×C4—41no—
235C32×C4—41no—
236C32×C4C2×D1211yescharacteristic
237C32×C4#23511yescharacteristic
238C32×C4#23712yes—
239C32×C4C2×#6712yes—
240C3⋊Dic3—21no—
241C3⋊Dic3—41no—
242C3⋊Dic3#21312yescomplemented
243C3×Dic3—34no—
244C3×Dic3—62no—
245C3×Dic3—62no—
246C3×Dic3—121no—
247C3×Dic3—122no—
248C3×Dic3—122no—
249C2×D8—92no—
250#127—271no—
251C4×D4—91no—
252C4⋊C8—271no—
253#102—272no—
254C33#45911yescharacteristic, complemented, Sylow
255C23×C3—22no—
256C23×C3—22no—
257C23×C3—21no—
258C23×C3—42no—
259C23×C3—42no—
260C23×C3—182no—
261C22×D3—62no—
262C22×D3—62no—
263C22×D3—122no—
264C3×D4—22no—
265C3×D4—22no—
266C3×D4—21no—
267C3×D4—22no—
268C3×D4—24no—
269C3×D4—42no—
270C3×D4—42no—
271C3×D4—42no—
272C3×D4—182no—
273C3×D4—182no—
274C3×D4#62512yescomplemented
275C3×D4#62614yescomplemented
276C2×C3×C4—22no—
277C2×C3×C4—21no—
278C2×C3×C4—41no—
279C2×C3×C4—91no—
280C2×C3×C4—91no—
281C2×C3×C4—181no—
282C2×C3×C4—362no—
283C2×C3×C4C2×D3212yes—
284C2×C3×C4C2×D3211yescharacteristic
285#67—364no—
286#67—364no—
287C2×Dic3—32no—
288C2×Dic3—32no—
289C2×Dic3—61no—
290C2×Dic3—61no—
291C2×Dic3—62no—
292C2×Dic3—62no—
293C2×Dic3—124no—
294C2×Dic3—124no—
295D12—64no—
296D12—62no—
297D12—62no—
298D12—62no—
299D12—62no—
300D12—122no—
301D12—362no—
302D12—362no—
303C3×C8—182no—
304C3⋊C8—64no—
305C3⋊C8—62no—
306C3⋊C8—62no—
307C2×C32—22no—
308C2×C32—22no—
309C2×C32—21no—
310C2×C32—21no—
311C2×C32—22no—
312C2×C32—21no—
313C2×C32—41no—
314C2×C32—41no—
315C2×C32—41no—
316C2×C32—42no—
317C2×C32—44no—
318C2×C32—44no—
319C2×C32—84no—
320C2×C32—84no—
321C2×C32D3×D812yes—
322C2×C32#91112yes—
323C2×C32#93011yescharacteristic
324C2×C32C2×#21311yescharacteristic
325C2×C32#93912yes—
326C2×C32#94911yescharacteristic
327C3⋊D3—82no—
328C3×D3—244no—
329C3×D3—242no—
330C3×D3—242no—
331C2×D4—92no—
332C2×D4—541no—
333C2×D4—541no—
334C2×D4#111311yescharacteristic
335C22×C4—181no—
336D8—184no—
337C2×C8—92no—
338C4⋊C4—91no—
339C22⋊C4—181no—
340C42—91no—
341C22×C3—24no—
342C22×C3—24no—
343C22×C3—22no—
344C22×C3—21no—
345C22×C3—22no—
346C22×C3—44no—
347C22×C3—44no—
348C22×C3—42no—
349C22×C3—42no—
350C22×C3—42no—
351C22×C3—42no—
352C22×C3—41no—
353C22×C3—41no—
354C22×C3—41no—
355C22×C3—82no—
356C22×C3—82no—
357C22×C3—82no—
358C22×C3—82no—
359C22×C3—362no—
360C22×C3—362no—
361C22×C3—362no—
362C22×C3D3×#6712yes—
363C22×C3#381111yescharacteristic
364C2×D3—124no—
365C2×D3—122no—
366C2×D3—122no—
367C2×D3—122no—
368C2×D3—122no—
369C2×D3—122no—
370C2×D3—122no—
371C2×D3—242no—
372C2×D3—242no—
373C3×C4—22no—
374C3×C4—21no—
375C3×C4—22no—
376C3×C4—21no—
377C3×C4—41no—
378C3×C4—41no—
379C3×C4—92no—
380C3×C4—181no—
381C3×C4—361no—
382C3×C4#380412yes—
383C3×C4#380511yescharacteristic
384C3×C4C2×#62511yescharacteristic
385C3×C4C2×#62612yes—
386Dic3—34no—
387Dic3—62no—
388Dic3—62no—
389Dic3—121no—
390Dic3—122no—
391Dic3—122no—
392C32—22no—
393C32—21no—
394C32—41no—
395C32#501511yescharacteristic, complemented
396C32#515012yescomplemented
397C23—21no—
398C23—182no—
399D4—21no—
400D4—182no—
401D4—182no—
402D4—1082no—
403D4—1082no—
404D4#629512yescomplemented
405C2×C4—91no—
406C2×C4—91no—
407C2×C4—181no—
408C2×C4—362no—
409C2×C4C2×#111311yescharacteristic
410C8—182no—
411C2×C3—21no—
412C2×C3—22no—
413C2×C3—21no—
414C2×C3—22no—
415C2×C3—22no—
416C2×C3—21no—
417C2×C3—44no—
418C2×C3—44no—
419C2×C3—42no—
420C2×C3—41no—
421C2×C3—41no—
422C2×C3—41no—
423C2×C3—84no—
424C2×C3—84no—
425C2×C3—722no—
426C2×C3D3×#21312yes—
427C2×C3#6392512yes—
428C2×C3#6392611yescharacteristic
429C2×C3#6392711yescharacteristic
430C2×C3#6397812yes—
431C2×C3#6398011yescharacteristic
432D3—244no—
433D3—242no—
434D3—242no—
435C22—22no—
436C22—41no—
437C22—41no—
438C22—362no—
439C22—362no—
440C22—362no—
441C22#8904911yescharacteristic
442C4—92no—
443C4—181no—
444C4—361no—
445C4#8904611yescharacteristic
446C4C2×#629511yescharacteristic
447C3—22no—
448C3—21no—
449C3—41no—
450C3#1059148112yescomplemented
451C3#1059148611yescharacteristic, complemented
452C2—42no—
453C2—722no—
454C2#1172370211yescharacteristic
455C2#1172370311yescharacteristic
456C2#1172371711yescharacteristic
subgroupG/H
centerC22#89049
commutator, cosocleC2×C33×C4C23
FrattiniC2×C4C2×#1113
FittingC2×C33×D4C22
socleC22×C33C2×D4
hypercenterC2×D4#1113
hypocenterC33#459
LCSG ▷ C2×C33×C4 ▷ C2×C33 ▷ C33
UCSC1 ◁ C22 ◁ C2×C4 ◁ C2×D4

Complements

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

NH ≅ G/Ntype
1. C3×D4×C3⋊Dic3C2semidirect
2. C2×C3×#3753C2semidirect
3. #11723697C2semidirect
4. #11722078C2semidirect
5. #11722075C2semidirect
18. D4×C3⋊Dic3D3semidirect
19. C2×#3753D3semidirect
35. C33×D4D4semidirect
92. C3×C3⋊Dic3D8semidirect
124. C32×D4D12semidirect
125. C32×D4#67semidirect
242. C3⋊Dic3#213semidirect
254. C33#459semidirect
274. C3×D4#625semidirect
275. C3×D4#626semidirect
395. C32#5015semidirect
396. C32#5150semidirect
404. D4#6295semidirect
450. C3#10591481semidirect
451. C3#10591486semidirect

Maximal subgroups

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

subgroupindex
C3×D4×C3⋊Dic32
C2×C3×#37532
#117236972
#117220782
#117220752
#105914863
#105914813