Finite Group Databasebeta

← #13161334

Subgroups — #13161334

Order 1728.

1400 conjugacy classes of subgroups; only the 162 normal subgroups are shown.

normal subgroups only

This group has 1400 conjugacy classes of subgroups — too many to draw. Only its 162 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×#88862C211yesmaximal, characteristic, complemented
3C3×#64155C211yesmaximal, characteristic, complemented
4C2×C32×#215C211yesmaximal, characteristic, complemented
5#11722036C211yesmaximal, characteristic, complemented
6#11722031C211yesmaximal, characteristic, complemented
7#11722023C211yesmaximal, characteristic, complemented
8#10592868—31nomaximal
9#10591651—34nomaximal
10C22×D3×C3⋊D3—21no—
11C2×C33×Q8C2211yescharacteristic, complemented
12C2×C4⋊(C3⋊(C3⋊D3))C2211yescharacteristic
13C2×C3×C4×C3⋊D3C2211yescharacteristic
14C2×C32×D12C2211yescharacteristic
15C2×#6291—21no—
16C3⋊D3×D12—22no—
17C3⋊D3×D12—22no—
18#88862—22no—
19C2×C3⋊(C3⋊(C3⋊C8))C2211yescharacteristic, complemented
20C3×#3752—21no—
21C3×C4⋊(C3⋊Dic3)C2211yescharacteristic, complemented
22C3×#3750—21no—
23C32×#215—22no—
24C2×C32×C3⋊C8C2211yescharacteristic, complemented
25C2×D4×C3⋊D3—31no—
26C2×D3×D12—34no—
27C2×C32×SD16—31no—
28#64155D311yescharacteristic, complemented
29C2×#3755—31no—
30C2×#3755—34no—
31C2×#3755—64no—
32C3×#946—34no—
33C2×C3×#215D314yescomplemented
34#63659—31no—
35#63648—31no—
36#63647—31no—
37#63564—34no—
38#63562—34no—
39(C2×D3)⋊(C3⋊C8)—34no—
40C22×C3⋊(C3⋊D3)—21no—
41C22×C3×C3⋊D3D411yescharacteristic
42C22×C32×D3—21no—
43C2×D3×C3⋊D3—24no—
44C2×D3×C3⋊D3—42no—
45C2×D3×C3⋊D3—42no—
46C33×Q8—21no—
47C33×Q8—21no—
48C2×C33×C4—21no—
49C2×C33×C4C2311yescharacteristic
50C4⋊(C3⋊(C3⋊D3))—21no—
51C4⋊(C3⋊(C3⋊D3))D412yes—
52C2×C3×C3⋊Dic3—21no—
53C2×C3×C3⋊Dic3D411yescharacteristic
54C3×C4×C3⋊D3—21no—
55C3×C4×C3⋊D3—21no—
56C32×D12—21no—
57C32×D12D412yes—
58#6291—24no—
59C3⋊(C3⋊(C3⋊C8))—21no—
60C32×C3⋊C8—21no—
61#5169—94no—
62#5046—91no—
63C23×C3⋊D3—61no—
64C22×D32—64no—
65C2×C32×Q8—24no—
66C2×C32×Q8C2×D314yescomplemented
67C2×C32×Q8C2×D311yescharacteristic, complemented
68C2×C32×D4—31no—
69C2×#638—61no—
70D4×C3⋊D3—62no—
71D4×C3⋊D3—62no—
72C2×C4⋊(C3⋊D3)—31no—
73C2×C4⋊(C3⋊D3)—34no—
74C2×C4⋊(C3⋊D3)—64no—
75C2×C4×C3⋊D3C2×D311yescharacteristic
76C2×C3×D12C2×D314yes—
77C2×C3×C4×D3—34no—
78C2×#626—64no—
79D3×D12—68no—
80D3×D12—68no—
81C32×SD16—62no—
82C2×C32×C8—31no—
83#3755—62no—
84#3755—68no—
85#3755—616no—
86#3752—21no—
87C4⋊(C3⋊Dic3)C2×D311yescharacteristic, complemented
88#3750—21no—
89C2×C3⋊(C3⋊C8)—31no—
90C2×C3⋊(C3⋊C8)—34no—
91C2×C3⋊(C3⋊C8)—64no—
92C3×#215—28no—
93C3×(C2×D3)⋊C4—64no—
94C3×C4⋊Dic3—34no—
95C3×Dic3⋊C4—64no—
96C2×C3×C3⋊C8C2×D314yescomplemented
97C22×C33C2×D411yescharacteristic
98C2×C3⋊(C3⋊D3)—22no—
99C2×C3⋊(C3⋊D3)—41no—
100C2×C3⋊(C3⋊D3)—41no—
101C2×C3×C3⋊D3—21no—
102C2×C3×C3⋊D3—21no—
103C2×C3×C3⋊D3SD1612yescomplemented
104C2×C32×D3—22no—
105C2×C32×D3—41no—
106C2×C32×D3—41no—
107D3×C3⋊D3—48no—
108C33×C4—41no—
109C33×C4C2×D411yescharacteristic
110C33×C4C2×D411yescharacteristic
111C3×C3⋊Dic3—21no—
112C3×C3⋊Dic3—41no—
113C2×D3×D4—94no—
114C22×D12—91no—
115C2×C3×SD16—34no—
116C3×C22⋊Q8—91no—
117#946—34no—
118C2×#215—94no—
119C2×#215—91no—
120C2×#215—184no—
121C2×#215C3⋊D311yescharacteristic, complemented
122#832—94no—
123#830—91no—
124#821—94no—
125(C2×D3)⋊C8—94no—
126#809—91no—
127#809—91no—
128C23×C32—61no—
129C22×C3⋊D3—64no—
130C22×C3⋊D3—61no—
131C22×C3⋊D3—64no—
132C22×C3⋊D3—124no—
133C22×C3⋊D3—122no—
134C22×C3⋊D3—122no—
135C22×C3⋊D3#6711yescharacteristic
136C22×C3×D3—24no—
137C22×C3×D3—34no—
138C2×D32—616no—
139C2×D32—128no—
140C2×D32—128no—
141C32×Q8—28no—
142C32×Q8—24no—
143C32×Q8—21no—
144C32×Q8—21no—
145C32×Q8—24no—
146C32×Q8—44no—
147C32×D4—32no—
148C32×D4—61no—
149C2×C32×C4—21no—
150C2×C32×C4—24no—
151C2×C32×C4—24no—
152C2×C32×C4—44no—
153C2×C32×C4C22×D314yes—
154C2×C32×C4C22×D311yescharacteristic
155#638—64no—
156C2×C3⋊Dic3—21no—
157C2×C3⋊Dic3#6711yescharacteristic
158C4⋊(C3⋊D3)—32no—
159C4⋊(C3⋊D3)—38no—
160C4⋊(C3⋊D3)—68no—
161C4⋊(C3⋊D3)—61no—
162C4⋊(C3⋊D3)—64no—
163C4⋊(C3⋊D3)—124no—
164C4×C3⋊D3—21no—
165C4×C3⋊D3—21no—
166C2×C3×Dic3—34no—
167C2×C3×Dic3—64no—
168C3×D12—24no—
169C3×D12#6718yes—
170C3×C4×D3—64no—
171C3×C4×D3—64no—
172#626—616no—
173C32×C8—61no—
174C3⋊(C3⋊C8)—68no—
175C3⋊(C3⋊C8)—64no—
176C3⋊(C3⋊C8)—61no—
177C3×C3⋊C8—24no—
178#450—271noSylow
179C2×C33#12011yescharacteristic
180C2×C33C2×SD1611yescharacteristic
181C2×C33AGL(1,ℤ/8)11yescharacteristic
182C3⋊(C3⋊D3)—42no—
183C3×C3⋊D3—22no—
184C32×D3—42no—
185C23×D3—181no—
186C23×D3—184no—
187C2×C3×Q8—24no—
188C2×C3×Q8D3214yescomplemented
189C2×C3×Q8C2×C3⋊D311yescharacteristic, complemented
190C2×C3×D4—34no—
191C22×C3×C4—91no—
192C2×#67—184no—
193D3×D4—188no—
194D3×D4—188no—
195C2×D12—94no—
196C2×D12—91no—
197C2×D12—184no—
198C2×D12—182no—
199C2×D12—181no—
200C2×D12—182no—
201C2×D12C2×C3⋊D311yescharacteristic
202C2×C4×D3—34no—
203C3×SD16—68no—
204C2×C3×C8—34no—
205C3×C4⋊C4—91no—
206C3×C4⋊C4—181no—
207C3×C22⋊C4—181no—
208#215—22no—
209#215—188no—
210#215—182no—
211#215—1816no—
212(C2×D3)⋊C4—64no—
213C4⋊Dic3—34no—
214Dic3⋊C4—64no—
215C2×C3⋊C8—94no—
216C2×C3⋊C8—91no—
217C2×C3⋊C8—184no—
218C2×C3⋊C8C2×C3⋊D311yescharacteristic, complemented
219C22×C32—24no—
220C22×C32—62no—
221C22×C32—121no—
222C22×C32—121no—
223C22×C32D3×D414yes—
224C22×C32C2×#6711yescharacteristic
225C2×C3⋊D3—21no—
226C2×C3⋊D3—21no—
227C2×C3⋊D3—62no—
228C2×C3⋊D3—68no—
229C2×C3⋊D3—128no—
230C2×C3⋊D3—128no—
231C2×C3⋊D3—121no—
232C2×C3⋊D3—121no—
233C2×C3⋊D3—124no—
234C2×C3⋊D3—124no—
235C2×C3⋊D3—244no—
236C2×C3⋊D3—244no—
237C2×C3⋊D3#21512yescomplemented
238C2×C3×D3—28no—
239C2×C3×D3—38no—
240C2×C3×D3—44no—
241C2×C3×D3—44no—
242C2×C3×D3—64no—
243C2×C3×D3—64no—
244D32—1232no—
245C32×C4—24no—
246C32×C4—24no—
247C32×C4—48no—
248C32×C4—44no—
249C32×C4—41no—
250C32×C4D3×D414yes—
251C32×C4D3×D411yescharacteristic
252C32×C4D3×D411yescharacteristic
253C32×C4C2×#6714yes—
254C3⋊Dic3—21no—
255C3⋊Dic3—41no—
256C3×Dic3—64no—
257C3×Dic3—124no—
258C22×D4—271no—
259C2×SD16—31no—
260C2×SD16—271no—
261C22⋊Q8—91no—
262#102—271no—
263#102—271no—
264C22⋊C8—271no—
265C33#45011yescharacteristic, complemented, Sylow
266C23×C3—64no—
267C23×C3—91no—
268C22×D3—21no—
269C22×D3—34no—
270C22×D3—184no—
271C22×D3—1816no—
272C22×D3—184no—
273C22×D3—181no—
274C22×D3—364no—
275C22×D3—368no—
276C22×D3—362no—
277C22×D3—362no—
278C22×D3—368no—
279C3×Q8—28no—
280C3×Q8—24no—
281C3×Q8—24no—
282C3×Q8—21no—
283C3×Q8—21no—
284C3×Q8—44no—
285C3×D4—38no—
286C3×D4—64no—
287C2×C3×C4—24no—
288C2×C3×C4—21no—
289C2×C3×C4—24no—
290C2×C3×C4—44no—
291C2×C3×C4—91no—
292C2×C3×C4—181no—
293C2×C3×C4—181no—
294C2×C3×C4—181no—
295C2×C3×C4C2×D3214yes—
296C2×C3×C4C22×C3⋊D311yescharacteristic
297#67—1816no—
298C2×Dic3—34no—
299C2×Dic3—64no—
300D12—21no—
301D12—98no—
302D12—92no—
303D12—188no—
304D12—184no—
305D12—184no—
306D12—181no—
307D12—364no—
308D12#63812yes—
309C4×D3—64no—
310C4×D3—64no—
311C3×C8—64no—
312C3⋊C8—21no—
313C3⋊C8—188no—
314C3⋊C8—184no—
315C3⋊C8—181no—
316C2×C32—24no—
317C2×C32—24no—
318C2×C32—24no—
319C2×C32—122no—
320C2×C32D3×SD1614yes—
321C2×C32#91414yes—
322C2×C32#93714yes—
323C2×C32#93711yescharacteristic
324C2×C32C2×#21511yescharacteristic
325C2×C32#94911yescharacteristic
326C3⋊D3—22no—
327C3⋊D3—122no—
328C3⋊D3—128no—
329C3⋊D3—248no—
330C3×D3—48no—
331C3×D3—68no—
332C24—541no—
333C2×Q8D3×C3⋊D311yescharacteristic, complemented
334C2×D4—31no—
335C2×D4—271no—
336C2×D4—541no—
337C2×D4—542no—
338C2×D4—542no—
339C22×C4—91no—
340SD16—62no—
341SD16—542no—
342C2×C8—31no—
343C2×C8—271no—
344C4⋊C4—91no—
345C4⋊C4—181no—
346C22⋊C4—181no—
347C22×C3—24no—
348C22×C3—68no—
349C22×C3—92no—
350C22×C3—124no—
351C22×C3—124no—
352C22×C3—181no—
353C22×C3—181no—
354C22×C3D3×#6714yes—
355C22×C3D4×C3⋊D311yescharacteristic
356C2×D3—22no—
357C2×D3—38no—
358C2×D3—41no—
359C2×D3—41no—
360C2×D3—64no—
361C2×D3—64no—
362C2×D3—182no—
363C2×D3—188no—
364C2×D3—368no—
365C2×D3—368no—
366C2×D3—3632no—
367C2×D3—361no—
368C2×D3—361no—
369C2×D3—364no—
370C2×D3—364no—
371C2×D3—724no—
372C2×D3—724no—
373C3×C4—24no—
374C3×C4—24no—
375C3×C4—48no—
376C3×C4—41no—
377C3×C4—44no—
378C3×C4—181no—
379C3×C4—361no—
380C3×C4D3×#6714yes—
381C3×C4#381114yes—
382C3×C4D4×C3⋊D311yescharacteristic
383C3×C4C2×#63811yescharacteristic
384Dic3—64no—
385Dic3—124no—
386C32—24no—
387C32#504611yescharacteristic, complemented
388C32#516914yescomplemented
389C23—61no—
390C23—91no—
391C23—544no—
392C23—541no—
393C23—1082no—
394C23—1082no—
395Q8—21no—
396Q8—21no—
397D4—32no—
398D4—61no—
399D4—272no—
400D4—544no—
401D4—541no—
402C2×C4—21no—
403C2×C4—91no—
404C2×C4—181no—
405C2×C4—181no—
406C2×C4—181no—
407C2×C4C2×D3×C3⋊D311yescharacteristic
408C8—61no—
409C8—541no—
410C2×C3—24no—
411C2×C3—24no—
412C2×C3—24no—
413C2×C3—128no—
414C2×C3—182no—
415C2×C3D3×#21514yes—
416C2×C3#6393914yes—
417C2×C3#6398314yes—
418C2×C3SD16×C3⋊D311yescharacteristic
419C2×C3#6412311yescharacteristic
420C2×C3#6414611yescharacteristic
421D3—42no—
422D3—68no—
423D3—362no—
424D3—368no—
425D3—728no—
426C22—62no—
427C22—92no—
428C22—121no—
429C22—121no—
430C22—181no—
431C22—181no—
432C22—542no—
433C22—1088no—
434C22—1081no—
435C22—1081no—
436C22C3⋊D3×#6711yescharacteristic
437C4—41no—
438C4—181no—
439C4—361no—
440C4D3×#63811yescharacteristic
441C4#8903911yescharacteristic
442C3—24no—
443C3#1059165114yescomplemented
444C3#1059286811yescharacteristic, complemented
445C2—122no—
446C2—182no—
447C2—1082no—
448C2C3⋊D3×#21511yescharacteristic
449C2#1172357011yescharacteristic
450C2#1172364311yescharacteristic
subgroupG/H
centerC22C3⋊D3×#67
commutator, cosocleC2×C33×C4C23
FrattiniC2×C4C2×D3×C3⋊D3
FittingC2×C33×Q8C22
socleC22×C33C2×D4
prime radicalC2×C3⋊D3×D12C2
hypercenterC2×Q8D3×C3⋊D3
hypocenterC33#450
LCSG ▷ C2×C33×C4 ▷ C2×C33 ▷ C33
UCSC1 ◁ C22 ◁ C2×C4 ◁ C2×Q8

Complements

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

NH ≅ G/Ntype
2. C2×#88862C2semidirect
3. C3×#64155C2semidirect
4. C2×C32×#215C2semidirect
5. #11722036C2semidirect
6. #11722031C2semidirect
7. #11722023C2semidirect
11. C2×C33×Q8C22semidirect
19. C2×C3⋊(C3⋊(C3⋊C8))C22semidirect
21. C3×C4⋊(C3⋊Dic3)C22semidirect
24. C2×C32×C3⋊C8C22semidirect
28. #64155D3semidirect
33. C2×C3×#215D3semidirect
66. C2×C32×Q8C2×D3semidirect
67. C2×C32×Q8C2×D3semidirect
87. C4⋊(C3⋊Dic3)C2×D3semidirect
96. C2×C3×C3⋊C8C2×D3semidirect
103. C2×C3×C3⋊D3SD16semidirect
121. C2×#215C3⋊D3semidirect
188. C2×C3×Q8D32semidirect
189. C2×C3×Q8C2×C3⋊D3semidirect
218. C2×C3⋊C8C2×C3⋊D3semidirect
237. C2×C3⋊D3#215semidirect
265. C33#450semidirect
333. C2×Q8D3×C3⋊D3semidirect
387. C32#5046semidirect
388. C32#5169semidirect
443. C3#10591651semidirect
444. C3#10592868semidirect

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×#888622
C3×#641552
C2×C32×#2152
#117220362
#117220312
#117220232
#105928683
#105916513