Finite Group Databasebeta

← #13161196

Subgroups — #13161196

Order 1728.

1572 conjugacy classes of subgroups; only the 170 normal subgroups are shown.

normal subgroups only

This group has 1572 conjugacy classes of subgroups — too many to draw. Only its 170 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×Q8×C3⋊(C3⋊D3)C211yesmaximal, characteristic, complemented
2C2×C3×#3755C211yesmaximal, characteristic, complemented
3#11723532C211yesmaximal, characteristic
4C2×#88796C211yesmaximal, characteristic, complemented
5C32×#835C211yesmaximal, characteristic, complemented
6#11722041C211yesmaximal, characteristic, complemented
7#11722024C211yesmaximal, characteristic, complemented
8#10592505—31nomaximal
9#10591592—34nomaximal
10C2×C33×Q8C2211yescharacteristic
11Q8×C3⋊(C3⋊D3)—22no—
12Q8×C3⋊(C3⋊D3)—22no—
13C2×C4×C3⋊(C3⋊D3)—21no—
14C2×C4×C3⋊(C3⋊D3)C2211yescharacteristic
15C2×#6308—21no—
16C2×#6308C2211yescharacteristic
17C2×C3×C4⋊(C3⋊D3)C2211yescharacteristic
18C2×#6292—21no—
19C3×#3755—22no—
20C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
21C32×C4⋊Dic3C2211yescharacteristic
22C2×C32×C3⋊C8C2211yescharacteristic
23#88803—21no—
24#88796—22no—
25C2×Q8×C3⋊D3—31no—
26C2×Q8×C3⋊D3—34no—
27C2×Q8×C3⋊D3—64no—
28C2×#3755D311yescharacteristic, complemented
29C2×#3744—31no—
30#64104—31no—
31C2×C3×#215—34no—
32#63916—34no—
33C2×#3716—34no—
34C32×#103—31no—
35#63647—31no—
36#63637—31no—
37C3×#835D314yescomplemented
38#63567—34no—
39#63557—34no—
40C22×C3⋊(C3⋊D3)D411yescharacteristic
41C22×C3×C3⋊D3—21no—
42C33×Q8—21no—
43C33×Q8D412yescomplemented
44C2×C33×C4—21no—
45C2×C33×C4C2311yescharacteristic
46C2×C3⋊(C3⋊Dic3)—21no—
47C2×C3⋊(C3⋊Dic3)D411yescharacteristic
48C4×C3⋊(C3⋊D3)—24no—
49C4×C3⋊(C3⋊D3)—21no—
50C4×C3⋊(C3⋊D3)—21no—
51#6308—24no—
52#6308—21no—
53#6308D412yescomplemented
54C3×C4⋊(C3⋊D3)—21no—
55C3×C4⋊(C3⋊D3)—21no—
56C2×C32×Dic3—21no—
57#6292—42no—
58C3×C3⋊(C3⋊C8)—21no—
59C32×C3⋊C8—21no—
60#5170—91no—
61#4806—94no—
62C2×C32×Q8—24no—
63C2×C32×Q8C2×D314yes—
64C2×C32×Q8C2×D311yescharacteristic
65Q8×C3⋊D3—62no—
66Q8×C3⋊D3—68no—
67Q8×C3⋊D3—616no—
68Q8×C3⋊D3—616no—
69Q8×C3⋊D3—68no—
70Q8×C3⋊D3—62no—
71C2×C4⋊(C3⋊D3)—61no—
72C2×C4⋊(C3⋊D3)C2×D311yescharacteristic
73C2×C4×C3⋊D3—31no—
74C2×C4×C3⋊D3—34no—
75C2×C4×C3⋊D3—61no—
76C2×C4×C3⋊D3—64no—
77C2×C4×C3⋊D3—64no—
78C2×C4×C3⋊D3—68no—
79C2×#634—31no—
80C2×#634—34no—
81C2×#634—61no—
82C2×#634—64no—
83C2×#634—64no—
84C2×#634—68no—
85C2×C3×D12—34no—
86C2×#626—64no—
87C2×C32×C8—31no—
88C32×C4⋊C4—31no—
89#3755—22no—
90#3752—61no—
91C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
92#3744—62no—
93C3×#215—68no—
94C3×C4⋊Dic3C2×D314yes—
95C2×C3×C3⋊C8—34no—
96C2×C3×C3⋊C8C2×D314yes—
97(C2×D3)⋊Dic3—64no—
98#3716—68no—
99C22×C33C2×D411yescharacteristic
100C2×C3⋊(C3⋊D3)—21no—
101C2×C3⋊(C3⋊D3)—21no—
102C2×C3⋊(C3⋊D3)SD1612yescomplemented
103C2×C3×C3⋊D3—41no—
104C2×C3×C3⋊D3—41no—
105C2×C3×C3⋊D3—41no—
106C33×C4—22no—
107C33×C4C2×D411yescharacteristic
108C33×C4C2×D411yescharacteristic
109C3⋊(C3⋊Dic3)—21no—
110C3⋊(C3⋊Dic3)—22no—
111C32×Dic3—41no—
112C2×D3×Q8—94no—
113C2×D3×Q8—91no—
114C2×D3×Q8—184no—
115C2×C3×SD16—91no—
116C2×#215—34no—
117#938—91no—
118C2×#214—91no—
119C2×#204—94no—
120#895—94no—
121C3×#103—34no—
122#835C3⋊D311yescharacteristic, complemented
123(C2×D3)⋊C8—94no—
124(C2×D3)⋊C8—91no—
125#816—91no—
126#810—94no—
127C22×C3⋊D3—21no—
128C22×C3⋊D3—31no—
129C22×C3⋊D3—34no—
130C22×C3⋊D3—64no—
131C22×C3×D3—64no—
132C32×Q8—28no—
133C32×Q8—24no—
134C32×Q8—21no—
135C32×Q8—28no—
136C32×Q8D1218yescomplemented
137C32×Q8#6712yescomplemented
138C2×C32×C4—28no—
139C2×C32×C4—24no—
140C2×C32×C4—21no—
141C2×C32×C4—24no—
142C2×C32×C4—61no—
143C2×C32×C4C22×D314yes—
144C2×C32×C4C22×D311yescharacteristic
145C2×C3⋊Dic3—31no—
146C2×C3⋊Dic3—34no—
147C2×C3⋊Dic3—64no—
148C2×C3⋊Dic3—68no—
149C2×C3⋊Dic3—64no—
150C2×C3⋊Dic3—61no—
151C4⋊(C3⋊D3)—21no—
152C4⋊(C3⋊D3)—21no—
153C4⋊(C3⋊D3)—122no—
154C4×C3⋊D3—632no—
155C4×C3⋊D3—616no—
156C4×C3⋊D3—64no—
157C4×C3⋊D3—68no—
158C4×C3⋊D3—68no—
159C4×C3⋊D3—64no—
160C4×C3⋊D3—64no—
161C4×C3⋊D3—61no—
162C4×C3⋊D3—61no—
163#634—32no—
164#634—38no—
165#634—68no—
166#634—632no—
167#634—616no—
168#634—64no—
169#634—68no—
170#634—64no—
171#634—61no—
172C2×C3×Dic3—24no—
173C3×D12—64no—
174C3×D12—64no—
175#626—128no—
176C32×C8—61no—
177C3⋊(C3⋊C8)—21no—
178C3×C3⋊C8—24no—
179C3×C3⋊C8—64no—
180#448—271noSylow
181C2×C33#12011yescharacteristic
182C2×C33C2×SD1611yescharacteristic
183C2×C33#13711yescharacteristic
184C3⋊(C3⋊D3)—22no—
185C3×C3⋊D3—81no—
186C2×C3×Q8—24no—
187C2×C3×Q8D3214yes—
188C2×C3×Q8C2×C3⋊D311yescharacteristic
189C2×C3×D4—91no—
190C2×#67—181no—
191D3×Q8—188no—
192D3×Q8—182no—
193D3×Q8—1816no—
194D3×Q8—1816no—
195D3×Q8—188no—
196D3×Q8—182no—
197C2×D12—34no—
198C2×D12—184no—
199C2×C4×D3—94no—
200C2×C4×D3—91no—
201C2×C4×D3—184no—
202C2×C4×D3—184no—
203C2×C4×D3—181no—
204C2×C4×D3—188no—
205C2×Dic6—94no—
206C2×Dic6—91no—
207C2×Dic6—184no—
208C2×Dic6—184no—
209C2×Dic6—181no—
210C2×Dic6—188no—
211C3×SD16—182no—
212C2×C3×C8—34no—
213C2×C3×C8—91no—
214C3×C4⋊C4—34no—
215#217—181no—
216#215—68no—
217#214—182no—
218(C2×D3)⋊C4—184no—
219C4⋊Dic3C2×C3⋊D311yescharacteristic
220C2×C3⋊C8—34no—
221C2×C3⋊C8C2×C3⋊D311yescharacteristic
222#204—188no—
223C22×C32—24no—
224C22×C32D3×D414yes—
225C22×C32D3×D411yescharacteristic
226C2×C3⋊D3—32no—
227C2×C3⋊D3—38no—
228C2×C3⋊D3—41no—
229C2×C3⋊D3—41no—
230C2×C3⋊D3—41no—
231C2×C3⋊D3—68no—
232C2×C3⋊D3—68no—
233C2×C3⋊D3—68no—
234C2×C3⋊D3—64no—
235C2×C3⋊D3—61no—
236C2×C3⋊D3—64no—
237C2×C3⋊D3—61no—
238C2×C3×D3—124no—
239C2×C3×D3—124no—
240C2×C3×D3—124no—
241C32×C4—216no—
242C32×C4—28no—
243C32×C4—22no—
244C32×C4—24no—
245C32×C4—24no—
246C32×C4—121no—
247C32×C4C2×D1214yes—
248C32×C4D3×D414yes—
249C32×C4D3×D411yescharacteristic
250C32×C4C2×#6711yescharacteristic
251C3⋊Dic3—616no—
252C3⋊Dic3—68no—
253C3⋊Dic3—68no—
254C3⋊Dic3—64no—
255C3⋊Dic3—62no—
256C3⋊Dic3—61no—
257C3×Dic3—44no—
258C22×Q8—271no—
259C2×SD16—91no—
260C2×SD16—271no—
261#121—271no—
262#103—31no—
263#103—271no—
264C22⋊C8—271no—
265C33#44811yescharacteristic, complemented, Sylow
266C23×C3—181no—
267C22×D3—64no—
268C22×D3—94no—
269C22×D3—91no—
270C22×D3—184no—
271C3×Q8—28no—
272C3×Q8—24no—
273C3×Q8—21no—
274C3×Q8—28no—
275C3×Q8#62618yescomplemented
276C3×Q8C4⋊(C3⋊D3)12yescomplemented
277C3×D4—181no—
278C3×D4—181no—
279C2×C3×C4—28no—
280C2×C3×C4—24no—
281C2×C3×C4—21no—
282C2×C3×C4—24no—
283C2×C3×C4—64no—
284C2×C3×C4C2×D3214yes—
285C2×C3×C4C22×C3⋊D311yescharacteristic
286#67—362no—
287C2×Dic3—21no—
288C2×Dic3—94no—
289C2×Dic3—91no—
290C2×Dic3—184no—
291C2×Dic3—188no—
292C2×Dic3—184no—
293C2×Dic3—181no—
294D12—64no—
295D12—64no—
296D12—368no—
297C4×D3—1832no—
298C4×D3—184no—
299C4×D3—1816no—
300C4×D3—188no—
301C4×D3—188no—
302C4×D3—184no—
303C4×D3—184no—
304C4×D3—181no—
305C4×D3—181no—
306Dic6—98no—
307Dic6—92no—
308Dic6—188no—
309Dic6—1832no—
310Dic6—184no—
311Dic6—1816no—
312Dic6—188no—
313Dic6—184no—
314Dic6—181no—
315C3×C8—64no—
316C3×C8—181no—
317C3⋊C8—21no—
318C3⋊C8—64no—
319C2×C32—24no—
320C2×C32—24no—
321C2×C32—24no—
322C2×C32#88214yes—
323C2×C32D3×SD1614yes—
324C2×C32D3×SD1611yescharacteristic
325C2×C32#91511yescharacteristic
326C2×C32#91814yes—
327C2×C32#93711yescharacteristic
328C3⋊D3—616no—
329C3⋊D3—62no—
330C3⋊D3—68no—
331C3⋊D3—81no—
332C3×D3—244no—
333C2×Q8—271no—
334C2×Q8—541no—
335C2×Q8—542no—
336C2×Q8—542no—
337C2×Q8D3×C3⋊D311yescharacteristic
338C2×D4—91no—
339C2×D4—541no—
340C22×C4—271no—
341C22×C4—541no—
342SD16—182no—
343SD16—542no—
344C2×C8—31no—
345C2×C8—91no—
346C4⋊C4—31no—
347C22⋊C4—541no—
348C22×C3—24no—
349C22×C3—361no—
350C22×C3—361no—
351C22×C3—361no—
352C22×C3#381114yes—
353C22×C3D4×C3⋊D311yescharacteristic
354C2×D3—92no—
355C2×D3—98no—
356C2×D3—124no—
357C2×D3—124no—
358C2×D3—124no—
359C2×D3—188no—
360C2×D3—188no—
361C2×D3—188no—
362C2×D3—181no—
363C2×D3—181no—
364C2×D3—184no—
365C2×D3—184no—
366C3×C4—216no—
367C3×C4—22no—
368C3×C4—28no—
369C3×C4—24no—
370C3×C4—24no—
371C3×C4—124no—
372C3×C4C2×#62614yes—
373C3×C4#381114yes—
374C3×C4C2×C4⋊(C3⋊D3)11yescharacteristic
375C3×C4D4×C3⋊D311yescharacteristic
376Dic3—41no—
377Dic3—188no—
378Dic3—1816no—
379Dic3—182no—
380Dic3—181no—
381Dic3—188no—
382Dic3—184no—
383C32—24no—
384C32#480614yescomplemented
385C32#517011yescharacteristic, complemented
386C23—181no—
387C23—271no—
388Q8—21no—
389Q8—272no—
390Q8—544no—
391Q8—541no—
392Q8#629212yescomplemented
393D4—181no—
394D4—181no—
395D4—1082no—
396C2×C4—21no—
397C2×C4—61no—
398C2×C4—271no—
399C2×C4—544no—
400C2×C4—541no—
401C2×C4—541no—
402C2×C4—541no—
403C2×C4C2×D3×C3⋊D311yescharacteristic
404C8—61no—
405C8—181no—
406C2×C3—24no—
407C2×C3—24no—
408C2×C3—24no—
409C2×C3—721no—
410C2×C3#6394014yes—
411C2×C3#6395114yes—
412C2×C3#6398114yes—
413C2×C3#6409111yescharacteristic
414C2×C3SD16×C3⋊D311yescharacteristic
415C2×C3#6412711yescharacteristic
416D3—1816no—
417D3—188no—
418D3—182no—
419D3—244no—
420C22—272no—
421C22—361no—
422C22—361no—
423C22—361no—
424C22—541no—
425C22—541no—
426C22#8903911yescharacteristic
427C4—22no—
428C4—121no—
429C4—541no—
430C4—542no—
431C4C2×#629211yescharacteristic
432C4#8903911yescharacteristic
433C3—24no—
434C3#1059159214yescomplemented
435C3#1059250511yescharacteristic, complemented
436C2—542no—
437C2—721no—
438C2#1172357111yescharacteristic
439C2#1172359011yescharacteristic
440C2#1172363911yescharacteristic
subgroupG/H
centerC22#89039
commutator, cosocleC2×C33×C4C23
FrattiniC2×C4C2×D3×C3⋊D3
FittingC2×C33×Q8C22
socleC22×C33C2×D4
prime radical#11723532C2
hypercenterC2×Q8D3×C3⋊D3
hypocenterC33#448
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
1. C2×Q8×C3⋊(C3⋊D3)C2semidirect
2. C2×C3×#3755C2semidirect
4. C2×#88796C2semidirect
5. C32×#835C2semidirect
6. #11722041C2semidirect
7. #11722024C2semidirect
28. C2×#3755D3semidirect
37. C3×#835D3semidirect
43. C33×Q8D4semidirect
53. #6308D4semidirect
102. C2×C3⋊(C3⋊D3)SD16semidirect
122. #835C3⋊D3semidirect
136. C32×Q8D12semidirect
137. C32×Q8#67semidirect
265. C33#448semidirect
275. C3×Q8#626semidirect
276. C3×Q8C4⋊(C3⋊D3)semidirect
384. C32#4806semidirect
385. C32#5170semidirect
392. Q8#6292semidirect
434. C3#10591592semidirect
435. C3#10592505semidirect

Maximal subgroups

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

subgroupindex
C2×Q8×C3⋊(C3⋊D3)2
C2×C3×#37552
#117235322
C2×#887962
C32×#8352
#117220412
#117220242
#105925053
#105915923