Finite Group Databasebeta

← #13161215

Subgroups — #13161215

Order 1728.

1512 conjugacy classes of subgroups; only the 170 normal subgroups are shown.

normal subgroups only

This group has 1512 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
2C2×C3×#3756C211yesmaximal, characteristic, complemented
3#11723449C211yesmaximal, characteristic
4C2×#88800C211yesmaximal, characteristic, complemented
5C32×#835C211yesmaximal, characteristic, complemented
6#11722041C211yesmaximal, characteristic, complemented
7#11722024C211yesmaximal, characteristic
8#10592508—31nomaximal
9#10591603—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×#634C2211yescharacteristic
18C3×#3756—22no—
19C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
20C32×C4⋊Dic3C2211yescharacteristic
21C2×C32×C3⋊C8C2211yescharacteristic
22Dic3⋊(C3⋊Dic3)—21no—
23#88804—21no—
24#88800—22no—
25C2×Q8×C3⋊D3—31no—
26C2×Q8×C3⋊D3—34no—
27C2×Q8×C3⋊D3—64no—
28C2×#3756D311yescharacteristic, complemented
29C2×#3746—31no—
30#64106—31no—
31C2×C3×#216—34no—
32#63871—34no—
33C2×#3719—34no—
34C32×#103—31no—
35#63647—31no—
36#63637—31no—
37C3×#835D314yescomplemented
38#63567—34no—
39#63557—34no—
40C22×C3⋊(C3⋊D3)D411yescharacteristic
41C33×Q8—21no—
42C33×Q8D412yes—
43C2×C33×C4—21no—
44C2×C33×C4C2311yescharacteristic
45C2×C3⋊(C3⋊Dic3)—21no—
46C2×C3⋊(C3⋊Dic3)D411yescharacteristic
47C4×C3⋊(C3⋊D3)—24no—
48C4×C3⋊(C3⋊D3)—21no—
49C4×C3⋊(C3⋊D3)—21no—
50#6308—24no—
51#6308—21no—
52#6308D412yes—
53C2×C3×C3⋊Dic3—21no—
54C3×#634—21no—
55C3×#634—21no—
56C2×C32×Dic3—21no—
57C3×C3⋊(C3⋊C8)—21no—
58C32×C3⋊C8—21no—
59#5186—91no—
60#4809—94no—
61C2×C32×Q8—24no—
62C2×C32×Q8C2×D314yes—
63C2×C32×Q8C2×D311yescharacteristic
64Q8×C3⋊D3—62no—
65Q8×C3⋊D3—68no—
66Q8×C3⋊D3—616no—
67Q8×C3⋊D3—616no—
68Q8×C3⋊D3—68no—
69Q8×C3⋊D3—62no—
70C2×C4×C3⋊D3—31no—
71C2×C4×C3⋊D3—34no—
72C2×C4×C3⋊D3—61no—
73C2×C4×C3⋊D3—64no—
74C2×C4×C3⋊D3—64no—
75C2×C4×C3⋊D3—68no—
76C2×#634—31no—
77C2×#634—34no—
78C2×#634—61no—
79C2×#634—64no—
80C2×#634—64no—
81C2×#634—68no—
82C2×#634C2×D311yescharacteristic
83C2×C3×Dic6—34no—
84C2×C32×C8—31no—
85C32×C4⋊C4—31no—
86#3756—22no—
87#3752—61no—
88C4⋊(C3⋊Dic3)—61no—
89C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
90#3746—62no—
91C3×#216—68no—
92C3×C4⋊Dic3C2×D314yes—
93C2×C3×C3⋊C8—34no—
94C2×C3×C3⋊C8C2×D314yes—
95Dic3⋊Dic3—64no—
96#3722—64no—
97#3719—68no—
98C22×C33C2×D411yescharacteristic
99C2×C3⋊(C3⋊D3)—21no—
100C2×C3⋊(C3⋊D3)—21no—
101C2×C3⋊(C3⋊D3)Q1612yescomplemented
102C33×C4—22no—
103C33×C4C2×D411yescharacteristic
104C33×C4C2×D411yescharacteristic
105C3⋊(C3⋊Dic3)—21no—
106C3⋊(C3⋊Dic3)—22no—
107C3×C3⋊Dic3—41no—
108C32×Dic3—41no—
109C2×D3×Q8—94no—
110C2×D3×Q8—91no—
111C2×D3×Q8—184no—
112C2×C3×Q16—91no—
113#946—91no—
114C2×#216—34no—
115C2×#216—91no—
116C2×Dic12—94no—
117#897—94no—
118C3×#103—34no—
119#835C3⋊D311yescharacteristic, complemented
120(C2×D3)⋊C8—94no—
121(C2×D3)⋊C8—91no—
122#816—91no—
123#810—94no—
124C22×C3⋊D3—31no—
125C22×C3⋊D3—34no—
126C22×C3⋊D3—64no—
127C32×Q8—28no—
128C32×Q8—24no—
129C32×Q8—21no—
130C32×Q8—28no—
131C32×Q8D1218yes—
132C32×Q8#6712yes—
133C2×C32×C4—28no—
134C2×C32×C4—24no—
135C2×C32×C4—21no—
136C2×C32×C4—24no—
137C2×C32×C4—61no—
138C2×C32×C4C22×D314yes—
139C2×C32×C4C22×D311yescharacteristic
140C2×C3⋊Dic3—21no—
141C2×C3⋊Dic3—31no—
142C2×C3⋊Dic3—34no—
143C2×C3⋊Dic3—64no—
144C2×C3⋊Dic3—68no—
145C2×C3⋊Dic3—64no—
146C2×C3⋊Dic3—61no—
147C4×C3⋊D3—632no—
148C4×C3⋊D3—616no—
149C4×C3⋊D3—64no—
150C4×C3⋊D3—68no—
151C4×C3⋊D3—68no—
152C4×C3⋊D3—64no—
153C4×C3⋊D3—64no—
154C4×C3⋊D3—61no—
155C4×C3⋊D3—61no—
156#634—21no—
157#634—21no—
158#634—32no—
159#634—38no—
160#634—68no—
161#634—632no—
162#634—616no—
163#634—64no—
164#634—68no—
165#634—64no—
166#634—61no—
167C2×C3×Dic3—24no—
168C2×C3×Dic3—64no—
169C3×Dic6—64no—
170C3×Dic6—64no—
171C32×C8—61no—
172C3⋊(C3⋊C8)—21no—
173C3×C3⋊C8—24no—
174C3×C3⋊C8—64no—
175#451—271noSylow
176C2×C33#12011yescharacteristic
177C2×C33C2×Q1611yescharacteristic
178C2×C33#13711yescharacteristic
179C3⋊(C3⋊D3)—22no—
180C2×C3×Q8—24no—
181C2×C3×Q8—91no—
182C2×C3×Q8D3214yes—
183C2×C3×Q8C2×C3⋊D311yescharacteristic
184D3×Q8—188no—
185D3×Q8—182no—
186D3×Q8—1816no—
187D3×Q8—1816no—
188D3×Q8—188no—
189D3×Q8—182no—
190C2×C4×D3—94no—
191C2×C4×D3—91no—
192C2×C4×D3—184no—
193C2×C4×D3—184no—
194C2×C4×D3—181no—
195C2×C4×D3—188no—
196C2×Dic6—34no—
197C2×Dic6—94no—
198C2×Dic6—91no—
199C2×Dic6—184no—
200C2×Dic6—184no—
201C2×Dic6—181no—
202C2×Dic6—188no—
203C3×Q16—182no—
204C2×C3×C8—34no—
205C2×C3×C8—91no—
206C3×C4⋊C4—34no—
207#216—68no—
208#216—182no—
209(C2×D3)⋊C4—181no—
210(C2×D3)⋊C4—184no—
211C4⋊Dic3—184no—
212C4⋊Dic3C2×C3⋊D311yescharacteristic
213Dic3⋊C4—181no—
214C2×C3⋊C8—34no—
215C2×C3⋊C8C2×C3⋊D311yescharacteristic
216Dic12—188no—
217C22×C32—24no—
218C22×C32D3×D414yes—
219C22×C32D3×D411yescharacteristic
220C2×C3⋊D3—32no—
221C2×C3⋊D3—38no—
222C2×C3⋊D3—68no—
223C2×C3⋊D3—68no—
224C2×C3⋊D3—68no—
225C2×C3⋊D3—64no—
226C2×C3⋊D3—64no—
227C2×C3⋊D3—61no—
228C2×C3⋊D3—61no—
229C32×C4—216no—
230C32×C4—28no—
231C32×C4—22no—
232C32×C4—24no—
233C32×C4—24no—
234C32×C4—121no—
235C32×C4C2×D1214yes—
236C32×C4D3×D414yes—
237C32×C4D3×D411yescharacteristic
238C32×C4C2×#6711yescharacteristic
239C3⋊Dic3—41no—
240C3⋊Dic3—616no—
241C3⋊Dic3—68no—
242C3⋊Dic3—68no—
243C3⋊Dic3—64no—
244C3⋊Dic3—62no—
245C3⋊Dic3—61no—
246C3×Dic3—44no—
247C3×Dic3—124no—
248C22×Q8—271no—
249C2×Q16—91no—
250C2×Q16—271no—
251C22⋊Q8—271no—
252#103—31no—
253#103—271no—
254C22⋊C8—271no—
255C33#45111yescharacteristic, complemented, Sylow
256C22×D3—94no—
257C22×D3—91no—
258C22×D3—184no—
259C3×Q8—28no—
260C3×Q8—24no—
261C3×Q8—21no—
262C3×Q8—28no—
263C3×Q8—181no—
264C3×Q8—181no—
265C3×Q8#62618yes—
266C3×Q8C4⋊(C3⋊D3)12yes—
267C2×C3×C4—28no—
268C2×C3×C4—24no—
269C2×C3×C4—21no—
270C2×C3×C4—24no—
271C2×C3×C4—64no—
272C2×C3×C4—181no—
273C2×C3×C4C2×D3214yes—
274C2×C3×C4C22×C3⋊D311yescharacteristic
275C2×Dic3—21no—
276C2×Dic3—64no—
277C2×Dic3—94no—
278C2×Dic3—91no—
279C2×Dic3—184no—
280C2×Dic3—188no—
281C2×Dic3—184no—
282C2×Dic3—181no—
283C4×D3—1832no—
284C4×D3—184no—
285C4×D3—1816no—
286C4×D3—188no—
287C4×D3—188no—
288C4×D3—184no—
289C4×D3—184no—
290C4×D3—181no—
291C4×D3—181no—
292Dic6—64no—
293Dic6—64no—
294Dic6—98no—
295Dic6—92no—
296Dic6—188no—
297Dic6—1832no—
298Dic6—184no—
299Dic6—1816no—
300Dic6—188no—
301Dic6—184no—
302Dic6—181no—
303C3×C8—64no—
304C3×C8—181no—
305C3⋊C8—21no—
306C3⋊C8—64no—
307C2×C32—24no—
308C2×C32—24no—
309C2×C32—24no—
310C2×C32#88214yes—
311C2×C32#91514yes—
312C2×C32D3×Q1614yes—
313C2×C32D3×Q1611yescharacteristic
314C2×C32#91811yescharacteristic
315C2×C32#93711yescharacteristic
316C3⋊D3—616no—
317C3⋊D3—62no—
318C3⋊D3—68no—
319C2×Q8—91no—
320C2×Q8—271no—
321C2×Q8—541no—
322C2×Q8—542no—
323C2×Q8—542no—
324C2×Q8D3×C3⋊D311yescharacteristic
325C22×C4—271no—
326C22×C4—541no—
327Q16—182no—
328Q16—542no—
329C2×C8—31no—
330C2×C8—91no—
331C4⋊C4—31no—
332C4⋊C4—541no—
333C22⋊C4—541no—
334C22×C3—24no—
335C22×C3#381114yes—
336C22×C3D4×C3⋊D311yescharacteristic
337C2×D3—92no—
338C2×D3—98no—
339C2×D3—188no—
340C2×D3—188no—
341C2×D3—188no—
342C2×D3—181no—
343C2×D3—181no—
344C2×D3—184no—
345C2×D3—184no—
346C3×C4—216no—
347C3×C4—22no—
348C3×C4—28no—
349C3×C4—24no—
350C3×C4—24no—
351C3×C4—124no—
352C3×C4—361no—
353C3×C4C2×#62614yes—
354C3×C4#381114yes—
355C3×C4C2×C4⋊(C3⋊D3)11yescharacteristic
356C3×C4D4×C3⋊D311yescharacteristic
357Dic3—41no—
358Dic3—124no—
359Dic3—188no—
360Dic3—1816no—
361Dic3—182no—
362Dic3—181no—
363Dic3—188no—
364Dic3—184no—
365C32—24no—
366C32#480914yescomplemented
367C32#518611yescharacteristic, complemented
368C23—271no—
369Q8—21no—
370Q8—181no—
371Q8—181no—
372Q8—272no—
373Q8—544no—
374Q8—541no—
375Q8#629212yes—
376C2×C4—21no—
377C2×C4—61no—
378C2×C4—181no—
379C2×C4—271no—
380C2×C4—544no—
381C2×C4—541no—
382C2×C4—541no—
383C2×C4—541no—
384C2×C4C2×D3×C3⋊D311yescharacteristic
385C8—61no—
386C8—181no—
387C2×C3—24no—
388C2×C3—24no—
389C2×C3—24no—
390C2×C3#6394414yes—
391C2×C3#6395114yes—
392C2×C3#6398114yes—
393C2×C3#6409111yescharacteristic
394C2×C3#6412411yescharacteristic
395C2×C3Q16×C3⋊D311yescharacteristic
396D3—1816no—
397D3—188no—
398D3—182no—
399C22—272no—
400C22—541no—
401C22—541no—
402C22#8903911yescharacteristic
403C4—22no—
404C4—121no—
405C4—361no—
406C4—541no—
407C4—542no—
408C4C2×#629211yescharacteristic
409C4#8903911yescharacteristic
410C3—24no—
411C3#1059160314yescomplemented
412C3#1059250811yescharacteristic, complemented
413C2—542no—
414C2#1172357711yescharacteristic
415C2#1172358811yescharacteristic
416C2#1172363911yescharacteristic
subgroupG/H
centerC22#89039
commutator, cosocleC2×C33×C4C23
FrattiniC2×C4C2×D3×C3⋊D3
FittingC2×C33×Q8C22
socleC22×C33C2×D4
prime radicalC22×C3⋊(C3⋊D3)D4
hypercenterC2×Q8D3×C3⋊D3
hypocenterC33#451
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×C3×#3756C2semidirect
4. C2×#88800C2semidirect
5. C32×#835C2semidirect
6. #11722041C2semidirect
28. C2×#3756D3semidirect
37. C3×#835D3semidirect
101. C2×C3⋊(C3⋊D3)Q16semidirect
119. #835C3⋊D3semidirect
255. C33#451semidirect
366. C32#4809semidirect
367. C32#5186semidirect
411. C3#10591603semidirect
412. C3#10592508semidirect

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×#37562
#117234492
C2×#888002
C32×#8352
#117220412
#117220242
#105925083
#105916033