Finite Group Databasebeta

← #13151446

Subgroups — #13151446

Order 1728.

1433 conjugacy classes of subgroups; only the 155 normal subgroups are shown.

normal subgroups only

This group has 1433 conjugacy classes of subgroups — too many to draw. Only its 155 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×D4×#272C211yesmaximal, characteristic
2C22×#6199C211yesmaximal, characteristic, complemented
3#64144⋊C3C212yesmaximal, complemented
4#11721320C212yesmaximal, complemented
5#11721320C211yesmaximal, characteristic, complemented
6#11721319C212yesmaximal, complemented
7#11721301C212yesmaximal, complemented
8#11721151C212yesmaximal, complemented
9#11721147C212yesmaximal, complemented
10D3×#935—32nomaximal
11C23×#272C2211yescharacteristic
12C23×#272C2211yescharacteristic
13C2×D4×H(3,𝔽3)C2211yescharacteristic
14C2×#6235C2211yescharacteristic
15C2×#6235C2211yescharacteristic
16D4×#272—24no—
17C2×C4⋊#272C2211yescharacteristic
18C2×C4×#272C2211yescharacteristic
19C22×(C3⋊Dic3)⋊C3C2212yescomplemented
20C2×#6199—24no—
21C2×#6199C2212yes—
22C2×#6199C2212yes—
23C2×#6199C2212yescomplemented
24#3757⋊C3C2212yescomplemented
25#3757⋊C3C2214yescomplemented
26#3750⋊C3C2214yescomplemented
27#88451C2214yescomplemented
28#88451C2212yescomplemented
29#88448C2214yescomplemented
30C2×C3×D3×D4—32no—
31C2×C3×D3×D4—61no—
32C22×D3×Dic3—32no—
33#64144D312yescomplemented
34C3×#935—32no—
35#63969—34no—
36D3×#217—32no—
37D3×#217—34no—
38#63962—32no—
39#63961—32no—
40#63884—34no—
41D3×Dic3⋊C4—34no—
42C23×H(3,𝔽3)C2311yescharacteristic
43C23×H(3,𝔽3)C2311yescharacteristic
44C22×#272—24no—
45C22×#272—22no—
46C22×#272—24no—
47C22×#272D414yes—
48C22×#272C2311yescharacteristic
49C22×#272C2311yescharacteristic
50C22×#272C2311yescharacteristic
51D4×H(3,𝔽3)—22no—
52C2×C4×H(3,𝔽3)C2311yescharacteristic
53#6235—22no—
54#6235—22no—
55C2×#1084C2311yescharacteristic
56C4⋊#272—22no—
57C4×#272—22no—
58C2×(C3⋊Dic3)⋊C3—24no—
59C2×(C3⋊Dic3)⋊C3C2314yes—
60C2×(C3⋊Dic3)⋊C3C2314yes—
61#6199—24no—
62#6199—24no—
63#6199—24no—
64C2×#935—91no—
65D3×#123—92no—
66C23×C3×D3—32no—
67C23×C3×D3—32no—
68C23×C3×D3—61no—
69C23×C3×D3—61no—
70C2×C32×D4—21no—
71C2×C32×D4C2×D312yes—
72C22×C3⋊Dic3C2×D312yescomplemented
73C2×C3×#67—32no—
74C2×C3×#67—32no—
75C2×C3×#67—61no—
76C2×C3×#67—61no—
77C22×C3×Dic3—32no—
78C3×D3×D4—68no—
79C3×D3×D4—68no—
80C2×C3×D12—32no—
81C2×C3×D12—61no—
82C2×C3×C4×D3—32no—
83C2×C3×C4×D3—61no—
84C2×D3×Dic3—34no—
85C2×D3×Dic3—34no—
86C2×D3×Dic3—34no—
87C2×D3×Dic3—68no—
88#3757C2×D312yescomplemented
89#3757C2×D314yescomplemented
90#3750C2×D314yescomplemented
91C3×#217—32no—
92C3×#217—34no—
93C3×Dic3⋊C4—34no—
94#3724—34no—
95Dic3⋊Dic3—34no—
96(C2×D3)⋊Dic3—34no—
97(C2×D3)⋊Dic3—34no—
98(C2×D3)⋊Dic3—34no—
99C22×H(3,𝔽3)—22no—
100C22×H(3,𝔽3)—21no—
101C22×H(3,𝔽3)—22no—
102C22×H(3,𝔽3)C2×D412yes—
103C22×H(3,𝔽3)C2411yescharacteristic
104C2×#272—22no—
105C2×#272—28no—
106C2×#272—21no—
107C2×#272—22no—
108C2×#272—44no—
109C2×#272C2×D412yes—
110C2×#272C2×D412yes—
111C2×#272#4114yes—
112C4×H(3,𝔽3)—21no—
113#1084—21no—
114(C3⋊Dic3)⋊C3—24no—
115(C3⋊Dic3)⋊C3—24no—
116C22×C3×D4—91no—
117C23×Dic3—91no—
118C2×D3×D4—32no—
119C2×D3×D4—61no—
120C22×C4×D3—92no—
121C3×#123—92no—
122C2×#217—91no—
123C2×#217—92no—
124#935—32no—
125#935—92no—
126#935—92no—
127#935—92no—
128#935—92no—
129#929—92no—
130C2×Dic3⋊C4—92no—
131#894—94no—
132D3×C4⋊C4—94no—
133#886—92no—
134#883—94no—
135D3×C22⋊C4—92no—
136D3×C22⋊C4—94no—
137C23×C32—21no—
138C23×C32—21no—
139C23×C32C22×D312yes—
140C23×C32C22×D312yes—
141C22×C3×D3—32no—
142C22×C3×D3—38no—
143C22×C3×D3—32no—
144C22×C3×D3—32no—
145C22×C3×D3—61no—
146C22×C3×D3—64no—
147C22×C3×D3—61no—
148C22×C3×D3—61no—
149C22×C3×D3—68no—
150C22×C3×D3—68no—
151C22×C3×D3—64no—
152C22×C3×D3—64no—
153C22×C3×D3—124no—
154C22×C3×D3—124no—
155C32×D4—24no—
156C32×D4—24no—
157C2×C32×C4—21no—
158C2×C32×C4C22×D312yes—
159C2×C3⋊Dic3—24no—
160C2×C3⋊Dic3C22×D314yes—
161C2×C3⋊Dic3C22×D314yes—
162C3×#67—64no—
163C3×#67—64no—
164C3×#67—64no—
165C3×#67—64no—
166C2×C3×Dic3—34no—
167C2×C3×Dic3—34no—
168C2×C3×Dic3—32no—
169C2×C3×Dic3—61no—
170C2×C3×Dic3—64no—
171C3×D12—64no—
172C3×D12—64no—
173C3×C4×D3—64no—
174C3×C4×D3—64no—
175D3×Dic3—68no—
176D3×Dic3—68no—
177D3×Dic3—68no—
178C2×#123—271noSylow
179C2×H(3,𝔽3)—22no—
180C2×H(3,𝔽3)—41no—
181C2×H(3,𝔽3)C22×D411yescharacteristic
182C2×H(3,𝔽3)C2×#4112yes—
183#272—22no—
184#272—41no—
185#272#12314yescomplemented
186C24×C3—91no—
187C24×C3—91no—
188C23×D3—32no—
189C23×D3—32no—
190C23×D3—61no—
191C23×D3—61no—
192C2×C3×D4—32no—
193C2×C3×D4—61no—
194C2×C3×D4—91no—
195C2×C3×D4—91no—
196C2×C3×D4—91no—
197C2×C3×D4—184no—
198C2×C3×D4D3211yescharacteristic
199C22×C3×C4—92no—
200C22×C3×C4—91no—
201C2×#67—32no—
202C2×#67—32no—
203C2×#67—61no—
204C2×#67—61no—
205C22×Dic3—32no—
206C22×Dic3—92no—
207C22×Dic3—92no—
208C22×Dic3—92no—
209C22×Dic3—92no—
210C22×Dic3—184no—
211D3×D4—68no—
212D3×D4—68no—
213C2×D12—32no—
214C2×D12—61no—
215C2×C4×D3—32no—
216C2×C4×D3—61no—
217C2×C4×D3—94no—
218C2×C4×D3—94no—
219C2×C4×D3—94no—
220C2×C4×D3—188no—
221C3×C4⋊C4—94no—
222C3×C22⋊C4—94no—
223C3×C22⋊C4—92no—
224#217—32no—
225#217—34no—
226#217—94no—
227#217—92no—
228#217—92no—
229#217—94no—
230(C2×D3)⋊C4—94no—
231(C2×D3)⋊C4—94no—
232(C2×D3)⋊C4—94no—
233C4⋊Dic3—94no—
234Dic3⋊C4—34no—
235Dic3⋊C4—94no—
236Dic3⋊C4—94no—
237Dic3⋊C4—94no—
238C22×C32—24no—
239C22×C32—22no—
240C22×C32—22no—
241C22×C32—24no—
242C22×C32—21no—
243C22×C32—22no—
244C22×C32—42no—
245C22×C32—42no—
246C22×C32D3×D414yes—
247C22×C32C23×D312yes—
248C2×C3×D3—34no—
249C2×C3×D3—38no—
250C2×C3×D3—34no—
251C2×C3×D3—62no—
252C2×C3×D3—64no—
253C2×C3×D3—62no—
254C2×C3×D3—616no—
255C2×C3×D3—64no—
256C2×C3×D3—62no—
257C2×C3×D3—62no—
258C2×C3×D3—64no—
259C2×C3×D3—128no—
260C2×C3×D3—122no—
261C2×C3×D3—122no—
262C2×C3×D3—128no—
263C2×C3×D3—128no—
264C32×C4—22no—
265C32×C4—22no—
266C3⋊Dic3—24no—
267C3⋊Dic3—24no—
268C3×Dic3—62no—
269C3×Dic3—64no—
270C3×Dic3—62no—
271C3×Dic3—64no—
272C22×D4—91no—
273C23×C4—271no—
274#123—92no—
275#123—272no—
276#123—272no—
277#123—272no—
278C2×C4⋊C4—272no—
279C2×C22⋊C4—271no—
280C2×C22⋊C4—272no—
281H(3,𝔽3)C2×#12311yescharacteristic, complemented, Sylow
282C23×C3—32no—
283C23×C3—32no—
284C23×C3—61no—
285C23×C3—61no—
286C23×C3—91no—
287C23×C3—91no—
288C23×C3—94no—
289C23×C3—91no—
290C23×C3—184no—
291C23×C3—182no—
292C23×C3—184no—
293C23×C3C2×D3211yescharacteristic
294C23×C3C2×D3211yescharacteristic
295C22×D3—32no—
296C22×D3—32no—
297C22×D3—38no—
298C22×D3—32no—
299C22×D3—61no—
300C22×D3—61no—
301C22×D3—61no—
302C22×D3—64no—
303C22×D3—68no—
304C22×D3—64no—
305C22×D3—68no—
306C22×D3—64no—
307C22×D3—124no—
308C22×D3—124no—
309C3×D4—22no—
310C3×D4—64no—
311C3×D4—64no—
312C3×D4—182no—
313C3×D4—182no—
314C3×D4—182no—
315C2×C3×C4—32no—
316C2×C3×C4—61no—
317C2×C3×C4—94no—
318C2×C3×C4—94no—
319C2×C3×C4—91no—
320C2×C3×C4—182no—
321C2×C3×C4—184no—
322C2×C3×C4C2×D3211yescharacteristic
323#67—64no—
324#67—64no—
325#67—64no—
326#67—64no—
327C2×Dic3—32no—
328C2×Dic3—34no—
329C2×Dic3—34no—
330C2×Dic3—61no—
331C2×Dic3—64no—
332C2×Dic3—94no—
333C2×Dic3—94no—
334C2×Dic3—184no—
335C2×Dic3—184no—
336C2×Dic3—184no—
337C2×Dic3—184no—
338D12—64no—
339D12—64no—
340C4×D3—64no—
341C4×D3—64no—
342C4×D3—188no—
343C4×D3—188no—
344C4×D3—188no—
345C2×C32—24no—
346C2×C32—22no—
347C2×C32—21no—
348C2×C32—42no—
349C2×C32—42no—
350C2×C32—42no—
351C2×C32C2×D3×D412yes—
352C2×C32D3×#4114yes—
353C3×D3—38no—
354C3×D3—64no—
355C3×D3—64no—
356C3×D3—122no—
357C3×D3—122no—
358C3×D3—122no—
359C24—91no—
360C24—91no—
361C2×D4—91no—
362C2×D4—91no—
363C2×D4—91no—
364C2×D4—184no—
365C2×D4((C3⋊D3)⋊C3)⋊C211yescharacteristic
366C22×C4—92no—
367C22×C4—91no—
368C22×C4—272no—
369C22×C4—272no—
370C22×C4—272no—
371C22×C4—544no—
372C4⋊C4—94no—
373C4⋊C4—274no—
374C22⋊C4—92no—
375C22⋊C4—94no—
376C22⋊C4—272no—
377C22⋊C4—274no—
378C22×C3—22no—
379C22×C3—22no—
380C22×C3—21no—
381C22×C3—32no—
382C22×C3—34no—
383C22×C3—64no—
384C22×C3—62no—
385C22×C3—64no—
386C22×C3—62no—
387C22×C3—61no—
388C22×C3—62no—
389C22×C3—94no—
390C22×C3—92no—
391C22×C3—92no—
392C22×C3—122no—
393C22×C3—122no—
394C22×C3—188no—
395C22×C3—182no—
396C22×C3—182no—
397C22×C3—181no—
398C22×C3—364no—
399C22×C3#381112yes—
400C22×C3C22×D3211yescharacteristic
401C2×D3—34no—
402C2×D3—38no—
403C2×D3—34no—
404C2×D3—62no—
405C2×D3—64no—
406C2×D3—62no—
407C2×D3—616no—
408C2×D3—64no—
409C2×D3—62no—
410C2×D3—64no—
411C2×D3—62no—
412C2×D3—128no—
413C2×D3—122no—
414C2×D3—122no—
415C2×D3—128no—
416C2×D3—128no—
417C3×C4—21no—
418C3×C4—62no—
419C3×C4—62no—
420C3×C4—184no—
421C3×C4—184no—
422C3×C4—181no—
423Dic3—62no—
424Dic3—62no—
425Dic3—64no—
426Dic3—64no—
427Dic3—184no—
428Dic3—184no—
429C32—21no—
430C32D3×#12312yescomplemented
431C23—91no—
432C23—91no—
433C23—94no—
434C23—91no—
435C23—184no—
436C23—182no—
437C23—184no—
438C23C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
439C23C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
440D4—22no—
441D4—182no—
442D4—182no—
443D4—182no—
444C2×C4—94no—
445C2×C4—94no—
446C2×C4—91no—
447C2×C4—182no—
448C2×C4—184no—
449C2×C4—544no—
450C2×C4—544no—
451C2×C4—544no—
452C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
453C2×C3—22no—
454C2×C3—34no—
455C2×C3—32no—
456C2×C3—41no—
457C2×C3—64no—
458C2×C3—62no—
459C2×C3—61no—
460C2×C3—94no—
461C2×C3—122no—
462C2×C3—122no—
463C2×C3—122no—
464C2×C3—182no—
465C2×C3—361no—
466C2×C3#6431212yes—
467C2×C3C2×#381111yescharacteristic
468D3—38no—
469D3—64no—
470D3—64no—
471D3—122no—
472D3—122no—
473D3—122no—
474C22—22no—
475C22—22no—
476C22—21no—
477C22—94no—
478C22—92no—
479C22—92no—
480C22—188no—
481C22—182no—
482C22—182no—
483C22—181no—
484C22—364no—
485C22#8867712yes—
486C22C22×((C3⋊D3)⋊C3)⋊C211yescharacteristic
487C4—21no—
488C4—184no—
489C4—184no—
490C4—181no—
491C3—32no—
492C3—61no—
493C3#1059516911yescharacteristic
494C2—22no—
495C2—41no—
496C2—94no—
497C2—182no—
498C2—361no—
499C2#1172250512yes—
500C2C2×#8867711yescharacteristic
subgroupG/H
centerC22C22×((C3⋊D3)⋊C3)⋊C2
commutator, cosocleC22×H(3,𝔽3)C24
Frattini, socleC22×C3C22×D32
FittingC2×D4×H(3,𝔽3)C22
prime radicalC2×D4×#272C2
hypercenterC2×D4((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)C2×#123
LCSG ▷ C22×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C22 ◁ C2×D4

Complements

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

NH ≅ G/Ntype
2. C22×#6199C2semidirect
3. #64144⋊C3C2semidirect
4. #11721320C2semidirect
5. #11721320C2semidirect
6. #11721319C2semidirect
7. #11721301C2semidirect
8. #11721151C2semidirect
9. #11721147C2semidirect
19. C22×(C3⋊Dic3)⋊C3C22semidirect
23. C2×#6199C22semidirect
24. #3757⋊C3C22semidirect
25. #3757⋊C3C22semidirect
26. #3750⋊C3C22semidirect
27. #88451C22semidirect
28. #88451C22semidirect
29. #88448C22semidirect
33. #64144D3semidirect
72. C22×C3⋊Dic3C2×D3semidirect
88. #3757C2×D3semidirect
89. #3757C2×D3semidirect
90. #3750C2×D3semidirect
185. #272#123semidirect
281. H(3,𝔽3)C2×#123semidirect
430. C32D3×#123semidirect

Maximal subgroups

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

subgroupindex
C2×D4×#2722
C22×#61992
#64144⋊C32
#117213202
#117213192
#117213012
#117211512
#117211472
D3×#9353