Finite Group Databasebeta

← #13140564

Subgroups — #13140564

Order 1728.

715 conjugacy classes of subgroups; only the 107 normal subgroups are shown.

normal subgroups only

This group has 715 conjugacy classes of subgroups — too many to draw. Only its 107 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×(C3⋊D3)⋊C3C211yesmaximal, characteristic
2#11721081C211yesmaximal, characteristic, complemented
3C2×#88430C211yesmaximal, characteristic, complemented
4#11720578C211yesmaximal, characteristic, complemented
5#63659⋊C3C211yesmaximal, characteristic, complemented
6#11720318C211yesmaximal, characteristic, complemented
7#11720308C211yesmaximal, characteristic, complemented
8#10591426—31nomaximal
9#10591421—31nomaximal
10C23×(C3⋊D3)⋊C3—21no—
11C2×D4×H(3,𝔽3)C2211yescharacteristic
12C2×#638⋊C3—21no—
13D4×(C3⋊D3)⋊C3—22no—
14D4×(C3⋊D3)⋊C3C414yescomplemented
15C2×C4⋊((C3⋊D3)⋊C3)C2211yescharacteristic
16C2×C4×(C3⋊D3)⋊C3C2211yescharacteristic
17C4⋊#1084C2211yescharacteristic, complemented
18C2×#6180C2211yescharacteristic, complemented
19C4⋊((C3⋊Dic3)⋊C3)C2211yescharacteristic, complemented
20C2×(C3⋊(C3⋊C8))⋊C3C2211yescharacteristic, complemented
21#88450—21no—
22#88447—21no—
23#88430—22no—
24C2×D4×C3⋊D3D311yescharacteristic
25C2×C3×D3×D4—31no—
26#63849—31no—
27#63841—31no—
28C2×#3712—31no—
29C2×#3711—31no—
30#63659D311yescharacteristic, complemented
31C3×#832—31no—
32C3×#832—31no—
33C3×#832—31no—
34C3×#832—61no—
35#63564—31no—
36#63564—31no—
37#63563—31no—
38#63562—31no—
39C23×H(3,𝔽3)—21no—
40C22×(C3⋊D3)⋊C3—21no—
41C22×(C3⋊D3)⋊C3—24no—
42C22×(C3⋊D3)⋊C3—42no—
43C22×(C3⋊D3)⋊C3—42no—
44C22×(C3⋊D3)⋊C3D411yescharacteristic
45D4×H(3,𝔽3)—21no—
46D4×H(3,𝔽3)C2×C412yes—
47C2×C4×H(3,𝔽3)C2311yescharacteristic
48C2×#1084—21no—
49#638⋊C3—24no—
50C2×(C3⋊Dic3)⋊C3—21no—
51C2×(C3⋊Dic3)⋊C3D411yescharacteristic
52C4⋊((C3⋊D3)⋊C3)—21no—
53C4⋊((C3⋊D3)⋊C3)C2×C412yes—
54C4×(C3⋊D3)⋊C3C2×C412yes—
55C4×(C3⋊D3)⋊C3D412yes—
56#6180—21no—
57(C3⋊(C3⋊C8))⋊C3—21no—
58#5215—91no—
59#4770—91no—
60#4770—91no—
61C23×C3⋊D3—21no—
62C23×C3×D3—61no—
63C2×C32×D4—21no—
64C2×C32×D4C2×D311yescharacteristic
65C2×C32×D4C2×D311yescharacteristic
66C2×#638—21no—
67D4×C3⋊D3—22no—
68D4×C3⋊D3Dic314yescomplemented
69C2×C4⋊(C3⋊D3)C2×D311yescharacteristic
70C2×C4×C3⋊D3C2×D311yescharacteristic
71C2×C3×#67—61no—
72C3×D3×D4—34no—
73C3×D3×D4—62no—
74C2×C3×D12—31no—
75C2×C3×C4×D3—31no—
76C4⋊(C3⋊Dic3)C2×D311yescharacteristic, complemented
77C2×C3⋊(C3⋊C8)C2×D311yescharacteristic, complemented
78C3×C4⋊Dic3—31no—
79C3×C4⋊Dic3—31no—
80C3×C4⋊Dic3—31no—
81C3×C4⋊Dic3—61no—
82C2×C3×C3⋊C8—31no—
83C2×C3×C3⋊C8—31no—
84C2×C3×C3⋊C8—31no—
85C2×C3×C3⋊C8—61no—
86#3722—61no—
87(C2×D3)⋊Dic3—61no—
88Dic32—61no—
89Dic32—61no—
90#3712—62no—
91#3711—62no—
92C22×H(3,𝔽3)—22no—
93C22×H(3,𝔽3)—41no—
94C22×H(3,𝔽3)—41no—
95C22×H(3,𝔽3)C2×D411yescharacteristic
96C2×(C3⋊D3)⋊C3—21no—
97C2×(C3⋊D3)⋊C3—21no—
98C2×(C3⋊D3)⋊C3—22no—
99C2×(C3⋊D3)⋊C3—48no—
100C2×(C3⋊D3)⋊C3—41no—
101C2×(C3⋊D3)⋊C3—41no—
102C2×(C3⋊D3)⋊C3C22⋊C412yes—
103C4×H(3,𝔽3)C22×C411yescharacteristic
104C4×H(3,𝔽3)C2×D411yescharacteristic
105#1084—41no—
106(C3⋊Dic3)⋊C3—41no—
107(C3⋊Dic3)⋊C3C22⋊C412yes—
108C22×C3×D4—91no—
109C2×D3×D4—31no—
110C2×D3×D4—91no—
111#926—91no—
112C2×#208—91no—
113C2×ΓL(1,𝔽25)—91no—
114C2×ΓL(1,𝔽25)—91no—
115#892—91no—
116#892—91no—
117C3×#102—91no—
118C3×#102—91no—
119#832—31no—
120#832—31no—
121#832—31no—
122#832—61no—
123#832—91no—
124#832—91no—
125#832—91no—
126#821—91no—
127#821—91no—
128#809—91no—
129#809—91no—
130C23×C32—21no—
131C23×C32—21no—
132C23×C32—41no—
133C22×C3⋊D3—21no—
134C22×C3⋊D3—24no—
135C22×C3⋊D3—42no—
136C22×C3⋊D3—42no—
137C22×C3⋊D3#6711yescharacteristic
138C22×C3×D3—31no—
139C22×C3×D3—64no—
140C22×C3×D3—61no—
141C22×C3×D3—122no—
142C22×C3×D3—122no—
143C32×D4—21no—
144C32×D4—21no—
145C32×D4—22no—
146C32×D4—41no—
147C32×D4C4×D312yes—
148C32×D4C2×Dic312yes—
149C2×C32×C4—21no—
150C2×C32×C4C22×D311yescharacteristic
151C2×C32×C4C22×D311yescharacteristic
152#638—24no—
153C2×C3⋊Dic3—21no—
154C2×C3⋊Dic3#6711yescharacteristic
155C4⋊(C3⋊D3)—21no—
156C4⋊(C3⋊D3)C2×Dic312yes—
157C4×C3⋊D3C2×Dic312yes—
158C4×C3⋊D3#6712yes—
159C3×#67—64no—
160C2×C3×Dic3—31no—
161C2×C3×Dic3—61no—
162C2×C3×Dic3—61no—
163C2×C3×Dic3—61no—
164C2×C3×Dic3—121no—
165C3×D12—32no—
166C3×D12—61no—
167C3×C4×D3—32no—
168C3×C4×D3—32no—
169C3⋊(C3⋊C8)—21no—
170C3×C3⋊C8—61no—
171C3×C3⋊C8—61no—
172C3×C3⋊C8—61no—
173C3×C3⋊C8—121no—
174#418—271noSylow
175C2×H(3,𝔽3)—42no—
176C2×H(3,𝔽3)C2×C22⋊C411yescharacteristic
177C2×H(3,𝔽3)AGL(1,ℤ/8)11yescharacteristic
178C2×H(3,𝔽3)AGL(1,ℤ/8)11yescharacteristic
179(C3⋊D3)⋊C3—22no—
180(C3⋊D3)⋊C3—42no—
181C24×C3—181no—
182C23×D3—61no—
183C23×D3—181no—
184C2×C3×D4—31no—
185C2×C3×D4—31no—
186C2×C3×D4—61no—
187C2×C3×D4—91no—
188C2×C3×D4—94no—
189C2×C3×D4—182no—
190C2×C3×D4—181no—
191C2×C3×D4D3211yescharacteristic
192C22×C3×C4—91no—
193C2×#67—61no—
194C2×#67—181no—
195D3×D4—34no—
196D3×D4—62no—
197D3×D4—94no—
198D3×D4—182no—
199C2×D12—31no—
200C2×D12—91no—
201C2×C4×D3—31no—
202C2×C4×D3—91no—
203C2×C3×C8—91no—
204C2×C3×C8—91no—
205C3×C4⋊C4—91no—
206C3×C4⋊C4—91no—
207#217—181no—
208(C2×D3)⋊C4—181no—
209(C2×D3)⋊C4—181no—
210C4⋊Dic3—31no—
211C4⋊Dic3—31no—
212C4⋊Dic3—31no—
213C4⋊Dic3—61no—
214C4⋊Dic3—91no—
215C4×Dic3—181no—
216C4×Dic3—181no—
217C4×Dic3—181no—
218#208—182no—
219C2×C3⋊C8—31no—
220C2×C3⋊C8—31no—
221C2×C3⋊C8—31no—
222C2×C3⋊C8—61no—
223C2×C3⋊C8—91no—
224ΓL(1,𝔽25)—182no—
225ΓL(1,𝔽25)—182no—
226C22×C32—22no—
227C22×C32—22no—
228C22×C32—21no—
229C22×C32—42no—
230C22×C32—41no—
231C22×C32—41no—
232C22×C32—41no—
233C22×C32—41no—
234C22×C32—81no—
235C22×C32—81no—
236C22×C32D3×D411yescharacteristic
237C22×C32C2×#6711yescharacteristic
238C2×C3⋊D3—21no—
239C2×C3⋊D3—21no—
240C2×C3⋊D3—22no—
241C2×C3⋊D3—48no—
242C2×C3⋊D3—41no—
243C2×C3⋊D3—41no—
244C2×C3⋊D3#21712yes—
245C2×C3×D3—32no—
246C2×C3×D3—61no—
247C2×C3×D3—61no—
248C2×C3×D3—62no—
249C2×C3×D3—128no—
250C2×C3×D3—121no—
251C2×C3×D3—121no—
252C32×C4—21no—
253C32×C4—21no—
254C32×C4C2×C4×D311yescharacteristic
255C32×C4D3×D411yescharacteristic
256C32×C4C22×Dic311yescharacteristic
257C32×C4C2×#6711yescharacteristic
258C3⋊Dic3—41no—
259C3⋊Dic3#21712yes—
260C3×Dic3—32no—
261C3×Dic3—121no—
262C3×Dic3—121no—
263C3×Dic3—121no—
264C3×Dic3—241no—
265C22×D4—91no—
266C2×M16—271no—
267#117—271no—
268#102—91no—
269#102—91no—
270#102—271no—
271#102—271no—
272H(3,𝔽3)#41811yescharacteristic, complemented, Sylow
273C23×C3—21no—
274C23×C3—61no—
275C23×C3—61no—
276C23×C3—91no—
277C23×C3—121no—
278C23×C3—181no—
279C23×C3—184no—
280C23×C3—362no—
281C23×C3—362no—
282C22×D3—31no—
283C22×D3—64no—
284C22×D3—61no—
285C22×D3—91no—
286C22×D3—122no—
287C22×D3—122no—
288C22×D3—184no—
289C22×D3—181no—
290C22×D3—362no—
291C22×D3—362no—
292C3×D4—21no—
293C3×D4—32no—
294C3×D4—32no—
295C3×D4—61no—
296C3×D4—61no—
297C3×D4—62no—
298C3×D4—92no—
299C3×D4—121no—
300C3×D4—181no—
301C3×D4—184no—
302C3×D4D3×Dic312yes—
303C2×C3×C4—31no—
304C2×C3×C4—31no—
305C2×C3×C4—61no—
306C2×C3×C4—92no—
307C2×C3×C4—92no—
308C2×C3×C4—91no—
309C2×C3×C4—181no—
310C2×C3×C4—181no—
311C2×C3×C4C2×D3211yescharacteristic
312#67—64no—
313#67—184no—
314C2×Dic3—31no—
315C2×Dic3—61no—
316C2×Dic3—61no—
317C2×Dic3—61no—
318C2×Dic3—91no—
319C2×Dic3—121no—
320C2×Dic3—181no—
321D12—32no—
322D12—61no—
323D12—92no—
324D12—181no—
325C4×D3—32no—
326C4×D3—32no—
327C4×D3—92no—
328C4×D3—92no—
329C3×C8—181no—
330C3×C8—181no—
331C3⋊C8—61no—
332C3⋊C8—61no—
333C3⋊C8—61no—
334C3⋊C8—121no—
335C3⋊C8—181no—
336C2×C32—21no—
337C2×C32—21no—
338C2×C32—21no—
339C2×C32—42no—
340C2×C32—42no—
341C2×C32—82no—
342C2×C32D3×C22⋊C411yescharacteristic
343C2×C32#91111yescharacteristic
344C2×C32#91411yescharacteristic
345C2×C32#93211yescharacteristic
346C2×C32#93211yescharacteristic
347C2×C32C2×#21711yescharacteristic
348C3⋊D3—22no—
349C3⋊D3—42no—
350C3×D3—62no—
351C3×D3—122no—
352C24—181no—
353C2×D4—94no—
354C2×D4—91no—
355C2×D4—182no—
356C2×D4—181no—
357C2×D4((C3⋊D3)⋊C3)⋊C211yescharacteristic
358C22×C4—91no—
359M16—542no—
360C2×C8—91no—
361C2×C8—91no—
362C4⋊C4—91no—
363C4⋊C4—91no—
364C22⋊C4—541no—
365C42—541no—
366C22×C3—22no—
367C22×C3—31no—
368C22×C3—31no—
369C22×C3—41no—
370C22×C3—41no—
371C22×C3—62no—
372C22×C3—62no—
373C22×C3—61no—
374C22×C3—92no—
375C22×C3—122no—
376C22×C3—121no—
377C22×C3—121no—
378C22×C3—121no—
379C22×C3—121no—
380C22×C3—181no—
381C22×C3—181no—
382C22×C3—182no—
383C22×C3—241no—
384C22×C3—241no—
385C22×C3—361no—
386C22×C3—361no—
387C22×C3—368no—
388C22×C3D3×#6711yescharacteristic
389C2×D3—32no—
390C2×D3—61no—
391C2×D3—61no—
392C2×D3—62no—
393C2×D3—92no—
394C2×D3—128no—
395C2×D3—121no—
396C2×D3—121no—
397C2×D3—181no—
398C2×D3—181no—
399C2×D3—182no—
400C2×D3—368no—
401C2×D3—361no—
402C2×D3—361no—
403C3×C4—31no—
404C3×C4—31no—
405C3×C4—31no—
406C3×C4—31no—
407C3×C4—61no—
408C3×C4—61no—
409C3×C4—92no—
410C3×C4—361no—
411C3×C4—361no—
412C3×C4C2×D3×Dic311yescharacteristic
413C3×C4D3×#6711yescharacteristic
414Dic3—32no—
415Dic3—92no—
416Dic3—121no—
417Dic3—121no—
418Dic3—121no—
419Dic3—241no—
420Dic3—361no—
421C32—21no—
422C32#477011yescharacteristic, complemented
423C32#521511yescharacteristic, complemented
424C23—21no—
425C23—91no—
426C23—184no—
427C23—181no—
428C23—362no—
429C23—362no—
430D4—21no—
431D4—92no—
432D4—184no—
433D4—181no—
434D4((C3⋊D3)⋊C3)⋊C412yes—
435C2×C4—91no—
436C2×C4—92no—
437C2×C4—92no—
438C2×C4—181no—
439C2×C4—181no—
440C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
441C8—181no—
442C8—181no—
443C2×C3—31no—
444C2×C3—31no—
445C2×C3—31no—
446C2×C3—31no—
447C2×C3—31no—
448C2×C3—31no—
449C2×C3—42no—
450C2×C3—61no—
451C2×C3—61no—
452C2×C3—61no—
453C2×C3—122no—
454C2×C3—122no—
455C2×C3—182no—
456C2×C3—242no—
457C2×C3—362no—
458C2×C3#6393211yescharacteristic
459C2×C3#6393411yescharacteristic
460C2×C3D3×#21711yescharacteristic
461D3—62no—
462D3—122no—
463D3—182no—
464D3—362no—
465C22—22no—
466C22—41no—
467C22—41no—
468C22—92no—
469C22—181no—
470C22—181no—
471C22—182no—
472C22—368no—
473C22—361no—
474C22—361no—
475C22#8867611yescharacteristic
476C4—92no—
477C4—361no—
478C4—361no—
479C4C2×((C3⋊D3)⋊C3)⋊C411yescharacteristic
480C4#8867611yescharacteristic
481C3—31no—
482C3—31no—
483C3—61no—
484C3#1059142111yescharacteristic
485C2—42no—
486C2—182no—
487C2—362no—
488C2#1172119611yescharacteristic
489C2#1172120011yescharacteristic
490C2#1172131011yescharacteristic
subgroupG/H
centerC22#88676
commutatorC4×H(3,𝔽3)C22×C4
FrattiniC2×C3×C4C2×D32
FittingC2×D4×H(3,𝔽3)C22
socleC22×C3D3×#67
prime radicalC2×D4×(C3⋊D3)⋊C3C2
cosocleC2×C4×H(3,𝔽3)C23
hypercenterC2×D4((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)#418
LCSG ▷ C4×H(3,𝔽3) ▷ C2×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C22 ◁ C2×C4 ◁ C2×D4

Complements

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

NH ≅ G/Ntype
2. #11721081C2semidirect
3. C2×#88430C2semidirect
4. #11720578C2semidirect
5. #63659⋊C3C2semidirect
6. #11720318C2semidirect
7. #11720308C2semidirect
14. D4×(C3⋊D3)⋊C3C4semidirect
17. C4⋊#1084C22semidirect
18. C2×#6180C22semidirect
19. C4⋊((C3⋊Dic3)⋊C3)C22semidirect
20. C2×(C3⋊(C3⋊C8))⋊C3C22semidirect
30. #63659D3semidirect
68. D4×C3⋊D3Dic3semidirect
76. C4⋊(C3⋊Dic3)C2×D3semidirect
77. C2×C3⋊(C3⋊C8)C2×D3semidirect
272. H(3,𝔽3)#418semidirect
422. C32#4770semidirect
423. C32#5215semidirect

Maximal subgroups

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

subgroupindex
C2×D4×(C3⋊D3)⋊C32
#117210812
C2×#884302
#117205782
#63659⋊C32
#117203182
#117203082
#105914263
#105914213