Finite Group Databasebeta

← #13138054

Subgroups — #13138054

Order 1728.

715 conjugacy classes of subgroups; only the 95 normal subgroups are shown.

normal subgroups only

This group has 715 conjugacy classes of subgroups — too many to draw. Only its 95 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×#88654C211yesmaximal, characteristic
2#64101⋊C3C211yesmaximal, characteristic, complemented
3C2×#88430C211yesmaximal, characteristic, complemented
4#11720531C211yesmaximal, characteristic, complemented
5#63636⋊C3C211yesmaximal, characteristic, complemented
6#11720324C211yesmaximal, characteristic, complemented
7#11720318C211yesmaximal, characteristic, complemented
8#10590594—31nomaximal
9#10590592—31nomaximal
10C22×((C3⋊D3)⋊C3)⋊C2—21no—
11C2×C4⋊#272C2211yescharacteristic
12C2×C4⋊((C3⋊D3)⋊C3)C2211yescharacteristic
13C2×C4×(C3⋊D3)⋊C3C2211yescharacteristic
14C2×#6200—21no—
15#88654—22no—
16#88654C414yescomplemented
17C4⋊C4×H(3,𝔽3)C2211yescharacteristic, complemented
18C2×#6180C2211yescharacteristic, complemented
19#3752⋊C3—21no—
20C4⋊((C3⋊Dic3)⋊C3)C2211yescharacteristic, complemented
21C4×(C3⋊Dic3)⋊C3—21no—
22C2×(C3⋊(C3⋊C8))⋊C3C2211yescharacteristic, complemented
23#88430—22no—
24C2×#3802—31no—
25C2×D3×D12—31no—
26#64101D311yescharacteristic, complemented
27C3×#892—31no—
28C2×#3712—31no—
29C2×#3711—31no—
30#63636D311yescharacteristic, complemented
31C3×#809—31no—
32C3×#809—31no—
33C3×#809—31no—
34C3×#809—61no—
35#63564—31no—
36#63563—31no—
37#63563—31no—
38#63562—31no—
39C22×#272—21no—
40C22×(C3⋊D3)⋊C3—21no—
41C22×(C3⋊D3)⋊C3D411yescharacteristic
42C2×((C3⋊D3)⋊C3)⋊C2—24no—
43C2×((C3⋊D3)⋊C3)⋊C2—42no—
44C2×((C3⋊D3)⋊C3)⋊C2—42no—
45C2×C4×H(3,𝔽3)—21no—
46C2×C4×H(3,𝔽3)C2311yescharacteristic
47C4⋊#272—21no—
48C4⋊#272C2×C412yes—
49C2×(C3⋊Dic3)⋊C3—21no—
50C2×(C3⋊Dic3)⋊C3D411yescharacteristic
51C4⋊((C3⋊D3)⋊C3)—21no—
52C4⋊((C3⋊D3)⋊C3)C2×C412yes—
53C4×(C3⋊D3)⋊C3C2×C412yes—
54C4×(C3⋊D3)⋊C3D412yes—
55#6200—24no—
56#6180—21no—
57(C3⋊(C3⋊C8))⋊C3—21no—
58#5001—91no—
59#4770—91no—
60#4770—91no—
61C22×D32—61no—
62C22×D32—61no—
63C2×C4⋊(C3⋊D3)C2×D311yescharacteristic
64C2×C4×C3⋊D3C2×D311yescharacteristic
65C2×C3×D12—31no—
66C2×C3×D12—31no—
67C2×C3×D12—31no—
68C2×C3×D12—61no—
69C2×C3×C4×D3—31no—
70C2×#626—61no—
71C2×#625—61no—
72#3802—34no—
73#3802—62no—
74D3×D12—34no—
75D3×D12—62no—
76C32×C4⋊C4—21no—
77C32×C4⋊C4C2×D311yescharacteristic, complemented
78C32×C4⋊C4C2×D311yescharacteristic, complemented
79#3752—21no—
80C4⋊(C3⋊Dic3)C2×D311yescharacteristic, complemented
81C4×C3⋊Dic3—21no—
82C2×C3⋊(C3⋊C8)C2×D311yescharacteristic, complemented
83C3×(C2×D3)⋊C4—61no—
84C3×C4⋊Dic3—31no—
85C3×C4×Dic3—61no—
86C2×C3×C3⋊C8—31no—
87C2×C3×C3⋊C8—31no—
88C2×C3×C3⋊C8—31no—
89C2×C3×C3⋊C8—61no—
90#3712—62no—
91#3711—62no—
92C22×H(3,𝔽3)C2×D411yescharacteristic
93C2×#272—22no—
94C2×#272—41no—
95C2×#272—41no—
96C2×(C3⋊D3)⋊C3—21no—
97C2×(C3⋊D3)⋊C3—21no—
98C2×(C3⋊D3)⋊C3—22no—
99C2×(C3⋊D3)⋊C3—41no—
100C2×(C3⋊D3)⋊C3—41no—
101C2×(C3⋊D3)⋊C3C22⋊C412yes—
102((C3⋊D3)⋊C3)⋊C2—48no—
103C4×H(3,𝔽3)—41no—
104C4×H(3,𝔽3)C22×C411yescharacteristic
105C4×H(3,𝔽3)C2×D411yescharacteristic
106(C3⋊Dic3)⋊C3—41no—
107(C3⋊Dic3)⋊C3C22⋊C412yes—
108C2×D3×D4—91no—
109C2×D3×D4—91no—
110C22×D12—91no—
111C3×#117—91no—
112C2×#208—91no—
113C2×ΓL(1,𝔽25)—91no—
114C2×ΓL(1,𝔽25)—91no—
115#892—31no—
116#892—91no—
117C3×#102—91no—
118C3×#102—91no—
119#832—91no—
120#832—91no—
121#821—91no—
122#821—91no—
123#809—31no—
124#809—31no—
125#809—31no—
126#809—61no—
127#809—91no—
128#809—91no—
129#809—91no—
130C22×C3⋊D3—21no—
131C22×C3⋊D3#6711yescharacteristic
132C22×C3×D3—31no—
133C22×C3×D3—61no—
134C22×C3×D3—61no—
135C22×C3×D3—61no—
136C22×C3×D3—121no—
137C2×D32—64no—
138C2×D32—64no—
139C2×D32—122no—
140C2×D32—122no—
141C2×D32—122no—
142C2×D32—122no—
143C2×C32×C4—21no—
144C2×C32×C4—21no—
145C2×C32×C4—21no—
146C2×C32×C4—41no—
147C2×C32×C4C22×D311yescharacteristic
148C2×C32×C4C22×D311yescharacteristic
149C2×C3⋊Dic3—21no—
150C2×C3⋊Dic3#6711yescharacteristic
151C4⋊(C3⋊D3)—21no—
152C4⋊(C3⋊D3)C4×D312yes—
153C4×C3⋊D3C4×D312yes—
154C4×C3⋊D3D1212yes—
155C2×C3×Dic3—31no—
156C2×C3×Dic3—61no—
157C3×D12—32no—
158C3×D12—32no—
159C3×D12—32no—
160C3×D12—62no—
161C3×D12—61no—
162C3×D12—61no—
163C3×D12—61no—
164C3×D12—121no—
165C3×C4×D3—32no—
166C3×C4×D3—32no—
167#626—64no—
168#625—64no—
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)C2×C22⋊C411yescharacteristic
176C2×H(3,𝔽3)AGL(1,ℤ/8)11yescharacteristic
177C2×H(3,𝔽3)AGL(1,ℤ/8)11yescharacteristic
178#272—42no—
179(C3⋊D3)⋊C3—22no—
180(C3⋊D3)⋊C3—42no—
181C23×D3—181no—
182C23×D3—181no—
183C23×D3—181no—
184C2×C3×D4—91no—
185C2×C3×D4—91no—
186C22×C3×C4—91no—
187C2×#67—181no—
188C2×#67—181no—
189D3×D4—94no—
190D3×D4—94no—
191D3×D4—182no—
192D3×D4—182no—
193C2×D12—31no—
194C2×D12—31no—
195C2×D12—31no—
196C2×D12—61no—
197C2×D12—91no—
198C2×D12—94no—
199C2×D12—182no—
200C2×D12—181no—
201C2×C4×D3—31no—
202C2×C4×D3—91no—
203C2×C3×C8—91no—
204C2×C3×C8—91no—
205C3×C4⋊C4—31no—
206C3×C4⋊C4—31no—
207C3×C4⋊C4—61no—
208C3×C4⋊C4—91no—
209C3×C4⋊C4D3211yescharacteristic
210C3×C22⋊C4—181no—
211C3×C42—181no—
212(C2×D3)⋊C4—61no—
213(C2×D3)⋊C4—181no—
214C4⋊Dic3—31no—
215C4⋊Dic3—91no—
216C4×Dic3—61no—
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—21no—
227C22×C32D3×D411yescharacteristic
228C22×C32C2×#6711yescharacteristic
229C2×C3⋊D3—21no—
230C2×C3⋊D3—21no—
231C2×C3⋊D3—22no—
232C2×C3⋊D3—41no—
233C2×C3⋊D3—41no—
234C2×C3⋊D3(C2×D3)⋊C412yes—
235C2×C3×D3—32no—
236C2×C3×D3—61no—
237C2×C3×D3—61no—
238C2×C3×D3—62no—
239C2×C3×D3—62no—
240C2×C3×D3—62no—
241C2×C3×D3—122no—
242C2×C3×D3—121no—
243C2×C3×D3—121no—
244C2×C3×D3—121no—
245C2×C3×D3—121no—
246C2×C3×D3—121no—
247C2×C3×D3—121no—
248C2×C3×D3—241no—
249C2×C3×D3—241no—
250D32—128no—
251D32—128no—
252C32×C4—21no—
253C32×C4—21no—
254C32×C4—41no—
255C32×C4—41no—
256C32×C4—81no—
257C32×C4C2×C4×D311yescharacteristic
258C32×C4C2×C4×D311yescharacteristic
259C32×C4C2×D1211yescharacteristic
260C32×C4D3×D411yescharacteristic
261C3⋊Dic3—41no—
262C3⋊Dic3(C2×D3)⋊C412yes—
263C3×Dic3—32no—
264C3×Dic3—121no—
265C22×D4—271no—
266C2×M16—271no—
267#117—91no—
268#102—91no—
269#102—91no—
270#102—271no—
271#102—271no—
272H(3,𝔽3)#41811yescharacteristic, complemented, Sylow
273C23×C3—91no—
274C23×C3—181no—
275C23×C3—181no—
276C22×D3—31no—
277C22×D3—61no—
278C22×D3—61no—
279C22×D3—61no—
280C22×D3—91no—
281C22×D3—121no—
282C22×D3—181no—
283C22×D3—184no—
284C22×D3—184no—
285C22×D3—184no—
286C22×D3—362no—
287C22×D3—362no—
288C22×D3—362no—
289C22×D3—362no—
290C22×D3—362no—
291C22×D3—362no—
292C3×D4—92no—
293C3×D4—92no—
294C3×D4—181no—
295C3×D4—181no—
296C2×C3×C4—21no—
297C2×C3×C4—31no—
298C2×C3×C4—31no—
299C2×C3×C4—61no—
300C2×C3×C4—61no—
301C2×C3×C4—61no—
302C2×C3×C4—92no—
303C2×C3×C4—92no—
304C2×C3×C4—91no—
305C2×C3×C4—121no—
306C2×C3×C4—181no—
307C2×C3×C4C2×D3211yescharacteristic
308#67—184no—
309#67—184no—
310C2×Dic3—31no—
311C2×Dic3—61no—
312C2×Dic3—91no—
313C2×Dic3—181no—
314D12—32no—
315D12—32no—
316D12—32no—
317D12—62no—
318D12—61no—
319D12—61no—
320D12—61no—
321D12—92no—
322D12—121no—
323D12—181no—
324D12—184no—
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×C32D3×C22⋊C411yescharacteristic
340C2×C32#91111yescharacteristic
341C2×C32#91411yescharacteristic
342C2×C32C2×(C2×D3)⋊C411yescharacteristic
343C2×C32#94911yescharacteristic
344C2×C32#94911yescharacteristic
345C3⋊D3—22no—
346C3⋊D3—42no—
347C3×D3—62no—
348C3×D3—122no—
349C3×D3—122no—
350C3×D3—122no—
351C3×D3—242no—
352C24—541no—
353C2×D4—91no—
354C2×D4—91no—
355C2×D4—274no—
356C2×D4—542no—
357C2×D4—541no—
358C22×C4—91no—
359M16—542no—
360C2×C8—91no—
361C2×C8—91no—
362C4⋊C4—91no—
363C4⋊C4((C3⋊D3)⋊C3)⋊C211yescharacteristic, complemented
364C22⋊C4—181no—
365C42—181no—
366C22×C3—31no—
367C22×C3—31no—
368C22×C3—61no—
369C22×C3—92no—
370C22×C3—181no—
371C22×C3—181no—
372C22×C3—182no—
373C22×C3—182no—
374C22×C3—361no—
375C22×C3—361no—
376C22×C3—361no—
377C22×C3—361no—
378C22×C3D3×#6711yescharacteristic
379C2×D3—32no—
380C2×D3—61no—
381C2×D3—61no—
382C2×D3—62no—
383C2×D3—62no—
384C2×D3—62no—
385C2×D3—92no—
386C2×D3—122no—
387C2×D3—121no—
388C2×D3—121no—
389C2×D3—121no—
390C2×D3—121no—
391C2×D3—121no—
392C2×D3—121no—
393C2×D3—181no—
394C2×D3—181no—
395C2×D3—182no—
396C2×D3—241no—
397C2×D3—241no—
398C2×D3—361no—
399C2×D3—361no—
400C2×D3—368no—
401C2×D3—368no—
402C2×D3—368no—
403C3×C4—31no—
404C3×C4—31no—
405C3×C4—31no—
406C3×C4—31no—
407C3×C4—41no—
408C3×C4—61no—
409C3×C4—61no—
410C3×C4—92no—
411C3×C4—121no—
412C3×C4—121no—
413C3×C4—241no—
414C3×C4—361no—
415C3×C4C4×D3211yescharacteristic
416C3×C4D3×D1211yescharacteristic
417Dic3—32no—
418Dic3—92no—
419Dic3—121no—
420Dic3—361no—
421C32—21no—
422C32#477011yescharacteristic, complemented
423C32#500111yescharacteristic, complemented
424C23—91no—
425C23—181no—
426C23—181no—
427C23—544no—
428C23—1082no—
429C23—1082no—
430D4—92no—
431D4—92no—
432D4—181no—
433D4—181no—
434D4—544no—
435C2×C4—21no—
436C2×C4—91no—
437C2×C4—92no—
438C2×C4—92no—
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—61no—
450C2×C3—61no—
451C2×C3—61no—
452C2×C3—182no—
453C2×C3—362no—
454C2×C3—362no—
455C2×C3D3×(C2×D3)⋊C411yescharacteristic
456C2×C3#6392511yescharacteristic
457C2×C3#6393911yescharacteristic
458D3—62no—
459D3—122no—
460D3—122no—
461D3—122no—
462D3—182no—
463D3—242no—
464D3—362no—
465C22—92no—
466C22—181no—
467C22—181no—
468C22—182no—
469C22—182no—
470C22—361no—
471C22—361no—
472C22—361no—
473C22—361no—
474C22—1088no—
475C22#8867611yescharacteristic
476C4—41no—
477C4—92no—
478C4—361no—
479C4C4×((C3⋊D3)⋊C3)⋊C211yescharacteristic
480C4#8865411yescharacteristic
481C3—31no—
482C3—31no—
483C3—61no—
484C3#1059059211yescharacteristic
485C2—182no—
486C2—362no—
487C2—362no—
488C2#1172116011yescharacteristic
489C2#1172119511yescharacteristic
490C2#1172123311yescharacteristic
subgroupG/H
centerC22#88676
commutatorC4×H(3,𝔽3)C22×C4
FrattiniC2×C3×C4C2×D32
FittingC4⋊C4×H(3,𝔽3)C22
socleC22×C3D3×#67
prime radicalC2×#88654C2
cosocleC2×C4×H(3,𝔽3)C23
hypercenterC4⋊C4((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)#418
LCSG ▷ C4×H(3,𝔽3) ▷ C2×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C22 ◁ C2×C4 ◁ C4⋊C4

Complements

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

NH ≅ G/Ntype
2. #64101⋊C3C2semidirect
3. C2×#88430C2semidirect
4. #11720531C2semidirect
5. #63636⋊C3C2semidirect
6. #11720324C2semidirect
7. #11720318C2semidirect
16. #88654C4semidirect
17. C4⋊C4×H(3,𝔽3)C22semidirect
18. C2×#6180C22semidirect
20. C4⋊((C3⋊Dic3)⋊C3)C22semidirect
22. C2×(C3⋊(C3⋊C8))⋊C3C22semidirect
26. #64101D3semidirect
30. #63636D3semidirect
77. C32×C4⋊C4C2×D3semidirect
78. C32×C4⋊C4C2×D3semidirect
80. C4⋊(C3⋊Dic3)C2×D3semidirect
82. C2×C3⋊(C3⋊C8)C2×D3semidirect
272. H(3,𝔽3)#418semidirect
363. C4⋊C4((C3⋊D3)⋊C3)⋊C2semidirect
422. C32#4770semidirect
423. C32#5001semidirect

Maximal subgroups

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

subgroupindex
C2×#886542
#64101⋊C32
C2×#884302
#117205312
#63636⋊C32
#117203242
#117203182
#105905943
#105905923