Finite Group Databasebeta

← #13139046

Subgroups — #13139046

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
2C2×#3743⋊C3C211yesmaximal, characteristic, complemented
3#11721081C211yesmaximal, characteristic, complemented
4#11720554C211yesmaximal, characteristic, complemented
5#63648⋊C3C211yesmaximal, characteristic, complemented
6#11720324C211yesmaximal, characteristic, complemented
7#11720308C211yesmaximal, characteristic, complemented
8#10590941—31nomaximal
9#10590939—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
17C2×C8×H(3,𝔽3)C2211yescharacteristic, complemented
18C4⋊#1084C2211yescharacteristic, complemented
19C4⋊((C3⋊Dic3)⋊C3)C2211yescharacteristic, complemented
20C2×(C3⋊(C3⋊C8))⋊C3C2211yescharacteristic, complemented
21#3743⋊C3—22no—
22#88450—21no—
23#88447—21no—
24C2×#3802—31no—
25C2×D3×D12—31no—
26C2×#3743D311yescharacteristic, complemented
27C2×C3×ΓL(1,𝔽25)—31no—
28#63849—31no—
29#63841—31no—
30#63648D311yescharacteristic, complemented
31C3×#821—31no—
32C3×#821—31no—
33C3×#821—31no—
34C3×#821—61no—
35#63564—31no—
36#63563—31no—
37#63562—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)C2311yescharacteristic
46C2×#1084—21no—
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—
56C8×H(3,𝔽3)—21no—
57(C3⋊(C3⋊C8))⋊C3—21no—
58#5131—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—
76C2×C32×C8—21no—
77C2×C32×C8C2×D311yescharacteristic, complemented
78C2×C32×C8C2×D311yescharacteristic, complemented
79C4⋊(C3⋊Dic3)C2×D311yescharacteristic, complemented
80C2×C3⋊(C3⋊C8)C2×D311yescharacteristic, complemented
81#3743—22no—
82C3×C4⋊Dic3—31no—
83C3×C4⋊Dic3—31no—
84C3×C4⋊Dic3—31no—
85C3×C4⋊Dic3—61no—
86C2×C3×C3⋊C8—31no—
87C3×ΓL(1,𝔽25)—62no—
88#3722—61no—
89(C2×D3)⋊Dic3—61no—
90Dic32—61no—
91Dic32—61no—
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)C22×C411yescharacteristic
104C4×H(3,𝔽3)C2×D411yescharacteristic
105#1084—41no—
106(C3⋊Dic3)⋊C3—41no—
107(C3⋊Dic3)⋊C3C22⋊C412yes—
108C2×D3×D4—91no—
109C2×D3×D4—91no—
110C22×D12—91no—
111C2×C3×M16—91no—
112#926—91no—
113C2×ΓL(1,𝔽25)—31no—
114C2×ΓL(1,𝔽25)—91no—
115#892—91no—
116#892—91no—
117C3×#102—91no—
118C3×#102—91no—
119#832—91no—
120#832—91no—
121#821—31no—
122#821—31no—
123#821—31no—
124#821—61no—
125#821—91no—
126#821—91no—
127#821—91no—
128#809—91no—
129#809—91no—
130C22×C3⋊D3—21no—
131C22×C3⋊D3D1211yescharacteristic
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×C4C22×D311yescharacteristic
145C2×C32×C4C22×D311yescharacteristic
146C2×C3⋊Dic3—21no—
147C2×C3⋊Dic3D1211yescharacteristic
148C4⋊(C3⋊D3)—21no—
149C4⋊(C3⋊D3)C4×D312yes—
150C4×C3⋊D3C4×D312yes—
151C4×C3⋊D3#6712yes—
152C2×C3×Dic3—31no—
153C2×C3×Dic3—61no—
154C2×C3×Dic3—61no—
155C2×C3×Dic3—61no—
156C2×C3×Dic3—121no—
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—
169C32×C8—21no—
170C32×C8—21no—
171C32×C8—41no—
172C3⋊(C3⋊C8)—21no—
173C3×C3⋊C8—61no—
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—
203C3×M16—182no—
204C2×C3×C8—31no—
205C2×C3×C8—31no—
206C2×C3×C8—61no—
207C2×C3×C8—91no—
208C2×C3×C8D3211yescharacteristic
209C3×C4⋊C4—91no—
210C3×C4⋊C4—91no—
211#217—181no—
212(C2×D3)⋊C4—181no—
213(C2×D3)⋊C4—181no—
214C4⋊Dic3—31no—
215C4⋊Dic3—31no—
216C4⋊Dic3—31no—
217C4⋊Dic3—61no—
218C4⋊Dic3—91no—
219C4×Dic3—181no—
220C4×Dic3—181no—
221C4×Dic3—181no—
222C2×C3⋊C8—31no—
223C2×C3⋊C8—91no—
224ΓL(1,𝔽25)—62no—
225ΓL(1,𝔽25)—182no—
226C22×C32—21no—
227C22×C32C2×D1211yescharacteristic
228C22×C32D3×D411yescharacteristic
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×C4C2×C4×D311yescharacteristic
255C32×C4C2×C4×D311yescharacteristic
256C32×C4D3×D411yescharacteristic
257C32×C4C2×#6711yescharacteristic
258C3⋊Dic3—41no—
259C3⋊Dic3(C2×D3)⋊C412yes—
260C3×Dic3—32no—
261C3×Dic3—121no—
262C3×Dic3—121no—
263C3×Dic3—121no—
264C3×Dic3—241no—
265C22×D4—271no—
266C2×M16—91no—
267#117—271no—
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—31no—
297C2×C3×C4—31no—
298C2×C3×C4—61no—
299C2×C3×C4—92no—
300C2×C3×C4—92no—
301C2×C3×C4—91no—
302C2×C3×C4—181no—
303C2×C3×C4—181no—
304C2×C3×C4C2×D3211yescharacteristic
305#67—184no—
306#67—184no—
307C2×Dic3—31no—
308C2×Dic3—61no—
309C2×Dic3—61no—
310C2×Dic3—61no—
311C2×Dic3—91no—
312C2×Dic3—121no—
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—21no—
330C3×C8—61no—
331C3×C8—61no—
332C3×C8—121no—
333C3×C8—181no—
334C3⋊C8—61no—
335C3⋊C8—181no—
336C2×C32—21no—
337C2×C32—21no—
338C2×C32—21no—
339C2×C32D3×C22⋊C411yescharacteristic
340C2×C32#90811yescharacteristic
341C2×C32#90811yescharacteristic
342C2×C32#91111yescharacteristic
343C2×C32#91411yescharacteristic
344C2×C32C2×(C2×D3)⋊C411yescharacteristic
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—182no—
360C2×C8—91no—
361C2×C8((C3⋊D3)⋊C3)⋊C211yescharacteristic, complemented
362C4⋊C4—91no—
363C4⋊C4—91no—
364C22⋊C4—541no—
365C42—541no—
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×D1211yescharacteristic
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—61no—
408C3×C4—61no—
409C3×C4—92no—
410C3×C4—361no—
411C3×C4—361no—
412C3×C4C4×D3211yescharacteristic
413C3×C4D3×#6711yescharacteristic
414Dic3—32no—
415Dic3—92no—
416Dic3—121no—
417Dic3—121no—
418Dic3—121no—
419Dic3—241no—
420Dic3—361no—
421C32—21no—
422C32#477011yescharacteristic, complemented
423C32#513111yescharacteristic, 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—91no—
436C2×C4—92no—
437C2×C4—92no—
438C2×C4—181no—
439C2×C4—181no—
440C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
441C8—21no—
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×C3#6379411yescharacteristic
456C2×C3#6379511yescharacteristic
457C2×C3D3×(C2×D3)⋊C411yescharacteristic
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#8865411yescharacteristic
476C4—92no—
477C4—361no—
478C4—361no—
479C4C4×((C3⋊D3)⋊C3)⋊C211yescharacteristic
480C4#8867611yescharacteristic
481C3—31no—
482C3—31no—
483C3—61no—
484C3#1059093911yescharacteristic
485C2—182no—
486C2—362no—
487C2—362no—
488C2#1172084811yescharacteristic
489C2#1172084911yescharacteristic
490C2#1172116011yescharacteristic
subgroupG/H
centerC22#88654
commutatorC4×H(3,𝔽3)C22×C4
FrattiniC2×C3×C4C2×D32
FittingC2×C8×H(3,𝔽3)C22
socleC22×C3D3×D12
prime radicalC2×#88654C2
cosocleC2×C4×H(3,𝔽3)C23
hypercenterC2×C8((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×C8

Complements

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

NH ≅ G/Ntype
2. C2×#3743⋊C3C2semidirect
3. #11721081C2semidirect
4. #11720554C2semidirect
5. #63648⋊C3C2semidirect
6. #11720324C2semidirect
7. #11720308C2semidirect
16. #88654C4semidirect
17. C2×C8×H(3,𝔽3)C22semidirect
18. C4⋊#1084C22semidirect
19. C4⋊((C3⋊Dic3)⋊C3)C22semidirect
20. C2×(C3⋊(C3⋊C8))⋊C3C22semidirect
26. C2×#3743D3semidirect
30. #63648D3semidirect
77. C2×C32×C8C2×D3semidirect
78. C2×C32×C8C2×D3semidirect
79. C4⋊(C3⋊Dic3)C2×D3semidirect
80. C2×C3⋊(C3⋊C8)C2×D3semidirect
272. H(3,𝔽3)#418semidirect
361. C2×C8((C3⋊D3)⋊C3)⋊C2semidirect
422. C32#4770semidirect
423. C32#5131semidirect

Maximal subgroups

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

subgroupindex
C2×#886542
C2×#3743⋊C32
#117210812
#117205542
#63648⋊C32
#117203242
#117203082
#105909413
#105909393