Finite Group Databasebeta

← #13140025

Subgroups — #13140025

Order 1728.

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

normal subgroups only

This group has 603 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×#3828⋊C3C211yesmaximal, characteristic
2C2×#88430C211yesmaximal, characteristic, complemented
3C2×((C3⋊D3)⋊C3)⋊C8C211yesmaximal, characteristic, complemented
4#11720573C211yesmaximal, characteristic, complemented
5#63657⋊C3C211yesmaximal, characteristic, complemented
6#11720317C211yesmaximal, characteristic, complemented
7#11720317C211yesmaximal, characteristic, complemented
8#10591153—31nomaximal
9#10591145—31nomaximal
10C22×C4×H(3,𝔽3)C2211yescharacteristic
11C2×#638⋊C3C411yescharacteristic
12C2×#638⋊C3C411yescharacteristic
13#3828⋊C3—22no—
14#3828⋊C3—22no—
15C2×C4⋊((C3⋊D3)⋊C3)C411yescharacteristic
16C2×C4×(C3⋊D3)⋊C3C2211yescharacteristic
17C2×C4×(C3⋊D3)⋊C3C2211yescharacteristic
18C2×#634⋊C3C411yescharacteristic
19C2×#6180C2211yescharacteristic
20C2×#6180C2211yescharacteristic
21C2×(C3⋊(C3⋊C8))⋊C3C2211yescharacteristic
22C2×(C3⋊(C3⋊C8))⋊C3C2211yescharacteristic
23#88430—22no—
24((C3⋊D3)⋊C3)⋊C8—22no—
25C2×#3828D311yescharacteristic
26C2×C3×#235—31no—
27C2×#3712—31no—
28C2×#3711—31no—
29C2×#3710—31no—
30C2×D3×C3⋊C8—31no—
31#63657D311yescharacteristic, complemented
32C3×#830—31no—
33C3×#830—31no—
34C3×#830—31no—
35C3×#830—61no—
36#63557—31no—
37#63557—31no—
38(C2×D3)⋊(C3⋊C8)—31no—
39(C2×D3)⋊(C3⋊C8)—31no—
40C23×H(3,𝔽3)C2×C411yescharacteristic
41C22×(C3⋊D3)⋊C3C2×C411yescharacteristic
42C22×(C3⋊D3)⋊C3C2×C411yescharacteristic
43C2×C4×H(3,𝔽3)—21no—
44C2×C4×H(3,𝔽3)—21no—
45C2×C4×H(3,𝔽3)C2×C411yescharacteristic
46C2×C4×H(3,𝔽3)C2311yescharacteristic
47#638⋊C3—22no—
48#638⋊C3—22no—
49C2×(C3⋊Dic3)⋊C3C2×C411yescharacteristic
50C2×(C3⋊Dic3)⋊C3C2×C411yescharacteristic
51C4⋊((C3⋊D3)⋊C3)—22no—
52C4×(C3⋊D3)⋊C3—21no—
53C4×(C3⋊D3)⋊C3—21no—
54C4×(C3⋊D3)⋊C3D412yes—
55C4×(C3⋊D3)⋊C3D412yes—
56#634⋊C3—22no—
57#6180—21no—
58#6180—21no—
59(C3⋊(C3⋊C8))⋊C3—21no—
60(C3⋊(C3⋊C8))⋊C3—21no—
61#5234—91no—
62#4727—91no—
63#4727—91no—
64C22×C32×C4—21no—
65C22×C32×C4C2×D311yescharacteristic
66C22×C32×C4C2×D311yescharacteristic
67C2×#638Dic311yescharacteristic
68C2×#638Dic311yescharacteristic
69#3828—22no—
70#3828—22no—
71C2×C4⋊(C3⋊D3)Dic311yescharacteristic
72C2×C4×C3⋊D3C2×D311yescharacteristic
73C2×C4×C3⋊D3C2×D311yescharacteristic
74C2×#634Dic311yescharacteristic
75C2×C3×#67—31no—
76C2×C3×#67—31no—
77C3×#235—62no—
78C3×#235—62no—
79C2×C3×D12—31no—
80C2×C3×C4×D3—31no—
81C2×C3×C4×D3—31no—
82C2×C3×Dic6—31no—
83C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
84C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
85C2×C3×C3⋊C8—31no—
86C2×C3×C3⋊C8—31no—
87C2×C3×C3⋊C8—31no—
88C2×C3×C3⋊C8—31no—
89C2×C3×C3⋊C8—31no—
90C2×C3×C3⋊C8—31no—
91C2×C3×C3⋊C8—61no—
92C2×C3×C3⋊C8—61no—
93#3712—62no—
94#3711—62no—
95#3710—62no—
96D3×C3⋊C8—62no—
97C22×H(3,𝔽3)—21no—
98C22×H(3,𝔽3)—21no—
99C22×H(3,𝔽3)—21no—
100C22×H(3,𝔽3)C22×C411yescharacteristic
101C2×(C3⋊D3)⋊C3—21no—
102C2×(C3⋊D3)⋊C3—21no—
103C2×(C3⋊D3)⋊C3—21no—
104C2×(C3⋊D3)⋊C3—21no—
105C2×(C3⋊D3)⋊C3—21no—
106C2×(C3⋊D3)⋊C3C22⋊C412yes—
107C4×H(3,𝔽3)—21no—
108C4×H(3,𝔽3)C2×D411yescharacteristic
109C4×H(3,𝔽3)C2×D411yescharacteristic
110(C3⋊Dic3)⋊C3—21no—
111(C3⋊Dic3)⋊C3C22⋊C412yes—
112C2×C3×#41—91no—
113C2×#235—31no—
114C2×#235—91no—
115C2×#208—91no—
116C22×C3⋊C8—91no—
117C2×ΓL(1,𝔽25)—91no—
118C2×ΓL(1,𝔽25)—91no—
119C2×D3×C8—91no—
120C2×D3×C8—91no—
121C3×C22⋊C8—91no—
122C3×C22⋊C8—91no—
123#830—31no—
124#830—31no—
125#830—31no—
126#830—61no—
127#830—91no—
128#830—91no—
129#830—91no—
130(C2×D3)⋊C8—91no—
131(C2×D3)⋊C8—91no—
132(C2×D3)⋊C8—91no—
133(C2×D3)⋊C8—91no—
134C23×C32—21no—
135C23×C32C4×D311yescharacteristic
136C23×C32C2×Dic311yescharacteristic
137C22×C3⋊D3C2×Dic311yescharacteristic
138C22×C3⋊D3C2×Dic311yescharacteristic
139C22×C3×D3—31no—
140C22×C3×D3—31no—
141C2×C32×C4—21no—
142C2×C32×C4—21no—
143C2×C32×C4—21no—
144C2×C32×C4—21no—
145C2×C32×C4—21no—
146C2×C32×C4—21no—
147C2×C32×C4—41no—
148C2×C32×C4—41no—
149C2×C32×C4C4×D311yescharacteristic
150C2×C32×C4C2×Dic311yescharacteristic
151C2×C32×C4C22×D311yescharacteristic
152C2×C32×C4C22×D311yescharacteristic
153#638—22no—
154#638—22no—
155C2×C3⋊Dic3C2×Dic311yescharacteristic
156C2×C3⋊Dic3C2×Dic311yescharacteristic
157C4⋊(C3⋊D3)—22no—
158C4×C3⋊D3—21no—
159C4×C3⋊D3—21no—
160C4×C3⋊D3#6712yes—
161C4×C3⋊D3#6712yes—
162#634—22no—
163C3×#67—62no—
164C3×#67—62no—
165C2×C3×Dic3—31no—
166C2×C3×Dic3—31no—
167C3×D12—62no—
168C3×C4×D3—32no—
169C3×C4×D3—32no—
170C3×C4×D3—61no—
171C3×C4×D3—61no—
172C3×Dic6—62no—
173C3⋊(C3⋊C8)—21no—
174C3⋊(C3⋊C8)—21no—
175C3×C3⋊C8—61no—
176C3×C3⋊C8—61no—
177C3×C3⋊C8—61no—
178C3×C3⋊C8—61no—
179C3×C3⋊C8—61no—
180C3×C3⋊C8—61no—
181C3×C3⋊C8—121no—
182C3×C3⋊C8—121no—
183#408—271noSylow
184C2×H(3,𝔽3)—41no—
185C2×H(3,𝔽3)C2×C22⋊C411yescharacteristic
186C2×H(3,𝔽3)#13111yescharacteristic
187C2×H(3,𝔽3)#13111yescharacteristic
188(C3⋊D3)⋊C3—22no—
189(C3⋊D3)⋊C3—41no—
190C3×#41—182no—
191C3×#41—182no—
192C2×C3×Q8—91no—
193C2×C3×D4—91no—
194C2×C3×D4—91no—
195C2×C3×D4—91no—
196C22×C3×C4—31no—
197C22×C3×C4—31no—
198C22×C3×C4—61no—
199C22×C3×C4—91no—
200C22×C3×C4—91no—
201C22×C3×C4D3211yescharacteristic
202C2×#67—31no—
203C2×#67—31no—
204C2×#67—91no—
205C2×#67—91no—
206#235—62no—
207#235—62no—
208#235—182no—
209#235—182no—
210C2×D12—31no—
211C2×D12—91no—
212C2×C4×D3—31no—
213C2×C4×D3—31no—
214C2×C4×D3—91no—
215C2×C4×D3—91no—
216C2×Dic6—31no—
217C2×Dic6—91no—
218C2×C3×C8—91no—
219C2×C3×C8—91no—
220C2×C3×C8—91no—
221C2×C3×C8—91no—
222#208—182no—
223C2×C3⋊C8—31no—
224C2×C3⋊C8—31no—
225C2×C3⋊C8—31no—
226C2×C3⋊C8—31no—
227C2×C3⋊C8—31no—
228C2×C3⋊C8—31no—
229C2×C3⋊C8—61no—
230C2×C3⋊C8—61no—
231C2×C3⋊C8—91no—
232C2×C3⋊C8—91no—
233C2×C3⋊C8—182no—
234ΓL(1,𝔽25)—182no—
235ΓL(1,𝔽25)—182no—
236D3×C8—182no—
237D3×C8—182no—
238C22×C32—22no—
239C22×C32—21no—
240C22×C32—21no—
241C22×C32—21no—
242C22×C32—21no—
243C22×C32—21no—
244C22×C32—21no—
245C22×C32—21no—
246C22×C32—41no—
247C22×C32—41no—
248C22×C32C2×C4×D311yescharacteristic
249C22×C32C22×Dic311yescharacteristic
250C2×C3⋊D3—21no—
251C2×C3⋊D3—21no—
252C2×C3⋊D3—21no—
253C2×C3⋊D3—21no—
254C2×C3⋊D3—21no—
255C2×C3⋊D3#21712yes—
256C2×C3×D3—32no—
257C2×C3×D3—61no—
258C2×C3×D3—61no—
259C2×C3×D3—61no—
260C2×C3×D3—61no—
261C2×C3×D3—61no—
262C32×C4—21no—
263C32×C4—21no—
264C32×C4—21no—
265C32×C4—21no—
266C32×C4—41no—
267C32×C4D3×D411yescharacteristic
268C32×C4D3×D411yescharacteristic
269C32×C4C2×#6711yescharacteristic
270C32×C4C2×#6711yescharacteristic
271C3⋊Dic3—21no—
272C3⋊Dic3#21712yes—
273C3×Dic3—32no—
274C3×Dic3—61no—
275C2×#41—91no—
276C2×M16—271no—
277C22×C8—271no—
278C22⋊C8—91no—
279C22⋊C8—91no—
280C22⋊C8—271no—
281C22⋊C8—271no—
282H(3,𝔽3)#40811yescharacteristic, complemented, Sylow
283C23×C3—31no—
284C23×C3—31no—
285C23×C3—61no—
286C23×C3—91no—
287C23×C3—91no—
288C23×C3D3×Dic311yescharacteristic
289C22×D3—31no—
290C22×D3—31no—
291C22×D3—91no—
292C22×D3—91no—
293C3×Q8—182no—
294C3×D4—182no—
295C3×D4—182no—
296C3×D4—182no—
297C2×C3×C4—21no—
298C2×C3×C4—21no—
299C2×C3×C4—31no—
300C2×C3×C4—31no—
301C2×C3×C4—31no—
302C2×C3×C4—31no—
303C2×C3×C4—61no—
304C2×C3×C4—61no—
305C2×C3×C4—61no—
306C2×C3×C4—61no—
307C2×C3×C4—61no—
308C2×C3×C4—61no—
309C2×C3×C4—92no—
310C2×C3×C4—92no—
311C2×C3×C4—91no—
312C2×C3×C4—91no—
313C2×C3×C4—121no—
314C2×C3×C4—121no—
315C2×C3×C4—181no—
316C2×C3×C4—181no—
317C2×C3×C4D3×Dic311yescharacteristic
318C2×C3×C4C2×D3211yescharacteristic
319#67—62no—
320#67—62no—
321#67—182no—
322#67—182no—
323C2×Dic3—31no—
324C2×Dic3—31no—
325C2×Dic3—91no—
326C2×Dic3—91no—
327D12—62no—
328D12—182no—
329C4×D3—32no—
330C4×D3—32no—
331C4×D3—61no—
332C4×D3—61no—
333C4×D3—92no—
334C4×D3—92no—
335C4×D3—181no—
336C4×D3—181no—
337Dic6—62no—
338Dic6—182no—
339C3×C8—181no—
340C3×C8—181no—
341C3×C8—181no—
342C3×C8—181no—
343C3⋊C8—61no—
344C3⋊C8—61no—
345C3⋊C8—61no—
346C3⋊C8—61no—
347C3⋊C8—61no—
348C3⋊C8—61no—
349C3⋊C8—121no—
350C3⋊C8—121no—
351C3⋊C8—181no—
352C3⋊C8—181no—
353C2×C32—21no—
354C2×C32—21no—
355C2×C32—21no—
356C2×C32—42no—
357C2×C32—41no—
358C2×C32—41no—
359C2×C32D3×C22⋊C411yescharacteristic
360C2×C32#90111yescharacteristic
361C2×C32#90711yescharacteristic
362C2×C32#94811yescharacteristic
363C2×C32#94811yescharacteristic
364C2×C32C2×#21711yescharacteristic
365C3⋊D3—22no—
366C3⋊D3—41no—
367C3×D3—62no—
368C3×D3—121no—
369#41—182no—
370#41—182no—
371C2×Q8—91no—
372C2×D4—91no—
373C2×D4—91no—
374C2×D4—91no—
375C22×C4—91no—
376C22×C4—91no—
377C22×C4((C3⋊D3)⋊C3)⋊C211yescharacteristic
378M16—542no—
379C2×C8—91no—
380C2×C8—91no—
381C2×C8—91no—
382C2×C8—91no—
383C2×C8—542no—
384C22×C3—21no—
385C22×C3—21no—
386C22×C3—21no—
387C22×C3—31no—
388C22×C3—31no—
389C22×C3—61no—
390C22×C3—61no—
391C22×C3—61no—
392C22×C3—61no—
393C22×C3—61no—
394C22×C3—61no—
395C22×C3—62no—
396C22×C3—61no—
397C22×C3—92no—
398C22×C3—121no—
399C22×C3—121no—
400C22×C3—181no—
401C22×C3—181no—
402C22×C3—181no—
403C22×C3—181no—
404C22×C3—181no—
405C22×C3C2×D3×Dic311yescharacteristic
406C2×D3—32no—
407C2×D3—61no—
408C2×D3—61no—
409C2×D3—61no—
410C2×D3—61no—
411C2×D3—61no—
412C2×D3—92no—
413C2×D3—181no—
414C2×D3—181no—
415C2×D3—181no—
416C2×D3—181no—
417C2×D3—181no—
418C3×C4—21no—
419C3×C4—31no—
420C3×C4—31no—
421C3×C4—31no—
422C3×C4—31no—
423C3×C4—61no—
424C3×C4—61no—
425C3×C4—61no—
426C3×C4—61no—
427C3×C4—92no—
428C3×C4—121no—
429C3×C4—181no—
430C3×C4D3×#6711yescharacteristic
431C3×C4D3×#6711yescharacteristic
432Dic3—32no—
433Dic3—61no—
434Dic3—92no—
435Dic3—181no—
436C32—21no—
437C32#472711yescharacteristic, complemented
438C32#523411yescharacteristic, complemented
439C23—91no—
440C23—91no—
441C23((C3⋊D3)⋊C3)⋊C411yescharacteristic
442Q8—182no—
443D4—182no—
444D4—182no—
445D4—182no—
446C2×C4—21no—
447C2×C4—21no—
448C2×C4—91no—
449C2×C4—92no—
450C2×C4—92no—
451C2×C4—91no—
452C2×C4—181no—
453C2×C4—181no—
454C2×C4((C3⋊D3)⋊C3)⋊C411yescharacteristic
455C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
456C8—181no—
457C8—181no—
458C8—181no—
459C8—181no—
460C2×C3—31no—
461C2×C3—31no—
462C2×C3—31no—
463C2×C3—31no—
464C2×C3—31no—
465C2×C3—31no—
466C2×C3—41no—
467C2×C3—61no—
468C2×C3—61no—
469C2×C3—61no—
470C2×C3—121no—
471C2×C3—121no—
472C2×C3—122no—
473C2×C3—182no—
474C2×C3—361no—
475C2×C3#6381411yescharacteristic
476C2×C3#6381511yescharacteristic
477C2×C3D3×#21711yescharacteristic
478D3—62no—
479D3—121no—
480D3—182no—
481D3—361no—
482C22—21no—
483C22—21no—
484C22—21no—
485C22—92no—
486C22—181no—
487C22—181no—
488C22—181no—
489C22—181no—
490C22—181no—
491C22C2×((C3⋊D3)⋊C3)⋊C411yescharacteristic
492C4—21no—
493C4—92no—
494C4—181no—
495C4#8867611yescharacteristic
496C4#8867611yescharacteristic
497C3—31no—
498C3—31no—
499C3—61no—
500C3#1059114511yescharacteristic
501C2—41no—
502C2—182no—
503C2—361no—
504C2#1172105411yescharacteristic
505C2#1172105511yescharacteristic
506C2#1172131011yescharacteristic
subgroupG/H
centerC2×C4C2×((C3⋊D3)⋊C3)⋊C2
commutatorC22×H(3,𝔽3)C22×C4
FrattiniC2×C3×C4C2×D32
FittingC22×C4×H(3,𝔽3)C22
socleC22×C3C2×D3×Dic3
prime radicalC2×#3828⋊C3C2
cosocleC2×C4×H(3,𝔽3)C23
hypercenterC22×C4((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)#408
LCSG ▷ C22×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C2×C4 ◁ C22×C4

Complements

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

NH ≅ G/Ntype
2. C2×#88430C2semidirect
3. C2×((C3⋊D3)⋊C3)⋊C8C2semidirect
4. #11720573C2semidirect
5. #63657⋊C3C2semidirect
6. #11720317C2semidirect
7. #11720317C2semidirect
31. #63657D3semidirect
282. H(3,𝔽3)#408semidirect
437. C32#4727semidirect
438. C32#5234semidirect

Maximal subgroups

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

subgroupindex
C2×#3828⋊C32
C2×#884302
C2×((C3⋊D3)⋊C3)⋊C82
#117205732
#63657⋊C32
#117203172
#105911533
#105911453