Finite Group Databasebeta

← #13140052

Subgroups — #13140052

Order 1728.

521 conjugacy classes of subgroups; only the 89 normal subgroups are shown.

normal subgroups only

This group has 521 conjugacy classes of subgroups — too many to draw. Only its 89 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
1C4×#6235C211yesmaximal, characteristic
2C2×#3748⋊C3C211yesmaximal, characteristic, complemented
3C2×#88435C211yesmaximal, characteristic, complemented
4#63657⋊C3C211yesmaximal, characteristic, complemented
5#11720323C211yesmaximal, characteristic, complemented
6#11720304C211yesmaximal, characteristic, complemented
7#11720302C211yesmaximal, characteristic, complemented
8#10591156—31nomaximal
9#10591067—31nomaximal
10C22×C4×H(3,𝔽3)C2211yescharacteristic
11C2×#6235C411yescharacteristic
12C2×C4×#272C2211yescharacteristic
13#88552C411yescharacteristic
14#88542C411yescharacteristic
15#88540C411yescharacteristic
16C4×#1084C2211yescharacteristic
17#3748⋊C3—22no—
18C2×(C3⋊(C3⋊C8))⋊C3C2211yescharacteristic
19C2×(C3⋊(C3⋊C8))⋊C3C2211yescharacteristic
20C2×(C3⋊(C3⋊C8))⋊C3C2211yescharacteristic
21C2×(C3⋊(C3⋊C8))⋊C3C2211yescharacteristic
22#88435—22no—
23C2×#3748D311yescharacteristic, complemented
24C3×C4×#67—31no—
25C3×C4×#67—31no—
26C3×C4×#67—61no—
27C2×C3×#208—31no—
28C2×#3711—31no—
29C2×#3711—31no—
30#63657D311yescharacteristic, complemented
31C3×#830—31no—
32Dic3⋊(C3⋊C8)—31no—
33Dic3⋊(C3⋊C8)—31no—
34(C2×D3)⋊(C3⋊C8)—31no—
35(C2×D3)⋊(C3⋊C8)—31no—
36Dic3×C3⋊C8—31no—
37Dic3×C3⋊C8—31no—
38C23×H(3,𝔽3)C2×C411yescharacteristic
39C22×#272C2×C411yescharacteristic
40C2×C4×H(3,𝔽3)—21no—
41C2×C4×H(3,𝔽3)—21no—
42C2×C4×H(3,𝔽3)C2×C411yescharacteristic
43C2×C4×H(3,𝔽3)C2311yescharacteristic
44#6235—22no—
45C2×#1084C2×C411yescharacteristic
46C2×#1084C2×C411yescharacteristic
47C2×#1084C2×C411yescharacteristic
48C4×#272—21no—
49C4×#272—21no—
50(C3⋊(C3⋊C8))⋊C3—21no—
51(C3⋊(C3⋊C8))⋊C3—21no—
52(C3⋊(C3⋊C8))⋊C3—21no—
53(C3⋊(C3⋊C8))⋊C3D412yescomplemented
54#5128—91no—
55#5012—91no—
56#4729—91no—
57C22×C32×C4—21no—
58C22×C32×C4C2×D311yescharacteristic
59C22×C32×C4C2×D311yescharacteristic
60C2×C3×#67—31no—
61C2×C3×#67—31no—
62C2×C3×#67—61no—
63C2×C3×C4×D3—31no—
64C2×C3×C4×D3—31no—
65C2×C3×C4×D3—61no—
66#3748—22no—
67C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
68C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
69C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
70C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
71C3×#217—31no—
72C3×#217—31no—
73C3×#217—61no—
74C3×(C2×D3)⋊C4—31no—
75C3×(C2×D3)⋊C4—31no—
76C3×(C2×D3)⋊C4—61no—
77C3×Dic3⋊C4—31no—
78C3×Dic3⋊C4—31no—
79C3×Dic3⋊C4—61no—
80C3×C4×Dic3—31no—
81C3×C4×Dic3—31no—
82C3×C4×Dic3—61no—
83C3×#208—62no—
84C2×C3×C3⋊C8—31no—
85C2×C3×C3⋊C8—31no—
86C2×C3×C3⋊C8—31no—
87C2×C3×C3⋊C8—31no—
88#3711—62no—
89#3711—62no—
90C22×H(3,𝔽3)—21no—
91C22×H(3,𝔽3)—21no—
92C22×H(3,𝔽3)—21no—
93C22×H(3,𝔽3)C22×C411yescharacteristic
94C2×#272—21no—
95C2×#272—21no—
96C2×#272—21no—
97C4×H(3,𝔽3)—21no—
98C4×H(3,𝔽3)C2×D411yescharacteristic
99C4×H(3,𝔽3)#4111yescharacteristic
100#1084—21no—
101#1084—21no—
102#1084M1612yescomplemented
103C2×C3×M16—91no—
104C3×C4×D4—91no—
105C4×#67—31no—
106C4×#67—31no—
107C4×#67—61no—
108C2×#208—31no—
109C2×#208—91no—
110C2×#208—91no—
111C2×ΓL(1,𝔽25)—91no—
112C2×ΓL(1,𝔽25)—91no—
113C3×C22⋊C8—91no—
114#830—31no—
115#830—91no—
116#830—91no—
117(C2×D3)⋊C8—91no—
118(C2×D3)⋊C8—91no—
119Dic3⋊C8—91no—
120Dic3⋊C8—91no—
121C8×Dic3—91no—
122C8×Dic3—91no—
123C4⋊(C3⋊C8)—91no—
124C4×C3⋊C8—91no—
125C23×C32—21no—
126C23×C32C4×D311yescharacteristic
127C23×C32C4×D311yescharacteristic
128C22×C3×D3—31no—
129C22×C3×D3—31no—
130C22×C3×D3—61no—
131C2×C32×C4—21no—
132C2×C32×C4—21no—
133C2×C32×C4—21no—
134C2×C32×C4—21no—
135C2×C32×C4—21no—
136C2×C32×C4—21no—
137C2×C32×C4—41no—
138C2×C32×C4—41no—
139C2×C32×C4C4×D311yescharacteristic
140C2×C32×C4C4×D311yescharacteristic
141C2×C32×C4C22×D311yescharacteristic
142C2×C32×C4C22×D311yescharacteristic
143C3×#67—62no—
144C3×#67—62no—
145C3×#67—64no—
146C2×C3×Dic3—31no—
147C2×C3×Dic3—31no—
148C2×C3×Dic3—31no—
149C2×C3×Dic3—31no—
150C2×C3×Dic3—31no—
151C2×C3×Dic3—31no—
152C2×C3×Dic3—61no—
153C2×C3×Dic3—61no—
154C2×C3×Dic3—61no—
155C3×C4×D3—61no—
156C3×C4×D3—61no—
157C3×C4×D3—61no—
158C3×C4×D3—61no—
159C3×C4×D3—121no—
160C3×C4×D3—121no—
161C3⋊(C3⋊C8)—21no—
162C3⋊(C3⋊C8)—21no—
163C3⋊(C3⋊C8)—21no—
164C3⋊(C3⋊C8)#6712yescomplemented
165C3×C3⋊C8—32no—
166C3×C3⋊C8—61no—
167C3×C3⋊C8—61no—
168C3×C3⋊C8—61no—
169#436—271noSylow
170C2×H(3,𝔽3)—41no—
171C2×H(3,𝔽3)C4×D411yescharacteristic
172C2×H(3,𝔽3)C2×M1611yescharacteristic
173C2×H(3,𝔽3)#13111yescharacteristic
174#272—41no—
175C2×C3×D4—91no—
176C22×C3×C4—31no—
177C22×C3×C4—31no—
178C22×C3×C4—61no—
179C22×C3×C4—91no—
180C22×C3×C4D3211yescharacteristic
181C2×#67—31no—
182C2×#67—31no—
183C2×#67—61no—
184C2×C4×D3—31no—
185C2×C4×D3—31no—
186C2×C4×D3—61no—
187C3×M16—182no—
188C2×C3×C8—91no—
189C2×C3×C8—91no—
190C2×C3×C8—91no—
191C2×C3×C8—91no—
192C3×C4⋊C4—91no—
193C3×C22⋊C4—91no—
194C3×C22⋊C4—91no—
195C3×C42—91no—
196#217—31no—
197#217—31no—
198#217—61no—
199(C2×D3)⋊C4—31no—
200(C2×D3)⋊C4—31no—
201(C2×D3)⋊C4—61no—
202Dic3⋊C4—31no—
203Dic3⋊C4—31no—
204Dic3⋊C4—61no—
205C4×Dic3—31no—
206C4×Dic3—31no—
207C4×Dic3—61no—
208#208—62no—
209#208—182no—
210#208—182no—
211C2×C3⋊C8—31no—
212C2×C3⋊C8—31no—
213C2×C3⋊C8—31no—
214C2×C3⋊C8—31no—
215C2×C3⋊C8—91no—
216C2×C3⋊C8—91no—
217C2×C3⋊C8—91no—
218C2×C3⋊C8—91no—
219ΓL(1,𝔽25)—182no—
220ΓL(1,𝔽25)—182no—
221C22×C32—22no—
222C22×C32—21no—
223C22×C32—21no—
224C22×C32—21no—
225C22×C32—21no—
226C22×C32—21no—
227C22×C32—21no—
228C22×C32—21no—
229C22×C32—41no—
230C22×C32—41no—
231C22×C32C2×C4×D311yescharacteristic
232C22×C32C2×C4×D311yescharacteristic
233C2×C3×D3—62no—
234C2×C3×D3—61no—
235C2×C3×D3—61no—
236C2×C3×D3—61no—
237C2×C3×D3—61no—
238C2×C3×D3—61no—
239C2×C3×D3—61no—
240C2×C3×D3—121no—
241C2×C3×D3—121no—
242C32×C4—21no—
243C32×C4—21no—
244C32×C4—21no—
245C32×C4—21no—
246C32×C4—41no—
247C32×C4#23511yescharacteristic
248C32×C4D3×D411yescharacteristic
249C32×C4#23711yescharacteristic
250C32×C4C2×#6711yescharacteristic
251C3×Dic3—32no—
252C3×Dic3—32no—
253C3×Dic3—62no—
254C3×Dic3—61no—
255C3×Dic3—61no—
256C3×Dic3—61no—
257C3×Dic3—61no—
258C3×Dic3—121no—
259C3×Dic3—121no—
260C2×M16—91no—
261C2×M16—271no—
262C4×D4—91no—
263C4⋊C8—271no—
264C22⋊C8—91no—
265C22⋊C8—271no—
266C4×C8—271no—
267H(3,𝔽3)#43611yescharacteristic, complemented, Sylow
268C23×C3—31no—
269C23×C3—31no—
270C23×C3—61no—
271C23×C3—91no—
272C23×C3#62411yescharacteristic
273C22×D3—31no—
274C22×D3—31no—
275C22×D3—61no—
276C3×D4—182no—
277C2×C3×C4—21no—
278C2×C3×C4—21no—
279C2×C3×C4—31no—
280C2×C3×C4—31no—
281C2×C3×C4—31no—
282C2×C3×C4—31no—
283C2×C3×C4—61no—
284C2×C3×C4—61no—
285C2×C3×C4—61no—
286C2×C3×C4—61no—
287C2×C3×C4—61no—
288C2×C3×C4—61no—
289C2×C3×C4—91no—
290C2×C3×C4—91no—
291C2×C3×C4—91no—
292C2×C3×C4—121no—
293C2×C3×C4—121no—
294C2×C3×C4—181no—
295C2×C3×C4—181no—
296C2×C3×C4#62411yescharacteristic
297C2×C3×C4C2×D3211yescharacteristic
298#67—62no—
299#67—62no—
300#67—64no—
301C2×Dic3—31no—
302C2×Dic3—31no—
303C2×Dic3—31no—
304C2×Dic3—31no—
305C2×Dic3—31no—
306C2×Dic3—31no—
307C2×Dic3—61no—
308C2×Dic3—61no—
309C2×Dic3—61no—
310C4×D3—61no—
311C4×D3—61no—
312C4×D3—61no—
313C4×D3—61no—
314C4×D3—121no—
315C4×D3—121no—
316C3×C8—92no—
317C3×C8—181no—
318C3×C8—181no—
319C3×C8—181no—
320C3⋊C8—32no—
321C3⋊C8—61no—
322C3⋊C8—61no—
323C3⋊C8—61no—
324C3⋊C8—92no—
325C3⋊C8—181no—
326C3⋊C8—181no—
327C3⋊C8—181no—
328C2×C32—21no—
329C2×C32—21no—
330C2×C32—21no—
331C2×C32—42no—
332C2×C32—41no—
333C2×C32—41no—
334C2×C32#67⋊C411yescharacteristic
335C2×C32#90111yescharacteristic
336C2×C32D3×M1611yescharacteristic
337C2×C32D3×M1611yescharacteristic
338C2×C32#90711yescharacteristic
339C2×C32C4×#6711yescharacteristic
340C3×D3—122no—
341C3×D3—121no—
342C3×D3—121no—
343C2×D4—91no—
344C22×C4—91no—
345C22×C4((C3⋊D3)⋊C3)⋊C211yescharacteristic
346M16—182no—
347M16—542no—
348C2×C8—91no—
349C2×C8—91no—
350C2×C8—91no—
351C2×C8—91no—
352C4⋊C4—91no—
353C22⋊C4—91no—
354C22⋊C4—91no—
355C42—91no—
356C22×C3—21no—
357C22×C3—21no—
358C22×C3—21no—
359C22×C3—31no—
360C22×C3—31no—
361C22×C3—61no—
362C22×C3—61no—
363C22×C3—61no—
364C22×C3—61no—
365C22×C3—61no—
366C22×C3—61no—
367C22×C3—62no—
368C22×C3—61no—
369C22×C3—121no—
370C22×C3—121no—
371C22×C3—181no—
372C22×C3—181no—
373C22×C3—181no—
374C22×C3C2×#62411yescharacteristic
375C2×D3—61no—
376C2×D3—61no—
377C2×D3—61no—
378C2×D3—61no—
379C2×D3—61no—
380C2×D3—61no—
381C2×D3—62no—
382C2×D3—121no—
383C2×D3—121no—
384C3×C4—21no—
385C3×C4—31no—
386C3×C4—31no—
387C3×C4—31no—
388C3×C4—31no—
389C3×C4—61no—
390C3×C4—61no—
391C3×C4—61no—
392C3×C4—61no—
393C3×C4—92no—
394C3×C4—121no—
395C3×C4—181no—
396C3×C4—181no—
397C3×C4#380411yescharacteristic
398C3×C4D3×#6711yescharacteristic
399Dic3—32no—
400Dic3—32no—
401Dic3—62no—
402Dic3—61no—
403Dic3—61no—
404Dic3—61no—
405Dic3—61no—
406Dic3—121no—
407Dic3—121no—
408C32—21no—
409C32#472911yescharacteristic, complemented
410C32#512811yescharacteristic, complemented
411C23—91no—
412C23#619911yescharacteristic
413D4—182no—
414C2×C4—21no—
415C2×C4—21no—
416C2×C4—91no—
417C2×C4—91no—
418C2×C4—91no—
419C2×C4—181no—
420C2×C4—181no—
421C2×C4#619911yescharacteristic
422C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
423C8—92no—
424C8—181no—
425C8—181no—
426C8—181no—
427C2×C3—31no—
428C2×C3—31no—
429C2×C3—31no—
430C2×C3—31no—
431C2×C3—31no—
432C2×C3—31no—
433C2×C3—41no—
434C2×C3—61no—
435C2×C3—61no—
436C2×C3—61no—
437C2×C3—121no—
438C2×C3—121no—
439C2×C3—122no—
440C2×C3—361no—
441C2×C3#6381711yescharacteristic
442C2×C3#6381811yescharacteristic
443C2×C3#6395211yescharacteristic
444D3—121no—
445D3—121no—
446D3—122no—
447C22—21no—
448C22—21no—
449C22—21no—
450C22—181no—
451C22—181no—
452C22—181no—
453C22C2×#619911yescharacteristic
454C4—21no—
455C4—92no—
456C4—181no—
457C4—181no—
458C4#8867111yescharacteristic
459C4#8867611yescharacteristic
460C3—31no—
461C3—31no—
462C3—61no—
463C3#1059105111yescharacteristic
464C2—41no—
465C2—361no—
466C2#1172106411yescharacteristic
467C2#1172106511yescharacteristic
468C2#1172129311yescharacteristic
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×#624
prime radicalC2×#6235C4
cosocleC2×C4×H(3,𝔽3)C23
hypercenterC22×C4((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)#436
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×#3748⋊C3C2semidirect
3. C2×#88435C2semidirect
4. #63657⋊C3C2semidirect
5. #11720323C2semidirect
6. #11720304C2semidirect
7. #11720302C2semidirect
23. C2×#3748D3semidirect
30. #63657D3semidirect
53. (C3⋊(C3⋊C8))⋊C3D4semidirect
102. #1084M16semidirect
164. C3⋊(C3⋊C8)#67semidirect
267. H(3,𝔽3)#436semidirect
409. C32#4729semidirect
410. C32#5128semidirect

Maximal subgroups

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

subgroupindex
C4×#62352
C2×#3748⋊C32
C2×#884352
#63657⋊C32
#117203232
#117203042
#117203022
#105911563
#105910673