Finite Group Databasebeta

← #11774656

Subgroups — #11774656

Order 1152.

573 conjugacy classes of subgroups.

The subgroup lattice has 481 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×C4×#237C211yesmaximal, characteristic
2#10590336C211yesmaximal, characteristic, complemented
3C4×#3711C211yesmaximal, characteristic, complemented
4D4⋊(C3⋊(C3⋊C8))C211yesmaximal, characteristic, complemented
5C3×D4⋊(C3⋊C8)C211yesmaximal, characteristic, complemented
6#10589084C211yesmaximal, characteristic, complemented
7#10589083C211yesmaximal, characteristic, complemented
8#71641—31nomaximal
9#68174—31nomaximal
10C2×C3×#237C411yescharacteristic
11C32×C4×D4C2211yescharacteristic
12C3×D4×Dic3C411yescharacteristic
13C3×C4×#67—21no—
14C2×C3×C4×Dic3—21no—
15C3×#892C411yescharacteristic
16C3×#887C411yescharacteristic
17C3×#67⋊C4—21no—
18C3×#877—21no—
19C3×C42×D3C2211yescharacteristic
20C3×C4×Dic6C2211yescharacteristic
21C2×#3711—21no—
22C2×#3711—21no—
23C4⋊(C3⋊(C3⋊C8))C2211yescharacteristic
24C4×C3⋊(C3⋊C8)C2211yescharacteristic
25C3×C4⋊(C3⋊C8)C2211yescharacteristic
26C3×C4×C3⋊C8C2211yescharacteristic
27Dic3⋊(C3⋊C8)—21no—
28#63554—21no—
29C3×C4×#41—31no—
30C4×#237D311yescharacteristic
31#4924—31no—
32C4×#208—31no—
33#4837—31no—
34C4×ΓL(1,𝔽25)—31no—
35C3×D4⋊C8—31no—
36#4535—31no—
37#4535—31no—
38D4⋊(C3⋊C8)—31no—
39D4⋊(C3⋊C8)—31no—
40D4⋊(C3⋊C8)—61no—
41D4⋊(C3⋊C8)D311yescharacteristic, complemented
42#4484—31no—
43#4456—31no—
44C2×C32×D4C2×C411yescharacteristic
45C22×C32×C4—21no—
46C2×C3×#67—21no—
47C22×C3×Dic3—21no—
48C22×C3×Dic3—21no—
49C3×#237—22no—
50C3×#237—22no—
51C2×C3×C4×D3C2×C411yescharacteristic
52C2×C3×C4×D3D411yescharacteristic
53C2×C3×C4×D3D411yescharacteristic
54C2×C3×Dic6C2×C411yescharacteristic
55C32×C4⋊C4C2×C411yescharacteristic
56C32×C22⋊C4—21no—
57C32×C42C2311yescharacteristic
58C2×C3⋊(C3⋊C8)—21no—
59C2×C3⋊(C3⋊C8)—21no—
60C3×#217—21no—
61C3×(C2×D3)⋊C4—21no—
62C3×C4⋊Dic3C2×C411yescharacteristic
63C3×Dic3⋊C4—21no—
64C3×Dic3⋊C4—21no—
65C3×C4×Dic3—21no—
66C3×C4×Dic3—21no—
67C3×C4×Dic3—21no—
68C3×C4×Dic3C2×C411yescharacteristic
69C3×C4×Dic3D411yescharacteristic
70C3×C4×Dic3D411yescharacteristic
71C2×C3×C3⋊C8—21no—
72C2×C3×C3⋊C8—21no—
73#3711—24no—
74#3711—42no—
75#1489—91noSylow
76C2×C3×#41—31no—
77C2×#237Dic311yescharacteristic
78C3×C4×Q8—31no—
79C3×C4×Q8—31no—
80C3×C4×D4—21no—
81C3×C4×D4—31no—
82C3×C4×D4—61no—
83C3×C4×D4—61no—
84C3×C4×D4C2×D311yescharacteristic
85C3×C4×D4C2×D311yescharacteristic
86C3×#117—31no—
87C3×#117—61no—
88C2×C3×C42—31no—
89C2×C3×C42—61no—
90D4×Dic3Dic311yescharacteristic
91C4×#67—21no—
92C2×C4×Dic3—21no—
93C2×#208—61no—
94C2×#208—61no—
95C2×ΓL(1,𝔽25)—61no—
96C2×ΓL(1,𝔽25)—61no—
97#892Dic311yescharacteristic
98#887Dic311yescharacteristic
99#67⋊C4—21no—
100#877—21no—
101C42×D3C2×D311yescharacteristic
102C4×Dic6C2×D311yescharacteristic
103C3×C4⋊C8—31no—
104C3×C4×C8—31no—
105#815—61no—
106Dic3⋊C8—61no—
107C4⋊(C3⋊C8)—31no—
108C4⋊(C3⋊C8)—31no—
109C4⋊(C3⋊C8)—61no—
110C4⋊(C3⋊C8)—61no—
111C4⋊(C3⋊C8)C2×D311yescharacteristic
112#803—61no—
113C4×C3⋊C8—31no—
114C4×C3⋊C8—31no—
115C4×C3⋊C8—61no—
116C4×C3⋊C8C2×D311yescharacteristic
117C23×C32—21no—
118C22×C3×D3C22⋊C411yescharacteristic
119C32×D4—21no—
120C32×D4M1612yescomplemented
121C2×C32×C4—21no—
122C2×C32×C4—41no—
123C2×C32×C4—41no—
124C2×C32×C4C22×C411yescharacteristic
125C2×C32×C4C2×D411yescharacteristic
126C2×C32×C4C2×D411yescharacteristic
127C3×#67—24no—
128C2×C3×Dic3—24no—
129C2×C3×Dic3—21no—
130C2×C3×Dic3—21no—
131C2×C3×Dic3—42no—
132C2×C3×Dic3C22⋊C411yescharacteristic
133C2×C3×Dic3C22⋊C411yescharacteristic
134C2×C3×Dic3C22⋊C411yescharacteristic
135C3×C4×D3—21no—
136C3×C4×D3—21no—
137C3×C4×D3—21no—
138C3×C4×D3—21no—
139C3×C4×D3—22no—
140C3×Dic6—21no—
141C3×Dic6M1612yescomplemented
142C3⋊(C3⋊C8)—22no—
143C3⋊(C3⋊C8)—41no—
144C3×C3⋊C8—22no—
145C3×C3⋊C8—41no—
146C4×#41—31no—
147#423—91no—
148C4×M16—91no—
149Q8⋊C8—91no—
150Q8⋊C8—91no—
151D4⋊C8—31no—
152D4⋊C8—91no—
153C3×#41—62no—
154C3×#41—62no—
155C2×C3×Q8—31no—
156C2×C3×D4—21no—
157C2×C3×D4—61no—
158C2×C3×D4C4×D311yescharacteristic
159C2×C3×D4C2×Dic311yescharacteristic
160C22×C3×C4—21no—
161C22×C3×C4—21no—
162C22×C3×C4—31no—
163C22×C3×C4—31no—
164C22×C3×C4—31no—
165C22×C3×C4—41no—
166C22×C3×C4—61no—
167C22×C3×C4—61no—
168C2×#67—21no—
169C22×Dic3—21no—
170C22×Dic3—21no—
171#237—22no—
172#237—22no—
173C2×C4×D3C2×Dic311yescharacteristic
174C2×C4×D3#6711yescharacteristic
175C2×C4×D3#6711yescharacteristic
176C2×Dic6C2×Dic311yescharacteristic
177C2×C3×C8—61no—
178C2×C3×C8—61no—
179C3×C4⋊C4—21no—
180C3×C4⋊C4—31no—
181C3×C4⋊C4—61no—
182C3×C4⋊C4—61no—
183C3×C4⋊C4C4×D311yescharacteristic
184C3×C4⋊C4C2×Dic311yescharacteristic
185C3×C22⋊C4—21no—
186C3×C22⋊C4—21no—
187C3×C22⋊C4—41no—
188C3×C22⋊C4—61no—
189C3×C22⋊C4—61no—
190C3×C42—21no—
191C3×C42—31no—
192C3×C42—31no—
193C3×C42—31no—
194C3×C42—61no—
195C3×C42—61no—
196C3×C42—61no—
197C3×C42C22×D311yescharacteristic
198C3×C42C22×D311yescharacteristic
199#217—21no—
200(C2×D3)⋊C4—21no—
201C4⋊Dic3C2×Dic311yescharacteristic
202Dic3⋊C4—21no—
203Dic3⋊C4—21no—
204C4×Dic3—21no—
205C4×Dic3—21no—
206C4×Dic3—21no—
207C4×Dic3C2×Dic311yescharacteristic
208C4×Dic3#6711yescharacteristic
209C4×Dic3#6711yescharacteristic
210#208—64no—
211#208—122no—
212C2×C3⋊C8—21no—
213C2×C3⋊C8—21no—
214C2×C3⋊C8—61no—
215C2×C3⋊C8—61no—
216C2×C3⋊C8—61no—
217C2×C3⋊C8—61no—
218C2×C3⋊C8—121no—
219C2×C3⋊C8—121no—
220ΓL(1,𝔽25)—64no—
221ΓL(1,𝔽25)—122no—
222C22×C32—22no—
223C22×C32—41no—
224C22×C32—41no—
225C22×C32C2×C22⋊C411yescharacteristic
226C2×C3×D3—21no—
227C2×C3×D3—21no—
228C2×C3×D3—21no—
229C32×C4—21no—
230C32×C4—41no—
231C32×C4C2×M1611yescharacteristic
232C32×C4C2×M1611yescharacteristic
233C32×C4AGL(1,ℤ/8)11yescharacteristic
234C32×C4#13711yescharacteristic
235C3×Dic3—22no—
236C3×Dic3—21no—
237C3×Dic3—21no—
238C3×Dic3—41no—
239C3×Dic3#10412yes—
240C2×#41—31no—
241C2×M16—181no—
242C2×M16—181no—
243C4×Q8—31no—
244C4×Q8—31no—
245C4×D4—31no—
246C4×D4—61no—
247C4×D4—61no—
248C4×D4D3211yescharacteristic
249#117—31no—
250#117—61no—
251C2×C42—31no—
252C2×C42—61no—
253C4⋊C8—31no—
254C4⋊C8—91no—
255C4⋊C8—181no—
256#97—181no—
257C4×C8—31no—
258C4×C8—91no—
259C23×C3—21no—
260C23×C3—21no—
261C23×C3—31no—
262C23×C3—41no—
263C22×D3#21711yescharacteristic
264C3×Q8—32no—
265C3×Q8—61no—
266C3×D4—21no—
267C3×D4—21no—
268C3×D4—22no—
269C3×D4—41no—
270C3×D4—64no—
271C3×D4ΓL(1,𝔽25)12yescomplemented
272C3×D4#20812yescomplemented
273C2×C3×C4—21no—
274C2×C3×C4—21no—
275C2×C3×C4—21no—
276C2×C3×C4—21no—
277C2×C3×C4—21no—
278C2×C3×C4—31no—
279C2×C3×C4—31no—
280C2×C3×C4—31no—
281C2×C3×C4—41no—
282C2×C3×C4—41no—
283C2×C3×C4—41no—
284C2×C3×C4—41no—
285C2×C3×C4—41no—
286C2×C3×C4—62no—
287C2×C3×C4—64no—
288C2×C3×C4—61no—
289C2×C3×C4—61no—
290C2×C3×C4—61no—
291C2×C3×C4—61no—
292C2×C3×C4—61no—
293C2×C3×C4—61no—
294C2×C3×C4—81no—
295C2×C3×C4—81no—
296C2×C3×C4—122no—
297C2×C3×C4C2×C4×D311yescharacteristic
298C2×C3×C4D3×D411yescharacteristic
299C2×C3×C4D3×D411yescharacteristic
300C2×C3×C4C22×Dic311yescharacteristic
301C2×C3×C4C2×#6711yescharacteristic
302C2×C3×C4C2×#6711yescharacteristic
303#67—24no—
304C2×Dic3—24no—
305C2×Dic3—21no—
306C2×Dic3—21no—
307C2×Dic3—42no—
308C2×Dic3#21711yescharacteristic
309C2×Dic3#21711yescharacteristic
310C2×Dic3#21711yescharacteristic
311C4×D3—21no—
312C4×D3—21no—
313C4×D3—21no—
314C4×D3—21no—
315C4×D3—22no—
316Dic6—21no—
317Dic6#20812yescomplemented
318C3×C8—62no—
319C3×C8—121no—
320C3⋊C8—22no—
321C3⋊C8—41no—
322C3⋊C8—62no—
323C3⋊C8—62no—
324C3⋊C8—122no—
325C3⋊C8—121no—
326C3⋊C8—121no—
327C3⋊C8—241no—
328C2×C32—42no—
329C2×C32#40711yescharacteristic
330C2×C32#41711yescharacteristic
331C2×C32C2×#10411yescharacteristic
332C3×D3—41no—
333#41—62no—
334#41—62no—
335C2×Q8—31no—
336C2×D4—61no—
337C2×D4D3×Dic311yescharacteristic
338C22×C4—21no—
339C22×C4—31no—
340C22×C4—31no—
341C22×C4—31no—
342C22×C4—61no—
343C22×C4—61no—
344M16—184no—
345M16—362no—
346C2×C8—61no—
347C2×C8—61no—
348C2×C8—181no—
349C2×C8—181no—
350C4⋊C4—31no—
351C4⋊C4—61no—
352C4⋊C4—61no—
353C4⋊C4D3×Dic311yescharacteristic
354C22⋊C4—21no—
355C22⋊C4—61no—
356C22⋊C4—61no—
357C42—31no—
358C42—31no—
359C42—31no—
360C42—61no—
361C42—61no—
362C42—61no—
363C42C2×D3211yescharacteristic
364C22×C3—22no—
365C22×C3—22no—
366C22×C3—21no—
367C22×C3—42no—
368C22×C3—41no—
369C22×C3—41no—
370C22×C3—41no—
371C22×C3—41no—
372C22×C3—61no—
373C22×C3—61no—
374C22×C3—61no—
375C22×C3—81no—
376C22×C3—81no—
377C22×C3D3×C22⋊C411yescharacteristic
378C22×C3C2×#21711yescharacteristic
379C2×D3—21no—
380C2×D3—21no—
381C2×D3—21no—
382C3×C4—21no—
383C3×C4—21no—
384C3×C4—21no—
385C3×C4—21no—
386C3×C4—21no—
387C3×C4—21no—
388C3×C4—32no—
389C3×C4—41no—
390C3×C4—41no—
391C3×C4—41no—
392C3×C4—61no—
393C3×C4—62no—
394C3×C4—61no—
395C3×C4—81no—
396C3×C4—121no—
397C3×C4C2×ΓL(1,𝔽25)11yescharacteristic
398C3×C4D3×M1611yescharacteristic
399C3×C4#91111yescharacteristic
400C3×C4#91511yescharacteristic
401C3×C4C2×#20811yescharacteristic
402C3×C4C2×#20811yescharacteristic
403C3×C4#94911yescharacteristic
404C3×C4#95111yescharacteristic
405Dic3—22no—
406Dic3—21no—
407Dic3—21no—
408Dic3—41no—
409Dic3#83712yes—
410C32#148911yescharacteristic, complemented, Sylow
411C23—21no—
412C23—31no—
413Q8—32no—
414Q8—61no—
415D4—21no—
416D4—64no—
417D4#371112yescomplemented
418C2×C4—21no—
419C2×C4—31no—
420C2×C4—31no—
421C2×C4—31no—
422C2×C4—41no—
423C2×C4—41no—
424C2×C4—62no—
425C2×C4—64no—
426C2×C4—61no—
427C2×C4—61no—
428C2×C4—61no—
429C2×C4—61no—
430C2×C4—61no—
431C2×C4—61no—
432C2×C4—122no—
433C2×C4C2×D3×Dic311yescharacteristic
434C2×C4D3×#6711yescharacteristic
435C2×C4D3×#6711yescharacteristic
436C8—62no—
437C8—121no—
438C8—182no—
439C8—361no—
440C2×C3—21no—
441C2×C3—21no—
442C2×C3—21no—
443C2×C3—42no—
444C2×C3—42no—
445C2×C3—82no—
446C2×C3—121no—
447C2×C3#472611yescharacteristic
448C2×C3#477111yescharacteristic
449C2×C3D3×#10411yescharacteristic
450C2×C3#520711yescharacteristic
451C2×C3#523211yescharacteristic
452C2×C3C2×#83711yescharacteristic
453D3—41no—
454C22—22no—
455C22—41no—
456C22—41no—
457C22—61no—
458C22—61no—
459C22—61no—
460C22D3×#21711yescharacteristic
461C4—21no—
462C4—32no—
463C4—41no—
464C4—61no—
465C4—62no—
466C4—61no—
467C4—121no—
468C4D3×#20811yescharacteristic
469C4C2×#371111yescharacteristic
470C4#6392511yescharacteristic
471C4#6392911yescharacteristic
472C3—21no—
473C3#6817411yescharacteristic, complemented
474C3#7164111yescharacteristic, complemented
475C2—42no—
476C2—121no—
477C2#1059115411yescharacteristic
478C2#1059141411yescharacteristic
479C2D3×#83711yescharacteristic
subgroupG/H
centerC2×C4D3×#67
commutatorC2×C32×C4C22×C4
FrattiniC42C2×D32
FittingC32×C4×D4C22
socleC22×C32C2×C22⋊C4
prime radicalC2×C3×#237C4
cosocleC32×C42C23
hypercenterC4×D4D32
hypocenterC32#1489
LCSG ▷ C2×C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C2×C4 ◁ C42 ◁ C4×D4

Complements

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

NH ≅ G/Ntype
2. #10590336C2semidirect
3. C4×#3711C2semidirect
4. D4⋊(C3⋊(C3⋊C8))C2semidirect
5. C3×D4⋊(C3⋊C8)C2semidirect
6. #10589084C2semidirect
7. #10589083C2semidirect
41. D4⋊(C3⋊C8)D3semidirect
120. C32×D4M16semidirect
141. C3×Dic6M16semidirect
271. C3×D4ΓL(1,𝔽25)semidirect
272. C3×D4#208semidirect
317. Dic6#208semidirect
410. C32#1489semidirect
417. D4#3711semidirect
473. C3#68174semidirect
474. C3#71641semidirect

Maximal subgroups

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

subgroupindex
C3×C4×#2372
#105903362
C4×#37112
D4⋊(C3⋊(C3⋊C8))2
C3×D4⋊(C3⋊C8)2
#105890842
#105890832
#716413
#681743