Finite Group Databasebeta

← #11774468

Subgroups — #11774468

Order 1152.

537 conjugacy classes of subgroups; only the 100 normal subgroups are shown.

normal subgroups only

This group has 537 conjugacy classes of subgroups — too many to draw. Only its 100 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
1#10593875C211yesmaximal, characteristic
2C3⋊D3×C4⋊C8C211yesmaximal, characteristic, complemented
3C4×#3712C211yesmaximal, characteristic, complemented
4C3×#4484C211yesmaximal, characteristic, complemented
5C3×#4483C211yesmaximal, characteristic, complemented
6#10589084C211yesmaximal, characteristic, complemented
7#10589081C211yesmaximal, characteristic, complemented
8#68234—31nomaximal
9#68170—31nomaximal
10C2×C8×C3⋊D3—21no—
11C42×C3⋊D3C2211yescharacteristic
12C3×C4×D12C2211yescharacteristic
13C3×C4×Dic6C2211yescharacteristic
14#63916C411yescharacteristic
15C4×#626—21no—
16#63872C411yescharacteristic
17#63871C411yescharacteristic
18#63863C411yescharacteristic
19#63862—21no—
20#63855—21no—
21#63853—21no—
22C2×#3712—21no—
23C32×C4⋊C8C2211yescharacteristic
24#63641—21no—
25C4⋊(C3⋊(C3⋊C8))C2211yescharacteristic
26C3×C4×C3⋊C8C2211yescharacteristic
27C3×C4×C3⋊C8C2211yescharacteristic
28#63555—21no—
29#5580—31no—
30#5557—31no—
31D3×C4⋊C8—31no—
32D3×C4⋊C8—31no—
33D3×C4⋊C8—61no—
34C4×ΓL(1,𝔽25)—31no—
35C4×ΓL(1,𝔽25)—31no—
36C3×Q8⋊C8—31no—
37C3×D4⋊C8—31no—
38#4535—31no—
39D4⋊(C3⋊C8)—31no—
40#4484D311yescharacteristic, complemented
41#4483D311yescharacteristic, complemented
42#4459—31no—
43#4456—31no—
44C2×C4×C3⋊D3C2×C411yescharacteristic
45C2×C4×C3⋊D3D411yescharacteristic
46C2×C4×C3⋊D3D411yescharacteristic
47C2×C3×D12C2×C411yescharacteristic
48C2×C3×C4×D3—21no—
49C2×C3×Dic6C2×C411yescharacteristic
50C2×#626—21no—
51C2×C32×C8—21no—
52C32×C42C2311yescharacteristic
53C4×C3⋊Dic3C2×C411yescharacteristic
54C4×C3⋊Dic3D411yescharacteristic
55C4×C3⋊Dic3D411yescharacteristic
56C2×C3⋊(C3⋊C8)—21no—
57C8×C3⋊D3—42no—
58C3×(C2×D3)⋊C4—21no—
59C3×C4⋊Dic3C2×C411yescharacteristic
60C3×C4⋊Dic3C2×C411yescharacteristic
61C3×Dic3⋊C4—21no—
62C3×C4×Dic3—21no—
63C2×C3×C3⋊C8—21no—
64C2×C3×C3⋊C8—21no—
65Dic3⋊Dic3—21no—
66Dic3⋊Dic3—21no—
67Dic3⋊Dic3—21no—
68#3722—21no—
69(C2×D3)⋊Dic3—21no—
70(C2×D3)⋊Dic3—21no—
71(C2×D3)⋊Dic3—21no—
72#3712—24no—
73#1483—91noSylow
74C3×C4×Q8—31no—
75C3×C4×D4—31no—
76#947—31no—
77#946—31no—
78#938—31no—
79#936—31no—
80C4×#67—61no—
81#926—61no—
82C2×ΓL(1,𝔽25)—61no—
83C2×ΓL(1,𝔽25)—61no—
84C2×D3×C8—61no—
85C2×D3×C8—61no—
86C2×D3×C8—62no—
87#898—61no—
88#897—31no—
89#895—31no—
90#894—61no—
91#890—31no—
92#890—31no—
93#883—61no—
94#879—61no—
95C4×D12—61no—
96C4×D12C2×D311yescharacteristic
97#872—61no—
98C42×D3—31no—
99C42×D3—31no—
100C42×D3—61no—
101C4×Dic6C2×D311yescharacteristic
102C3×C4⋊C8—21no—
103C3×C4⋊C8C2×D311yescharacteristic
104C3×C4⋊C8C2×D311yescharacteristic
105C3×C4×C8—31no—
106C3×C4×C8—31no—
107#815—61no—
108#815—61no—
109Dic3⋊C8—61no—
110Dic3⋊C8—61no—
111Dic3⋊C8—62no—
112C4⋊(C3⋊C8)—31no—
113C4⋊(C3⋊C8)—31no—
114C4⋊(C3⋊C8)—61no—
115C4×C3⋊C8C2×D311yescharacteristic
116C4×C3⋊C8C2×D311yescharacteristic
117C22×C3⋊D3C22⋊C411yescharacteristic
118C22×C3×D3—21no—
119C2×C32×C4C22×C411yescharacteristic
120C2×C32×C4C2×D411yescharacteristic
121C2×C32×C4C2×D411yescharacteristic
122C2×C3⋊Dic3C22⋊C411yescharacteristic
123C2×C3⋊Dic3C22⋊C411yescharacteristic
124C2×C3⋊Dic3C22⋊C411yescharacteristic
125C4×C3⋊D3—21no—
126C4×C3⋊D3—21no—
127C4×C3⋊D3—21no—
128C4×C3⋊D3—21no—
129C4×C3⋊D3—22no—
130C2×C3×Dic3—21no—
131C2×C3×Dic3—21no—
132C2×C3×Dic3—21no—
133C3×D12—21no—
134C3×D12—21no—
135C3×C4×D3—41no—
136C3×C4×D3—41no—
137C3×Dic6—21no—
138C3×Dic6—21no—
139#626—42no—
140C32×C8—41no—
141C3⋊(C3⋊C8)—41no—
142C3×C3⋊C8—22no—
143C3×C3⋊C8—22no—
144#529—91no—
145C2×C4⋊C8—91no—
146C4×M16—91no—
147Q8⋊C8—31no—
148Q8⋊C8—91no—
149D4⋊C8—31no—
150D4⋊C8—91no—
151C2×C3×Q8—31no—
152C2×C3×D4—31no—
153C22×C3×C4—61no—
154C2×#67—61no—
155C2×D12—61no—
156C2×D12C4×D311yescharacteristic
157C2×C4×D3—21no—
158C2×C4×D3—31no—
159C2×C4×D3—31no—
160C2×C4×D3—31no—
161C2×C4×D3—31no—
162C2×C4×D3—31no—
163C2×C4×D3—31no—
164C2×C4×D3—61no—
165C2×C4×D3—61no—
166C2×C4×D3—61no—
167C2×Dic6C4×D311yescharacteristic
168C2×C3×C8—22no—
169C2×C3×C8—21no—
170C2×C3×C8—21no—
171C2×C3×C8—61no—
172C2×C3×C8—61no—
173C3×C4⋊C4—31no—
174C3×C4⋊C4—31no—
175C3×C4⋊C4—61no—
176C3×C22⋊C4—61no—
177C3×C42—21no—
178C3×C42—61no—
179C3×C42C22×D311yescharacteristic
180C3×C42C22×D311yescharacteristic
181#217—61no—
182#217—61no—
183#217—61no—
184(C2×D3)⋊C4—21no—
185(C2×D3)⋊C4—61no—
186(C2×D3)⋊C4—61no—
187(C2×D3)⋊C4—61no—
188(C2×D3)⋊C4—61no—
189(C2×D3)⋊C4—61no—
190C4⋊Dic3—61no—
191C4⋊Dic3—61no—
192C4⋊Dic3—61no—
193C4⋊Dic3C4×D311yescharacteristic
194C4⋊Dic3C4×D311yescharacteristic
195Dic3⋊C4—21no—
196Dic3⋊C4—61no—
197Dic3⋊C4—61no—
198Dic3⋊C4—61no—
199C4×Dic3—21no—
200C4×Dic3—31no—
201C4×Dic3—31no—
202C4×Dic3—31no—
203C4×Dic3—31no—
204C4×Dic3—31no—
205C4×Dic3—31no—
206C4×Dic3—61no—
207C4×Dic3—61no—
208C4×Dic3—61no—
209C2×C3⋊C8—21no—
210C2×C3⋊C8—21no—
211C2×C3⋊C8—62no—
212C2×C3⋊C8—61no—
213C2×C3⋊C8—61no—
214ΓL(1,𝔽25)—64no—
215ΓL(1,𝔽25)—64no—
216D3×C8—124no—
217D3×C8—122no—
218D3×C8—122no—
219C22×C32C2×C22⋊C411yescharacteristic
220C2×C3⋊D3—21no—
221C2×C3⋊D3—21no—
222C2×C3⋊D3#10412yes—
223C2×C3×D3—41no—
224C2×C3×D3—41no—
225C2×C3×D3—41no—
226C32×C4—21no—
227C32×C4#13111yescharacteristic
228C32×C4#13111yescharacteristic
229C32×C4AGL(1,ℤ/8)11yescharacteristic
230C32×C4#13711yescharacteristic
231C3⋊Dic3—21no—
232C3⋊Dic3—21no—
233C3⋊Dic3—21no—
234C3×Dic3—41no—
235C3×Dic3—41no—
236C3×Dic3—41no—
237C2×M16—181no—
238C22×C8—181no—
239#126—181no—
240#125—91no—
241#124—91no—
242#123—181no—
243C22⋊Q8—91no—
244#121—91no—
245C4×Q8—31no—
246C4×D4—31no—
247C4×D4—181no—
248#117—181no—
249C2×C42—91no—
250C4⋊C8—91no—
251C4⋊C8—181no—
252C4⋊C8D3211yescharacteristic
253#97—181no—
254C4×C8—31no—
255C4×C8—31no—
256C23×C3—61no—
257C22×D3—21no—
258C22×D3—31no—
259C22×D3—31no—
260C22×D3—61no—
261C3×Q8—61no—
262C3×Q8—61no—
263C3×D4—61no—
264C3×D4—61no—
265C2×C3×C4—21no—
266C2×C3×C4—21no—
267C2×C3×C4—21no—
268C2×C3×C4—61no—
269C2×C3×C4—61no—
270C2×C3×C4—61no—
271C2×C3×C4—121no—
272C2×C3×C4—121no—
273C2×C3×C4C2×C4×D311yescharacteristic
274C2×C3×C4C2×C4×D311yescharacteristic
275C2×C3×C4D3×D411yescharacteristic
276C2×C3×C4D3×D411yescharacteristic
277C2×C3×C4D3×D411yescharacteristic
278C2×C3×C4D3×D411yescharacteristic
279#67—122no—
280C2×Dic3—21no—
281C2×Dic3—21no—
282C2×Dic3—21no—
283C2×Dic3—31no—
284C2×Dic3—31no—
285C2×Dic3—31no—
286C2×Dic3—31no—
287C2×Dic3—31no—
288C2×Dic3—31no—
289C2×Dic3—61no—
290C2×Dic3—61no—
291C2×Dic3—61no—
292D12—21no—
293D12—21no—
294D12—122no—
295C4×D3—41no—
296C4×D3—41no—
297C4×D3—62no—
298C4×D3—62no—
299C4×D3—62no—
300C4×D3—62no—
301C4×D3—62no—
302C4×D3—62no—
303C4×D3—61no—
304C4×D3—61no—
305C4×D3—61no—
306C4×D3—61no—
307C4×D3—61no—
308C4×D3—61no—
309C4×D3—61no—
310C4×D3—61no—
311C4×D3—122no—
312Dic6—21no—
313Dic6—21no—
314C3×C8—42no—
315C3×C8—41no—
316C3×C8—41no—
317C3×C8—62no—
318C3×C8—62no—
319C3⋊C8—22no—
320C3⋊C8—22no—
321C3⋊C8—122no—
322C3⋊C8—121no—
323C3⋊C8—121no—
324C2×C32#40811yescharacteristic
325C2×C32#41711yescharacteristic
326C2×C32C2×#10411yescharacteristic
327C3⋊D3—22no—
328C3×D3—81no—
329C2×Q8—31no—
330C2×D4—31no—
331C2×D4—181no—
332C22×C4—61no—
333C22×C4—91no—
334C22×C4—91no—
335C22×C4—91no—
336M16—184no—
337C2×C8—21no—
338C2×C8—61no—
339C2×C8—61no—
340C2×C8—181no—
341C2×C8—362no—
342C4⋊C4—31no—
343C4⋊C4—31no—
344C4⋊C4—61no—
345C4⋊C4—181no—
346C4⋊C4—181no—
347C4⋊C4—181no—
348C22⋊C4—61no—
349C22⋊C4—181no—
350C22⋊C4—181no—
351C22⋊C4—181no—
352C22⋊C4—181no—
353C42—61no—
354C42—91no—
355C42—91no—
356C42—91no—
357C42C2×D3211yescharacteristic
358C22×C3—21no—
359C22×C3—121no—
360C22×C3—121no—
361C22×C3—121no—
362C22×C3D3×C22⋊C411yescharacteristic
363C22×C3D3×C22⋊C411yescharacteristic
364C2×D3—32no—
365C2×D3—32no—
366C2×D3—41no—
367C2×D3—41no—
368C2×D3—41no—
369C2×D3—62no—
370C2×D3—62no—
371C2×D3—62no—
372C2×D3—61no—
373C2×D3—61no—
374C2×D3—61no—
375C2×D3—61no—
376C3×C4—21no—
377C3×C4—21no—
378C3×C4—21no—
379C3×C4—21no—
380C3×C4—21no—
381C3×C4—21no—
382C3×C4—41no—
383C3×C4—121no—
384C3×C4—121no—
385C3×C4—121no—
386C3×C4#90111yescharacteristic
387C3×C4#90111yescharacteristic
388C3×C4#90711yescharacteristic
389C3×C4#90711yescharacteristic
390C3×C4#91111yescharacteristic
391C3×C4#91411yescharacteristic
392C3×C4#91511yescharacteristic
393C3×C4#91811yescharacteristic
394Dic3—41no—
395Dic3—41no—
396Dic3—41no—
397Dic3—62no—
398Dic3—62no—
399Dic3—61no—
400Dic3—61no—
401Dic3—61no—
402Dic3—61no—
403Dic3—61no—
404Dic3—61no—
405Dic3—121no—
406C32#148311yescharacteristic, complemented, Sylow
407C23—61no—
408C23—91no—
409Q8—61no—
410Q8—61no—
411D4—61no—
412D4—61no—
413D4—362no—
414C2×C4—61no—
415C2×C4—61no—
416C2×C4—61no—
417C2×C4—91no—
418C2×C4—91no—
419C2×C4—91no—
420C2×C4—121no—
421C2×C4—121no—
422C2×C4—182no—
423C2×C4—181no—
424C2×C4—181no—
425C2×C4—181no—
426C2×C4—181no—
427C2×C4C4×D3211yescharacteristic
428C2×C4#380211yescharacteristic
429C2×C4#381111yescharacteristic
430C8—41no—
431C8—62no—
432C8—62no—
433C8—361no—
434C2×C3—21no—
435C2×C3—21no—
436C2×C3—21no—
437C2×C3—241no—
438C2×C3#472711yescharacteristic
439C2×C3#472711yescharacteristic
440C2×C3#477111yescharacteristic
441C2×C3#480311yescharacteristic
442C2×C3D3×#10411yescharacteristic
443C2×C3D3×#10411yescharacteristic
444D3—64no—
445D3—62no—
446D3—62no—
447D3—81no—
448C22—92no—
449C22—121no—
450C22—121no—
451C22—121no—
452C22—181no—
453C22—181no—
454C22#6392111yescharacteristic
455C4—21no—
456C4—121no—
457C4—121no—
458C4—121no—
459C4—181no—
460C4—181no—
461C4—181no—
462C4#6380311yescharacteristic
463C4#6380511yescharacteristic
464C4#6393711yescharacteristic
465C4#6395111yescharacteristic
466C3—21no—
467C3#6817011yescharacteristic, complemented
468C3#6823411yescharacteristic, complemented
469C2—182no—
470C2—241no—
471C2#1059061211yescharacteristic
472C2#1059095811yescharacteristic
473C2#1059100811yescharacteristic
subgroupG/H
centerC2×C4#3811
commutatorC2×C32×C4C22×C4
FrattiniC42C2×D32
FittingC32×C4⋊C8C22
socleC22×C32C2×C22⋊C4
prime radical#63916C4
cosocleC32×C42C23
hypercenterC4⋊C8D32
hypocenterC32#1483
LCSG ▷ C2×C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C2×C4 ◁ C42 ◁ C4⋊C8

Complements

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

NH ≅ G/Ntype
2. C3⋊D3×C4⋊C8C2semidirect
3. C4×#3712C2semidirect
4. C3×#4484C2semidirect
5. C3×#4483C2semidirect
6. #10589084C2semidirect
7. #10589081C2semidirect
40. #4484D3semidirect
41. #4483D3semidirect
406. C32#1483semidirect
467. C3#68170semidirect
468. C3#68234semidirect

Maximal subgroups

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

subgroupindex
#105938752
C3⋊D3×C4⋊C82
C4×#37122
C3×#44842
C3×#44832
#105890842
#105890812
#682343
#681703