Finite Group Databasebeta

← #11778484

Subgroups — #11778484

Order 1152.

505 conjugacy classes of subgroups; only the 80 normal subgroups are shown.

normal subgroups only

This group has 505 conjugacy classes of subgroups — too many to draw. Only its 80 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#10594981C211yesmaximal, characteristic
2C3×#4721C211yesmaximal, characteristic, complemented
3#10591326C211yesmaximal, characteristic
4#10591101C211yesmaximal, characteristic, complemented
5#10589334C211yesmaximal, characteristic
6C3×#4467C211yesmaximal, characteristic, complemented
7#10589153C211yesmaximal, characteristic, complemented
8#71172—31nomaximal
9#68320—31nomaximal
10C2×#3752C2211yescharacteristic
11C2×C3×C4⋊Dic3C2211yescharacteristic
12C3×#924C2211yescharacteristic
13C3×#924C2211yescharacteristic
14#63982—21no—
15#63974—21no—
16C2×#3722—21no—
17#63895—21no—
18#63895—21no—
19#63888—21no—
20#63888—21no—
21C32×C22⋊C8C2211yescharacteristic
22#63657C2211yescharacteristic
23C3×C3⋊#95C2211yescharacteristic
24C3×#817—21no—
25C3×#816—21no—
26#63579—21no—
27#63574—21no—
28#63565—21no—
29#5634—31no—
30#5634—31no—
31C3×#483—31no—
32#5043—31no—
33#4992—31no—
34#4971—31no—
35#4721D311yescharacteristic, complemented
36#4674—31no—
37C3×#328—31no—
38#4538—31no—
39#4468—31no—
40#4468—31no—
41#4468—61no—
42#4467D311yescharacteristic, complemented
43#4452—31no—
44C23×C3⋊D3D411yescharacteristic
45C22×C32×C4C2311yescharacteristic
46C22×C3⋊Dic3D411yescharacteristic
47C22×C3×Dic3—21no—
48C22×C3×Dic3—21no—
49C2×C3×Dic6D411yescharacteristic
50C2×C3×Dic6D411yescharacteristic
51C2×#627—21no—
52C2×#624—21no—
53C2×#624—21no—
54C2×C32×C8—21no—
55#3752—21no—
56#3752—21no—
57C2×C3⋊(C3⋊C8)—21no—
58C3×#217—21no—
59C3×#217—21no—
60C3×C4⋊Dic3—21no—
61C3×C4⋊Dic3—21no—
62C3×C4⋊Dic3D411yescharacteristic
63C3×C4⋊Dic3D411yescharacteristic
64C3×Dic3⋊C4—21no—
65C3×Dic3⋊C4—21no—
66#3724—21no—
67Dic3⋊Dic3—21no—
68Dic3⋊Dic3—21no—
69#3722—21no—
70#3722—21no—
71#3722—42no—
72#1608—91noSylow
73C3×C22⋊Q8—31no—
74C3×C22⋊Q8—31no—
75C2×C3×C4⋊C4—31no—
76#946—61no—
77#946—61no—
78C2×(C2×D3)⋊C4—31no—
79C2×(C2×D3)⋊C4—31no—
80C2×(C2×D3)⋊C4—61no—
81C2×(C2×D3)⋊C4—61no—
82C2×(C2×D3)⋊C4—61no—
83C2×C4⋊Dic3C2×D311yescharacteristic
84#924C2×D311yescharacteristic
85#924C2×D311yescharacteristic
86#897—61no—
87#897—61no—
88#896—61no—
89#896—61no—
90#896—61no—
91#896—61no—
92D3×C22⋊C4—61no—
93D3×C22⋊C4—61no—
94#878—61no—
95#878—61no—
96C3×#106—61no—
97C3×#103—61no—
98C3×C22⋊C8—21no—
99C3×C22⋊C8C2×D311yescharacteristic
100C3×C22⋊C8C2×D311yescharacteristic
101C3×#95—31no—
102#835—61no—
103C3⋊#95—61no—
104C3⋊#95—61no—
105C3⋊#95C2×D311yescharacteristic
106#830—31no—
107#830—31no—
108#830—61no—
109#817—21no—
110#816—21no—
111#810—61no—
112#808—61no—
113#808—61no—
114C23×C32C2×D411yescharacteristic
115C22×C3⋊D3—21no—
116C22×C3⋊D3—21no—
117C22×C3⋊D3—41no—
118C22×C3⋊D3—41no—
119C22×C3⋊D3SD1612yes—
120C2×C32×C4—41no—
121C2×C32×C4C2×D411yescharacteristic
122C2×C32×C4C2×D411yescharacteristic
123C2×C3⋊Dic3—21no—
124C2×C3⋊Dic3—41no—
125C2×C3×Dic3—21no—
126C2×C3×Dic3—21no—
127C2×C3×Dic3—21no—
128C2×C3×Dic3—21no—
129C2×C3×Dic3—41no—
130C2×C3×Dic3—41no—
131C2×C3×Dic3—41no—
132C2×C3×Dic3—41no—
133C2×C3×Dic3—41no—
134C2×C3×Dic3—41no—
135C3×Dic6—41no—
136C3×Dic6—41no—
137#627—81no—
138#624—81no—
139#624—81no—
140C32×C8—41no—
141C3⋊(C3⋊C8)—41no—
142#543—91no—
143#483—31no—
144#483—91no—
145#397—91no—
146#328—31no—
147#328—91no—
148#323—91no—
149C23×D3—31no—
150C23×D3—31no—
151C23×D3—61no—
152C2×C3×Q8—31no—
153C2×C3×Q8—31no—
154C22×C3×C4—21no—
155C22×C3×C4—61no—
156C22×C3×C4—61no—
157C22×C3×C4C22×D311yescharacteristic
158C22×C3×C4C22×D311yescharacteristic
159C22×Dic3—21no—
160C22×Dic3—21no—
161C22×Dic3—31no—
162C22×Dic3—31no—
163C22×Dic3—61no—
164C2×C4×D3—61no—
165C2×C4×D3—61no—
166C2×C4×D3—61no—
167C2×C4×D3—61no—
168C2×Dic6—61no—
169C2×Dic6—61no—
170C2×Dic6D1211yescharacteristic
171C2×Dic6#6711yescharacteristic
172C2×C3×C8—21no—
173C2×C3×C8—21no—
174C2×C3×C8—41no—
175C3×C4⋊C4—31no—
176C3×C4⋊C4—31no—
177C3×C4⋊C4—61no—
178C3×C4⋊C4—61no—
179C3×C4⋊C4—61no—
180C3×C4⋊C4—61no—
181C3×C22⋊C4—61no—
182C3×C22⋊C4—61no—
183#217—21no—
184#217—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—
190(C2×D3)⋊C4—61no—
191(C2×D3)⋊C4—61no—
192(C2×D3)⋊C4—61no—
193(C2×D3)⋊C4—121no—
194(C2×D3)⋊C4—121no—
195(C2×D3)⋊C4—122no—
196(C2×D3)⋊C4—122no—
197C4⋊Dic3—21no—
198C4⋊Dic3—21no—
199C4⋊Dic3—61no—
200C4⋊Dic3—61no—
201C4⋊Dic3D1211yescharacteristic
202C4⋊Dic3#6711yescharacteristic
203Dic3⋊C4—21no—
204Dic3⋊C4—21no—
205Dic3⋊C4—61no—
206Dic3⋊C4—61no—
207Dic3⋊C4—61no—
208Dic3⋊C4—61no—
209C2×C3⋊C8—61no—
210C2×C3⋊C8—61no—
211C2×C3⋊C8—121no—
212C22×C32—21no—
213C22×C32—21no—
214C22×C32#12011yescharacteristic
215C22×C32C2×SD1611yescharacteristic
216C22×C32#13711yescharacteristic
217C2×C3⋊D3—22no—
218C2×C3⋊D3—41no—
219C2×C3⋊D3—41no—
220C2×C3⋊D3—42no—
221C2×C3⋊D3—81no—
222C32×C4—21no—
223C32×C4—21no—
224C3⋊Dic3—41no—
225C3×Dic3—41no—
226C3×Dic3—41no—
227C3×Dic3—41no—
228C3×Dic3—41no—
229C3×Dic3—81no—
230C3×Dic3—81no—
231C22⋊Q8—31no—
232C22⋊Q8—31no—
233C22⋊Q8—181no—
234C22⋊Q8—181no—
235C22⋊Q8—181no—
236C22⋊Q8—181no—
237C22⋊Q8—181no—
238C2×C4⋊C4—31no—
239C2×C22⋊C4—91no—
240C2×C22⋊C4—181no—
241C2×C22⋊C4—181no—
242#106—61no—
243#106—181no—
244#103—61no—
245#103—181no—
246C22⋊C8—91no—
247C22⋊C8D3211yescharacteristic
248#95—31no—
249#95—181no—
250C23×C3—21no—
251C23×C3D3×D411yescharacteristic
252C23×C3D3×D411yescharacteristic
253C22×D3—32no—
254C22×D3—32no—
255C22×D3—62no—
256C22×D3—61no—
257C22×D3—61no—
258C22×D3—61no—
259C22×D3—61no—
260C22×D3—121no—
261C22×D3—121no—
262C22×D3—121no—
263C22×D3—121no—
264C22×D3—121no—
265C22×D3—121no—
266C22×D3—241no—
267C22×D3—241no—
268C3×Q8—121no—
269C3×Q8—121no—
270C2×C3×C4—21no—
271C2×C3×C4—21no—
272C2×C3×C4—41no—
273C2×C3×C4—41no—
274C2×C3×C4—61no—
275C2×C3×C4—61no—
276C2×C3×C4—61no—
277C2×C3×C4—61no—
278C2×C3×C4—81no—
279C2×C3×C4—121no—
280C2×C3×C4—121no—
281C2×C3×C4—121no—
282C2×C3×C4—121no—
283C2×C3×C4—121no—
284C2×C3×C4—121no—
285C2×C3×C4C2×D1211yescharacteristic
286C2×C3×C4D3×D411yescharacteristic
287C2×C3×C4D3×D411yescharacteristic
288C2×C3×C4C2×#6711yescharacteristic
289C2×Dic3—21no—
290C2×Dic3—21no—
291C2×Dic3—21no—
292C2×Dic3—21no—
293C2×Dic3—41no—
294C2×Dic3—41no—
295C2×Dic3—41no—
296C2×Dic3—41no—
297C2×Dic3—41no—
298C2×Dic3—41no—
299C2×Dic3—61no—
300C2×Dic3—61no—
301C2×Dic3—121no—
302C2×Dic3—121no—
303C2×Dic3—121no—
304C2×Dic3—241no—
305C4×D3—241no—
306C4×D3—241no—
307C4×D3—241no—
308C4×D3—241no—
309Dic6—41no—
310Dic6—41no—
311Dic6—241no—
312Dic6—241no—
313C3×C8—41no—
314C3×C8—41no—
315C3×C8—81no—
316C3⋊C8—121no—
317C3⋊C8—121no—
318C3⋊C8—241no—
319C2×C32—21no—
320C2×C32—21no—
321C2×C32#44811yescharacteristic
322C2×C32#45511yescharacteristic
323C2×C32#45811yescharacteristic
324C3⋊D3—42no—
325C24—91no—
326C2×Q8—31no—
327C2×Q8—31no—
328C2×Q8—181no—
329C22×C4—61no—
330C22×C4—61no—
331C22×C4—91no—
332C22×C4—181no—
333C22×C4—181no—
334C22×C4C2×D3211yescharacteristic
335C2×C8—21no—
336C2×C8—181no—
337C4⋊C4—31no—
338C4⋊C4—31no—
339C4⋊C4—61no—
340C4⋊C4—61no—
341C4⋊C4—61no—
342C4⋊C4—61no—
343C4⋊C4—181no—
344C4⋊C4—181no—
345C4⋊C4—181no—
346C22⋊C4—61no—
347C22⋊C4—61no—
348C22⋊C4—181no—
349C22⋊C4—181no—
350C22⋊C4—181no—
351C22⋊C4—181no—
352C22⋊C4—362no—
353C22×C3—21no—
354C22×C3—21no—
355C22×C3—21no—
356C22×C3—21no—
357C22×C3—21no—
358C22×C3—21no—
359C22×C3—21no—
360C22×C3—41no—
361C22×C3—41no—
362C22×C3#88211yescharacteristic
363C22×C3D3×SD1611yescharacteristic
364C22×C3D3×SD1611yescharacteristic
365C22×C3#91511yescharacteristic
366C22×C3#91811yescharacteristic
367C22×C3#93711yescharacteristic
368C2×D3—62no—
369C2×D3—62no—
370C2×D3—122no—
371C2×D3—121no—
372C2×D3—121no—
373C2×D3—121no—
374C2×D3—121no—
375C2×D3—122no—
376C2×D3—122no—
377C2×D3—241no—
378C2×D3—241no—
379C2×D3—242no—
380C2×D3—241no—
381C2×D3—241no—
382C2×D3—242no—
383C3×C4—21no—
384C3×C4—21no—
385C3×C4—21no—
386C3×C4—21no—
387C3×C4—41no—
388C3×C4—41no—
389C3×C4—121no—
390C3×C4—121no—
391C3×C4—121no—
392C3×C4—121no—
393C3×C4—241no—
394C3×C4—241no—
395Dic3—41no—
396Dic3—41no—
397Dic3—41no—
398Dic3—41no—
399Dic3—81no—
400Dic3—81no—
401Dic3—121no—
402Dic3—121no—
403Dic3—241no—
404C32#160811yescharacteristic, complemented, Sylow
405C23—92no—
406C23—181no—
407C23—181no—
408C23—361no—
409C23—361no—
410C23#380211yescharacteristic
411Q8—121no—
412Q8—121no—
413Q8—721no—
414C2×C4—41no—
415C2×C4—61no—
416C2×C4—61no—
417C2×C4—61no—
418C2×C4—61no—
419C2×C4—121no—
420C2×C4—121no—
421C2×C4—121no—
422C2×C4—121no—
423C2×C4—121no—
424C2×C4—121no—
425C2×C4—181no—
426C2×C4—361no—
427C2×C4—721no—
428C2×C4—721no—
429C2×C4D3×D1211yescharacteristic
430C2×C4D3×#6711yescharacteristic
431C8—41no—
432C8—361no—
433C2×C3—21no—
434C2×C3—21no—
435C2×C3—21no—
436C2×C3—21no—
437C2×C3—21no—
438C2×C3—21no—
439C2×C3—21no—
440C2×C3—41no—
441C2×C3—41no—
442C2×C3#474211yescharacteristic
443C2×C3#480611yescharacteristic
444C2×C3#482511yescharacteristic
445C2×C3#505311yescharacteristic
446C2×C3#517011yescharacteristic
447C2×C3#520211yescharacteristic
448D3—122no—
449D3—122no—
450D3—242no—
451C22—21no—
452C22—21no—
453C22—182no—
454C22—362no—
455C22—361no—
456C22—361no—
457C22—721no—
458C22#6379611yescharacteristic
459C22#6381111yescharacteristic
460C22#6392211yescharacteristic
461C4—21no—
462C4—21no—
463C4—121no—
464C4—121no—
465C4—121no—
466C4—121no—
467C4—241no—
468C4—241no—
469C4—361no—
470C3—21no—
471C3#6832011yescharacteristic, complemented
472C3#7117211yescharacteristic, complemented
473C2—21no—
474C2—21no—
475C2—362no—
476C2#1059040811yescharacteristic
477C2#1059082411yescharacteristic
478C2#1059101111yescharacteristic
subgroupG/H
centerC22#63922
commutator, cosocleC22×C32×C4C23
FrattiniC22×C4C2×D32
FittingC32×C22⋊C8C22
socleC22×C32#120
prime radicalC23×C3⋊D3D4
hypercenterC22⋊C8D32
hypocenterC32#1608
LCSG ▷ C22×C32×C4 ▷ C22×C32 ▷ C32
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22⋊C8

Complements

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

NH ≅ G/Ntype
2. C3×#4721C2semidirect
4. #10591101C2semidirect
6. C3×#4467C2semidirect
7. #10589153C2semidirect
35. #4721D3semidirect
42. #4467D3semidirect
404. C32#1608semidirect
471. C3#68320semidirect
472. C3#71172semidirect

Maximal subgroups

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

subgroupindex
#105949812
C3×#47212
#105913262
#105911012
#105893342
C3×#44672
#105891532
#711723
#683203