Finite Group Databasebeta

← #11791852

Subgroups — #11791852

Order 1152.

809 conjugacy classes of subgroups.

The subgroup lattice has 495 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#64133C211yesmaximal, characteristic
2C3×#5212C211yesmaximal, characteristic
3C3×C4×#217C211yesmaximal, characteristic
4#10589187C214yesmaximal
5#69529—31nomaximal
6#69457—31nomaximal
7C23×C32×C4C2211yescharacteristic
8C22×#634C411yescharacteristic, complemented
9C2×#3757C2212yes—
10C2×C4⋊(C3⋊Dic3)C411yescharacteristic, complemented
11#64133—21no—
12#64133—21no—
13#64133—21no—
14#64133—21no—
15C2×#3750C412yescomplemented
16C2×C3×#217C2212yes—
17C2×C3×#217C2212yes—
18C2×C3×C4⋊Dic3—21no—
19C2×C3×C4×Dic3—21no—
20C3×C3⋊#95—21no—
21C3×C3⋊#95—21no—
22C2×#924—31no—
23C2×#924—31no—
24C2×#924—61no—
25C3×#380—31no—
26C3×C4×C22⋊C4—31no—
27#5212D311yescharacteristic
28C4×#217D311yescharacteristic
29#4527—34no—
30#4527—34no—
31C24×C32C2311yescharacteristic
32C22×C32×C4—42no—
33C22×C32×C4C2×C411yescharacteristic
34C22×C32×C4C2×C411yescharacteristic
35C22×C32×C4D411yescharacteristic
36C22×C32×C4D411yescharacteristic
37C22×C32×C4D411yescharacteristic
38C22×C32×C4D411yescharacteristic
39C22×C3⋊Dic3C2×C412yes—
40C22×C3⋊Dic3C2×C412yes—
41C2×#634—21no—
42C2×#634—21no—
43C2×#634—42no—
44C22×C3×Dic3—22no—
45C22×C3×Dic3—22no—
46#3757—22no—
47#3757—22no—
48C4⋊(C3⋊Dic3)—21no—
49C4⋊(C3⋊Dic3)—21no—
50#3750—22no—
51#3750—22no—
52C3×#217—22no—
53C3×#217—24no—
54C3×#217Q814yescomplemented
55C3×C4⋊Dic3—42no—
56C3×C4×Dic3—42no—
57#1996—91noSylow
58C23×C3×C4—21no—
59C23×C3×C4C2×D311yescharacteristic
60C23×C3×C4C2×D311yescharacteristic
61C22×Dic6—31no—
62C22×Dic6—31no—
63C22×Dic6—61no—
64C2×C3×C4⋊C4—61no—
65C2×C3×C22⋊C4—32no—
66C2×C3×C22⋊C4—32no—
67C2×C3×C42—61no—
68C2×#217—32no—
69C2×#217—32no—
70C2×#217—62no—
71C2×#217C2×D312yes—
72C2×#217C2×D312yes—
73C2×C4⋊Dic3—21no—
74C2×C4⋊Dic3—31no—
75C2×C4⋊Dic3—31no—
76C2×C4⋊Dic3—61no—
77#924—62no—
78#924—62no—
79#924—62no—
80#924—62no—
81#924—61no—
82#924—61no—
83#924—61no—
84#924—61no—
85#924—61no—
86#924—61no—
87#924—61no—
88#924—61no—
89C2×Dic3⋊C4—32no—
90C2×Dic3⋊C4—32no—
91C2×Dic3⋊C4—62no—
92C2×C4×Dic3—21no—
93C3×#95—61no—
94C3×#95—61no—
95C3⋊#95—21no—
96C3⋊#95—21no—
97C23×C32—41no—
98C23×C32—41no—
99C23×C32C22×C411yescharacteristic
100C23×C32C2×D411yescharacteristic
101C23×C32C2×D411yescharacteristic
102C23×C32C2×Q812yes—
103C23×C32#4112yes—
104C2×C32×C4—22no—
105C2×C32×C4—22no—
106C2×C32×C4—22no—
107C2×C32×C4—41no—
108C2×C32×C4—41no—
109C2×C32×C4—41no—
110C2×C32×C4C22⋊C411yescharacteristic
111C2×C32×C4C22⋊C411yescharacteristic
112C2×C32×C4C22⋊C411yescharacteristic
113C2×C32×C4C22⋊C411yescharacteristic
114C2×C3⋊Dic3—22no—
115C2×C3⋊Dic3—22no—
116C2×C3⋊Dic3—42no—
117C2×C3⋊Dic3—42no—
118#634—42no—
119#634—42no—
120C2×C3×Dic3—24no—
121C2×C3×Dic3—42no—
122C2×C3×Dic3—42no—
123C2×C3×Dic3—42no—
124C2×C3×Dic3—42no—
125C2×C3×Dic3—42no—
126C2×C22⋊Q8—91no—
127#380—31no—
128C4×C22⋊C4—31no—
129#342—94no—
130C24×C3—21no—
131C24×C3C22×D311yescharacteristic
132C24×C3C22×D311yescharacteristic
133C22×C3×C4—21no—
134C22×C3×C4—21no—
135C22×C3×C4—21no—
136C22×C3×C4—21no—
137C22×C3×C4—21no—
138C22×C3×C4—21no—
139C22×C3×C4—42no—
140C22×C3×C4—42no—
141C22×C3×C4—44no—
142C22×C3×C4—62no—
143C22×C3×C4—62no—
144C22×C3×C4C4×D311yescharacteristic
145C22×C3×C4C4×D311yescharacteristic
146C22×C3×C4C4×D311yescharacteristic
147C22×C3×C4C4×D311yescharacteristic
148C22×C3×C4D1211yescharacteristic
149C22×C3×C4D1211yescharacteristic
150C22×C3×C4D1211yescharacteristic
151C22×C3×C4D1211yescharacteristic
152C22×C3×C4#6711yescharacteristic
153C22×C3×C4#6711yescharacteristic
154C22×C3×C4#6711yescharacteristic
155C22×C3×C4#6711yescharacteristic
156C22×Dic3—22no—
157C22×Dic3—22no—
158C22×Dic3—32no—
159C22×Dic3—32no—
160C22×Dic3—32no—
161C22×Dic3—32no—
162C22×Dic3—62no—
163C22×Dic3—62no—
164C2×Dic6—62no—
165C2×Dic6—62no—
166C2×Dic6—61no—
167C2×Dic6—61no—
168C2×Dic6—61no—
169C2×Dic6—61no—
170C2×Dic6—124no—
171C2×Dic6—122no—
172C2×Dic6—122no—
173C3×C4⋊C4—122no—
174C3×C22⋊C4—34no—
175C3×C22⋊C4—62no—
176C3×C22⋊C4—64no—
177C3×C42—122no—
178#217—22no—
179#217—24no—
180#217—64no—
181#217—64no—
182#217—62no—
183#217—62no—
184#217—62no—
185#217—62no—
186#217Dic614yescomplemented
187C4⋊Dic3—42no—
188C4⋊Dic3—62no—
189C4⋊Dic3—62no—
190C4⋊Dic3—61no—
191C4⋊Dic3—61no—
192C4⋊Dic3—61no—
193C4⋊Dic3—61no—
194Dic3⋊C4—64no—
195Dic3⋊C4—64no—
196Dic3⋊C4—62no—
197Dic3⋊C4—62no—
198Dic3⋊C4—62no—
199Dic3⋊C4—62no—
200C4×Dic3—42no—
201C22×C32—21no—
202C22×C32—21no—
203C22×C32—21no—
204C22×C32—21no—
205C22×C32—21no—
206C22×C32—21no—
207C22×C32—42no—
208C22×C32—42no—
209C22×C32C2×C22⋊C411yescharacteristic
210C22×C32C4×Q812yes—
211C22×C32C22⋊Q812yes—
212C22×C32#12411yescharacteristic
213C22×C32C4⋊Q811yescharacteristic
214C32×C4—21no—
215C32×C4—21no—
216C32×C4—21no—
217C32×C4#9912yes—
218C3⋊Dic3—42no—
219C3⋊Dic3—42no—
220C3×Dic3—44no—
221C3×Dic3—82no—
222C22×Q8—91no—
223C23×C4D3211yescharacteristic
224C22⋊Q8—181no—
225C22⋊Q8—181no—
226C22⋊Q8—181no—
227C22⋊Q8—181no—
228C2×C4⋊C4—61no—
229C2×C4⋊C4—92no—
230C2×C4⋊C4—91no—
231C2×C22⋊C4—32no—
232C2×C22⋊C4—32no—
233C2×C22⋊C4—92no—
234C2×C42—61no—
235#95—61no—
236#95—61no—
237C23×C3—22no—
238C23×C3—22no—
239C23×C3—21no—
240C23×C3—21no—
241C23×C3—21no—
242C23×C3—41no—
243C23×C3—41no—
244C23×C3—41no—
245C23×C3—41no—
246C23×C3—42no—
247C23×C3—42no—
248C23×C3C2×Dic612yes—
249C23×C3C2×C4×D311yescharacteristic
250C23×C3C2×C4×D311yescharacteristic
251C23×C3C2×D1211yescharacteristic
252C23×C3C2×D1211yescharacteristic
253C23×C3#23512yes—
254C23×C3#23712yes—
255C23×C3D3×Q812yes—
256C23×C3C2×#6711yescharacteristic
257C23×C3C2×#6711yescharacteristic
258C2×C3×C4—24no—
259C2×C3×C4—22no—
260C2×C3×C4—22no—
261C2×C3×C4—22no—
262C2×C3×C4—22no—
263C2×C3×C4—22no—
264C2×C3×C4—22no—
265C2×C3×C4—21no—
266C2×C3×C4—21no—
267C2×C3×C4—21no—
268C2×C3×C4—21no—
269C2×C3×C4—42no—
270C2×C3×C4—42no—
271C2×C3×C4—41no—
272C2×C3×C4—41no—
273C2×C3×C4—41no—
274C2×C3×C4—41no—
275C2×C3×C4—41no—
276C2×C3×C4—41no—
277C2×C3×C4—42no—
278C2×C3×C4—42no—
279C2×C3×C4—42no—
280C2×C3×C4—64no—
281C2×C3×C4—122no—
282C2×C3×C4—122no—
283C2×C3×C4—122no—
284C2×C3×C4—122no—
285C2×C3×C4—122no—
286C2×C3×C4(C2×D3)⋊C411yescharacteristic
287C2×C3×C4(C2×D3)⋊C411yescharacteristic
288C2×C3×C4(C2×D3)⋊C411yescharacteristic
289C2×C3×C4(C2×D3)⋊C411yescharacteristic
290C2×C3×C4(C2×D3)⋊C411yescharacteristic
291C2×C3×C4(C2×D3)⋊C411yescharacteristic
292C2×C3×C4(C2×D3)⋊C411yescharacteristic
293C2×C3×C4(C2×D3)⋊C411yescharacteristic
294C2×Dic3—24no—
295C2×Dic3—42no—
296C2×Dic3—42no—
297C2×Dic3—42no—
298C2×Dic3—42no—
299C2×Dic3—42no—
300C2×Dic3—64no—
301C2×Dic3—64no—
302C2×Dic3—62no—
303C2×Dic3—62no—
304C2×Dic3—62no—
305C2×Dic3—62no—
306C2×Dic3—124no—
307C2×Dic3—124no—
308C2×Dic3—122no—
309C2×Dic3—122no—
310C2×Dic3—122no—
311C2×Dic3—122no—
312Dic6—124no—
313Dic6—122no—
314Dic6—122no—
315Dic6—124no—
316Dic6—122no—
317Dic6—122no—
318C2×C32—22no—
319C2×C32—22no—
320C2×C32—21no—
321C2×C32—21no—
322C2×C32#39111yescharacteristic
323C2×C32C2×#9911yescharacteristic
324C2×C32#41011yescharacteristic
325C24C2×D3211yescharacteristic
326C2×Q8—181no—
327C2×Q8—181no—
328C2×Q8—362no—
329C22×C4—42no—
330C22×C4—62no—
331C22×C4—62no—
332C22×C4—92no—
333C22×C4—92no—
334C22×C4#62411yescharacteristic
335C22×C4#62411yescharacteristic
336C22×C4#62611yescharacteristic
337C22×C4#62611yescharacteristic
338C22×C4#62611yescharacteristic
339C22×C4#62611yescharacteristic
340C4⋊C4—122no—
341C4⋊C4—182no—
342C4⋊C4—182no—
343C4⋊C4—181no—
344C4⋊C4—181no—
345C22⋊C4—34no—
346C22⋊C4—62no—
347C22⋊C4—64no—
348C22⋊C4—182no—
349C22⋊C4—182no—
350C42—122no—
351C22×C3—22no—
352C22×C3—22no—
353C22×C3—21no—
354C22×C3—21no—
355C22×C3—21no—
356C22×C3—21no—
357C22×C3—21no—
358C22×C3—21no—
359C22×C3—21no—
360C22×C3—21no—
361C22×C3—21no—
362C22×C3—21no—
363C22×C3—21no—
364C22×C3—21no—
365C22×C3—22no—
366C22×C3—22no—
367C22×C3—21no—
368C22×C3—21no—
369C22×C3—21no—
370C22×C3—41no—
371C22×C3—41no—
372C22×C3—41no—
373C22×C3—41no—
374C22×C3—42no—
375C22×C3—42no—
376C22×C3—42no—
377C22×C3—42no—
378C22×C3—44no—
379C22×C3—44no—
380C22×C3C4×Dic612yes—
381C22×C3#86911yescharacteristic
382C22×C3#87511yescharacteristic
383C22×C3#88712yes—
384C22×C3#89712yes—
385C22×C3#92412yes—
386C22×C3C2×(C2×D3)⋊C411yescharacteristic
387C22×C3C2×(C2×D3)⋊C411yescharacteristic
388C22×C3#93611yescharacteristic
389C22×C3#94411yescharacteristic
390C3×C4—22no—
391C3×C4—21no—
392C3×C4—21no—
393C3×C4—21no—
394C3×C4—21no—
395C3×C4—21no—
396C3×C4—21no—
397C3×C4—22no—
398C3×C4—41no—
399C3×C4—41no—
400C3×C4—124no—
401C3×C4—242no—
402C3×C4#80612yes—
403C3×C4#80612yes—
404Dic3—44no—
405Dic3—82no—
406Dic3—124no—
407Dic3—124no—
408Dic3—122no—
409Dic3—122no—
410Dic3—122no—
411Dic3—122no—
412C32#199611yescharacteristic, complemented, Sylow
413C23—41no—
414C23—41no—
415C23D3×Dic612yes—
416C23#379512yes—
417C23C2×#62411yescharacteristic
418C23C2×#62611yescharacteristic
419C23C2×#62611yescharacteristic
420Q8—362no—
421Q8—362no—
422C2×C4—22no—
423C2×C4—22no—
424C2×C4—22no—
425C2×C4—41no—
426C2×C4—41no—
427C2×C4—41no—
428C2×C4—64no—
429C2×C4—122no—
430C2×C4—122no—
431C2×C4—122no—
432C2×C4—122no—
433C2×C4—122no—
434C2×C4—182no—
435C2×C4—182no—
436C2×C4—362no—
437C2×C4—362no—
438C2×C4#372211yescharacteristic
439C2×C4#372211yescharacteristic
440C2×C4#372211yescharacteristic
441C2×C4#372211yescharacteristic
442C2×C3—24no—
443C2×C3—22no—
444C2×C3—21no—
445C2×C3—22no—
446C2×C3—21no—
447C2×C3—21no—
448C2×C3—22no—
449C2×C3—22no—
450C2×C3—21no—
451C2×C3—21no—
452C2×C3—21no—
453C2×C3—21no—
454C2×C3—42no—
455C2×C3—41no—
456C2×C3—41no—
457C2×C3#492911yescharacteristic
458C2×C3C2×#80611yescharacteristic
459C2×C3C2×#80611yescharacteristic
460C2×C3#497311yescharacteristic
461C2×C3#500611yescharacteristic
462C2×C3#500611yescharacteristic
463C22—21no—
464C22—21no—
465C22—21no—
466C22—21no—
467C22—21no—
468C22—21no—
469C22—42no—
470C22—42no—
471C22#6384412yes—
472C22#6385712yes—
473C22#6386011yescharacteristic
474C22#6386611yescharacteristic
475C22C2×#372211yescharacteristic
476C4—21no—
477C4—21no—
478C4—21no—
479C4—124no—
480C4—242no—
481C4—362no—
482C4—362no—
483C4#6358112yes—
484C3—21no—
485C3#6945711yescharacteristic, complemented
486C3#6952911yescharacteristic, complemented
487C2—22no—
488C2—21no—
489C2—22no—
490C2—21no—
491C2#1059121311yescharacteristic
492C2#1059123211yescharacteristic
493C2C2×#6358111yescharacteristic
subgroupG/H
centerC22C2×#3722
commutatorC23×C32C22×C4
FrattiniC24C2×D32
FittingC23×C32×C4C22
socleC22×C32C2×C22⋊C4
prime radical, cosocleC24×C32C23
hypercenterC23×C4D32
hypocenterC32#1996
LCSG ▷ C23×C32 ▷ C2×C32 ▷ C32
UCSC1 ◁ C22 ◁ C23×C4

Complements

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

NH ≅ G/Ntype
8. C22×#634C4semidirect
10. C2×C4⋊(C3⋊Dic3)C4semidirect
15. C2×#3750C4semidirect
54. C3×#217Q8semidirect
186. #217Dic6semidirect
412. C32#1996semidirect
485. C3#69457semidirect
486. C3#69529semidirect

Maximal subgroups

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

subgroupindex
C2×#641332
C3×#52122
C3×C4×#2172
#105891872
#695293
#694573