Finite Group Databasebeta

← #11793911

Subgroups — #11793911

Order 1152.

643 conjugacy classes of subgroups.

The subgroup lattice has 527 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#63830C211yesmaximal, characteristic
2#10592828C211yesmaximal, characteristic, complemented
3C2×C3×#823C211yesmaximal, characteristic, complemented
4C2×#63578C211yesmaximal, characteristic, complemented
5#10591106C211yesmaximal, characteristic, complemented
6C2×#63562C211yesmaximal, characteristic, complemented
7C3×#4554C211yesmaximal, characteristic, complemented
8#70092—31nomaximal
9#68986—31nomaximal
10C2×#3828C2211yescharacteristic
11C22×C3×D12C2211yescharacteristic
12C2×C32×M16C2211yescharacteristic
13C22×C3⋊(C3⋊C8)C2211yescharacteristic
14C2×C3×C4⋊Dic3C2211yescharacteristic
15C2×C3×#208C2211yescharacteristic
16C2×C3×#208C2211yescharacteristic
17#63830—22no—
18#63830—22no—
19C2×#3716—21no—
20C2×#3714—21no—
21#63647—22no—
22C3×#823—22no—
23#63578—22no—
24#63567—21no—
25#63564—21no—
26#63562—22no—
27C2×#932—31no—
28C2×#908—31no—
29C2×C3×#100—31no—
30C2×#832—31no—
31#5133—31no—
32C2×#823D311yescharacteristic, complemented
33#5129—31no—
34#5129—31no—
35#5129—61no—
36C2×#822—31no—
37C2×#822—31no—
38#4966—31no—
39C2×#809—31no—
40C3×#343—31no—
41#4554D311yescharacteristic, complemented
42C23×C3×D3—21no—
43C22×C32×C4C2311yescharacteristic
44C2×#638—21no—
45#3828—21no—
46#3828—21no—
47#3828—21no—
48#3828—21no—
49C2×C4⋊(C3⋊D3)D411yescharacteristic
50C2×C4×C3⋊D3—21no—
51C2×#634D411yescharacteristic
52C22×C3×Dic3—21no—
53C2×C3×D12—21no—
54C2×C3×D12—21no—
55C2×C3×D12—21no—
56C2×C3×D12—21no—
57C2×C3×D12—21no—
58C2×C3×D12D412yes—
59C32×M16—21no—
60C32×M16—21no—
61C2×C32×C8—21no—
62C2×C3⋊(C3⋊C8)—21no—
63C2×C3⋊(C3⋊C8)—21no—
64C2×C3⋊(C3⋊C8)D412yes—
65C3×C4⋊Dic3—21no—
66C3×C4⋊Dic3—21no—
67C3×#208—21no—
68C3×#208—21no—
69C3×#208—21no—
70C3×#208D412yes—
71C2×C3×C3⋊C8—21no—
72C2×C3×C3⋊C8—21no—
73#3716—24no—
74#3714—24no—
75#2051—91noSylow
76C22×C3×D4—31no—
77C2×#235—31no—
78C2×#235—31no—
79C2×#235—61no—
80C22×D12C2×D311yescharacteristic
81C2×C3×M16—21no—
82C2×C3×M16—31no—
83C2×C3×M16—31no—
84C2×C3×M16C2×D311yescharacteristic
85C2×C3×M16C2×D311yescharacteristic
86C2×C3×C4⋊C4—31no—
87C2×#214—61no—
88#932—62no—
89#932—62no—
90C2×#213—61no—
91C2×C4⋊Dic3C2×D311yescharacteristic
92C2×#208C2×D311yescharacteristic
93C2×#208C2×D311yescharacteristic
94C22×C3⋊C8—31no—
95C22×C3⋊C8—31no—
96C22×C3⋊C8—61no—
97#908—62no—
98#908—62no—
99C2×D24—61no—
100C2×#204—61no—
101C3×#100—62no—
102#832—62no—
103#823—22no—
104#822—62no—
105#822—62no—
106#821—61no—
107(C2×D3)⋊C8—62no—
108(C2×D3)⋊C8—62no—
109(C2×D3)⋊C8—64no—
110#816—61no—
111#810—61no—
112#809—61no—
113#809—62no—
114C23×C32#4111yescharacteristic
115C22×C3⋊D3—21no—
116C22×C3×D3—22no—
117C22×C3×D3—41no—
118C22×C3×D3—41no—
119C22×C3×D3—81no—
120C22×C3×D3—81no—
121C2×C32×C4C2×D411yescharacteristic
122C2×C32×C4C2×D411yescharacteristic
123C2×C32×C4C2×D411yescharacteristic
124C2×C32×C4C2×D411yescharacteristic
125C2×C32×C4#4111yescharacteristic
126C2×C32×C4#4111yescharacteristic
127#638—41no—
128#638—41no—
129C2×C3⋊Dic3—21no—
130C4⋊(C3⋊D3)—21no—
131C4⋊(C3⋊D3)—21no—
132C4×C3⋊D3—41no—
133C4×C3⋊D3—41no—
134#634—21no—
135#634—21no—
136C2×C3×Dic3—41no—
137C2×C3×Dic3—81no—
138C3×D12—22no—
139C3×D12—22no—
140C3×D12—41no—
141C3×D12—41no—
142C32×C8—41no—
143C3⋊(C3⋊C8)—22no—
144C3×C3⋊C8—22no—
145C3×C3⋊C8—41no—
146C2×AGL(1,ℤ/8)—91no—
147C2×#108—91no—
148#417—91no—
149C2×#102—91no—
150C2×#100—31no—
151#408—91no—
152#343—31no—
153C24×C3—61no—
154C23×D3—21no—
155C2×C3×D4—32no—
156C2×C3×D4—61no—
157C2×C3×D4—61no—
158C2×C3×D4—61no—
159C2×C3×D4—61no—
160C2×C3×D4—61no—
161C22×C3×C4—21no—
162C22×C3×C4—61no—
163C22×C3×C4C22×D311yescharacteristic
164C22×C3×C4C22×D311yescharacteristic
165C2×#67—62no—
166C2×#67—61no—
167C2×#67—61no—
168C22×Dic3—21no—
169#235—61no—
170#235—61no—
171#235—61no—
172#235—61no—
173#235—61no—
174#235—61no—
175#235—61no—
176#235—61no—
177#235—121no—
178#235—121no—
179#235—121no—
180#235—121no—
181C2×D12—21no—
182C2×D12—21no—
183C2×D12—21no—
184C2×D12—21no—
185C2×D12—21no—
186C2×D12—31no—
187C2×D12—31no—
188C2×D12—61no—
189C2×D12#6712yes—
190C2×C4×D3—62no—
191C2×C4×D3—61no—
192C2×C4×D3—61no—
193C2×Dic6—31no—
194C2×Dic6—31no—
195C2×Dic6—61no—
196C3×M16—21no—
197C3×M16—21no—
198C3×M16—21no—
199C3×M16—21no—
200C3×M16—32no—
201C3×M16—41no—
202C3×M16—41no—
203C3×M16—61no—
204C3×M16—61no—
205C3×M16—61no—
206C2×C3×C8—22no—
207C2×C3×C8—21no—
208C2×C3×C8—21no—
209C2×C3×C8—61no—
210C2×C3×C8—61no—
211C3×C4⋊C4—61no—
212C3×C4⋊C4—61no—
213#214—64no—
214#213—64no—
215C4⋊Dic3—21no—
216C4⋊Dic3—21no—
217#208—21no—
218#208—21no—
219#208—21no—
220#208D1212yes—
221C2×C3⋊C8—21no—
222C2×C3⋊C8—21no—
223C2×C3⋊C8—32no—
224C2×C3⋊C8—32no—
225C2×C3⋊C8—62no—
226C2×C3⋊C8—61no—
227C2×C3⋊C8—61no—
228C2×C3⋊C8—61no—
229C2×C3⋊C8—61no—
230C2×C3⋊C8—121no—
231C2×C3⋊C8—121no—
232D24—64no—
233#204—64no—
234C22×C32—21no—
235C22×C32—21no—
236C22×C32#12111yescharacteristic
237C22×C32#12311yescharacteristic
238C22×C32#12411yescharacteristic
239C2×C3⋊D3—41no—
240C2×C3⋊D3—41no—
241C2×C3⋊D3—41no—
242C2×C3×D3—42no—
243C2×C3×D3—82no—
244C2×C3×D3—81no—
245C2×C3×D3—81no—
246C2×C3×D3—82no—
247C32×C4#12011yescharacteristic
248C32×C4#12111yescharacteristic
249C32×C4#12111yescharacteristic
250C32×C4#12311yescharacteristic
251C3⋊Dic3—41no—
252C3×Dic3—81no—
253C2×#41—91no—
254C22×D4—31no—
255AGL(1,ℤ/8)—182no—
256AGL(1,ℤ/8)—182no—
257C2×SD16—181no—
258C2×D8—181no—
259C2×M16—31no—
260C2×M16—31no—
261C2×M16D3211yescharacteristic
262C22×C8—91no—
263C2×C4⋊C4—31no—
264#108—182no—
265#103—181no—
266#102—181no—
267#102—182no—
268#100—62no—
269C22⋊C8—182no—
270C23×C3—21no—
271C23×C3—62no—
272C23×C3—121no—
273C23×C3—121no—
274C23×C3—241no—
275C23×C3—241no—
276C23×C3#23511yescharacteristic
277C23×C3#23511yescharacteristic
278C22×D3—22no—
279C22×D3—41no—
280C22×D3—41no—
281C22×D3—62no—
282C22×D3—61no—
283C22×D3—61no—
284C22×D3—81no—
285C22×D3—81no—
286C3×D4—62no—
287C3×D4—62no—
288C3×D4—121no—
289C3×D4—121no—
290C2×C3×C4—21no—
291C2×C3×C4—21no—
292C2×C3×C4—21no—
293C2×C3×C4—21no—
294C2×C3×C4—21no—
295C2×C3×C4—21no—
296C2×C3×C4—121no—
297C2×C3×C4—241no—
298C2×C3×C4C2×D1211yescharacteristic
299C2×C3×C4#23511yescharacteristic
300C2×C3×C4#23511yescharacteristic
301C2×C3×C4D3×D411yescharacteristic
302C2×C3×C4D3×D411yescharacteristic
303C2×C3×C4D3×D411yescharacteristic
304C2×C3×C4D3×D411yescharacteristic
305C2×C3×C4D3×D411yescharacteristic
306C2×C3×C4D3×D411yescharacteristic
307C2×C3×C4#23711yescharacteristic
308C2×C3×C4#23911yescharacteristic
309C2×C3×C4C2×#6711yescharacteristic
310#67—122no—
311#67—122no—
312#67—121no—
313#67—121no—
314#67—121no—
315#67—121no—
316C2×Dic3—41no—
317C2×Dic3—62no—
318C2×Dic3—61no—
319C2×Dic3—61no—
320C2×Dic3—81no—
321D12—22no—
322D12—22no—
323D12—41no—
324D12—41no—
325D12—61no—
326D12—61no—
327D12—61no—
328D12—61no—
329D12—121no—
330D12—121no—
331C4×D3—122no—
332C4×D3—122no—
333C4×D3—121no—
334C4×D3—121no—
335C4×D3—121no—
336C4×D3—121no—
337Dic6—61no—
338Dic6—61no—
339Dic6—61no—
340Dic6—61no—
341Dic6—121no—
342Dic6—121no—
343C3×C8—42no—
344C3×C8—41no—
345C3×C8—41no—
346C3×C8—62no—
347C3×C8—121no—
348C3⋊C8—22no—
349C3⋊C8—41no—
350C3⋊C8—62no—
351C3⋊C8—62no—
352C3⋊C8—122no—
353C2×C32—21no—
354C2×C32—21no—
355C2×C32#39411yescharacteristic
356C2×C32#47111yescharacteristic
357C2×C32#47211yescharacteristic
358C3⋊D3—81no—
359C3×D3—82no—
360C24—61no—
361#41—181no—
362#41—181no—
363#41—181no—
364#41—181no—
365C2×Q8—91no—
366C2×D4—32no—
367C2×D4—61no—
368C2×D4—61no—
369C2×D4—61no—
370C2×D4—61no—
371C2×D4—61no—
372C2×D4—91no—
373C2×D4—181no—
374C22×C4—61no—
375C22×C4—181no—
376C22×C4C2×D3211yescharacteristic
377SD16—184no—
378D8—184no—
379M16—21no—
380M16—21no—
381M16—32no—
382M16—61no—
383M16—61no—
384M16—61no—
385C2×C8—21no—
386C2×C8—61no—
387C2×C8—61no—
388C2×C8—92no—
389C2×C8—181no—
390C2×C8—181no—
391C4⋊C4—61no—
392C4⋊C4—61no—
393C22×C3—21no—
394C22×C3—21no—
395C22×C3—21no—
396C22×C3—21no—
397C22×C3—21no—
398C22×C3—21no—
399C22×C3—21no—
400C22×C3—41no—
401C22×C3—41no—
402C22×C3—122no—
403C22×C3—242no—
404C22×C3—242no—
405C22×C3—241no—
406C22×C3—241no—
407C22×C3#87511yescharacteristic
408C22×C3#88411yescharacteristic
409C22×C3#88411yescharacteristic
410C22×C3#88511yescharacteristic
411C22×C3#89411yescharacteristic
412C22×C3#92911yescharacteristic
413C2×D3—42no—
414C2×D3—82no—
415C2×D3—81no—
416C2×D3—81no—
417C2×D3—82no—
418C2×D3—122no—
419C2×D3—122no—
420C2×D3—122no—
421C2×D3—121no—
422C2×D3—121no—
423C2×D3—121no—
424C2×D3—121no—
425C2×D3—121no—
426C2×D3—121no—
427C3×C4—21no—
428C3×C4—21no—
429C3×C4—21no—
430C3×C4—21no—
431C3×C4—241no—
432C3×C4#88211yescharacteristic
433C3×C4#88311yescharacteristic
434C3×C4#88411yescharacteristic
435C3×C4#88411yescharacteristic
436C3×C4#89411yescharacteristic
437C3×C4#89511yescharacteristic
438C3×C4#93711yescharacteristic
439C3×C4#93911yescharacteristic
440Dic3—81no—
441Dic3—122no—
442Dic3—121no—
443Dic3—121no—
444C32#205111yescharacteristic, complemented, Sylow
445C23—62no—
446C23—121no—
447C23—121no—
448C23—181no—
449C23—241no—
450C23—241no—
451C23#379811yescharacteristic
452Q8—181no—
453Q8—181no—
454D4—62no—
455D4—62no—
456D4—121no—
457D4—121no—
458D4—181no—
459D4—181no—
460D4—361no—
461D4—361no—
462C2×C4—121no—
463C2×C4—181no—
464C2×C4—241no—
465C2×C4—361no—
466C2×C4—361no—
467C2×C4#379911yescharacteristic
468C2×C4D3×D1211yescharacteristic
469C2×C4#380211yescharacteristic
470C2×C4#380411yescharacteristic
471C2×C4D3×#6711yescharacteristic
472C2×C4#381111yescharacteristic
473C8—41no—
474C8—62no—
475C8—121no—
476C8—182no—
477C2×C3—21no—
478C2×C3—21no—
479C2×C3—21no—
480C2×C3—21no—
481C2×C3—21no—
482C2×C3—21no—
483C2×C3—21no—
484C2×C3—41no—
485C2×C3—41no—
486C2×C3—242no—
487C2×C3#467211yescharacteristic
488C2×C3#474811yescharacteristic
489C2×C3#474911yescharacteristic
490C2×C3#474911yescharacteristic
491C2×C3#475411yescharacteristic
492C2×C3#495911yescharacteristic
493D3—82no—
494D3—242no—
495D3—241no—
496D3—241no—
497C22—21no—
498C22—21no—
499C22—122no—
500C22—242no—
501C22—242no—
502C22—241no—
503C22—241no—
504C22—361no—
505C22—361no—
506C22—361no—
507C22#6385011yescharacteristic
508C22#6385211yescharacteristic
509C22#6391211yescharacteristic
510C4—241no—
511C4—361no—
512C4#6385411yescharacteristic
513C4#6391011yescharacteristic
514C4#6392211yescharacteristic
515C4#6397111yescharacteristic
516C3—21no—
517C3#6898611yescharacteristic, complemented
518C3#7009211yescharacteristic, complemented
519C2—21no—
520C2—21no—
521C2—242no—
522C2—721no—
523C2#1059098811yescharacteristic
524C2#1059099111yescharacteristic
525C2#1059129211yescharacteristic
subgroupG/H
centerC22#63912
commutator, cosocleC22×C32×C4C23
FrattiniC22×C4C2×D32
FittingC2×C32×M16C22
socleC22×C32#121
prime radicalC2×#63830C2
hypercenterC2×M16D32
hypocenterC32#2051
LCSG ▷ C22×C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×M16

Complements

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

NH ≅ G/Ntype
2. #10592828C2semidirect
3. C2×C3×#823C2semidirect
4. C2×#63578C2semidirect
5. #10591106C2semidirect
6. C2×#63562C2semidirect
7. C3×#4554C2semidirect
32. C2×#823D3semidirect
41. #4554D3semidirect
444. C32#2051semidirect
517. C3#68986semidirect
518. C3#70092semidirect

Maximal subgroups

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

subgroupindex
C2×#638302
#105928282
C2×C3×#8232
C2×#635782
#105911062
C2×#635622
C3×#45542
#700923
#689863