Finite Group Databasebeta

← #11859346

Subgroups — #11859346

Order 1152.

1209 conjugacy classes of subgroups.

The subgroup lattice has 509 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×C3⋊(C3⋊C8)C211yesmaximal, characteristic, complemented
2C3×#5791C211yesmaximal, characteristic
3C2×#63815C211yesmaximal, characteristic, complemented
4C3×#5207C211yesmaximal, characteristic, complemented
5#10591158C214yesmaximal, complemented
6#67⋊(C3⋊C8)C214yesmaximal, complemented
7#10591117C211yesmaximal, characteristic, complemented
8#10591071C211yesmaximal, characteristic, complemented
9#10591057C211yesmaximal, characteristic, complemented
10#82916—31nomaximal
11#78626—31nomaximal
12C23×C32×C4C2211yescharacteristic
13C2×C3×#235C2211yescharacteristic
14C22×C3⋊(C3⋊C8)—24no—
15C22×C3⋊(C3⋊C8)C2211yescharacteristic
16C22×C3⋊(C3⋊C8)C2212yes—
17C22×C3⋊(C3⋊C8)C2212yes—
18C22×C3⋊(C3⋊C8)C2211yescharacteristic
19C3×#953C412yes—
20C3×#930C412yes—
21C3×#929C412yes—
22C3×C4×#67C2214yes—
23C3×#926C2211yescharacteristic
24C3×#924C412yes—
25C2×C3×#208C2211yescharacteristic
26C2×C3×#208C2211yescharacteristic
27C2×#3711C2212yescomplemented
28#63815—24no—
29C2×D3×C3⋊C8C2212yescomplemented
30C3×#830C2214yes—
31Dic3⋊(C3⋊C8)C2214yescomplemented
32(C2×D3)⋊(C3⋊C8)C2214yescomplemented
33#63554C2212yescomplemented
34Dic3×C3⋊C8C2212yescomplemented
35C3×#525—31no—
36C2×#948—31no—
37#5791D311yescharacteristic
38C23×C3⋊C8—31no—
39C23×C3⋊C8—31no—
40C23×C3⋊C8—61no—
41C2×#907—31no—
42C3×#407—31no—
43#5234—31no—
44#5207D311yescharacteristic, complemented
45#5129—31no—
46#5122—31no—
47#5119—31no—
48#5011—34no—
49D4×C3⋊C8—34no—
50#4999—31no—
51#4997—31no—
52#4730—34no—
53#67⋊C8—34no—
54C24×C32C2×C411yescharacteristic
55C22×C32×C4—22no—
56C22×C32×C4—22no—
57C22×C32×C4C2×C412yes—
58C22×C32×C4C2×C411yescharacteristic
59C22×C32×C4C2312yes—
60C22×C32×C4C2311yescharacteristic
61C2×C3×#67C2×C414yes—
62C2×C3×#67C2×C412yes—
63C3×#235—22no—
64C3×#235—22no—
65C2×C3×D12C2×C411yescharacteristic
66C2×C3×C4×D3C2312yes—
67C2×C3×Dic6C2×C411yescharacteristic
68C2×C3⋊(C3⋊C8)—22no—
69C2×C3⋊(C3⋊C8)—24no—
70C2×C3⋊(C3⋊C8)—24no—
71C2×C3⋊(C3⋊C8)D414yescomplemented
72C2×C3⋊(C3⋊C8)C2312yes—
73C2×C3⋊(C3⋊C8)C2311yescharacteristic
74C2×C3⋊(C3⋊C8)C2311yescharacteristic
75C3×#217C2×C414yes—
76C3×#217C2×C412yes—
77C3×(C2×D3)⋊C4C2×C414yes—
78C3×C4⋊Dic3C2×C412yes—
79C3×Dic3⋊C4C2×C414yes—
80C3×C4×Dic3C2312yes—
81C3×#208—22no—
82C3×#208—22no—
83C2×C3×C3⋊C8C2312yes—
84C2×C3×C3⋊C8C2312yes—
85#3711—24no—
86D3×C3⋊C8—24no—
87#2929—91noSylow
88C2×C3×#41—31no—
89C23×C3×C4—21no—
90C23×C3×C4C2×D311yescharacteristic
91C23×C3×C4C2×D311yescharacteristic
92C2×#235C2×D311yescharacteristic
93C2×C3×M16—31no—
94C2×C3×M16—31no—
95C3×#123—32no—
96C3×C22⋊Q8—32no—
97C3×#121—32no—
98C3×#120—32no—
99C3×C4×D4—34no—
100C3×#117—31no—
101#953Dic312yes—
102#948—64no—
103#930Dic312yes—
104#929Dic312yes—
105C4×#67C2×D314yes—
106#926C2×D311yescharacteristic
107#924Dic312yes—
108C2×#208—32no—
109C2×#208C2×D311yescharacteristic
110C2×#208C2×D311yescharacteristic
111C22×C3⋊C8—31no—
112C22×C3⋊C8—31no—
113C22×C3⋊C8—31no—
114C22×C3⋊C8—31no—
115C22×C3⋊C8—32no—
116C22×C3⋊C8—32no—
117C22×C3⋊C8—32no—
118C22×C3⋊C8—32no—
119C22×C3⋊C8—32no—
120C22×C3⋊C8—61no—
121C22×C3⋊C8—61no—
122C22×C3⋊C8—62no—
123C22×C3⋊C8—62no—
124C22×C3⋊C8—64no—
125C22×C3⋊C8—64no—
126C22×C3⋊C8—68no—
127#907—64no—
128C2×ΓL(1,𝔽25)—32no—
129C2×D3×C8—32no—
130C3×C22⋊C8—34no—
131#830—34no—
132#830C2×D314yes—
133(C2×D3)⋊C8—34no—
134#815—32no—
135Dic3⋊C8—34no—
136C8×Dic3—32no—
137C4⋊(C3⋊C8)—34no—
138#803—32no—
139C4×C3⋊C8—32no—
140C23×C32—22no—
141C23×C32—24no—
142C23×C32C22×C411yescharacteristic
143C23×C32C22×C412yes—
144C22×C3×D3C22×C412yes—
145C2×C32×C4—22no—
146C2×C32×C4—24no—
147C2×C32×C4—22no—
148C2×C32×C4—22no—
149C2×C32×C4C22×C412yes—
150C2×C32×C4C22×C411yescharacteristic
151C2×C32×C4C2×D412yes—
152C2×C32×C4#4112yes—
153C2×C32×C4C2411yescharacteristic
154C3×#67—24no—
155C3×#67—28no—
156C2×C3×Dic3C22×C414yes—
157C2×C3×Dic3C22×C412yes—
158C3×D12—22no—
159C3×C4×D3—22no—
160C3×C4×D3—22no—
161C3×Dic6—22no—
162C3⋊(C3⋊C8)—22no—
163C3⋊(C3⋊C8)C2×D412yes—
164C3⋊(C3⋊C8)C2×D412yes—
165C3×C3⋊C8—22no—
166C3×C3⋊C8—22no—
167C2×#131—91no—
168C23×C8—91no—
169#525—31no—
170#435—94no—
171C8×D4—94no—
172#424—91no—
173#408—91no—
174#407—31no—
175#405—91no—
176C24×C3—21no—
177C24×C3C4×D311yescharacteristic
178C24×C3C2×Dic311yescharacteristic
179C3×#41—62no—
180C3×#41—62no—
181C2×C3×Q8—31no—
182C2×C3×D4—34no—
183C2×C3×D4—32no—
184C2×C3×D4—31no—
185C22×C3×C4—24no—
186C22×C3×C4—24no—
187C22×C3×C4—22no—
188C22×C3×C4—22no—
189C22×C3×C4—22no—
190C22×C3×C4—22no—
191C22×C3×C4—22no—
192C22×C3×C4—21no—
193C22×C3×C4—22no—
194C22×C3×C4—21no—
195C22×C3×C4—32no—
196C22×C3×C4C4×D312yes—
197C22×C3×C4C4×D311yescharacteristic
198C22×C3×C4C2×Dic312yes—
199C22×C3×C4C2×Dic311yescharacteristic
200C22×C3×C4C22×D312yes—
201C22×C3×C4C22×D312yes—
202C22×C3×C4C22×D311yescharacteristic
203C22×C3×C4C22×D311yescharacteristic
204C2×#67C2×Dic314yes—
205C2×#67C2×Dic312yes—
206#235—22no—
207#235—22no—
208C2×D12C2×Dic311yescharacteristic
209C2×C4×D3C22×D312yes—
210C2×Dic6C2×Dic311yescharacteristic
211C3×M16—62no—
212C3×M16—62no—
213C2×C3×C8—32no—
214C2×C3×C8—32no—
215C3×C4⋊C4—34no—
216C3×C4⋊C4—32no—
217C3×C22⋊C4—32no—
218C3×C22⋊C4—34no—
219C3×C22⋊C4—34no—
220C3×C42—32no—
221#217C2×Dic314yes—
222#217C2×Dic312yes—
223(C2×D3)⋊C4C2×Dic314yes—
224C4⋊Dic3C2×Dic312yes—
225Dic3⋊C4C2×Dic314yes—
226C4×Dic3C22×D312yes—
227#208—22no—
228#208—22no—
229#208—64no—
230C2×C3⋊C8—34no—
231C2×C3⋊C8—34no—
232C2×C3⋊C8—31no—
233C2×C3⋊C8—31no—
234C2×C3⋊C8—31no—
235C2×C3⋊C8—31no—
236C2×C3⋊C8—32no—
237C2×C3⋊C8—32no—
238C2×C3⋊C8—64no—
239C2×C3⋊C8—61no—
240C2×C3⋊C8—61no—
241C2×C3⋊C8—62no—
242C2×C3⋊C8—64no—
243C2×C3⋊C8—64no—
244C2×C3⋊C8—64no—
245C2×C3⋊C8—64no—
246C2×C3⋊C8—64no—
247C2×C3⋊C8—68no—
248C2×C3⋊C8—68no—
249C2×C3⋊C8—64no—
250C2×C3⋊C8—62no—
251C2×C3⋊C8—62no—
252C2×C3⋊C8C22×D312yes—
253C2×C3⋊C8C22×D312yes—
254ΓL(1,𝔽25)—64no—
255D3×C8—64no—
256C22×C32—28no—
257C22×C32—22no—
258C22×C32—22no—
259C22×C32—22no—
260C22×C32C4×D412yes—
261C22×C32#13114yescomplemented
262C22×C32C23×C411yescharacteristic
263C2×C3×D3—22no—
264C2×C3×D3—24no—
265C32×C4—22no—
266C32×C4C4×D412yes—
267C32×C4C22×D411yescharacteristic
268C32×C4C2×#4111yescharacteristic
269C3×Dic3—22no—
270C3×Dic3—24no—
271C2×#41—31no—
272C23×C4D3211yescharacteristic
273#131—184no—
274C2×M16—31no—
275C2×M16—31no—
276C2×M16—92no—
277C22×C8—92no—
278C22×C8—91no—
279C22×C8—91no—
280C22×C8—92no—
281C22×C8—92no—
282C22×C8—184no—
283#123—32no—
284C22⋊Q8—32no—
285#121—32no—
286#120—32no—
287C4×D4—34no—
288#117—31no—
289C4⋊C8—94no—
290C22⋊C8—34no—
291C22⋊C8—94no—
292#97—92no—
293C4×C8—92no—
294C23×C3—24no—
295C23×C3—28no—
296C23×C3—22no—
297C23×C3—24no—
298C23×C3—22no—
299C23×C3—24no—
300C23×C3—22no—
301C23×C3—21no—
302C23×C3—32no—
303C23×C3C2×C4×D311yescharacteristic
304C23×C3C2×C4×D312yes—
305C23×C3C22×Dic311yescharacteristic
306C23×C3C22×Dic312yes—
307C22×D3C22×Dic312yes—
308C3×Q8—62no—
309C3×D4—64no—
310C3×D4—68no—
311C3×D4—62no—
312C2×C3×C4—24no—
313C2×C3×C4—24no—
314C2×C3×C4—24no—
315C2×C3×C4—28no—
316C2×C3×C4—22no—
317C2×C3×C4—22no—
318C2×C3×C4—24no—
319C2×C3×C4—24no—
320C2×C3×C4—22no—
321C2×C3×C4—22no—
322C2×C3×C4—22no—
323C2×C3×C4—22no—
324C2×C3×C4—21no—
325C2×C3×C4—22no—
326C2×C3×C4—21no—
327C2×C3×C4—22no—
328C2×C3×C4—22no—
329C2×C3×C4—34no—
330C2×C3×C4—32no—
331C2×C3×C4—62no—
332C2×C3×C4—62no—
333C2×C3×C4C2×C4×D312yes—
334C2×C3×C4C2×C4×D311yescharacteristic
335C2×C3×C4D3×D412yes—
336C2×C3×C4D3×D412yes—
337C2×C3×C4#23712yes—
338C2×C3×C4#23712yes—
339C2×C3×C4C22×Dic312yes—
340C2×C3×C4C22×Dic311yescharacteristic
341C2×C3×C4C23×D311yescharacteristic
342C2×C3×C4C23×D311yescharacteristic
343#67—24no—
344#67—28no—
345C2×Dic3C22×Dic314yes—
346C2×Dic3C22×Dic312yes—
347D12—22no—
348C4×D3—22no—
349C4×D3—22no—
350Dic6—22no—
351C3×C8—62no—
352C3×C8—62no—
353C3⋊C8—22no—
354C3⋊C8—22no—
355C3⋊C8—32no—
356C3⋊C8—32no—
357C3⋊C8—32no—
358C3⋊C8—32no—
359C3⋊C8—62no—
360C3⋊C8—62no—
361C3⋊C8—64no—
362C3⋊C8—62no—
363C3⋊C8—62no—
364C2×C32—22no—
365C2×C32—24no—
366C2×C32C2×C4×D411yescharacteristic
367C2×C32C2×#13112yes—
368C3×D3—42no—
369C24D3×Dic311yescharacteristic
370#41—62no—
371#41—62no—
372C2×Q8—31no—
373C2×D4—34no—
374C2×D4—32no—
375C2×D4—31no—
376C22×C4—22no—
377C22×C4—22no—
378C22×C4—32no—
379C22×C4D3×Dic312yes—
380C22×C4D3×Dic311yescharacteristic
381C22×C4C2×D3212yes—
382C22×C4C2×D3211yescharacteristic
383M16—62no—
384M16—62no—
385M16—184no—
386C2×C8—32no—
387C2×C8—32no—
388C2×C8—91no—
389C2×C8—91no—
390C2×C8—94no—
391C2×C8—92no—
392C2×C8—184no—
393C2×C8—184no—
394C2×C8—184no—
395C2×C8—182no—
396C4⋊C4—32no—
397C4⋊C4—34no—
398C22⋊C4—32no—
399C22⋊C4—34no—
400C22⋊C4—34no—
401C42—32no—
402C22×C3—24no—
403C22×C3—216no—
404C22×C3—28no—
405C22×C3—28no—
406C22×C3—24no—
407C22×C3—24no—
408C22×C3—22no—
409C22×C3—22no—
410C22×C3—22no—
411C22×C3—22no—
412C22×C3—22no—
413C22×C3—22no—
414C22×C3—21no—
415C22×C3—24no—
416C22×C3—22no—
417C22×C3—62no—
418C22×C3—64no—
419C22×C3#67⋊C412yes—
420C22×C3#90714yescomplemented
421C22×C3D4×Dic312yes—
422C22×C3#94814yescomplemented
423C22×C3C22×C4×D311yescharacteristic
424C22×C3C23×Dic311yescharacteristic
425C2×D3—22no—
426C2×D3—24no—
427C3×C4—24no—
428C3×C4—22no—
429C3×C4—22no—
430C3×C4—21no—
431C3×C4—21no—
432C3×C4—22no—
433C3×C4—62no—
434C3×C4—64no—
435C3×C4#67⋊C412yes—
436C3×C4D4×Dic312yes—
437C3×C4C2×D3×D411yescharacteristic
438C3×C4C2×D3×D411yescharacteristic
439C3×C4C2×#23711yescharacteristic
440C3×C4C2×#23711yescharacteristic
441Dic3—22no—
442Dic3—24no—
443C32#292911yescharacteristic, complemented, Sylow
444C23—24no—
445C23—22no—
446C23—32no—
447C23C2×D3×Dic312yes—
448C23C2×D3×Dic311yescharacteristic
449Q8—62no—
450D4—68no—
451D4—64no—
452D4—62no—
453C2×C4—22no—
454C2×C4—22no—
455C2×C4—22no—
456C2×C4—24no—
457C2×C4—34no—
458C2×C4—32no—
459C2×C4—62no—
460C2×C4—62no—
461C2×C4C2×D3×Dic312yes—
462C2×C4C2×D3×Dic311yescharacteristic
463C2×C4#380512yes—
464C2×C4#381112yes—
465C2×C4C22×D3211yescharacteristic
466C8—62no—
467C8—62no—
468C8—92no—
469C8—92no—
470C8—182no—
471C2×C3—28no—
472C2×C3—24no—
473C2×C3—22no—
474C2×C3—24no—
475C2×C3—22no—
476C2×C3—24no—
477C2×C3—21no—
478C2×C3—22no—
479C2×C3—122no—
480C2×C3C2×#67⋊C411yescharacteristic
481C2×C3C2×#90712yes—
482C2×C3C2×D4×Dic311yescharacteristic
483C2×C3C2×#94812yes—
484D3—42no—
485C22—28no—
486C22—22no—
487C22—22no—
488C22—22no—
489C22—62no—
490C22—64no—
491C22#6381514yescomplemented
492C22#67⋊Dic312yes—
493C22C22×D3×Dic311yescharacteristic
494C4—22no—
495C4—62no—
496C4—64no—
497C4#67⋊Dic312yes—
498C4C2×#380511yescharacteristic
499C4C2×#381111yescharacteristic
500C3—21no—
501C3#7862611yescharacteristic, complemented
502C3#8291611yescharacteristic, complemented
503C2—24no—
504C2—22no—
505C2—122no—
506C2C2×#6381512yes—
507C2C2×#67⋊Dic311yescharacteristic
subgroupG/H
center, FrattiniC2×C4C22×D32
commutator, socleC22×C32C23×C4
FittingC23×C32×C4C22
prime radicalC3×#5791C2
cosocleC2×C32×C4C24
hypercenterC23×C4D32
hypocenterC32#2929
LCSG ▷ C22×C32 ▷ C32
UCSC1 ◁ C2×C4 ◁ C23×C4

Complements

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

NH ≅ G/Ntype
1. C23×C3⋊(C3⋊C8)C2semidirect
3. C2×#63815C2semidirect
4. C3×#5207C2semidirect
5. #10591158C2semidirect
6. #67⋊(C3⋊C8)C2semidirect
7. #10591117C2semidirect
8. #10591071C2semidirect
9. #10591057C2semidirect
27. C2×#3711C22semidirect
29. C2×D3×C3⋊C8C22semidirect
31. Dic3⋊(C3⋊C8)C22semidirect
32. (C2×D3)⋊(C3⋊C8)C22semidirect
33. #63554C22semidirect
34. Dic3×C3⋊C8C22semidirect
44. #5207D3semidirect
71. C2×C3⋊(C3⋊C8)D4semidirect
261. C22×C32#131semidirect
420. C22×C3#907semidirect
422. C22×C3#948semidirect
443. C32#2929semidirect
491. C22#63815semidirect
501. C3#78626semidirect
502. C3#82916semidirect

Maximal subgroups

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

subgroupindex
C23×C3⋊(C3⋊C8)2
C3×#57912
C2×#638152
C3×#52072
#105911582
#67⋊(C3⋊C8)2
#105911172
#105910712
#105910572
#829163
#786263