Finite Group Databasebeta

← #11859347

Subgroups — #11859347

Order 1152.

1205 conjugacy classes of subgroups.

The subgroup lattice has 528 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#10596262C211yesmaximal, characteristic
2C23×C3×C3⋊C8C211yesmaximal, characteristic, complemented
3C2×#63817C211yesmaximal, characteristic, complemented
4C3×#5207C211yesmaximal, characteristic, complemented
5#10591157C214yesmaximal, complemented
6#10591148C214yesmaximal, complemented
7#10591119C211yesmaximal, characteristic, complemented
8#10591073C211yesmaximal, characteristic, complemented
9#10591059C211yesmaximal, characteristic, complemented
10#83734—31nomaximal
11#78626—31nomaximal
12C23×C32×C4C2211yescharacteristic
13C2×#3828C2211yescharacteristic
14#64162C412yes—
15#64139C412yes—
16#64138C412yes—
17C4×#638C2214yes—
18#64135C2211yescharacteristic
19#64133C412yes—
20C2×C3×#208C2211yescharacteristic
21C2×C3×#208C2211yescharacteristic
22C22×C3×C3⋊C8—24no—
23C22×C3×C3⋊C8C2211yescharacteristic
24C22×C3×C3⋊C8C2212yes—
25C22×C3×C3⋊C8C2212yes—
26C22×C3×C3⋊C8C2211yescharacteristic
27C2×#3712C2212yescomplemented
28#63817—24no—
29C2×#3710C2212yescomplemented
30C3×#830C2214yes—
31#63572C2214yescomplemented
32#63557C2214yescomplemented
33#63555C2212yescomplemented
34#63553C2212yescomplemented
35C23×C3×C8—31no—
36#5791—31no—
37#5791—31no—
38#5791—61no—
39C23×C3⋊C8D311yescharacteristic, complemented
40C2×#907—31no—
41C2×#901—31no—
42C3×#407—31no—
43#5207D311yescharacteristic, complemented
44#5129—31no—
45#5122—31no—
46#5119—31no—
47#5111—34no—
48#5110—31no—
49C8×#67—34no—
50#5102—31no—
51#5101—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×#638C2×C412yes—
62C2×#638C2×C414yes—
63#3828—22no—
64#3828—22no—
65C2×C4⋊(C3⋊D3)C2×C411yescharacteristic
66C2×C4×C3⋊D3C2312yes—
67C2×#634C2×C411yescharacteristic
68#3757C2×C414yes—
69#3757C2×C412yes—
70#3752C2×C414yes—
71C4⋊(C3⋊Dic3)C2×C412yes—
72#3750C2×C414yes—
73C4×C3⋊Dic3C2312yes—
74C3×#208—22no—
75C3×#208—22no—
76C2×C3×C3⋊C8—22no—
77C2×C3×C3⋊C8—24no—
78C2×C3×C3⋊C8—24no—
79C2×C3×C3⋊C8D414yescomplemented
80C2×C3×C3⋊C8C2312yes—
81C2×C3×C3⋊C8C2312yes—
82C2×C3×C3⋊C8C2312yes—
83C2×C3×C3⋊C8C2311yescharacteristic
84C2×C3×C3⋊C8C2311yescharacteristic
85#3712—24no—
86#3710—24no—
87#2929—91noSylow
88C23×C3×C4—21no—
89C23×C3×C4C2×D311yescharacteristic
90C23×C3×C4C2×D311yescharacteristic
91C2×#235—31no—
92C2×#235—31no—
93C2×#235—61no—
94C2×C3×M16—31no—
95C2×C3×M16—31no—
96C22×C3×C8—31no—
97C22×C3×C8—31no—
98C22×C3×C8—32no—
99C22×C3×C8—32no—
100C22×C3×C8—64no—
101#953—32no—
102#953—32no—
103#953—62no—
104#930—32no—
105#930—32no—
106#930—62no—
107#929—32no—
108#929—32no—
109#929—62no—
110C4×#67—34no—
111C4×#67—34no—
112C4×#67—64no—
113#926—31no—
114#926—31no—
115#926—61no—
116#924—32no—
117#924—32no—
118#924—62no—
119C2×#208C2×D311yescharacteristic
120C2×#208C2×D311yescharacteristic
121C22×C3⋊C8—24no—
122C22×C3⋊C8C2×D311yescharacteristic
123C22×C3⋊C8C2×D312yes—
124C22×C3⋊C8C2×D312yes—
125C22×C3⋊C8C2×D311yescharacteristic
126#907—64no—
127#901—64no—
128C2×ΓL(1,𝔽25)—32no—
129C2×ΓL(1,𝔽25)—32no—
130C2×D3×C8—32no—
131C2×D3×C8—32no—
132C3×C22⋊C8—34no—
133#830C2×D314yes—
134(C2×D3)⋊C8—34no—
135(C2×D3)⋊C8—34no—
136#815—32no—
137#815—32no—
138Dic3⋊C8—34no—
139Dic3⋊C8—34no—
140C8×Dic3—32no—
141C8×Dic3—32no—
142C23×C32—22no—
143C23×C32—24no—
144C23×C32C22×C411yescharacteristic
145C23×C32C22×C412yes—
146C22×C3⋊D3C22×C412yes—
147C2×C32×C4—22no—
148C2×C32×C4—24no—
149C2×C32×C4—22no—
150C2×C32×C4—22no—
151C2×C32×C4C22×C412yes—
152C2×C32×C4C22×C411yescharacteristic
153C2×C32×C4C2×D412yes—
154C2×C32×C4#4112yes—
155C2×C32×C4C2411yescharacteristic
156#638—24no—
157#638—28no—
158C2×C3⋊Dic3C22×C414yes—
159C2×C3⋊Dic3C22×C412yes—
160C4⋊(C3⋊D3)—22no—
161C4×C3⋊D3—22no—
162C4×C3⋊D3—22no—
163#634—22no—
164C3×C3⋊C8—22no—
165C3×C3⋊C8—22no—
166C3×C3⋊C8—22no—
167C3×C3⋊C8C2×D412yes—
168C3×C3⋊C8C2×D412yes—
169C2×#131—91no—
170C23×C8—31no—
171#525—91no—
172#435—94no—
173C8×D4—94no—
174#424—91no—
175#408—91no—
176#407—31no—
177#405—91no—
178C24×C3—21no—
179C24×C3C4×D311yescharacteristic
180C24×C3C4×D311yescharacteristic
181C22×C3×C4—22no—
182C22×C3×C4—22no—
183C22×C3×C4—22no—
184C22×C3×C4—22no—
185C22×C3×C4—22no—
186C22×C3×C4—21no—
187C22×C3×C4—21no—
188C22×C3×C4—22no—
189C22×C3×C4—42no—
190C22×C3×C4—42no—
191C22×C3×C4C4×D312yes—
192C22×C3×C4C4×D312yes—
193C22×C3×C4C4×D311yescharacteristic
194C22×C3×C4C4×D311yescharacteristic
195C22×C3×C4C22×D312yes—
196C22×C3×C4C22×D312yes—
197C22×C3×C4C22×D311yescharacteristic
198C22×C3×C4C22×D311yescharacteristic
199C2×#67—34no—
200C2×#67—34no—
201C2×#67—32no—
202C2×#67—32no—
203C2×#67—64no—
204C2×#67—62no—
205#235—62no—
206#235—62no—
207#235—62no—
208#235—62no—
209#235—122no—
210#235—122no—
211C2×D12—31no—
212C2×D12—31no—
213C2×D12—61no—
214C2×C4×D3—32no—
215C2×C4×D3—32no—
216C2×C4×D3—62no—
217C2×Dic6—31no—
218C2×Dic6—31no—
219C2×Dic6—61no—
220C3×M16—62no—
221C3×M16—62no—
222C2×C3×C8—34no—
223C2×C3×C8—31no—
224C2×C3×C8—31no—
225C2×C3×C8—32no—
226C2×C3×C8—32no—
227C2×C3×C8—32no—
228C2×C3×C8—64no—
229C2×C3×C8—64no—
230C2×C3×C8—62no—
231#217—34no—
232#217—34no—
233#217—32no—
234#217—32no—
235#217—64no—
236#217—62no—
237(C2×D3)⋊C4—34no—
238(C2×D3)⋊C4—34no—
239(C2×D3)⋊C4—64no—
240C4⋊Dic3—32no—
241C4⋊Dic3—32no—
242C4⋊Dic3—62no—
243Dic3⋊C4—34no—
244Dic3⋊C4—34no—
245Dic3⋊C4—64no—
246C4×Dic3—32no—
247C4×Dic3—32no—
248C4×Dic3—62no—
249#208—22no—
250#208—22no—
251C2×C3⋊C8—22no—
252C2×C3⋊C8—24no—
253C2×C3⋊C8—24no—
254C2×C3⋊C8#6714yescomplemented
255C2×C3⋊C8C22×D312yes—
256C2×C3⋊C8C22×D312yes—
257C2×C3⋊C8C22×D312yes—
258C2×C3⋊C8C22×D311yescharacteristic
259C2×C3⋊C8C22×D311yescharacteristic
260ΓL(1,𝔽25)—64no—
261ΓL(1,𝔽25)—64no—
262D3×C8—64no—
263D3×C8—64no—
264C22×C32—28no—
265C22×C32—22no—
266C22×C32—22no—
267C22×C32—22no—
268C22×C32C4×D412yes—
269C22×C32#13114yescomplemented
270C22×C32C23×C411yescharacteristic
271C2×C3⋊D3—22no—
272C2×C3⋊D3—24no—
273C32×C4—22no—
274C32×C4C4×D412yes—
275C32×C4C22×D411yescharacteristic
276C32×C4C2×#4111yescharacteristic
277C3⋊Dic3—22no—
278C3⋊Dic3—24no—
279C2×#41—91no—
280C23×C4D3211yescharacteristic
281#131—184no—
282C2×M16—31no—
283C2×M16—31no—
284C2×M16—92no—
285C22×C8—31no—
286C22×C8—31no—
287C22×C8—32no—
288C22×C8—32no—
289C22×C8—64no—
290C22×C8—92no—
291#123—92no—
292C22⋊Q8—92no—
293#121—92no—
294#120—92no—
295C4×D4—94no—
296#117—91no—
297C4⋊C8—94no—
298C22⋊C8—34no—
299C22⋊C8—94no—
300#97—92no—
301C4×C8—92no—
302C23×C3—22no—
303C23×C3—24no—
304C23×C3—24no—
305C23×C3—22no—
306C23×C3—22no—
307C23×C3—21no—
308C23×C3—42no—
309C23×C3—44no—
310C23×C3C2×C4×D311yescharacteristic
311C23×C3C2×C4×D312yes—
312C23×C3C2×C4×D312yes—
313C23×C3C2×C4×D311yescharacteristic
314C22×D3—32no—
315C22×D3—32no—
316C22×D3—62no—
317C2×C3×C4—22no—
318C2×C3×C4—22no—
319C2×C3×C4—24no—
320C2×C3×C4—24no—
321C2×C3×C4—22no—
322C2×C3×C4—22no—
323C2×C3×C4—22no—
324C2×C3×C4—22no—
325C2×C3×C4—21no—
326C2×C3×C4—22no—
327C2×C3×C4—21no—
328C2×C3×C4—22no—
329C2×C3×C4—22no—
330C2×C3×C4—42no—
331C2×C3×C4—42no—
332C2×C3×C4—44no—
333C2×C3×C4—42no—
334C2×C3×C4C2×C4×D312yes—
335C2×C3×C4C2×C4×D312yes—
336C2×C3×C4C2×C4×D311yescharacteristic
337C2×C3×C4C2×C4×D311yescharacteristic
338C2×C3×C4#23512yes—
339C2×C3×C4D3×D412yes—
340C2×C3×C4#23712yes—
341C2×C3×C4C2×#6712yes—
342C2×C3×C4C23×D311yescharacteristic
343C2×C3×C4C23×D311yescharacteristic
344#67—64no—
345#67—68no—
346#67—68no—
347#67—64no—
348#67—124no—
349#67—128no—
350C2×Dic3—34no—
351C2×Dic3—34no—
352C2×Dic3—32no—
353C2×Dic3—32no—
354C2×Dic3—64no—
355C2×Dic3—62no—
356D12—62no—
357D12—62no—
358D12—122no—
359C4×D3—62no—
360C4×D3—62no—
361C4×D3—62no—
362C4×D3—62no—
363C4×D3—122no—
364C4×D3—122no—
365Dic6—62no—
366Dic6—62no—
367Dic6—122no—
368C3×C8—32no—
369C3×C8—32no—
370C3×C8—62no—
371C3×C8—62no—
372C3×C8—62no—
373C3⋊C8—22no—
374C3⋊C8—22no—
375C3⋊C8—22no—
376C3⋊C8C2×#6712yes—
377C3⋊C8C2×#6712yes—
378C2×C32—22no—
379C2×C32—24no—
380C2×C32C2×C4×D411yescharacteristic
381C2×C32C2×#13112yes—
382C3⋊D3—42no—
383C24#62411yescharacteristic
384#41—182no—
385#41—182no—
386C2×Q8—91no—
387C2×D4—94no—
388C2×D4—92no—
389C2×D4—91no—
390C22×C4—22no—
391C22×C4—22no—
392C22×C4—92no—
393C22×C4#62412yes—
394C22×C4#62411yescharacteristic
395C22×C4C2×D3212yes—
396C22×C4C2×D3211yescharacteristic
397M16—62no—
398M16—62no—
399M16—184no—
400C2×C8—31no—
401C2×C8—31no—
402C2×C8—34no—
403C2×C8—32no—
404C2×C8—32no—
405C2×C8—32no—
406C2×C8—64no—
407C2×C8—64no—
408C2×C8—62no—
409C2×C8—184no—
410C4⋊C4—92no—
411C4⋊C4—94no—
412C22⋊C4—92no—
413C22⋊C4—94no—
414C22⋊C4—94no—
415C42—92no—
416C22×C3—28no—
417C22×C3—28no—
418C22×C3—22no—
419C22×C3—22no—
420C22×C3—22no—
421C22×C3—22no—
422C22×C3—22no—
423C22×C3—22no—
424C22×C3—21no—
425C22×C3—24no—
426C22×C3—22no—
427C22×C3—42no—
428C22×C3—42no—
429C22×C3—42no—
430C22×C3—48no—
431C22×C3#67⋊C412yes—
432C22×C3#90114yescomplemented
433C22×C3#90714yescomplemented
434C22×C3C4×#6712yes—
435C22×C3C22×C4×D311yescharacteristic
436C22×C3C22×C4×D311yescharacteristic
437C2×D3—64no—
438C2×D3—62no—
439C2×D3—64no—
440C2×D3—62no—
441C2×D3—122no—
442C2×D3—124no—
443C3×C4—22no—
444C3×C4—22no—
445C3×C4—21no—
446C3×C4—21no—
447C3×C4—22no—
448C3×C4—42no—
449C3×C4#67⋊C412yes—
450C3×C4C4×#6712yes—
451C3×C4C2×#23511yescharacteristic
452C3×C4C2×D3×D411yescharacteristic
453C3×C4C2×#23711yescharacteristic
454C3×C4C22×#6711yescharacteristic
455Dic3—64no—
456Dic3—62no—
457Dic3—64no—
458Dic3—62no—
459Dic3—122no—
460Dic3—124no—
461C32#292911yescharacteristic, complemented, Sylow
462C23—24no—
463C23—22no—
464C23—92no—
465C23C2×#62412yes—
466C23C2×#62411yescharacteristic
467Q8—182no—
468D4—188no—
469D4—184no—
470D4—182no—
471C2×C4—22no—
472C2×C4—22no—
473C2×C4—22no—
474C2×C4—24no—
475C2×C4—94no—
476C2×C4—92no—
477C2×C4—182no—
478C2×C4—182no—
479C2×C4#380412yes—
480C2×C4C2×#62412yes—
481C2×C4C2×#62411yescharacteristic
482C2×C4D3×#6712yes—
483C2×C4C22×D3211yescharacteristic
484C8—32no—
485C8—32no—
486C8—62no—
487C8—62no—
488C8—62no—
489C2×C3—24no—
490C2×C3—22no—
491C2×C3—22no—
492C2×C3—24no—
493C2×C3—21no—
494C2×C3—22no—
495C2×C3—44no—
496C2×C3—42no—
497C2×C3C2×#67⋊C411yescharacteristic
498C2×C3C2×#90112yes—
499C2×C3C2×#90712yes—
500C2×C3C2×C4×#6711yescharacteristic
501D3—122no—
502D3—122no—
503D3—242no—
504C22—28no—
505C22—22no—
506C22—22no—
507C22—22no—
508C22—182no—
509C22—184no—
510C22#6381714yescomplemented
511C22#6395212yes—
512C22C22×#62411yescharacteristic
513C4—22no—
514C4—182no—
515C4—184no—
516C4#6395212yes—
517C4C2×#380411yescharacteristic
518C4C2×D3×#6711yescharacteristic
519C3—21no—
520C3#7862611yescharacteristic, complemented
521C3#8373411yescharacteristic, complemented
522C2—24no—
523C2—22no—
524C2—362no—
525C2C2×#6381712yes—
526C2C2×#6395211yescharacteristic
subgroupG/H
center, FrattiniC2×C4C22×D32
commutator, socleC22×C32C23×C4
FittingC23×C32×C4C22
prime radical#10596262C2
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
2. C23×C3×C3⋊C8C2semidirect
3. C2×#63817C2semidirect
4. C3×#5207C2semidirect
5. #10591157C2semidirect
6. #10591148C2semidirect
7. #10591119C2semidirect
8. #10591073C2semidirect
9. #10591059C2semidirect
27. C2×#3712C22semidirect
29. C2×#3710C22semidirect
31. #63572C22semidirect
32. #63557C22semidirect
33. #63555C22semidirect
34. #63553C22semidirect
39. C23×C3⋊C8D3semidirect
43. #5207D3semidirect
79. C2×C3×C3⋊C8D4semidirect
254. C2×C3⋊C8#67semidirect
269. C22×C32#131semidirect
432. C22×C3#901semidirect
433. C22×C3#907semidirect
461. C32#2929semidirect
510. C22#63817semidirect
520. C3#78626semidirect
521. C3#83734semidirect

Maximal subgroups

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

subgroupindex
#105962622
C23×C3×C3⋊C82
C2×#638172
C3×#52072
#105911572
#105911482
#105911192
#105910732
#105910592
#837343
#786263