Finite Group Databasebeta

← #11792209

Subgroups — #11792209

Order 1152.

779 conjugacy classes of subgroups.

The subgroup lattice has 544 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#909C211yesmaximal, characteristic
2C2×#63873C211yesmaximal, characteristic, complemented
3#10594855C211yesmaximal, characteristic
4C2×C3×#824C211yesmaximal, characteristic, complemented
5C2×#63568C211yesmaximal, characteristic
6#10591100C211yesmaximal, characteristic, complemented
7#10589421C211yesmaximal, characteristic, complemented
8#72108—31nomaximal
9#70119—31nomaximal
10C2×C3×#235C2211yescharacteristic
11C22×C3×Dic6C2211yescharacteristic
12C22×C3×Dic6C2211yescharacteristic
13C2×C32×M16C2211yescharacteristic
14#64135C2211yescharacteristic
15C2×C4×C3⋊Dic3C2211yescharacteristic
16C2×C3×#208C2211yescharacteristic
17C3×#909—22no—
18C3×#909—22no—
19C2×C3×Dic12—21no—
20C2×C3×#204—21no—
21C2×#3724—22no—
22#63961—22no—
23#63873—21no—
24#63873—22no—
25#63873—22no—
26#63863—21no—
27#63856—22no—
28C3×#824—22no—
29#63568—22no—
30#63566—22no—
31C2×C3×#137—31no—
32#5825—31no—
33#5816—31no—
34C2×#944—31no—
35C2×#944—31no—
36C2×#909D311yescharacteristic
37C2×C3×#101—31no—
38C2×#837—31no—
39#5225—31no—
40C2×#825—31no—
41C2×#824D311yescharacteristic, complemented
42#5120—31no—
43#4555—31no—
44#4555—31no—
45#4555—61no—
46C22×C32×C4C2311yescharacteristic
47C22×C3⋊Dic3—21no—
48C2×C3×#67—21no—
49C22×C3×Dic3—21no—
50C22×C3×Dic3D412yes—
51C3×#235—21no—
52C3×#235—21no—
53C3×#235—21no—
54C3×#235—21no—
55C2×C3×D12D411yescharacteristic
56C2×C3×C4×D3—21no—
57C2×C3×Dic6—21no—
58C2×C3×Dic6—21no—
59C2×C3×Dic6—21no—
60C2×C3×Dic6—21no—
61C2×C3×Dic6—21no—
62C2×C3×Dic6—21no—
63C2×C3×Dic6—21no—
64C2×C3×Dic6—21no—
65C2×C3×Dic6—21no—
66C2×C3×Dic6—21no—
67C2×C3×Dic6—21no—
68C2×C3×Dic6D411yescharacteristic
69C2×C3×Dic6D412yes—
70C32×M16—21no—
71C32×M16D412yes—
72C2×C32×C8—21no—
73#3757—21no—
74C4⋊(C3⋊Dic3)D412yes—
75C4×C3⋊Dic3—21no—
76C4×C3⋊Dic3—21no—
77C4×C3⋊Dic3D412yes—
78C3×#208—21no—
79C3×#208—21no—
80C2×C3×C3⋊C8—21no—
81C3×Dic12—24no—
82C3×#204—24no—
83#3724—24no—
84#3724—28no—
85(C2×D3)⋊Dic3—24no—
86#2005—91noSylow
87C2×C3×#41—31no—
88C22×C3×Q8—31no—
89C22×C3×Q8—31no—
90C2×#235C2×D311yescharacteristic
91C22×Dic6C2×D311yescharacteristic
92C22×Dic6C2×D311yescharacteristic
93C3×#137—62no—
94C3×#137—62no—
95C2×C3×Q16—61no—
96C2×C3×SD16—61no—
97C2×C3×M16—21no—
98C2×C3×M16—31no—
99C2×C3×M16C2×D311yescharacteristic
100C2×C3×M16C2×D311yescharacteristic
101#947—61no—
102#946—62no—
103#944—62no—
104#944—62no—
105#944—62no—
106#944—61no—
107#944—62no—
108#944—61no—
109#936—61no—
110#935—62no—
111#929—62no—
112#926—31no—
113#926—31no—
114#926—61no—
115#924—62no—
116C2×Dic3⋊C4—62no—
117C2×Dic3⋊C4—62no—
118C2×C4×Dic3—31no—
119C2×C4×Dic3—31no—
120C2×C4×Dic3—61no—
121C2×#208C2×D311yescharacteristic
122#909—22no—
123#909—22no—
124C2×Dic12—21no—
125C2×#204—21no—
126C3×#101—62no—
127#837—62no—
128#835—62no—
129#825—62no—
130#824—22no—
131#816—62no—
132C23×C32C2×D411yescharacteristic
133C22×C3×D3—21no—
134C2×C32×C4C2×D411yescharacteristic
135C2×C32×C4C2×D411yescharacteristic
136C2×C32×C4C2×D411yescharacteristic
137C2×C32×C4C2×D411yescharacteristic
138C2×C32×C4C2×D411yescharacteristic
139C2×C32×C4#4111yescharacteristic
140C2×C3⋊Dic3—24no—
141C2×C3⋊Dic3—22no—
142C2×C3⋊Dic3—22no—
143C3×#67—41no—
144C3×#67—41no—
145C2×C3×Dic3—22no—
146C2×C3×Dic3—22no—
147C2×C3×Dic3—22no—
148C2×C3×Dic3—21no—
149C2×C3×Dic3—22no—
150C2×C3×Dic3—41no—
151C2×C3×Dic3—41no—
152C3×D12—21no—
153C3×D12—21no—
154C3×C4×D3—41no—
155C3×C4×D3—41no—
156C3×Dic6—22no—
157C3×Dic6—21no—
158C3×Dic6—22no—
159C3×Dic6—21no—
160C3×Dic6—41no—
161C3×Dic6—41no—
162C3×Dic6—41no—
163C3×Dic6—41no—
164C3×Dic6—41no—
165C3×Dic6—41no—
166C32×C8—22no—
167C3×C3⋊C8—41no—
168C2×#137—31no—
169#536—91no—
170C2×C4⋊Q8—91no—
171C2×#104—91no—
172#419—91no—
173C2×#101—31no—
174#339—91no—
175C3×#41—61no—
176C3×#41—61no—
177C3×#41—61no—
178C3×#41—61no—
179C2×C3×Q8—31no—
180C2×C3×Q8—32no—
181C2×C3×Q8—61no—
182C2×C3×Q8—61no—
183C2×C3×Q8—61no—
184C2×C3×Q8—61no—
185C2×C3×Q8—61no—
186C2×C3×Q8—61no—
187C2×C3×Q8—61no—
188C2×C3×Q8—61no—
189C2×C3×Q8—61no—
190C2×C3×Q8—61no—
191C2×C3×Q8—61no—
192C2×C3×D4—31no—
193C2×C3×D4—61no—
194C22×C3×C4—21no—
195C22×C3×C4—32no—
196C22×C3×C4—61no—
197C22×C3×C4—61no—
198C22×C3×C4C22×D311yescharacteristic
199C22×C3×C4C22×D311yescharacteristic
200C2×#67—21no—
201C22×Dic3—21no—
202C22×Dic3—61no—
203C22×Dic3—61no—
204C22×Dic3—121no—
205C22×Dic3D1212yes—
206#235—21no—
207#235—21no—
208#235—21no—
209#235—21no—
210C2×D12#6711yescharacteristic
211C2×C4×D3—21no—
212C2×Dic6—21no—
213C2×Dic6—21no—
214C2×Dic6—21no—
215C2×Dic6—21no—
216C2×Dic6—21no—
217C2×Dic6—21no—
218C2×Dic6—21no—
219C2×Dic6—21no—
220C2×Dic6—21no—
221C2×Dic6—21no—
222C2×Dic6—21no—
223C2×Dic6#6711yescharacteristic
224C2×Dic6#6712yes—
225C3×Q16—64no—
226C3×SD16—64no—
227C3×M16—21no—
228C3×M16—21no—
229C3×M16—22no—
230C3×M16—41no—
231C3×M16—61no—
232C3×M16—61no—
233C3×M16#6712yes—
234C3×M16#6712yes—
235C2×C3×C8—21no—
236C2×C3×C8—21no—
237C2×C3×C8—41no—
238C2×C3×C8—61no—
239#217—64no—
240#217—61no—
241#217—61no—
242#217—121no—
243(C2×D3)⋊C4—64no—
244C4⋊Dic3—32no—
245C4⋊Dic3—32no—
246C4⋊Dic3—62no—
247Dic3⋊C4—68no—
248Dic3⋊C4—68no—
249Dic3⋊C4—64no—
250Dic3⋊C4—64no—
251C4×Dic3—32no—
252C4×Dic3—32no—
253C4×Dic3—62no—
254C4×Dic3—61no—
255C4×Dic3—61no—
256C4×Dic3—61no—
257C4×Dic3—61no—
258C4×Dic3—121no—
259C4×Dic3—121no—
260#208—21no—
261#208—21no—
262C2×C3⋊C8—21no—
263Dic12—24no—
264#204—24no—
265C22×C32—22no—
266C22×C32#12011yescharacteristic
267C22×C32#12011yescharacteristic
268C22×C32#12111yescharacteristic
269C2×C3×D3—41no—
270C2×C3×D3—42no—
271C32×C4#12011yescharacteristic
272C32×C4#12111yescharacteristic
273C32×C4#12111yescharacteristic
274C32×C4#12711yescharacteristic
275C3⋊Dic3—24no—
276C3⋊Dic3—42no—
277C3×Dic3—42no—
278C3×Dic3—41no—
279C3×Dic3—42no—
280C2×#41—31no—
281C22×Q8—31no—
282C22×Q8—31no—
283#137—62no—
284#137—62no—
285C2×Q16—61no—
286C2×SD16—61no—
287C2×M16—31no—
288C2×M16D3211yescharacteristic
289C4⋊Q8—182no—
290C4⋊Q8—182no—
291C4⋊Q8—181no—
292#124—181no—
293#123—182no—
294C22⋊Q8—182no—
295#117—91no—
296C2×C4⋊C4—182no—
297C2×C42—91no—
298#104—182no—
299#103—182no—
300#101—62no—
301C23×C3—21no—
302C23×C3—61no—
303C23×C3C2×D1211yescharacteristic
304C23×C3D3×D411yescharacteristic
305C22×D3—21no—
306C3×Q8—62no—
307C3×Q8—61no—
308C3×Q8—62no—
309C3×Q8—61no—
310C3×Q8—121no—
311C3×Q8—121no—
312C3×Q8—121no—
313C3×Q8—121no—
314C3×Q8—121no—
315C3×Q8—121no—
316C3×D4—61no—
317C3×D4—61no—
318C3×D4—121no—
319C3×D4—121no—
320C2×C3×C4—21no—
321C2×C3×C4—21no—
322C2×C3×C4—21no—
323C2×C3×C4—21no—
324C2×C3×C4—21no—
325C2×C3×C4—21no—
326C2×C3×C4—61no—
327C2×C3×C4—62no—
328C2×C3×C4—62no—
329C2×C3×C4—62no—
330C2×C3×C4—62no—
331C2×C3×C4—121no—
332C2×C3×C4—121no—
333C2×C3×C4—121no—
334C2×C3×C4—121no—
335C2×C3×C4#23511yescharacteristic
336C2×C3×C4D3×D411yescharacteristic
337C2×C3×C4D3×D411yescharacteristic
338C2×C3×C4D3×D411yescharacteristic
339C2×C3×C4D3×D411yescharacteristic
340C2×C3×C4D3×D411yescharacteristic
341C2×C3×C4D3×D411yescharacteristic
342C2×C3×C4#23711yescharacteristic
343C2×C3×C4C2×#6711yescharacteristic
344C2×C3×C4C2×#6711yescharacteristic
345C2×C3×C4C2×#6711yescharacteristic
346C2×C3×C4C2×#6711yescharacteristic
347#67—41no—
348#67—41no—
349C2×Dic3—22no—
350C2×Dic3—22no—
351C2×Dic3—22no—
352C2×Dic3—22no—
353C2×Dic3—21no—
354C2×Dic3—41no—
355C2×Dic3—41no—
356C2×Dic3—64no—
357C2×Dic3—64no—
358C2×Dic3—62no—
359C2×Dic3—62no—
360C2×Dic3—62no—
361C2×Dic3—62no—
362C2×Dic3—122no—
363C2×Dic3—124no—
364C2×Dic3—122no—
365D12—21no—
366D12—21no—
367C4×D3—41no—
368C4×D3—41no—
369Dic6—22no—
370Dic6—21no—
371Dic6—22no—
372Dic6—21no—
373Dic6—41no—
374Dic6—41no—
375Dic6—41no—
376Dic6—41no—
377Dic6—41no—
378Dic6—41no—
379C3×C8—22no—
380C3×C8—22no—
381C3×C8—42no—
382C3×C8—121no—
383C3⋊C8—41no—
384C2×C32—21no—
385C2×C32—21no—
386C2×C32#39211yescharacteristic
387C2×C32#45612yes—
388C3×D3—81no—
389#41—61no—
390#41—61no—
391#41—61no—
392#41—61no—
393C2×Q8—31no—
394C2×Q8—32no—
395C2×Q8—61no—
396C2×Q8—61no—
397C2×Q8—61no—
398C2×Q8—61no—
399C2×Q8—61no—
400C2×Q8—61no—
401C2×Q8—61no—
402C2×Q8—61no—
403C2×Q8—61no—
404C2×Q8—61no—
405C2×Q8—61no—
406C2×D4—31no—
407C2×D4—61no—
408C22×C4—32no—
409C22×C4—61no—
410C22×C4—61no—
411C22×C4—181no—
412C22×C4C2×D3211yescharacteristic
413Q16—64no—
414SD16—64no—
415M16—21no—
416M16—61no—
417M16—61no—
418M16#62512yes—
419C2×C8—21no—
420C2×C8—61no—
421C4⋊C4—92no—
422C4⋊C4—188no—
423C4⋊C4—184no—
424C22⋊C4—184no—
425C22⋊C4—181no—
426C42—92no—
427C42—181no—
428C42—181no—
429C22×C3—22no—
430C22×C3—22no—
431C22×C3—21no—
432C22×C3—21no—
433C22×C3—21no—
434C22×C3—42no—
435C22×C3—121no—
436C22×C3—122no—
437C22×C3#88211yescharacteristic
438C22×C3#88211yescharacteristic
439C22×C3#93011yescharacteristic
440C22×C3#93711yescharacteristic
441C22×C3#93711yescharacteristic
442C22×C3#93811yescharacteristic
443C2×D3—42no—
444C2×D3—41no—
445C3×C4—21no—
446C3×C4—21no—
447C3×C4—21no—
448C3×C4—21no—
449C3×C4—121no—
450C3×C4—122no—
451C3×C4—122no—
452C3×C4#88411yescharacteristic
453C3×C4#88411yescharacteristic
454C3×C4#93711yescharacteristic
455C3×C4#93911yescharacteristic
456C3×C4#93911yescharacteristic
457C3×C4#94011yescharacteristic
458C3×C4#94011yescharacteristic
459C3×C4#95311yescharacteristic
460Dic3—42no—
461Dic3—42no—
462Dic3—41no—
463Dic3—64no—
464Dic3—64no—
465Dic3—124no—
466Dic3—122no—
467Dic3—122no—
468Dic3—242no—
469C32#200511yescharacteristic, complemented, Sylow
470C23—61no—
471C23D3×D1211yescharacteristic
472Q8—62no—
473Q8—61no—
474Q8—62no—
475Q8—61no—
476Q8—121no—
477Q8—121no—
478Q8—121no—
479Q8—121no—
480Q8—121no—
481Q8—121no—
482D4—61no—
483D4—61no—
484D4—121no—
485D4—121no—
486C2×C4—62no—
487C2×C4—62no—
488C2×C4—62no—
489C2×C4—62no—
490C2×C4—61no—
491C2×C4—121no—
492C2×C4—121no—
493C2×C4—121no—
494C2×C4—121no—
495C2×C4—184no—
496C2×C4—182no—
497C2×C4—182no—
498C2×C4#379511yescharacteristic
499C2×C4#380211yescharacteristic
500C2×C4#380211yescharacteristic
501C2×C4C2×#62511yescharacteristic
502C2×C4D3×#6711yescharacteristic
503C2×C4D3×#6711yescharacteristic
504C8—22no—
505C8—121no—
506C2×C3—21no—
507C2×C3—21no—
508C2×C3—21no—
509C2×C3—21no—
510C2×C3—22no—
511C2×C3—21no—
512C2×C3—41no—
513C2×C3—41no—
514C2×C3—241no—
515C2×C3#475212yes—
516C2×C3#496011yescharacteristic
517C2×C3#520512yes—
518C2×C3#522311yescharacteristic
519D3—81no—
520C22—22no—
521C22—122no—
522C22—121no—
523C22#6390811yescharacteristic
524C22#6392211yescharacteristic
525C22#6392211yescharacteristic
526C4—122no—
527C4—121no—
528C4—122no—
529C4—184no—
530C4—362no—
531C4#6388511yescharacteristic
532C4#6388511yescharacteristic
533C4#6391411yescharacteristic
534C4#6397611yescharacteristic
535C3—21no—
536C3#7011911yescharacteristic, complemented
537C3#7210811yescharacteristic, complemented
538C2—21no—
539C2—21no—
540C2—241no—
541C2#1059101812yes—
542C2#1059138911yescharacteristic
subgroupG/H
centerC22#63922
commutator, cosocleC22×C32×C4C23
FrattiniC22×C4C2×D32
FittingC2×C32×M16C22
socleC22×C32#120
prime radicalC2×C3×#235C22
hypercenterC2×M16D32
hypocenterC32#2005
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. C2×#63873C2semidirect
4. C2×C3×#824C2semidirect
6. #10591100C2semidirect
7. #10589421C2semidirect
41. C2×#824D3semidirect
469. C32#2005semidirect
536. C3#70119semidirect
537. C3#72108semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#9092
C2×#638732
#105948552
C2×C3×#8242
C2×#635682
#105911002
#105894212
#721083
#701193