Finite Group Databasebeta

← #11791184

Subgroups — #11791184

Order 1152.

653 conjugacy classes of subgroups.

The subgroup lattice has 520 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#63822C211yesmaximal, characteristic
2C2×#63649C211yesmaximal, characteristic, complemented
3C2×C3×#825C211yesmaximal, characteristic, complemented
4C3×#5129C211yesmaximal, characteristic, complemented
5C2×#63569C211yesmaximal, characteristic, complemented
6#10591070C211yesmaximal, characteristic, complemented
7C2×Dic3×C3⋊C8C211yesmaximal, characteristic, complemented
8#70373—31nomaximal
9#69999—31nomaximal
10C2×C3×#235C2211yescharacteristic
11C2×C3×#235C2211yescharacteristic
12C2×C32×M16C2211yescharacteristic
13C2×#3748C2211yescharacteristic
14C22×C3⋊(C3⋊C8)C2211yescharacteristic
15C2×C3×C4×Dic3C2211yescharacteristic
16C22×C3×C3⋊C8C2211yescharacteristic
17C2×#3718C411yescharacteristic, complemented
18C2×#3715C411yescharacteristic, complemented
19C2×#3715C411yescharacteristic, complemented
20#63822—22no—
21#63822—22no—
22C2×#3713C411yescharacteristic, complemented
23#63649—22no—
24C3×#825—22no—
25C3×(C2×D3)⋊C8—22no—
26#63569—22no—
27(C2×D3)⋊(C3⋊C8)—22no—
28Dic3×C3⋊C8—22no—
29C2×#950—31no—
30C2×#950—31no—
31C2×C3×#104—31no—
32C3×#408—31no—
33C2×#837—31no—
34#5234—31no—
35C2×#825D311yescharacteristic, complemented
36#5129D311yescharacteristic, complemented
37C2×#822—31no—
38C2×#822—31no—
39C2×#822—61no—
40#5110—31no—
41C2×C8×Dic3—31no—
42C2×#805—31no—
43C2×C4×C3⋊C8—31no—
44C22×C32×C4C2311yescharacteristic
45C2×C3×#67—21no—
46C2×C3×#67—21no—
47C22×C3×Dic3—21no—
48C3×#235—21no—
49C3×#235—21no—
50C3×#235—21no—
51C3×#235—21no—
52C3×#235—21no—
53C3×#235—21no—
54C3×#235—21no—
55C3×#235—21no—
56C2×C3×D12C2×C411yescharacteristic
57C2×C3×D12C2×C411yescharacteristic
58C2×C3×C4×D3—21no—
59C2×C3×C4×D3—21no—
60C2×C3×Dic6C2×C411yescharacteristic
61C2×C3×Dic6C2×C411yescharacteristic
62C32×M16—21no—
63C32×M16—21no—
64C2×C32×C8—21no—
65#3748—21no—
66#3748—21no—
67C2×C3⋊(C3⋊C8)—21no—
68C2×C3⋊(C3⋊C8)C2×C411yescharacteristic
69C2×C3⋊(C3⋊C8)C2×C411yescharacteristic
70C2×C3⋊(C3⋊C8)D412yes—
71C2×C3⋊(C3⋊C8)D412yes—
72C3×C4×Dic3—21no—
73C3×C4×Dic3D412yes—
74C2×C3×C3⋊C8—21no—
75C2×C3×C3⋊C8—21no—
76C2×C3×C3⋊C8D412yes—
77#3718—22no—
78#3715—22no—
79#3715—22no—
80#3713—22no—
81#1974—91noSylow
82C2×C3×#41—31no—
83C2×C3×#41—31no—
84C2×#235C2×D311yescharacteristic
85C2×#235C2×D311yescharacteristic
86C2×C3×M16—21no—
87C2×C3×M16C2×D311yescharacteristic
88C2×C3×M16C2×D311yescharacteristic
89C22×C3×C8—31no—
90C2×C3×C42—31no—
91#950—62no—
92#950—62no—
93#950—62no—
94#950—62no—
95C2×#216—31no—
96C2×#216—31no—
97C2×#215—31no—
98C2×#215—31no—
99C2×#214—31no—
100C2×#214—31no—
101C2×#213—31no—
102C2×#213—31no—
103C2×C4×Dic3C2×D311yescharacteristic
104C2×#208—31no—
105C2×#208—31no—
106C2×#208—61no—
107C22×C3⋊C8—31no—
108C22×C3⋊C8—31no—
109C22×C3⋊C8—61no—
110C22×C3⋊C8C2×D311yescharacteristic
111C3×#104—62no—
112C3×C22⋊C8—62no—
113#837—62no—
114#830—62no—
115#825—22no—
116#822—62no—
117#822—62no—
118#822—64no—
119(C2×D3)⋊C8—22no—
120(C2×D3)⋊C8—62no—
121C8×Dic3—62no—
122#805—62no—
123C4×C3⋊C8—62no—
124C23×C32#4111yescharacteristic
125C22×C3×D3—21no—
126C22×C3×D3—21no—
127C2×C32×C4C22×C411yescharacteristic
128C2×C32×C4C2×D411yescharacteristic
129C2×C32×C4C2×D411yescharacteristic
130C2×C32×C4C2×D411yescharacteristic
131C2×C32×C4C2×D411yescharacteristic
132C2×C32×C4#4111yescharacteristic
133C3×#67—41no—
134C3×#67—41no—
135C3×#67—41no—
136C3×#67—41no—
137C2×C3×Dic3—24no—
138C2×C3×Dic3—22no—
139C2×C3×Dic3—21no—
140C2×C3×Dic3—21no—
141C3×D12—21no—
142C3×D12—21no—
143C3×D12—21no—
144C3×D12—21no—
145C3×C4×D3—41no—
146C3×C4×D3—41no—
147C3×C4×D3—41no—
148C3×C4×D3—41no—
149C3×Dic6—21no—
150C3×Dic6—21no—
151C3×Dic6—21no—
152C3×Dic6—21no—
153C32×C8—41no—
154C3⋊(C3⋊C8)—41no—
155C3⋊(C3⋊C8)C22⋊C412yes—
156C3⋊(C3⋊C8)C22⋊C412yes—
157C3×C3⋊C8—22no—
158C2×#135—91no—
159C2×#108—91no—
160C2×#104—31no—
161C2×#104—91no—
162#408—31no—
163#408—91no—
164C2×C4×C8—91no—
165C3×#41—61no—
166C3×#41—61no—
167C3×#41—61no—
168C3×#41—61no—
169C3×#41—61no—
170C3×#41—61no—
171C3×#41—61no—
172C3×#41—61no—
173C2×C3×Q8—31no—
174C2×C3×Q8—31no—
175C2×C3×D4—31no—
176C2×C3×D4—31no—
177C2×C3×D4—61no—
178C2×C3×D4—61no—
179C22×C3×C4—21no—
180C22×C3×C4—61no—
181C22×C3×C4—61no—
182C22×C3×C4—61no—
183C22×C3×C4C22×D311yescharacteristic
184C22×C3×C4C22×D311yescharacteristic
185C2×#67—21no—
186C2×#67—21no—
187C22×Dic3—21no—
188#235—21no—
189#235—21no—
190#235—21no—
191#235—21no—
192#235—21no—
193#235—21no—
194#235—21no—
195#235—21no—
196C2×D12C4×D311yescharacteristic
197C2×D12C4×D311yescharacteristic
198C2×C4×D3—21no—
199C2×C4×D3—21no—
200C2×Dic6C4×D311yescharacteristic
201C2×Dic6C4×D311yescharacteristic
202C3×M16—22no—
203C3×M16—21no—
204C3×M16—21no—
205C3×M16—21no—
206C3×M16—21no—
207C3×M16—41no—
208C2×C3×C8—21no—
209C2×C3×C8—21no—
210C2×C3×C8—32no—
211C2×C3×C8—41no—
212C2×C3×C8—61no—
213C2×C3×C8—61no—
214C3×C42—32no—
215C3×C42—61no—
216#216—62no—
217#216—62no—
218#215—62no—
219#215—62no—
220#214—62no—
221#214—62no—
222#213—62no—
223#213—62no—
224C4×Dic3—21no—
225C4×Dic3#6712yes—
226#208—62no—
227#208—61no—
228#208—61no—
229#208—61no—
230#208—61no—
231#208—121no—
232C2×C3⋊C8—21no—
233C2×C3⋊C8—21no—
234C2×C3⋊C8—31no—
235C2×C3⋊C8—31no—
236C2×C3⋊C8—31no—
237C2×C3⋊C8—31no—
238C2×C3⋊C8—32no—
239C2×C3⋊C8—32no—
240C2×C3⋊C8—32no—
241C2×C3⋊C8—32no—
242C2×C3⋊C8—62no—
243C2×C3⋊C8—61no—
244C2×C3⋊C8—61no—
245C2×C3⋊C8—62no—
246C2×C3⋊C8—61no—
247C2×C3⋊C8—61no—
248C2×C3⋊C8—121no—
249C2×C3⋊C8#6712yes—
250C22×C32—22no—
251C22×C32C4×D411yescharacteristic
252C22×C32#12111yescharacteristic
253C22×C32#12411yescharacteristic
254C2×C3×D3—41no—
255C2×C3×D3—41no—
256C2×C3×D3—42no—
257C2×C3×D3—42no—
258C32×C4C2×C22⋊C411yescharacteristic
259C32×C4C4×D411yescharacteristic
260C32×C4#12111yescharacteristic
261C32×C4#12711yescharacteristic
262C3×Dic3—24no—
263C3×Dic3—41no—
264C3×Dic3—41no—
265C2×#41—31no—
266C2×#41—31no—
267#135—182no—
268#135—182no—
269C2×Q16—91no—
270C2×SD16—91no—
271C2×SD16—91no—
272C2×D8—91no—
273C2×M16—91no—
274C2×M16D3211yescharacteristic
275C22×C8—31no—
276C22×C8—91no—
277C2×C42—31no—
278#108—182no—
279#104—62no—
280#104—182no—
281C22⋊C8—62no—
282C22⋊C8—182no—
283C4×C8—182no—
284C23×C3—21no—
285C23×C3—61no—
286C23×C3—61no—
287C23×C3#23511yescharacteristic
288C23×C3#23711yescharacteristic
289C22×D3—21no—
290C22×D3—21no—
291C3×Q8—61no—
292C3×Q8—61no—
293C3×Q8—61no—
294C3×Q8—61no—
295C3×D4—61no—
296C3×D4—61no—
297C3×D4—61no—
298C3×D4—61no—
299C3×D4—121no—
300C3×D4—121no—
301C3×D4—121no—
302C3×D4—121no—
303C2×C3×C4—21no—
304C2×C3×C4—21no—
305C2×C3×C4—21no—
306C2×C3×C4—21no—
307C2×C3×C4—21no—
308C2×C3×C4—21no—
309C2×C3×C4—64no—
310C2×C3×C4—62no—
311C2×C3×C4—61no—
312C2×C3×C4—61no—
313C2×C3×C4—121no—
314C2×C3×C4—121no—
315C2×C3×C4—121no—
316C2×C3×C4—121no—
317C2×C3×C4C2×C4×D311yescharacteristic
318C2×C3×C4C2×C4×D311yescharacteristic
319C2×C3×C4#23511yescharacteristic
320C2×C3×C4D3×D411yescharacteristic
321C2×C3×C4D3×D411yescharacteristic
322C2×C3×C4D3×D411yescharacteristic
323C2×C3×C4D3×D411yescharacteristic
324C2×C3×C4D3×D411yescharacteristic
325C2×C3×C4D3×D411yescharacteristic
326C2×C3×C4#23711yescharacteristic
327C2×C3×C4C2×#6711yescharacteristic
328C2×C3×C4C2×#6711yescharacteristic
329#67—41no—
330#67—41no—
331#67—41no—
332#67—41no—
333C2×Dic3—24no—
334C2×Dic3—22no—
335C2×Dic3—21no—
336C2×Dic3—21no—
337D12—21no—
338D12—21no—
339D12—21no—
340D12—21no—
341C4×D3—41no—
342C4×D3—41no—
343C4×D3—41no—
344C4×D3—41no—
345Dic6—21no—
346Dic6—21no—
347Dic6—21no—
348Dic6—21no—
349C3×C8—42no—
350C3×C8—41no—
351C3×C8—41no—
352C3×C8—62no—
353C3⋊C8—22no—
354C3⋊C8—32no—
355C3⋊C8—32no—
356C3⋊C8—32no—
357C3⋊C8—32no—
358C3⋊C8—62no—
359C3⋊C8—62no—
360C3⋊C8—122no—
361C3⋊C8—121no—
362C3⋊C8—121no—
363C2×C32—21no—
364C2×C32—21no—
365C2×C32#39011yescharacteristic
366C2×C32#44312yes—
367C3×D3—81no—
368C3×D3—81no—
369#41—61no—
370#41—61no—
371#41—61no—
372#41—61no—
373#41—61no—
374#41—61no—
375#41—61no—
376#41—61no—
377C2×Q8—31no—
378C2×Q8—31no—
379C2×D4—31no—
380C2×D4—31no—
381C2×D4—61no—
382C2×D4—61no—
383C22×C4—61no—
384C22×C4—61no—
385C22×C4—61no—
386C22×C4C2×D3211yescharacteristic
387Q16—182no—
388SD16—182no—
389SD16—182no—
390D8—182no—
391M16—21no—
392M16—21no—
393M16—181no—
394M16—181no—
395C2×C8—21no—
396C2×C8—32no—
397C2×C8—61no—
398C2×C8—61no—
399C2×C8—91no—
400C2×C8—91no—
401C2×C8—92no—
402C2×C8—92no—
403C2×C8—181no—
404C42—32no—
405C42—61no—
406C22×C3—22no—
407C22×C3—22no—
408C22×C3—21no—
409C22×C3—21no—
410C22×C3—21no—
411C22×C3—42no—
412C22×C3—122no—
413C22×C3—121no—
414C22×C3—121no—
415C22×C3—122no—
416C22×C3#67⋊C411yescharacteristic
417C22×C3#88411yescharacteristic
418C22×C3#88511yescharacteristic
419C22×C3#88511yescharacteristic
420C22×C3C4×#6711yescharacteristic
421C22×C3#93911yescharacteristic
422C2×D3—42no—
423C2×D3—41no—
424C2×D3—41no—
425C2×D3—42no—
426C3×C4—21no—
427C3×C4—21no—
428C3×C4—21no—
429C3×C4—21no—
430C3×C4—64no—
431C3×C4—121no—
432C3×C4—121no—
433C3×C4D3×C22⋊C411yescharacteristic
434C3×C4D3×C22⋊C411yescharacteristic
435C3×C4#67⋊C411yescharacteristic
436C3×C4#88411yescharacteristic
437C3×C4C4×#6711yescharacteristic
438C3×C4#93911yescharacteristic
439C3×C4#94011yescharacteristic
440C3×C4#94011yescharacteristic
441Dic3—24no—
442Dic3—41no—
443Dic3—41no—
444C32#197411yescharacteristic, complemented, Sylow
445C23—61no—
446C23—61no—
447C23#379511yescharacteristic
448Q8—61no—
449Q8—61no—
450Q8—61no—
451Q8—61no—
452D4—61no—
453D4—61no—
454D4—61no—
455D4—61no—
456D4—121no—
457D4—121no—
458D4—121no—
459D4—121no—
460C2×C4—64no—
461C2×C4—62no—
462C2×C4—61no—
463C2×C4—61no—
464C2×C4—121no—
465C2×C4—121no—
466C2×C4—121no—
467C2×C4—121no—
468C2×C4#379511yescharacteristic
469C2×C4C4×D3211yescharacteristic
470C2×C4#380211yescharacteristic
471C2×C4D3×#6711yescharacteristic
472C2×C4D3×#6711yescharacteristic
473C2×C4#381111yescharacteristic
474C8—41no—
475C8—62no—
476C8—92no—
477C8—92no—
478C8—361no—
479C2×C3—21no—
480C2×C3—21no—
481C2×C3—21no—
482C2×C3—21no—
483C2×C3—21no—
484C2×C3—22no—
485C2×C3—41no—
486C2×C3—41no—
487C2×C3—241no—
488C2×C3—241no—
489C2×C3#482412yes—
490C2×C3#495711yescharacteristic
491C2×C3#495711yescharacteristic
492C2×C3#514412yes—
493D3—81no—
494D3—81no—
495C22—22no—
496C22—121no—
497C22—121no—
498C22—122no—
499C22—122no—
500C22#6387911yescharacteristic
501C22#6388511yescharacteristic
502C22#6391311yescharacteristic
503C4—64no—
504C4—121no—
505C4—121no—
506C4#6387911yescharacteristic
507C4#6388511yescharacteristic
508C4#6392111yescharacteristic
509C4#6397911yescharacteristic
510C3—21no—
511C3#6999911yescharacteristic, complemented
512C3#7037311yescharacteristic, complemented
513C2—21no—
514C2—21no—
515C2—241no—
516C2—241no—
517C2#1059097512yes—
518C2#1059138611yescharacteristic
subgroupG/H
centerC2×C4D3×#67
commutatorC2×C32×C4C22×C4
FrattiniC22×C4C2×D32
FittingC2×C32×M16C22
socleC22×C32C4×D4
prime radicalC2×#63822C2
cosocleC22×C32×C4C23
hypercenterC2×M16D32
hypocenterC32#1974
LCSG ▷ C2×C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C2×C4 ◁ C22×C4 ◁ C2×M16

Complements

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

NH ≅ G/Ntype
2. C2×#63649C2semidirect
3. C2×C3×#825C2semidirect
4. C3×#5129C2semidirect
5. C2×#63569C2semidirect
6. #10591070C2semidirect
7. C2×Dic3×C3⋊C8C2semidirect
17. C2×#3718C4semidirect
18. C2×#3715C4semidirect
19. C2×#3715C4semidirect
22. C2×#3713C4semidirect
35. C2×#825D3semidirect
36. #5129D3semidirect
444. C32#1974semidirect
511. C3#69999semidirect
512. C3#70373semidirect

Maximal subgroups

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

subgroupindex
C2×#638222
C2×#636492
C2×C3×#8252
C3×#51292
C2×#635692
#105910702
C2×Dic3×C3⋊C82
#703733
#699993