Finite Group Databasebeta

← #11793948

Subgroups — #11793948

Order 1152.

597 conjugacy classes of subgroups.

The subgroup lattice has 491 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#942C211yesmaximal, characteristic
2C2×#63663C211yesmaximal, characteristic, complemented
3C2×C3×#835C211yesmaximal, characteristic, complemented
4C2×#63577C211yesmaximal, characteristic, complemented
5#10591087C211yesmaximal, characteristic
6#10591070C211yesmaximal, characteristic
7#10589167C211yesmaximal, characteristic, complemented
8#72105—31nomaximal
9#70264—31nomaximal
10C22×C32×Q8C2211yescharacteristic
11C2×C3×#235C2211yescharacteristic
12C2×#3748C2211yescharacteristic
13C2×#3748C2211yescharacteristic
14C2×C3×#216—21no—
15C3×#942—22no—
16C3×#942—22no—
17C2×C3×#215—21no—
18C2×C3×C4⋊Dic3C2211yescharacteristic
19C2×C3×#208C2211yescharacteristic
20C22×C3×C3⋊C8C2211yescharacteristic
21#63663—22no—
22C3×#835—22no—
23#63577—22no—
24#63565—21no—
25#63562—21no—
26(C2×D3)⋊(C3⋊C8)—22no—
27C2×C3×#137—31no—
28C2×#942D311yescharacteristic
29C2×C3×#103—31no—
30#5234—31no—
31#5232—31no—
32C2×#836—31no—
33C2×#836—31no—
34C2×#836—61no—
35C2×#835D311yescharacteristic, complemented
36C2×#822—31no—
37#5110—31no—
38C2×#819—31no—
39#4966—31no—
40#4554—31no—
41#4530—31no—
42C2×C32×Q8—21no—
43C2×C32×Q8—21no—
44C2×C32×Q8—21no—
45C2×C32×Q8—21no—
46C2×C32×Q8—21no—
47C2×C32×Q8D412yes—
48C22×C32×C4—21no—
49C22×C32×C4C2311yescharacteristic
50C2×C3×#67—21no—
51C22×C3×Dic3—21no—
52C3×#235—21no—
53C3×#235—21no—
54C3×#235—21no—
55C3×#235—21no—
56C2×C3×D12D411yescharacteristic
57C2×C3×C4×D3—21no—
58C2×C3×Dic6D411yescharacteristic
59#3748—21no—
60#3748—21no—
61#3748—21no—
62#3748—21no—
63C2×C3⋊(C3⋊C8)—21no—
64C2×C3⋊(C3⋊C8)—21no—
65C3×#216—24no—
66C3×#215—24no—
67C3×C4⋊Dic3—21no—
68C3×C4⋊Dic3—21no—
69C3×#208—21no—
70C3×#208D412yes—
71C2×C3×C3⋊C8—21no—
72C2×C3×C3⋊C8—21no—
73C2×C3×C3⋊C8—21no—
74C2×C3×C3⋊C8D412yes—
75#2052—91noSylow
76C2×C3×#41—31no—
77C22×C3×Q8—21no—
78C22×C3×Q8C2×D311yescharacteristic
79C22×C3×Q8C2×D311yescharacteristic
80C2×#235C2×D311yescharacteristic
81C3×#137—62no—
82C3×#137—62no—
83C2×C3×Q16—61no—
84C2×C3×SD16—61no—
85C2×C3×M16—31no—
86C22×C3×C8—31no—
87C2×C3×C4⋊C4—31no—
88C2×#216—21no—
89#942—22no—
90#942—22no—
91C2×#215—21no—
92C2×C4⋊Dic3C2×D311yescharacteristic
93C2×#208—31no—
94C2×#208—31no—
95C2×#208—31no—
96C2×#208—31no—
97C2×#208—61no—
98C2×#208—61no—
99C2×#208C2×D311yescharacteristic
100C22×C3⋊C8C2×D311yescharacteristic
101C3×#103—62no—
102#836—64no—
103#836—62no—
104#836—62no—
105#835—22no—
106#835—61no—
107#832—61no—
108#830—62no—
109#822—62no—
110(C2×D3)⋊C8—62no—
111#819—62no—
112#810—61no—
113#809—61no—
114C23×C32#4111yescharacteristic
115C22×C3×D3—21no—
116C32×Q8—22no—
117C32×Q8—22no—
118C32×Q8—41no—
119C32×Q8—41no—
120C2×C32×C4—22no—
121C2×C32×C4—41no—
122C2×C32×C4—41no—
123C2×C32×C4C2×D411yescharacteristic
124C2×C32×C4C2×D411yescharacteristic
125C2×C32×C4C2×D411yescharacteristic
126C2×C32×C4C2×D411yescharacteristic
127C2×C32×C4#4111yescharacteristic
128C2×C32×C4#4111yescharacteristic
129C3×#67—41no—
130C3×#67—41no—
131C2×C3×Dic3—21no—
132C2×C3×Dic3—41no—
133C2×C3×Dic3—81no—
134C3×D12—21no—
135C3×D12—21no—
136C3×C4×D3—41no—
137C3×C4×D3—41no—
138C3×Dic6—21no—
139C3×Dic6—21no—
140C3⋊(C3⋊C8)—41no—
141C3⋊(C3⋊C8)—41no—
142C3×C3⋊C8—22no—
143C3×C3⋊C8—22no—
144C2×#137—31no—
145C2×#108—91no—
146#417—91no—
147C2×#103—31no—
148C2×#101—91no—
149#408—91no—
150#343—91no—
151C3×#41—61no—
152C3×#41—61no—
153C3×#41—61no—
154C3×#41—61no—
155C2×C3×Q8—22no—
156C2×C3×Q8—21no—
157C2×C3×Q8—21no—
158C2×C3×Q8—21no—
159C2×C3×Q8—21no—
160C2×C3×Q8—21no—
161C2×C3×Q8—21no—
162C2×C3×Q8—21no—
163C2×C3×Q8—21no—
164C2×C3×Q8—21no—
165C2×C3×Q8—21no—
166C2×C3×Q8—22no—
167C2×C3×Q8—31no—
168C2×C3×Q8—41no—
169C2×C3×Q8—41no—
170C2×C3×Q8—41no—
171C2×C3×Q8—41no—
172C2×C3×Q8D1212yes—
173C2×C3×Q8#6712yes—
174C2×C3×D4—31no—
175C2×C3×D4—61no—
176C22×C3×C4—22no—
177C22×C3×C4—21no—
178C22×C3×C4—21no—
179C22×C3×C4—21no—
180C22×C3×C4—61no—
181C22×C3×C4—61no—
182C22×C3×C4C22×D311yescharacteristic
183C22×C3×C4C22×D311yescharacteristic
184C2×#67—21no—
185C22×Dic3—21no—
186#235—21no—
187#235—21no—
188#235—21no—
189#235—21no—
190C2×D12#6711yescharacteristic
191C2×C4×D3—21no—
192C2×Dic6#6711yescharacteristic
193C3×Q16—64no—
194C3×SD16—64no—
195C3×M16—32no—
196C3×M16—61no—
197C2×C3×C8—32no—
198C2×C3×C8—61no—
199C2×C3×C8—61no—
200C2×C3×C8—61no—
201C3×C4⋊C4—61no—
202C3×C4⋊C4—61no—
203#216—24no—
204#215—24no—
205C4⋊Dic3—21no—
206C4⋊Dic3—21no—
207#208—21no—
208#208—62no—
209#208—62no—
210#208—61no—
211#208—61no—
212#208—61no—
213#208—61no—
214#208—61no—
215#208—61no—
216#208—61no—
217#208—61no—
218#208—121no—
219#208—121no—
220#208#6712yes—
221C2×C3⋊C8—21no—
222C2×C3⋊C8—21no—
223C2×C3⋊C8—21no—
224C2×C3⋊C8—61no—
225C2×C3⋊C8—61no—
226C2×C3⋊C8—61no—
227C2×C3⋊C8—61no—
228C2×C3⋊C8—121no—
229C2×C3⋊C8—121no—
230C2×C3⋊C8#6712yes—
231C22×C32—21no—
232C22×C32—21no—
233C22×C32#12111yescharacteristic
234C22×C32#12311yescharacteristic
235C22×C32#12411yescharacteristic
236C2×C3×D3—41no—
237C2×C3×D3—41no—
238C2×C3×D3—41no—
239C32×C4—42no—
240C32×C4#12011yescharacteristic
241C32×C4#12111yescharacteristic
242C32×C4#12111yescharacteristic
243C32×C4#12311yescharacteristic
244C3×Dic3—41no—
245C3×Dic3—81no—
246C2×#41—31no—
247C22×Q8D3211yescharacteristic
248#137—62no—
249#137—62no—
250C2×Q16—61no—
251C2×SD16—61no—
252C2×M16—31no—
253C2×M16—91no—
254C2×M16—91no—
255C22×C8—31no—
256C2×C4⋊C4—31no—
257#108—182no—
258#103—62no—
259#103—181no—
260#102—181no—
261#101—182no—
262C22⋊C8—182no—
263C23×C3—21no—
264C23×C3—61no—
265C23×C3#23711yescharacteristic
266C23×C3#23711yescharacteristic
267C22×D3—21no—
268C3×Q8—22no—
269C3×Q8—22no—
270C3×Q8—22no—
271C3×Q8—22no—
272C3×Q8—42no—
273C3×Q8—42no—
274C3×Q8—42no—
275C3×Q8—42no—
276C3×Q8—41no—
277C3×Q8—41no—
278C3×Q8—41no—
279C3×Q8—41no—
280C3×Q8—61no—
281C3×Q8—61no—
282C3×D4—61no—
283C3×D4—61no—
284C3×D4—121no—
285C3×D4—121no—
286C2×C3×C4—24no—
287C2×C3×C4—22no—
288C2×C3×C4—22no—
289C2×C3×C4—21no—
290C2×C3×C4—21no—
291C2×C3×C4—21no—
292C2×C3×C4—21no—
293C2×C3×C4—21no—
294C2×C3×C4—21no—
295C2×C3×C4—42no—
296C2×C3×C4—42no—
297C2×C3×C4—41no—
298C2×C3×C4—41no—
299C2×C3×C4—41no—
300C2×C3×C4—41no—
301C2×C3×C4—61no—
302C2×C3×C4—121no—
303C2×C3×C4—121no—
304C2×C3×C4—121no—
305C2×C3×C4—241no—
306C2×C3×C4C2×D1211yescharacteristic
307C2×C3×C4#23511yescharacteristic
308C2×C3×C4D3×D411yescharacteristic
309C2×C3×C4D3×D411yescharacteristic
310C2×C3×C4D3×D411yescharacteristic
311C2×C3×C4#23711yescharacteristic
312C2×C3×C4#23711yescharacteristic
313C2×C3×C4#23711yescharacteristic
314C2×C3×C4C2×#6711yescharacteristic
315C2×C3×C4C2×#6711yescharacteristic
316C2×C3×C4C2×#6711yescharacteristic
317C2×C3×C4C2×#6711yescharacteristic
318#67—41no—
319#67—41no—
320C2×Dic3—21no—
321C2×Dic3—41no—
322C2×Dic3—81no—
323D12—21no—
324D12—21no—
325C4×D3—41no—
326C4×D3—41no—
327Dic6—21no—
328Dic6—21no—
329C3×C8—62no—
330C3×C8—62no—
331C3⋊C8—22no—
332C3⋊C8—22no—
333C3⋊C8—122no—
334C3⋊C8—122no—
335C3⋊C8—121no—
336C3⋊C8—121no—
337C3⋊C8—121no—
338C3⋊C8—121no—
339C2×C32—21no—
340C2×C32—21no—
341C2×C32#39411yescharacteristic
342C2×C32#47111yescharacteristic
343C2×C32#47311yescharacteristic
344C3×D3—81no—
345#41—61no—
346#41—61no—
347#41—61no—
348#41—61no—
349C2×Q8—21no—
350C2×Q8—21no—
351C2×Q8—21no—
352C2×Q8—21no—
353C2×Q8—21no—
354C2×Q8—31no—
355C2×Q8#62612yes—
356C2×D4—31no—
357C2×D4—61no—
358C22×C4—21no—
359C22×C4—61no—
360C22×C4—61no—
361C22×C4C2×D3211yescharacteristic
362Q16—64no—
363SD16—64no—
364M16—32no—
365M16—61no—
366M16—181no—
367M16—181no—
368M16—181no—
369M16—181no—
370C2×C8—32no—
371C2×C8—61no—
372C2×C8—61no—
373C2×C8—61no—
374C2×C8—181no—
375C2×C8—181no—
376C4⋊C4—61no—
377C4⋊C4—61no—
378C22×C3—21no—
379C22×C3—21no—
380C22×C3—21no—
381C22×C3—21no—
382C22×C3—21no—
383C22×C3—21no—
384C22×C3—21no—
385C22×C3—41no—
386C22×C3—41no—
387C22×C3—121no—
388C22×C3—121no—
389C22×C3—121no—
390C22×C3#88511yescharacteristic
391C22×C3#88611yescharacteristic
392C22×C3#93511yescharacteristic
393C22×C3#93611yescharacteristic
394C22×C3#93811yescharacteristic
395C22×C3#93911yescharacteristic
396C2×D3—41no—
397C2×D3—41no—
398C2×D3—41no—
399C3×C4—21no—
400C3×C4—21no—
401C3×C4—21no—
402C3×C4—21no—
403C3×C4—44no—
404C3×C4—42no—
405C3×C4—42no—
406C3×C4—121no—
407C3×C4—241no—
408C3×C4#88211yescharacteristic
409C3×C4#88311yescharacteristic
410C3×C4#93011yescharacteristic
411C3×C4#93511yescharacteristic
412C3×C4#93811yescharacteristic
413C3×C4#93911yescharacteristic
414C3×C4#93911yescharacteristic
415C3×C4#95311yescharacteristic
416Dic3—41no—
417Dic3—81no—
418C32#205211yescharacteristic, complemented, Sylow
419C23—61no—
420C23#380511yescharacteristic
421Q8—22no—
422Q8—22no—
423Q8—41no—
424Q8—41no—
425Q8—61no—
426Q8—61no—
427D4—61no—
428D4—61no—
429D4—121no—
430D4—121no—
431C2×C4—22no—
432C2×C4—41no—
433C2×C4—41no—
434C2×C4—61no—
435C2×C4—121no—
436C2×C4—121no—
437C2×C4—121no—
438C2×C4—241no—
439C2×C4#380411yescharacteristic
440C2×C4#380511yescharacteristic
441C2×C4C2×#62611yescharacteristic
442C2×C4D3×#6711yescharacteristic
443C2×C4D3×#6711yescharacteristic
444C2×C4D3×#6711yescharacteristic
445C8—62no—
446C8—62no—
447C8—361no—
448C8—361no—
449C2×C3—21no—
450C2×C3—21no—
451C2×C3—21no—
452C2×C3—21no—
453C2×C3—21no—
454C2×C3—21no—
455C2×C3—21no—
456C2×C3—41no—
457C2×C3—41no—
458C2×C3—241no—
459C2×C3#496111yescharacteristic
460C2×C3#517711yescharacteristic
461C2×C3#519211yescharacteristic
462C2×C3#520311yescharacteristic
463C2×C3#520411yescharacteristic
464C2×C3#522211yescharacteristic
465D3—81no—
466C22—21no—
467C22—21no—
468C22—121no—
469C22—121no—
470C22—121no—
471C22#6395911yescharacteristic
472C22#6396211yescharacteristic
473C22#6397011yescharacteristic
474C4—42no—
475C4—121no—
476C4—241no—
477C4#6396911yescharacteristic
478C4#6397011yescharacteristic
479C4#6397711yescharacteristic
480C4#6397811yescharacteristic
481C3—21no—
482C3#7026411yescharacteristic, complemented
483C3#7210511yescharacteristic, complemented
484C2—21no—
485C2—21no—
486C2—241no—
487C2#1059164111yescharacteristic
488C2#1059164211yescharacteristic
489C2#1059170911yescharacteristic
subgroupG/H
centerC22#63970
commutator, cosocleC22×C32×C4C23
FrattiniC22×C4C2×D32
FittingC22×C32×Q8C22
socleC22×C32#121
prime radicalC2×C3×#235C22
hypercenterC22×Q8D32
hypocenterC32#2052
LCSG ▷ C22×C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22×Q8

Complements

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

NH ≅ G/Ntype
2. C2×#63663C2semidirect
3. C2×C3×#835C2semidirect
4. C2×#63577C2semidirect
7. #10589167C2semidirect
35. C2×#835D3semidirect
418. C32#2052semidirect
482. C3#70264semidirect
483. C3#72105semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#9422
C2×#636632
C2×C3×#8352
C2×#635772
#105910872
#105910702
#105891672
#721053
#702643