Finite Group Databasebeta

← #11868746

Subgroups — #11868746

Order 1152.

901 conjugacy classes of subgroups.

The subgroup lattice has 503 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C4×Q8×C3⋊D3C211yesmaximal, characteristic
2#10593854C211yesmaximal, characteristic
3C3×#5022C212yesmaximal, complemented
4#10591631C211yesmaximal, characteristic, complemented
5#10591630C211yesmaximal, characteristic, complemented
6#10591562C212yesmaximal
7#10590583C212yesmaximal
8#10590432C211yesmaximal, characteristic, complemented
9#10590431C211yesmaximal, characteristic, complemented
10#10590340C211yesmaximal, characteristic
11#10590292C212yesmaximal, complemented
12#81230—32nomaximal
13C2×Q8×C3⋊D3C2211yescharacteristic
14C32×C4×Q8C2211yescharacteristic
15Q8×C3⋊Dic3C2211yescharacteristic
16C4⋊C4×C3⋊D3—21no—
17C4⋊C4×C3⋊D3C2211yescharacteristic
18#64096—21no—
19#64096C2211yescharacteristic
20C42×C3⋊D3—21no—
21C42×C3⋊D3C2211yescharacteristic
22C4×#634—21no—
23C4×#634C2211yescharacteristic
24C3×#870C2212yes—
25#63919C2211yescharacteristic
26#63919C2211yescharacteristic
27#63865—21no—
28#63865—21no—
29#63849C2212yes—
30#63838—21no—
31#63838—21no—
32C2×#3712C2212yes—
33C3×#835C2212yes—
34C3×#835C2212yes—
35C3×#808C2212yes—
36C3×#807C2212yes—
37C3×C4⋊(C3⋊C8)C2212yes—
38#63575C2212yes—
39#63573C2212yes—
40#63572C2211yescharacteristic
41#63572C2211yescharacteristic
42#63567C2212yes—
43#63567C2212yes—
44#5688—32no—
45C4×D3×Q8—32no—
46C4×D3×Q8—61no—
47C3×#479—32no—
48#5063—32no—
49#5022D312yescomplemented
50#4881—32no—
51#4875—32no—
52#4861—32no—
53#4854—32no—
54#4837—32no—
55#4829—32no—
56#4802—32no—
57#4802—32no—
58#4793—32no—
59#4791—32no—
60C2×C32×Q8C2311yescharacteristic
61Q8×C3⋊D3—22no—
62Q8×C3⋊D3Q814yescomplemented
63C2×C4×C3⋊D3—21no—
64C2×C4×C3⋊D3—21no—
65C2×C4×C3⋊D3D411yescharacteristic
66C2×C4×C3⋊D3D411yescharacteristic
67C2×C4×C3⋊D3C2311yescharacteristic
68C2×#634—21no—
69C2×#634C2311yescharacteristic
70C2×#627—21no—
71C2×#627—21no—
72C32×C4⋊C4—21no—
73C32×C4⋊C4C2311yescharacteristic
74C32×C42—21no—
75C32×C42C2311yescharacteristic
76C4⋊(C3⋊Dic3)—21no—
77C4⋊(C3⋊Dic3)C2311yescharacteristic
78#3750—21no—
79#3750—21no—
80#3750—21no—
81C4×C3⋊Dic3—21no—
82C4×C3⋊Dic3—21no—
83C4×C3⋊Dic3—21no—
84C4×C3⋊Dic3D411yescharacteristic
85C4×C3⋊Dic3D411yescharacteristic
86C4×C3⋊Dic3C2311yescharacteristic
87C3×C4⋊Dic3C2312yes—
88C3×C4⋊Dic3C2312yes—
89C3×Dic3⋊C4—22no—
90C3×Dic3⋊C4—22no—
91C2×C3×C3⋊C8C2312yes—
92C2×C3×C3⋊C8C2312yes—
93#3724—21no—
94#3724—21no—
95#3722—22no—
96Dic32—22no—
97#3712—24no—
98#3078—91noSylow
99C2×D3×Q8—32no—
100C2×D3×Q8—61no—
101C3×#125—32no—
102C3×C4×Q8—21no—
103C3×C4×Q8C2×D312yes—
104Q8×Dic3—32no—
105Q8×Dic3—61no—
106C2×ΓL(1,𝔽25)—32no—
107C2×ΓL(1,𝔽25)—32no—
108#896—62no—
109#896—62no—
110#892—32no—
111#892—32no—
112D3×C4⋊C4—32no—
113D3×C4⋊C4—61no—
114D3×C4⋊C4—62no—
115D3×C4⋊C4—62no—
116#888—32no—
117#888—32no—
118#887—32no—
119#887—61no—
120#887—62no—
121#887—62no—
122#887—62no—
123#887—62no—
124C42×D3—32no—
125C42×D3—61no—
126C42×D3—62no—
127C42×D3—62no—
128#870C2×D312yes—
129C4×Dic6—32no—
130C4×Dic6—61no—
131C4×Dic6—62no—
132C4×Dic6—62no—
133C3×#107—32no—
134C3×#106—32no—
135C3×C4⋊C8—32no—
136C3×#103—32no—
137C3×#103—32no—
138#835C2×D312yes—
139#835C2×D312yes—
140#818—32no—
141#817—32no—
142#816—32no—
143#816—32no—
144Dic3⋊C8—32no—
145Dic3⋊C8—32no—
146#810—32no—
147#810—32no—
148#808—32no—
149#808C2×D312yes—
150#807—32no—
151#807C2×D312yes—
152C4⋊(C3⋊C8)C2×D312yes—
153C22×C3⋊D3C2×D411yescharacteristic
154C32×Q8—21no—
155C32×Q8C2×Q812yes—
156C2×C32×C4—21no—
157C2×C32×C4—21no—
158C2×C32×C4C2×D411yescharacteristic
159C2×C32×C4C2×D411yescharacteristic
160C2×C32×C4C2411yescharacteristic
161C2×C3⋊Dic3—21no—
162C2×C3⋊Dic3—21no—
163C2×C3⋊Dic3C2×D411yescharacteristic
164C2×C3⋊Dic3C2×D411yescharacteristic
165C2×C3⋊Dic3C2×D411yescharacteristic
166C4×C3⋊D3—24no—
167C4×C3⋊D3—22no—
168C4×C3⋊D3—41no—
169C4×C3⋊D3—42no—
170C4×C3⋊D3C2×Q812yes—
171C4×C3⋊D3#4112yes—
172#634—21no—
173#634—24no—
174#634C2×Q812yes—
175C2×C3×Dic3—22no—
176C2×C3×Dic3—22no—
177#627—42no—
178#627—42no—
179C3×C3⋊C8—22no—
180C3×C3⋊C8—22no—
181#533—91no—
182C2×C4×Q8—91no—
183#479—32no—
184#479—92no—
185#477—91no—
186#477—91no—
187#475—91no—
188#475—91no—
189#428—92no—
190#423—91no—
191#419—92no—
192C2×C3×Q8—21no—
193C2×C3×Q8C22×D312yes—
194D3×Q8—38no—
195D3×Q8—64no—
196D3×Q8—64no—
197D3×Q8—64no—
198C2×C4×D3—32no—
199C2×C4×D3—32no—
200C2×C4×D3—32no—
201C2×C4×D3—61no—
202C2×C4×D3—61no—
203C2×C4×D3—61no—
204C2×C4×D3—62no—
205C2×C4×D3—62no—
206C2×C4×D3—62no—
207C2×C4×D3—62no—
208C2×Dic6—32no—
209C2×Dic6—61no—
210C2×Dic6—62no—
211C2×Dic6—62no—
212C2×Dic6—62no—
213C2×Dic6—62no—
214C2×C3×C8—32no—
215C2×C3×C8—32no—
216C3×C4⋊C4—22no—
217C3×C4⋊C4—22no—
218C3×C4⋊C4—21no—
219C3×C4⋊C4—32no—
220C3×C4⋊C4—32no—
221C3×C4⋊C4—62no—
222C3×C4⋊C4—62no—
223C3×C4⋊C4C22×D312yes—
224C3×C42—22no—
225C3×C42—22no—
226C3×C42—21no—
227C3×C42C22×D312yes—
228(C2×D3)⋊C4—62no—
229(C2×D3)⋊C4—62no—
230C4⋊Dic3—32no—
231C4⋊Dic3—61no—
232C4⋊Dic3—62no—
233C4⋊Dic3—62no—
234C4⋊Dic3C22×D312yes—
235C4⋊Dic3C22×D312yes—
236Dic3⋊C4—22no—
237Dic3⋊C4—22no—
238Dic3⋊C4—62no—
239Dic3⋊C4—62no—
240Dic3⋊C4—62no—
241Dic3⋊C4—62no—
242Dic3⋊C4—62no—
243Dic3⋊C4—62no—
244Dic3⋊C4—62no—
245Dic3⋊C4—62no—
246C4×Dic3—32no—
247C4×Dic3—32no—
248C4×Dic3—32no—
249C4×Dic3—61no—
250C4×Dic3—61no—
251C4×Dic3—61no—
252C4×Dic3—62no—
253C4×Dic3—62no—
254C4×Dic3—62no—
255C4×Dic3—62no—
256C4×Dic3—62no—
257C4×Dic3—62no—
258C4×Dic3—62no—
259C4×Dic3—62no—
260C2×C3⋊C8C22×D312yes—
261C2×C3⋊C8C22×D312yes—
262ΓL(1,𝔽25)—64no—
263ΓL(1,𝔽25)—64no—
264C22×C32C22×D411yescharacteristic
265C2×C3⋊D3—21no—
266C2×C3⋊D3—21no—
267C2×C3⋊D3C22⋊Q812yes—
268C32×C4—22no—
269C32×C4—21no—
270C32×C4—21no—
271C32×C4—41no—
272C32×C4C22×Q811yescharacteristic
273C32×C4C2×#4111yescharacteristic
274C3⋊Dic3—22no—
275C3⋊Dic3—21no—
276C3⋊Dic3—41no—
277C3⋊Dic3C22⋊Q812yes—
278C3⋊Dic3#13712yes—
279C3×Dic3—42no—
280C3×Dic3—42no—
281C22×Q8—91no—
282C2×M16—92no—
283C4⋊Q8—91no—
284C4⋊Q8—91no—
285#125—32no—
286C22⋊Q8—181no—
287C22⋊Q8—181no—
288C4×Q8—91no—
289C4×Q8—91no—
290C4×Q8—91no—
291C4×Q8—181no—
292C4×Q8—181no—
293C4×Q8—181no—
294C4×Q8—181no—
295C4×Q8D3211yescharacteristic
296#117—92no—
297C2×C4⋊C4—91no—
298C2×C4⋊C4—181no—
299C2×C42—91no—
300C2×C42—181no—
301#107—32no—
302#107—92no—
303#106—32no—
304#106—92no—
305C4⋊C8—32no—
306C4⋊C8—91no—
307C4⋊C8—91no—
308#103—32no—
309#103—32no—
310#103—92no—
311#103—92no—
312C22×D3—32no—
313C22×D3—61no—
314C3×Q8—22no—
315C3×Q8—22no—
316C3×Q8—22no—
317C3×Q8D3×Q814yes—
318C2×C3×C4—22no—
319C2×C3×C4—22no—
320C2×C3×C4—22no—
321C2×C3×C4—22no—
322C2×C3×C4—21no—
323C2×C3×C4—21no—
324C2×C3×C4—21no—
325C2×C3×C4—62no—
326C2×C3×C4—62no—
327C2×C3×C4D3×D412yes—
328C2×C3×C4D3×D412yes—
329C2×C3×C4C23×D312yes—
330C2×Dic3—22no—
331C2×Dic3—22no—
332C2×Dic3—32no—
333C2×Dic3—32no—
334C2×Dic3—32no—
335C2×Dic3—61no—
336C2×Dic3—61no—
337C2×Dic3—61no—
338C2×Dic3—62no—
339C2×Dic3—62no—
340C2×Dic3—62no—
341C2×Dic3—62no—
342C4×D3—34no—
343C4×D3—34no—
344C4×D3—62no—
345C4×D3—62no—
346C4×D3—64no—
347C4×D3—68no—
348C4×D3—64no—
349C4×D3—68no—
350C4×D3—122no—
351C4×D3—122no—
352C4×D3—124no—
353C4×D3—124no—
354Dic6—34no—
355Dic6—62no—
356Dic6—68no—
357Dic6—68no—
358Dic6—62no—
359Dic6—62no—
360Dic6—124no—
361Dic6—124no—
362C3×C8—62no—
363C3×C8—62no—
364C3⋊C8—22no—
365C3⋊C8—22no—
366C2×C32C2×C22⋊Q811yescharacteristic
367C2×C32C2×#13711yescharacteristic
368C2×C32#57511yescharacteristic
369C3⋊D3—22no—
370C2×Q8—94no—
371C2×Q8—91no—
372C2×Q8—181no—
373C2×Q8—181no—
374C2×Q8—181no—
375C2×Q8—182no—
376C2×Q8C2×D3211yescharacteristic
377C22×C4—91no—
378C22×C4—91no—
379C22×C4—91no—
380C22×C4—181no—
381C22×C4—181no—
382M16—184no—
383C2×C8—32no—
384C2×C8—32no—
385C4⋊C4—21no—
386C4⋊C4—32no—
387C4⋊C4—32no—
388C4⋊C4—62no—
389C4⋊C4—62no—
390C4⋊C4—91no—
391C4⋊C4—181no—
392C4⋊C4—181no—
393C4⋊C4—181no—
394C4⋊C4—181no—
395C4⋊C4—181no—
396C4⋊C4—181no—
397C4⋊C4C2×D3211yescharacteristic
398C22⋊C4—182no—
399C42—21no—
400C42—91no—
401C42—91no—
402C42—91no—
403C42—181no—
404C42—181no—
405C42—182no—
406C42—181no—
407C42C2×D3211yescharacteristic
408C22×C3—21no—
409C22×C3C2×D3×D412yes—
410C2×D3—34no—
411C2×D3—62no—
412C2×D3—62no—
413C2×D3—62no—
414C2×D3—62no—
415C2×D3—62no—
416C3×C4—24no—
417C3×C4—22no—
418C3×C4—22no—
419C3×C4—22no—
420C3×C4—24no—
421C3×C4—21no—
422C3×C4—21no—
423C3×C4—41no—
424C3×C4—42no—
425C3×C4—42no—
426C3×C4—122no—
427C3×C4—122no—
428C3×C4C2×D3×Q812yes—
429C3×C4D3×#4112yes—
430Dic3—34no—
431Dic3—34no—
432Dic3—42no—
433Dic3—42no—
434Dic3—62no—
435Dic3—62no—
436Dic3—64no—
437Dic3—62no—
438Dic3—64no—
439Dic3—121no—
440Dic3—122no—
441Dic3—122no—
442C32#307811yescharacteristic, complemented, Sylow
443C23—91no—
444Q8—21no—
445Q8—92no—
446Q8—184no—
447Q8—181no—
448Q8—362no—
449Q8—362no—
450Q8#379712yes—
451C2×C4—21no—
452C2×C4—21no—
453C2×C4—62no—
454C2×C4—62no—
455C2×C4—91no—
456C2×C4—91no—
457C2×C4—92no—
458C2×C4—92no—
459C2×C4—91no—
460C2×C4—182no—
461C2×C4—184no—
462C2×C4—181no—
463C2×C4—181no—
464C2×C4—361no—
465C2×C4—362no—
466C2×C4#381111yescharacteristic
467C2×C4#381111yescharacteristic
468C2×C4C22×D3211yescharacteristic
469C8—62no—
470C8—62no—
471C2×C3—21no—
472C2×C3—21no—
473C2×C3—21no—
474C2×C3D3×C22⋊Q812yes—
475C2×C3#576812yes—
476C2×C3D3×#13712yes—
477D3—64no—
478D3—64no—
479C22—92no—
480C22—181no—
481C22—181no—
482C22C2×#381111yescharacteristic
483C4—22no—
484C4—21no—
485C4—21no—
486C4—41no—
487C4—92no—
488C4—92no—
489C4—122no—
490C4—122no—
491C4—182no—
492C4—181no—
493C4—361no—
494C4C2×#379711yescharacteristic
495C4#6430611yescharacteristic
496C3—21no—
497C3#8123012yescomplemented
498C2—182no—
499C2#1059492511yescharacteristic
500C2#1059532211yescharacteristic
501C2#1059546011yescharacteristic
subgroupG/H
centerC22C2×#3811
commutator, cosocleC2×C32×C4C24
FrattiniC2×C4C22×D32
FittingC32×C4×Q8C22
socleC22×C32C22×D4
prime radicalC22×C3⋊D3C2×D4
hypercenterC4×Q8D32
hypocenterC32#3078
LCSG ▷ C2×C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C22 ◁ C42 ◁ C4×Q8

Complements

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

NH ≅ G/Ntype
3. C3×#5022C2semidirect
4. #10591631C2semidirect
5. #10591630C2semidirect
8. #10590432C2semidirect
9. #10590431C2semidirect
11. #10590292C2semidirect
49. #5022D3semidirect
62. Q8×C3⋊D3Q8semidirect
442. C32#3078semidirect
497. C3#81230semidirect

Maximal subgroups

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

subgroupindex
C4×Q8×C3⋊D32
#105938542
C3×#50222
#105916312
#105916302
#105915622
#105905832
#105904322
#105904312
#105903402
#105902922
#812303