Finite Group Databasebeta

← #11860451

Subgroups — #11860451

Order 1152.

788 conjugacy classes of subgroups; only the 153 normal subgroups are shown.

normal subgroups only

This group has 788 conjugacy classes of subgroups — too many to draw. Only its 153 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#64239C211yesmaximal, characteristic
2C4×#3805C211yesmaximal, characteristic
3C2×#63838C211yesmaximal, characteristic, complemented
4C4×#3792C211yesmaximal, characteristic, complemented
5#10593507C211yesmaximal, characteristic, complemented
6#10593503C211yesmaximal, characteristic, complemented
7#10593496C211yesmaximal, characteristic
8#10593445C212yesmaximal, complemented
9#10593433C212yesmaximal, complemented
10#10593431C212yesmaximal, complemented
11#10593369C212yesmaximal, complemented
12C22×#627C2211yescharacteristic
13C2×#3805C2211yescharacteristic
14C2×#3792C2211yescharacteristic
15C2×#3792C2211yescharacteristic
16C2×#3776C2212yes—
17#64239C418yescomplemented
18C2×#3775C2212yes—
19C2×C4×C3⋊Dic3C2211yescharacteristic
20C3×C4×#67—21no—
21C2×C3×Dic3⋊C4—21no—
22#67⋊Dic3—21no—
23C2×#3724C2211yescharacteristic
24#63957C2211yescharacteristic
25C2×Dic32—21no—
26C4×#627C2211yescharacteristic
27#63902C2211yescharacteristic
28C4×#625C2211yescharacteristic
29#63878—21no—
30C4×D3×Dic3—21no—
31#63838—21no—
32#63838—21no—
33#63838C2211yescharacteristic
34#63838C2212yes—
35#63838C2212yes—
36#63777C2212yes—
37C4×#622C2212yes—
38#63743C2212yes—
39#63742C2212yes—
40#63740C2212yes—
41#63740C2212yes—
42#63740C2212yes—
43#63739C2212yes—
44C4×#237—61no—
45C2×#887—61no—
46C22×C32×C4D411yescharacteristic
47C22×C3⋊Dic3D411yescharacteristic
48C22×C3⋊Dic3C2311yescharacteristic
49C2×C3×#67—21no—
50C22×C3×Dic3—21no—
51C22×C3×Dic3—21no—
52C2×C3×C4×D3—21no—
53C2×#627—21no—
54C2×#627—22no—
55C2×#627C2×C414yescomplemented
56C2×#627C2311yescharacteristic
57C2×#627C2312yes—
58C2×#625C2311yescharacteristic
59#3805—22no—
60#3805C2×C414yes—
61C2×D3×Dic3—21no—
62C2×D3×Dic3—21no—
63#3792—22no—
64#3792C2×C414yescomplemented
65C2×#622C2312yes—
66C2×#622C2312yes—
67#3776C2×C418yescomplemented
68#3775C2×C418yescomplemented
69C4×C3⋊Dic3D411yescharacteristic
70C4×C3⋊Dic3D411yescharacteristic
71C4×C3⋊Dic3C2311yescharacteristic
72C4×C3⋊Dic3C2311yescharacteristic
73C3×#217—21no—
74C3×(C2×D3)⋊C4—21no—
75C3×Dic3⋊C4—22no—
76C3×Dic3⋊C4—22no—
77C3×Dic3⋊C4—21no—
78C3×C4×Dic3—21no—
79#3724—21no—
80#3724C2312yes—
81#3724C2311yescharacteristic
82#3724C2311yescharacteristic
83Dic3⋊Dic3—21no—
84(C2×D3)⋊Dic3—21no—
85Dic32—22no—
86Dic32—22no—
87Dic32—21no—
88Dic32—21no—
89C4×#137—91nomaximal, Sylow
90C2×#237—61no—
91C22×Dic6—61no—
92C3×C4×D4—61no—
93C2×C3×C4⋊C4—61no—
94D4×Dic3—61no—
95C4×#67—21no—
96C4×#67—61no—
97C2×Dic3⋊C4—21no—
98C2×Dic3⋊C4—61no—
99C2×C4×Dic3—61no—
100C2×C4×Dic3—61no—
101C2×C4×Dic3—61no—
102C2×C4×Dic3—61no—
103C2×C4×Dic3—61no—
104#892—61no—
105#887—62no—
106#887—62no—
107#887—61no—
108#887—62no—
109#887—62no—
110#67⋊C4—61no—
111#67⋊C4—61no—
112#877—61no—
113#877—61no—
114C42×D3—61no—
115C4×Dic6—61no—
116C23×C32C2×D411yescharacteristic
117C22×C3×D3—21no—
118C2×C32×C4—41no—
119C2×C32×C4C2×D411yescharacteristic
120C2×C32×C4C2×D411yescharacteristic
121C2×C3⋊Dic3—22no—
122C2×C3⋊Dic3C22×C412yes—
123C2×C3⋊Dic3C2×D411yescharacteristic
124C2×C3⋊Dic3C2×D411yescharacteristic
125C2×C3⋊Dic3C2×D411yescharacteristic
126C2×C3⋊Dic3#4112yes—
127C2×C3⋊Dic3C2411yescharacteristic
128C3×#67—24no—
129C2×C3×Dic3—24no—
130C2×C3×Dic3—22no—
131C2×C3×Dic3—21no—
132C2×C3×Dic3—21no—
133C2×C3×Dic3—22no—
134C2×C3×Dic3—21no—
135C2×C3×Dic3—42no—
136C3×C4×D3—42no—
137#627—24no—
138#627—21no—
139#627—22no—
140#627C22×C414yes—
141#627C22×C412yes—
142#625—21no—
143#625C22×C412yes—
144D3×Dic3—24no—
145D3×Dic3—42no—
146#622—22no—
147#622C22×C414yes—
148C2×#137—91no—
149C4×#41—91no—
150C2×C4×Q8—91no—
151#441—92no—
152#440—92no—
153C4×Q16—92no—
154C4×SD16—92no—
155#428—91no—
156#419—91no—
157#417—91no—
158C4×M16—91no—
159C2×C3×D4—61no—
160C22×C3×C4—21no—
161C22×C3×C4—21no—
162C22×C3×C4—61no—
163C22×C3×C4—61no—
164C22×C3×C4—61no—
165C2×#67—21no—
166C2×#67—61no—
167C22×Dic3—21no—
168C22×Dic3—21no—
169C22×Dic3—61no—
170C22×Dic3—61no—
171C22×Dic3—61no—
172C22×Dic3—61no—
173C22×Dic3—61no—
174C22×Dic3—61no—
175#237—64no—
176#237—64no—
177C2×C4×D3—21no—
178C2×C4×D3—61no—
179C2×C4×D3—61no—
180C2×Dic6—61no—
181C2×Dic6—62no—
182C2×Dic6—64no—
183C2×Dic6—62no—
184C2×Dic6—64no—
185C3×C4⋊C4—62no—
186C3×C4⋊C4—62no—
187C3×C4⋊C4—61no—
188C3×C22⋊C4—61no—
189C3×C22⋊C4—61no—
190C3×C42—61no—
191#217—21no—
192#217—61no—
193(C2×D3)⋊C4—21no—
194(C2×D3)⋊C4—61no—
195C4⋊Dic3—61no—
196Dic3⋊C4—21no—
197Dic3⋊C4—22no—
198Dic3⋊C4—22no—
199Dic3⋊C4—61no—
200Dic3⋊C4—61no—
201Dic3⋊C4—62no—
202Dic3⋊C4—61no—
203Dic3⋊C4—62no—
204C4×Dic3—21no—
205C4×Dic3—61no—
206C4×Dic3—61no—
207C4×Dic3—61no—
208C4×Dic3—61no—
209C4×Dic3—61no—
210C4×Dic3—61no—
211C4×Dic3—61no—
212C4×Dic3—61no—
213C4×Dic3—61no—
214C4×Dic3—61no—
215C4×Dic3—61no—
216C4×Dic3—61no—
217C4×Dic3—62no—
218C4×Dic3—62no—
219C4×Dic3—62no—
220C4×Dic3—62no—
221C22×C32—21no—
222C22×C32—21no—
223C22×C32C4×D412yes—
224C22×C32C22×D411yescharacteristic
225C2×C3×D3—22no—
226C2×C3×D3—41no—
227C2×C3×D3—41no—
228C32×C4—21no—
229C32×C4—21no—
230C3⋊Dic3—21no—
231C3⋊Dic3C4×D412yes—
232C3⋊Dic3#13712yescomplemented
233C3⋊Dic3C23×C411yescharacteristic
234C3⋊Dic3C2×#4111yescharacteristic
235C3×Dic3—22no—
236C3×Dic3—24no—
237C3×Dic3—41no—
238C3×Dic3—41no—
239C3×Dic3—42no—
240C2×#41—91no—
241C22×Q8—91no—
242#137—98no—
243C2×Q16—92no—
244C2×SD16—92no—
245C2×M16—91no—
246C2×M16—91no—
247C4×Q8—91no—
248C4×Q8—92no—
249C4×Q8—92no—
250C4×Q8—91no—
251C4×Q8—181no—
252C4×Q8—181no—
253C4×D4—61no—
254C4×D4—91no—
255C4×D4—91no—
256C4×D4—181no—
257#117—91no—
258#117—181no—
259C2×C4⋊C4—61no—
260C2×C4⋊C4—91no—
261C2×C42—91no—
262C2×C42—181no—
263C2×C42—181no—
264#107—92no—
265#106—92no—
266#103—92no—
267#103—92no—
268#103—92no—
269#102—92no—
270#97—92no—
271C4×C8—92no—
272C23×C3—21no—
273C23×C3—21no—
274C23×C3—61no—
275C22×D3—21no—
276C3×D4—64no—
277C2×C3×C4—21no—
278C2×C3×C4—21no—
279C2×C3×C4—21no—
280C2×C3×C4—21no—
281C2×C3×C4—42no—
282C2×C3×C4—42no—
283C2×C3×C4—62no—
284C2×C3×C4—61no—
285C2×C3×C4—64no—
286C2×C3×C4—62no—
287C2×C3×C4—61no—
288C2×C3×C4—61no—
289C2×C3×C4—122no—
290C2×C3×C4—122no—
291#67—24no—
292#67—62no—
293#67—62no—
294C2×Dic3—21no—
295C2×Dic3—22no—
296C2×Dic3—24no—
297C2×Dic3—21no—
298C2×Dic3—22no—
299C2×Dic3—21no—
300C2×Dic3—42no—
301C2×Dic3—62no—
302C2×Dic3—62no—
303C2×Dic3—61no—
304C2×Dic3—61no—
305C2×Dic3—61no—
306C2×Dic3—61no—
307C2×Dic3—61no—
308C2×Dic3—62no—
309C2×Dic3—62no—
310C2×Dic3—61no—
311C2×Dic3—61no—
312C2×Dic3—61no—
313C2×Dic3—64no—
314C2×Dic3—64no—
315C2×Dic3—122no—
316C2×Dic3—122no—
317C4×D3—42no—
318C4×D3—64no—
319C4×D3—122no—
320Dic6—64no—
321Dic6—62no—
322Dic6—68no—
323Dic6—64no—
324Dic6—62no—
325C2×C32—22no—
326C2×C32C2×C4×D411yescharacteristic
327C2×C32C2×#13711yescharacteristic
328C2×C32#57511yescharacteristic
329C3×D3—42no—
330#41—94no—
331#41—182no—
332C2×Q8—92no—
333C2×Q8—91no—
334C2×Q8—94no—
335C2×Q8—181no—
336C2×Q8—182no—
337C2×D4—61no—
338C2×D4—91no—
339C22×C4—61no—
340C22×C4—61no—
341C22×C4—61no—
342C22×C4—91no—
343C22×C4—91no—
344C22×C4—181no—
345C22×C4—181no—
346C22×C4GO+(4,𝔽2)11yescharacteristic
347Q16—98no—
348SD16—98no—
349M16—94no—
350M16—182no—
351C2×C8—92no—
352C2×C8—92no—
353C4⋊C4—62no—
354C4⋊C4—62no—
355C4⋊C4—61no—
356C4⋊C4—92no—
357C4⋊C4—91no—
358C4⋊C4—91no—
359C4⋊C4—181no—
360C4⋊C4—181no—
361C22⋊C4—61no—
362C22⋊C4—61no—
363C22⋊C4—181no—
364C42—61no—
365C42—91no—
366C42—91no—
367C42—91no—
368C42—91no—
369C42—181no—
370C42—181no—
371C42—182no—
372C42—182no—
373C22×C3—22no—
374C22×C3—22no—
375C22×C3—21no—
376C22×C3—21no—
377C22×C3—22no—
378C22×C3—22no—
379C22×C3—41no—
380C22×C3—41no—
381C22×C3—62no—
382C22×C3—121no—
383C22×C3—121no—
384C2×D3—22no—
385C2×D3—41no—
386C2×D3—41no—
387C3×C4—22no—
388C3×C4—41no—
389C3×C4—41no—
390C3×C4—41no—
391C3×C4—64no—
392C3×C4—62no—
393C3×C4—121no—
394C3×C4—122no—
395C3×C4—121no—
396Dic3—24no—
397Dic3—22no—
398Dic3—41no—
399Dic3—42no—
400Dic3—41no—
401Dic3—62no—
402Dic3—61no—
403Dic3—61no—
404Dic3—61no—
405Dic3—61no—
406Dic3—62no—
407Dic3—62no—
408Dic3—62no—
409Dic3—62no—
410Dic3—121no—
411C32C4×#13711yescharacteristic, complemented, Sylow
412C23—61no—
413C23C2×GO+(4,𝔽2)11yescharacteristic
414Q8—94no—
415Q8—92no—
416Q8—182no—
417Q8—184no—
418Q8—181no—
419D4—64no—
420D4—92no—
421D4—181no—
422C2×C4—41no—
423C2×C4—61no—
424C2×C4—62no—
425C2×C4—64no—
426C2×C4—61no—
427C2×C4—61no—
428C2×C4—62no—
429C2×C4—91no—
430C2×C4—91no—
431C2×C4—91no—
432C2×C4—92no—
433C2×C4—92no—
434C2×C4—91no—
435C2×C4—122no—
436C2×C4—122no—
437C2×C4—184no—
438C2×C4—182no—
439C2×C4—362no—
440C2×C4C2×GO+(4,𝔽2)11yescharacteristic
441C2×C4C2×GO+(4,𝔽2)11yescharacteristic
442C8—94no—
443C8—182no—
444C2×C3—24no—
445C2×C3—21no—
446C2×C3—21no—
447C2×C3—21no—
448C2×C3—21no—
449C2×C3—21no—
450C2×C3—21no—
451C2×C3—42no—
452C2×C3—122no—
453D3—42no—
454C22—21no—
455C22—21no—
456C22—62no—
457C22—121no—
458C22—121no—
459C22C4×GO+(4,𝔽2)12yes—
460C22C22×GO+(4,𝔽2)11yescharacteristic
461C4—21no—
462C4—21no—
463C4—64no—
464C4—62no—
465C4—92no—
466C4—91no—
467C4—91no—
468C4—92no—
469C4—121no—
470C4—121no—
471C4—122no—
472C4—181no—
473C3—21no—
474C3—21no—
475C2—22no—
476C2—122no—
477C2#1059650511yescharacteristic
478C2C2×C4×GO+(4,𝔽2)11yescharacteristic
479C2#1059653211yescharacteristic
subgroupG/H
center, FrattiniC22C22×GO+(4,𝔽2)
commutatorC3⋊Dic3C23×C4
FittingC22×C32×C4D4
socleC22×C32C22×D4
prime radicalC2×#3805C22
cosocleC2×C3⋊Dic3C24
hypercenterC22×C4GO+(4,𝔽2)
hypocenterC32C4×#137
LCSG ▷ C3⋊Dic3 ▷ C2×C32 ▷ C32
UCSC1 ◁ C22 ◁ C22×C4

Complements

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

NH ≅ G/Ntype
3. C2×#63838C2semidirect
4. C4×#3792C2semidirect
5. #10593507C2semidirect
6. #10593503C2semidirect
8. #10593445C2semidirect
9. #10593433C2semidirect
10. #10593431C2semidirect
11. #10593369C2semidirect
17. #64239C4semidirect
55. C2×#627C2×C4semidirect
64. #3792C2×C4semidirect
67. #3776C2×C4semidirect
68. #3775C2×C4semidirect
232. C3⋊Dic3#137semidirect
411. C32C4×#137semidirect

Maximal subgroups

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

subgroupindex
C2×#642392
C4×#38052
C2×#638382
C4×#37922
#105935072
#105935032
#105934962
#105934452
#105934332
#105934312
#105933692
C4×#1379