Finite Group Databasebeta

← #11787158

Subgroups — #11787158

Order 1152.

746 conjugacy classes of subgroups; only the 93 normal subgroups are shown.

normal subgroups only

This group has 746 conjugacy classes of subgroups — too many to draw. Only its 93 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
1C22×#3787C211yesmaximal, characteristic, complemented
2C4×#3811C211yesmaximal, characteristic
3C2×D32⋊C8C212yesmaximal, complemented
4#10591152C211yesmaximal, characteristic, complemented
5#10590043C212yesmaximal, complemented
6C22×C4×C3⋊D3C2211yescharacteristic
7C2×#3811C411yescharacteristic
8C2×C4×D32C2211yescharacteristic
9C22×#622—21no—
10C2×#3787—21no—
11C2×#3787—24no—
12C2×#3787C2212yes—
13C3×C4×#67—21no—
14#63974C411yescharacteristic
15#67⋊Dic3—21no—
16#63921C411yescharacteristic
17C4×#626—21no—
18#63883C411yescharacteristic
19C4×#624C2211yescharacteristic
20#63881—21no—
21C2×#3710C2212yes—
22D32⋊C8—24no—
23C3×#830—21no—
24#63557—21no—
25C4×D3×D4—61no—
26D3×C22⋊C8—61no—
27C23×C3⋊D3C2×C411yescharacteristic
28C22×D32C2×C411yescharacteristic
29C22×C32×C4D411yescharacteristic
30C22×C3⋊Dic3D411yescharacteristic
31C2×C4×C3⋊D3—22no—
32C2×C4×C3⋊D3—22no—
33C2×C4×C3⋊D3C2×C411yescharacteristic
34C2×C4×C3⋊D3D411yescharacteristic
35C2×C4×C3⋊D3D411yescharacteristic
36C2×C4×C3⋊D3C2311yescharacteristic
37C2×C3×#67—21no—
38C2×C3×C4×D3—21no—
39#3811—22no—
40#3811C814yescomplemented
41C2×#626—21no—
42C2×#624C2×C411yescharacteristic
43C2×#624C2×C411yescharacteristic
44C2×#624C2×C411yescharacteristic
45C2×D3×Dic3—21no—
46C4×D32—22no—
47C4×D32—22no—
48C2×#622—24no—
49C2×#622—22no—
50#3787—24no—
51#3787—22no—
52#3787D414yescomplemented
53C3×#217—21no—
54C3×(C2×D3)⋊C4—21no—
55C3×Dic3⋊C4—21no—
56C3×C4×Dic3—21no—
57C2×C3×C3⋊C8—22no—
58Dic3⋊Dic3—21no—
59#3722—21no—
60(C2×D3)⋊Dic3—21no—
61Dic32—21no—
62#3710—22no—
63#3710—22no—
64#1869—91nomaximal, Sylow
65C2×D3×D4—61no—
66C22×C4×D3—61no—
67C22×C4×D3—61no—
68C22×C4×D3—61no—
69C3×C4×D4—61no—
70D4×Dic3—61no—
71C4×#67—21no—
72C4×#67—61no—
73C2×D3×C8—62no—
74#893—61no—
75D3×C4⋊C4—61no—
76#67⋊C4—61no—
77#67⋊C4—61no—
78D3×C22⋊C4—61no—
79D3×C22⋊C4—61no—
80C4×D12—61no—
81C42×D3—61no—
82C3×C22⋊C8—61no—
83#830—21no—
84(C2×D3)⋊C8—62no—
85C23×C32C22⋊C411yescharacteristic
86C22×C3⋊D3—22no—
87C22×C3⋊D3—22no—
88C22×C3⋊D3C22⋊C411yescharacteristic
89C22×C3⋊D3C2×C812yes—
90C22×C3⋊D3M1612yes—
91C22×C3⋊D3C22×C411yescharacteristic
92C22×C3×D3—21no—
93C2×D32—21no—
94C2×D32—22no—
95C2×D32—22no—
96C2×D32C2×C812yescomplemented
97C2×C32×C4—22no—
98C2×C32×C4C22⋊C411yescharacteristic
99C2×C32×C4C2×D411yescharacteristic
100C2×C3⋊Dic3—21no—
101C2×C3⋊Dic3—21no—
102C2×C3⋊Dic3C22⋊C411yescharacteristic
103C2×C3⋊Dic3C2×D411yescharacteristic
104C4×C3⋊D3—22no—
105C4×C3⋊D3—22no—
106C4×C3⋊D3—21no—
107C4×C3⋊D3—21no—
108C4×C3⋊D3C2×D412yes—
109C4×C3⋊D3#4112yes—
110C3×#67—24no—
111C2×C3×Dic3—21no—
112C2×C3×Dic3—21no—
113C2×C3×Dic3—21no—
114C3×C4×D3—42no—
115#626—24no—
116#624—21no—
117#624—22no—
118#624—22no—
119#624C2×C812yescomplemented
120D3×Dic3—41no—
121D3×Dic3—41no—
122#622—24no—
123C3×C3⋊C8—42no—
124C23×C8—91no—
125C2×C4×D4—91no—
126C2×C22⋊C8—91no—
127C2×C22⋊C8—92no—
128#336—92no—
129C23×D3—61no—
130C23×D3—61no—
131C23×D3—61no—
132C2×C3×D4—61no—
133C22×C3×C4—21no—
134C22×C3×C4—21no—
135C22×C3×C4—61no—
136C2×#67—21no—
137C2×#67—61no—
138C22×Dic3—61no—
139C22×Dic3—61no—
140C22×Dic3—61no—
141D3×D4—64no—
142D3×D4—64no—
143C2×D12—61no—
144C2×C4×D3—21no—
145C2×C4×D3—61no—
146C2×C4×D3—61no—
147C2×C4×D3—61no—
148C2×C4×D3—61no—
149C2×C4×D3—61no—
150C2×C4×D3—61no—
151C2×C4×D3—61no—
152C2×C4×D3—61no—
153C2×C4×D3—61no—
154C2×C4×D3—61no—
155C2×C4×D3—61no—
156C2×C4×D3—61no—
157C2×C4×D3—122no—
158C2×C4×D3—122no—
159C2×C4×D3—122no—
160C2×C4×D3—122no—
161C2×C4×D3—122no—
162C2×C4×D3—122no—
163C2×C3×C8—62no—
164C3×C4⋊C4—61no—
165C3×C22⋊C4—61no—
166C3×C22⋊C4—61no—
167C3×C42—61no—
168#217—21no—
169#217—61no—
170(C2×D3)⋊C4—21no—
171(C2×D3)⋊C4—61no—
172(C2×D3)⋊C4—61no—
173(C2×D3)⋊C4—61no—
174C4⋊Dic3—61no—
175Dic3⋊C4—21no—
176Dic3⋊C4—61no—
177C4×Dic3—21no—
178C4×Dic3—61no—
179C4×Dic3—61no—
180C2×C3⋊C8—22no—
181D3×C8—122no—
182D3×C8—122no—
183C22×C32—21no—
184C22×C32—21no—
185C22×C32C22⋊C812yescomplemented
186C22×C32C2×C22⋊C411yescharacteristic
187C2×C3⋊D3—21no—
188C2×C3⋊D3—21no—
189C2×C3⋊D3—24no—
190C2×C3⋊D3—24no—
191C2×C3⋊D3C22⋊C812yescomplemented
192C2×C3⋊D3C4×D412yes—
193C2×C3⋊D3C22×C811yescharacteristic
194C2×C3⋊D3C2×M1611yescharacteristic
195C2×C3⋊D3#13112yes—
196C2×C3×D3—22no—
197C2×C3×D3—41no—
198C2×C3×D3—41no—
199D32—24no—
200D32—42no—
201C32×C4—21no—
202C32×C4#12112yes—
203C3⋊Dic3—21no—
204C3⋊Dic3#12011yescharacteristic
205C3⋊Dic3#12311yescharacteristic
206C3×Dic3—22no—
207C3×Dic3—41no—
208C3×Dic3—41no—
209C22×D4—91no—
210C23×C4—91no—
211C23×C4—91no—
212C22×C8—92no—
213C22×C8—92no—
214C22×C8—184no—
215C22×C8—181no—
216C22×C8—181no—
217C4×D4—61no—
218C4×D4—181no—
219C4×D4—181no—
220C4×D4—181no—
221C2×C4⋊C4—91no—
222C2×C22⋊C4—91no—
223C2×C22⋊C4—91no—
224C2×C42—91no—
225C22⋊C8—61no—
226C22⋊C8—181no—
227C22⋊C8—184no—
228C23×C3—21no—
229C23×C3—21no—
230C23×C3—61no—
231C22×D3—21no—
232C22×D3—62no—
233C22×D3—61no—
234C22×D3—61no—
235C22×D3—61no—
236C22×D3—61no—
237C22×D3—62no—
238C22×D3—62no—
239C22×D3—62no—
240C22×D3—62no—
241C22×D3—62no—
242C22×D3—122no—
243C22×D3—122no—
244C22×D3—122no—
245C22×D3—122no—
246C22×D3—122no—
247C22×D3—122no—
248C3×D4—64no—
249C2×C3×C4—21no—
250C2×C3×C4—21no—
251C2×C3×C4—21no—
252C2×C3×C4—21no—
253C2×C3×C4—42no—
254C2×C3×C4—42no—
255C2×C3×C4—61no—
256C2×C3×C4—61no—
257C2×C3×C4—61no—
258C2×C3×C4—122no—
259#67—24no—
260#67—64no—
261C2×Dic3—21no—
262C2×Dic3—21no—
263C2×Dic3—21no—
264C2×Dic3—61no—
265C2×Dic3—61no—
266C2×Dic3—61no—
267C2×Dic3—61no—
268C2×Dic3—121no—
269C2×Dic3—121no—
270C2×Dic3—121no—
271C2×Dic3—121no—
272C2×Dic3—121no—
273C2×Dic3—121no—
274D12—64no—
275C4×D3—42no—
276C4×D3—62no—
277C4×D3—62no—
278C4×D3—62no—
279C4×D3—62no—
280C4×D3—62no—
281C4×D3—62no—
282C4×D3—122no—
283C4×D3—122no—
284C4×D3—121no—
285C4×D3—121no—
286C4×D3—121no—
287C4×D3—121no—
288C4×D3—122no—
289C4×D3—122no—
290C4×D3—122no—
291C4×D3—122no—
292C4×D3—121no—
293C4×D3—121no—
294C3×C8—122no—
295C3⋊C8—42no—
296C2×C32—22no—
297C2×C32#38611yescharacteristic
298C2×C32C2×C22⋊C811yescharacteristic
299C2×C32#40811yescharacteristic
300C3⋊D3—22no—
301C3⋊D3C8×D412yes—
302C3⋊D3#43512yes—
303C3×D3—42no—
304C24—91no—
305C24—91no—
306C2×D4—61no—
307C2×D4—94no—
308C2×D4—181no—
309C2×D4—182no—
310C22×C4—61no—
311C22×C4—91no—
312C22×C4—91no—
313C22×C4—91no—
314C22×C4—91no—
315C22×C4—91no—
316C22×C4—91no—
317C22×C4—91no—
318C22×C4—91no—
319C22×C4—182no—
320C22×C4—182no—
321C22×C4—181no—
322C22×C4—182no—
323C22×C4—182no—
324C22×C4GO+(4,𝔽2)11yescharacteristic
325C2×C8—62no—
326C2×C8—94no—
327C2×C8—182no—
328C2×C8—182no—
329C2×C8—182no—
330C2×C8—182no—
331C2×C8—184no—
332C2×C8—184no—
333C4⋊C4—61no—
334C4⋊C4—181no—
335C22⋊C4—61no—
336C22⋊C4—61no—
337C22⋊C4—181no—
338C22⋊C4—181no—
339C42—61no—
340C42—181no—
341C22×C3—21no—
342C22×C3—21no—
343C22×C3—22no—
344C22×C3—22no—
345C22×C3—41no—
346C22×C3—41no—
347C22×C3—41no—
348C22×C3—41no—
349C22×C3—62no—
350C22×C3—121no—
351C22×C3—121no—
352C2×D3—22no—
353C2×D3—41no—
354C2×D3—41no—
355C2×D3—62no—
356C2×D3—62no—
357C2×D3—61no—
358C2×D3—61no—
359C2×D3—62no—
360C2×D3—61no—
361C2×D3—61no—
362C2×D3—62no—
363C2×D3—62no—
364C2×D3—62no—
365C2×D3—124no—
366C2×D3—124no—
367C2×D3—124no—
368C2×D3—124no—
369C2×D3—124no—
370C2×D3—121no—
371C2×D3—121no—
372C2×D3—121no—
373C2×D3—121no—
374C2×D3—124no—
375C3×C4—22no—
376C3×C4—22no—
377C3×C4—41no—
378C3×C4—41no—
379C3×C4—62no—
380C3×C4—121no—
381C3×C4—121no—
382Dic3—22no—
383Dic3—41no—
384Dic3—41no—
385Dic3—61no—
386Dic3—61no—
387Dic3—61no—
388Dic3—61no—
389Dic3—121no—
390Dic3—121no—
391C32#186911yescharacteristic, complemented, Sylow
392C23—61no—
393C23—92no—
394C23—91no—
395C23—91no—
396C23—92no—
397C23—92no—
398C23—182no—
399C23—182no—
400C23—181no—
401C23—182no—
402C23—182no—
403C23D32⋊C411yescharacteristic
404D4—64no—
405D4—184no—
406C2×C4—22no—
407C2×C4—61no—
408C2×C4—61no—
409C2×C4—61no—
410C2×C4—92no—
411C2×C4—91no—
412C2×C4—92no—
413C2×C4—92no—
414C2×C4—91no—
415C2×C4—122no—
416C2×C4—181no—
417C2×C4—181no—
418C2×C4—182no—
419C2×C4—181no—
420C2×C4—181no—
421C2×C4—181no—
422C2×C4—182no—
423C2×C4—182no—
424C2×C4—182no—
425C2×C4—361no—
426C2×C4—361no—
427C2×C4D32⋊C411yescharacteristic
428C2×C4C2×GO+(4,𝔽2)11yescharacteristic
429C8—122no—
430C8—184no—
431C2×C3—21no—
432C2×C3—21no—
433C2×C3—21no—
434C2×C3—21no—
435C2×C3—21no—
436C2×C3—21no—
437C2×C3—42no—
438C2×C3—42no—
439C2×C3—122no—
440D3—42no—
441D3—62no—
442D3—62no—
443D3—62no—
444D3—62no—
445D3—122no—
446D3—122no—
447C22—21no—
448C22—21no—
449C22—62no—
450C22—91no—
451C22—91no—
452C22—92no—
453C22—92no—
454C22—92no—
455C22—121no—
456C22—121no—
457C22—184no—
458C22—184no—
459C22—184no—
460C22—181no—
461C22—181no—
462C22—362no—
463C22D32⋊C812yescomplemented
464C22C2×D32⋊C411yescharacteristic
465C4—21no—
466C4—62no—
467C4—91no—
468C4—91no—
469C4—121no—
470C4—121no—
471C4—181no—
472C4#6424212yes—
473C3—21no—
474C3—21no—
475C2—22no—
476C2—92no—
477C2—92no—
478C2—122no—
479C2—182no—
480C2C2×D32⋊C811yescharacteristic
481C2#1059330911yescharacteristic
482C2#1059351511yescharacteristic
subgroupG/H
center, FrattiniC2×C4C2×GO+(4,𝔽2)
commutatorC2×C3⋊D3C22×C8
FittingC22×C32×C4D4
socleC22×C32C2×C22⋊C4
prime radicalC2×#3811C4
cosocleC2×C4×C3⋊D3C23
hypercenterC22×C4GO+(4,𝔽2)
hypocenterC32#1869
LCSG ▷ C2×C3⋊D3 ▷ C32
UCSC1 ◁ C2×C4 ◁ C22×C4

Complements

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

NH ≅ G/Ntype
1. C22×#3787C2semidirect
3. C2×D32⋊C8C2semidirect
4. #10591152C2semidirect
5. #10590043C2semidirect
40. #3811C8semidirect
52. #3787D4semidirect
96. C2×D32C2×C8semidirect
119. #624C2×C8semidirect
185. C22×C32C22⋊C8semidirect
191. C2×C3⋊D3C22⋊C8semidirect
391. C32#1869semidirect
463. C22D32⋊C8semidirect

Maximal subgroups

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

subgroupindex
C22×#37872
C4×#38112
C2×D32⋊C82
#105911522
#105900432
#18699