Finite Group Databasebeta

← #11787157

Subgroups — #11787157

Order 1152.

702 conjugacy classes of subgroups; only the 89 normal subgroups are shown.

normal subgroups only

This group has 702 conjugacy classes of subgroups — too many to draw. Only its 89 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×C4×GO+(4,𝔽2)C211yesmaximal, characteristic
2C2×D32×C8C211yesmaximal, characteristic, complemented
3C2×D32⋊C8C211yesmaximal, characteristic, complemented
4C2×D32⋊C8C211yesmaximal, characteristic, complemented
5#10590935C211yesmaximal, characteristic, complemented
6#10590181C211yesmaximal, characteristic, complemented
7#10590044C211yesmaximal, characteristic, complemented
8C22×GO+(4,𝔽2)C411yescharacteristic
9C2×C4×D32C2211yescharacteristic
10C2×C4×D32C2211yescharacteristic
11C2×C4×#157C2211yescharacteristic
12C2×#3787C2211yescharacteristic
13C2×#3773C411yescharacteristic
14C2×D32⋊C4C411yescharacteristic
15C2×D32⋊C4C411yescharacteristic
16C4×GO+(4,𝔽2)—22no—
17C4×GO+(4,𝔽2)—22no—
18C2×C8×C3⋊D3C2211yescharacteristic
19C2×C3×D3×C8—21no—
20C2×#3710C2211yescharacteristic
21C2×#3710C2211yescharacteristic
22C2×D3×C3⋊C8—21no—
23D32×C8—22no—
24D32×C8—22no—
25D32⋊C8—22no—
26D32⋊C8—22no—
27C3×(C2×D3)⋊C8—21no—
28(C2×D3)⋊(C3⋊C8)—21no—
29C22×D3×C8—61no—
30D3×C22⋊C8—61no—
31C22×D32C2×C411yescharacteristic
32C22×D32C2×C411yescharacteristic
33C22×#157C2×C411yescharacteristic
34C22×#157C2×C411yescharacteristic
35C2×GO+(4,𝔽2)—22no—
36C2×GO+(4,𝔽2)—22no—
37C2×GO+(4,𝔽2)C814yescomplemented
38C2×C4×C3⋊D3C2311yescharacteristic
39C2×C3×C4×D3—21no—
40C2×C3×C4×D3—21no—
41C2×#624C2×C411yescharacteristic
42C2×#624C2×C411yescharacteristic
43C2×D3×Dic3—21no—
44C2×D3×Dic3—21no—
45C4×D32—21no—
46C4×D32—21no—
47C4×D32—21no—
48C4×D32—21no—
49C4×D32—21no—
50C4×D32—21no—
51C4×D32D412yes—
52C4×D32D412yes—
53C2×#622—21no—
54C4×#157—21no—
55C4×#157—21no—
56#3787—21no—
57#3787—21no—
58#3773—22no—
59D32⋊C4—22no—
60D32⋊C4—22no—
61C2×C32×C8D411yescharacteristic, complemented
62C2×C3⋊(C3⋊C8)D411yescharacteristic, complemented
63C8×C3⋊D3—21no—
64C8×C3⋊D3—21no—
65C2×C3×C3⋊C8—21no—
66C2×C3×C3⋊C8—21no—
67C3×D3×C8—24no—
68#3710—21no—
69#3710—21no—
70#3710—21no—
71#3710D412yescomplemented
72D3×C3⋊C8—24no—
73#1869—91nomaximal, Sylow
74C22×C4×D3—61no—
75C22×C4×D3—61no—
76C22×C3×C8—61no—
77C22×C3⋊C8—61no—
78C2×D3×C8—21no—
79C2×D3×C8—61no—
80C2×D3×C8—61no—
81C2×D3×C8—61no—
82C2×D3×C8—61no—
83C2×D3×C8—64no—
84C2×D3×C8—64no—
85C2×D3×C8—61no—
86C3×C22⋊C8—61no—
87#830—61no—
88(C2×D3)⋊C8—21no—
89(C2×D3)⋊C8—61no—
90C22×C3⋊D3C22×C411yescharacteristic
91C22×C3×D3—21no—
92C22×C3×D3—21no—
93C2×D32—21no—
94C2×D32—21no—
95C2×D32—21no—
96C2×D32—21no—
97C2×D32—21no—
98C2×D32—21no—
99C2×D32—21no—
100C2×D32—21no—
101C2×D32C22⋊C412yes—
102C2×D32C2×C812yes—
103C2×D32C2×C812yescomplemented
104C2×D32M1612yes—
105C2×#157—21no—
106C2×#157—21no—
107C2×#157—21no—
108C2×#157—21no—
109C2×#157—21no—
110C2×#157C2×C812yescomplemented
111GO+(4,𝔽2)—28no—
112C2×C32×C4C2×D411yescharacteristic
113C2×C3⋊Dic3#4111yescharacteristic
114C4×C3⋊D3C2×D411yescharacteristic
115C4×C3⋊D3C2×D411yescharacteristic
116C4×C3⋊D3C2×D411yescharacteristic
117C4×C3⋊D3#4111yescharacteristic
118C2×C3×Dic3—21no—
119C2×C3×Dic3—21no—
120C3×C4×D3—22no—
121C3×C4×D3—22no—
122C3×C4×D3—41no—
123C3×C4×D3—41no—
124#624—21no—
125#624—21no—
126#624—21no—
127#624C22⋊C412yes—
128D3×Dic3—22no—
129D3×Dic3—22no—
130D3×Dic3—41no—
131D3×Dic3—41no—
132#622—41no—
133C32×C8—21no—
134C3⋊(C3⋊C8)—21no—
135C3×C3⋊C8—22no—
136C3×C3⋊C8—41no—
137C23×C8—91no—
138C2×C4×D4—91no—
139C2×C22⋊C8—91no—
140C2×C22⋊C8—91no—
141C2×C22⋊C8—91no—
142#336—91no—
143#336—91no—
144C23×D3—61no—
145C23×D3—61no—
146C22×C3×C4—61no—
147C22×C3×C4—61no—
148C22×Dic3—61no—
149C22×Dic3—61no—
150C2×C4×D3—21no—
151C2×C4×D3—21no—
152C2×C4×D3—62no—
153C2×C4×D3—62no—
154C2×C4×D3—61no—
155C2×C4×D3—61no—
156C2×C4×D3—61no—
157C2×C4×D3—61no—
158C2×C4×D3—62no—
159C2×C4×D3—62no—
160C2×C4×D3—61no—
161C2×C4×D3—61no—
162C2×C4×D3—121no—
163C2×C4×D3—121no—
164C2×C4×D3—121no—
165C2×C4×D3—121no—
166C2×C3×C8—21no—
167C2×C3×C8—21no—
168C2×C3×C8—64no—
169C2×C3×C8—61no—
170C2×C3×C8—61no—
171C2×C3⋊C8—21no—
172C2×C3⋊C8—21no—
173C2×C3⋊C8—61no—
174C2×C3⋊C8—61no—
175C2×C3⋊C8—64no—
176D3×C8—24no—
177D3×C8—62no—
178D3×C8—62no—
179D3×C8—62no—
180D3×C8—62no—
181D3×C8—64no—
182D3×C8—121no—
183D3×C8—121no—
184D3×C8—121no—
185D3×C8—121no—
186C22×C32C4×D411yescharacteristic
187C2×C3⋊D3C2×C22⋊C411yescharacteristic
188C2×C3⋊D3C4×D411yescharacteristic
189C2×C3⋊D3C22×C811yescharacteristic
190C2×C3⋊D3C2×M1611yescharacteristic
191C2×C3⋊D3#13111yescharacteristic
192C2×C3⋊D3#13111yescharacteristic
193C2×C3×D3—22no—
194C2×C3×D3—22no—
195C2×C3×D3—22no—
196C2×C3×D3—22no—
197C2×C3×D3—41no—
198C2×C3×D3—41no—
199D32—22no—
200D32—22no—
201D32—22no—
202D32—22no—
203D32—22no—
204D32—41no—
205D32—41no—
206D32C22⋊C814yescomplemented
207#157—22no—
208#157—41no—
209C32×C4#12011yescharacteristic
210C32×C4#12111yescharacteristic
211C3⋊Dic3#12111yescharacteristic
212C3⋊Dic3#12311yescharacteristic
213C3×Dic3—22no—
214C3×Dic3—41no—
215C22×D4—91no—
216C23×C4—91no—
217C23×C4—91no—
218C22×C8—61no—
219C22×C8—91no—
220C22×C8—91no—
221C22×C8—91no—
222C22×C8—91no—
223C22×C8—182no—
224C22×C8—182no—
225C22×C8—181no—
226C4×D4—182no—
227C4×D4—182no—
228C2×C4⋊C4—91no—
229C2×C22⋊C4—91no—
230C2×C22⋊C4—91no—
231C2×C42—91no—
232C22⋊C8—61no—
233C22⋊C8—181no—
234C22⋊C8—182no—
235C22⋊C8—182no—
236C23×C3—61no—
237C23×C3—61no—
238C22×D3—21no—
239C22×D3—21no—
240C22×D3—62no—
241C22×D3—62no—
242C22×D3—62no—
243C22×D3—62no—
244C22×D3—61no—
245C22×D3—61no—
246C22×D3—62no—
247C22×D3—62no—
248C22×D3—62no—
249C22×D3—62no—
250C22×D3—121no—
251C22×D3—121no—
252C22×D3—121no—
253C22×D3—121no—
254C2×C3×C4—21no—
255C2×C3×C4—21no—
256C2×C3×C4—62no—
257C2×C3×C4—62no—
258C2×C3×C4—61no—
259C2×C3×C4—61no—
260C2×C3×C4—121no—
261C2×C3×C4—121no—
262C2×Dic3—21no—
263C2×Dic3—21no—
264C2×Dic3—61no—
265C2×Dic3—61no—
266C2×Dic3—62no—
267C2×Dic3—62no—
268C2×Dic3—121no—
269C2×Dic3—121no—
270C4×D3—22no—
271C4×D3—22no—
272C4×D3—41no—
273C4×D3—41no—
274C4×D3—62no—
275C4×D3—61no—
276C4×D3—61no—
277C4×D3—61no—
278C4×D3—61no—
279C4×D3—61no—
280C4×D3—61no—
281C4×D3—61no—
282C4×D3—61no—
283C4×D3—62no—
284C4×D3—62no—
285C4×D3—62no—
286C4×D3—121no—
287C4×D3—121no—
288C4×D3—121no—
289C4×D3—121no—
290C3×C8—22no—
291C3×C8—41no—
292C3×C8—62no—
293C3×C8—121no—
294C3⋊C8—22no—
295C3⋊C8—41no—
296C3⋊C8—62no—
297C3⋊C8—121no—
298C2×C32#38611yescharacteristic
299C2×C32C8×D411yescharacteristic
300C2×C32#43511yescharacteristic
301C3⋊D3C2×C22⋊C811yescharacteristic
302C3⋊D3#40811yescharacteristic
303C3⋊D3C8×D411yescharacteristic
304C3⋊D3#43511yescharacteristic
305C3×D3—24no—
306C3×D3—42no—
307C24—91no—
308C24—91no—
309C2×D4—94no—
310C2×D4—182no—
311C2×D4—182no—
312C22×C4—61no—
313C22×C4—61no—
314C22×C4—92no—
315C22×C4—91no—
316C22×C4—92no—
317C22×C4—91no—
318C22×C4—91no—
319C22×C4—91no—
320C22×C4—91no—
321C22×C4—181no—
322C22×C4—181no—
323C22×C4—181no—
324C22×C4—181no—
325C22×C4—181no—
326C22×C4—181no—
327C22×C4—181no—
328C22×C4—181no—
329C2×C8—64no—
330C2×C8—61no—
331C2×C8—61no—
332C2×C8—92no—
333C2×C8—91no—
334C2×C8—181no—
335C2×C8—181no—
336C2×C8—181no—
337C2×C8—184no—
338C2×C8—181no—
339C2×C8—181no—
340C2×C8—181no—
341C2×C8—181no—
342C2×C8—181no—
343C2×C8GO+(4,𝔽2)11yescharacteristic, complemented
344C4⋊C4—182no—
345C22⋊C4—182no—
346C22⋊C4—182no—
347C42—181no—
348C42—181no—
349C22×C3—21no—
350C22×C3—21no—
351C22×C3—62no—
352C22×C3—62no—
353C22×C3—62no—
354C22×C3—62no—
355C22×C3—121no—
356C22×C3—121no—
357C2×D3—22no—
358C2×D3—22no—
359C2×D3—22no—
360C2×D3—22no—
361C2×D3—41no—
362C2×D3—41no—
363C2×D3—64no—
364C2×D3—61no—
365C2×D3—61no—
366C2×D3—61no—
367C2×D3—61no—
368C2×D3—61no—
369C2×D3—61no—
370C2×D3—61no—
371C2×D3—61no—
372C2×D3—61no—
373C2×D3—61no—
374C2×D3—61no—
375C2×D3—61no—
376C2×D3—64no—
377C2×D3—64no—
378C2×D3—64no—
379C2×D3—122no—
380C2×D3—122no—
381C2×D3—122no—
382C2×D3—122no—
383C3×C4—21no—
384C3×C4—21no—
385C3×C4—21no—
386C3×C4—21no—
387C3×C4—62no—
388C3×C4—121no—
389Dic3—22no—
390Dic3—41no—
391Dic3—61no—
392Dic3—61no—
393Dic3—61no—
394Dic3—61no—
395C32#186911yescharacteristic, complemented, Sylow
396C23—61no—
397C23—61no—
398C23—92no—
399C23—92no—
400C23—92no—
401C23—92no—
402C23—91no—
403C23—181no—
404C23—181no—
405C23—181no—
406C23—181no—
407C23—181no—
408C23—181no—
409C23—181no—
410C23—181no—
411D4—188no—
412C2×C4—61no—
413C2×C4—62no—
414C2×C4—62no—
415C2×C4—61no—
416C2×C4—92no—
417C2×C4—91no—
418C2×C4—91no—
419C2×C4—91no—
420C2×C4—91no—
421C2×C4—91no—
422C2×C4—92no—
423C2×C4—121no—
424C2×C4—121no—
425C2×C4—181no—
426C2×C4—182no—
427C2×C4—182no—
428C2×C4—181no—
429C2×C4—181no—
430C2×C4—181no—
431C2×C4—181no—
432C2×C4—181no—
433C2×C4—181no—
434C2×C4—181no—
435C2×C4—361no—
436C2×C4—361no—
437C2×C4C2×GO+(4,𝔽2)11yescharacteristic
438C8—21no—
439C8—62no—
440C8—121no—
441C8—181no—
442C8—361no—
443C2×C3—21no—
444C2×C3—21no—
445C2×C3—21no—
446C2×C3—21no—
447C2×C3—21no—
448C2×C3—21no—
449C2×C3—64no—
450C2×C3—122no—
451D3—24no—
452D3—42no—
453D3—61no—
454D3—61no—
455D3—61no—
456D3—61no—
457D3—61no—
458D3—61no—
459D3—61no—
460D3—61no—
461C22—62no—
462C22—62no—
463C22—62no—
464C22—62no—
465C22—94no—
466C22—91no—
467C22—91no—
468C22—91no—
469C22—91no—
470C22—91no—
471C22—91no—
472C22—121no—
473C22—121no—
474C22—182no—
475C22—182no—
476C22—182no—
477C22—182no—
478C22—182no—
479C22—361no—
480C22—361no—
481C22C4×GO+(4,𝔽2)11yescharacteristic
482C4—62no—
483C4—91no—
484C4—91no—
485C4—121no—
486C4—182no—
487C4—361no—
488C4#6423011yescharacteristic
489C4#6424111yescharacteristic
490C3—21no—
491C3—21no—
492C2—64no—
493C2—91no—
494C2—91no—
495C2—91no—
496C2—91no—
497C2—122no—
498C2C8×GO+(4,𝔽2)11yescharacteristic
499C2#1059328211yescharacteristic
500C2#1059345311yescharacteristic
subgroupG/H
center, FrattiniC2×C4C2×GO+(4,𝔽2)
commutatorC2×C3⋊D3C22×C8
FittingC2×C32×C8D4
socleC22×C32C4×D4
prime radicalC22×GO+(4,𝔽2)C4
cosocleC2×C4×C3⋊D3C23
hypercenterC2×C8GO+(4,𝔽2)
hypocenterC32#1869
LCSG ▷ C2×C3⋊D3 ▷ C32
UCSC1 ◁ C2×C4 ◁ C2×C8

Complements

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

NH ≅ G/Ntype
2. C2×D32×C8C2semidirect
3. C2×D32⋊C8C2semidirect
4. C2×D32⋊C8C2semidirect
5. #10590935C2semidirect
6. #10590181C2semidirect
7. #10590044C2semidirect
37. C2×GO+(4,𝔽2)C8semidirect
61. C2×C32×C8D4semidirect
62. C2×C3⋊(C3⋊C8)D4semidirect
71. #3710D4semidirect
103. C2×D32C2×C8semidirect
110. C2×#157C2×C8semidirect
206. D32C22⋊C8semidirect
343. C2×C8GO+(4,𝔽2)semidirect
395. C32#1869semidirect

Maximal subgroups

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

subgroupindex
C2×C4×GO+(4,𝔽2)2
C2×D32×C82
C2×D32⋊C82
#105909352
#105901812
#105900442
#18699