Finite Group Databasebeta

← #11787156

Subgroups — #11787156

Order 1152.

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

normal subgroups only

This group has 814 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
1C4×#3811C211yesmaximal, characteristic
2C22×#3710C211yesmaximal, characteristic, complemented
3#10593800C211yesmaximal, characteristic, complemented
4C2×D32⋊C8C212yesmaximal, complemented
5#10590043C212yesmaximal, complemented
6C22×C4×C3⋊D3C2211yescharacteristic
7C2×#3811C411yescharacteristic
8C2×C4×D32C2211yescharacteristic
9C2×#3787C2212yes—
10C3×C4×#67—21no—
11C22×C3×C3⋊C8—21no—
12#63974C411yescharacteristic
13#67⋊Dic3—21no—
14#63921C411yescharacteristic
15C4×#626—21no—
16#63883C411yescharacteristic
17C4×#624C2211yescharacteristic
18#63881—21no—
19C2×#3710—21no—
20C2×#3710—24no—
21C2×#3710C2212yes—
22C22⋊#622—21no—
23#63778—21no—
24D32⋊C8—24no—
25C22×D3×C8—61no—
26C4×D3×D4—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—22no—
49#3787—22no—
50#3787—22no—
51C3×#217—21no—
52C3×(C2×D3)⋊C4—21no—
53C3×Dic3⋊C4—21no—
54C3×C4×Dic3—21no—
55C2×C3×C3⋊C8—24no—
56C2×C3×C3⋊C8—22no—
57Dic3⋊Dic3—21no—
58#3722—21no—
59(C2×D3)⋊Dic3—21no—
60Dic32—21no—
61#3710—24no—
62#3710—22no—
63#3710D414yescomplemented
64#1869—91nomaximal, Sylow
65C2×D3×D4—61no—
66C22×C4×D3—61no—
67C22×C4×D3—61no—
68C22×C4×D3—61no—
69C22×C3×C8—61no—
70C3×C4×D4—61no—
71D4×Dic3—61no—
72C4×#67—21no—
73C4×#67—61no—
74C22×C3⋊C8—21no—
75C2×D3×C8—62no—
76C2×D3×C8—68no—
77C2×D3×C8—62no—
78#893—61no—
79D3×C4⋊C4—61no—
80#67⋊C4—61no—
81#67⋊C4—61no—
82D3×C22⋊C4—61no—
83D3×C22⋊C4—61no—
84C4×D12—61no—
85C42×D3—61no—
86C23×C32C22⋊C411yescharacteristic
87C22×C3⋊D3—22no—
88C22×C3⋊D3—22no—
89C22×C3⋊D3C22⋊C411yescharacteristic
90C22×C3⋊D3C2×C812yes—
91C22×C3⋊D3M1612yes—
92C22×C3⋊D3C22×C411yescharacteristic
93C22×C3×D3—21no—
94C2×D32—21no—
95C2×D32—22no—
96C2×D32—22no—
97C2×D32C2×C812yescomplemented
98C2×C32×C4—21no—
99C2×C32×C4—21no—
100C2×C32×C4C22⋊C411yescharacteristic
101C2×C32×C4C2×D411yescharacteristic
102C2×C3⋊Dic3—22no—
103C2×C3⋊Dic3C22⋊C411yescharacteristic
104C2×C3⋊Dic3C2×D411yescharacteristic
105C4×C3⋊D3—22no—
106C4×C3⋊D3—21no—
107C4×C3⋊D3—21no—
108C4×C3⋊D3—22no—
109C4×C3⋊D3C2×D412yes—
110C4×C3⋊D3#4112yes—
111C3×#67—24no—
112C2×C3×Dic3—21no—
113C2×C3×Dic3—21no—
114C2×C3×Dic3—21no—
115C3×C4×D3—41no—
116C3×C4×D3—41no—
117#626—24no—
118#624—21no—
119#624—22no—
120#624—22no—
121#624C2×C812yescomplemented
122D3×Dic3—42no—
123#622—42no—
124C3×C3⋊C8—24no—
125C23×C8—91no—
126C2×C4×D4—91no—
127C2×C22⋊C8—91no—
128C2×C22⋊C8—92no—
129#336—92no—
130C23×D3—61no—
131C23×D3—61no—
132C23×D3—61no—
133C2×C3×D4—61no—
134C22×C3×C4—21no—
135C22×C3×C4—21no—
136C22×C3×C4—61no—
137C2×#67—21no—
138C2×#67—61no—
139C22×Dic3—61no—
140C22×Dic3—61no—
141C22×Dic3—61no—
142D3×D4—64no—
143D3×D4—64no—
144C2×D12—61no—
145C2×C4×D3—21no—
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—64no—
158C2×C4×D3—64no—
159C2×C4×D3—61no—
160C2×C4×D3—122no—
161C2×C4×D3—122no—
162C2×C4×D3—122no—
163C2×C4×D3—122no—
164C2×C3×C8—64no—
165C2×C3×C8—62no—
166C3×C4⋊C4—61no—
167C3×C22⋊C4—61no—
168C3×C22⋊C4—61no—
169C3×C42—61no—
170#217—21no—
171#217—61no—
172(C2×D3)⋊C4—21no—
173(C2×D3)⋊C4—61no—
174(C2×D3)⋊C4—61no—
175(C2×D3)⋊C4—61no—
176C4⋊Dic3—61no—
177Dic3⋊C4—21no—
178Dic3⋊C4—61no—
179C4×Dic3—21no—
180C4×Dic3—61no—
181C4×Dic3—61no—
182C2×C3⋊C8—24no—
183C2×C3⋊C8—22no—
184D3×C8—64no—
185D3×C8—64no—
186D3×C8—68no—
187C22×C32—21no—
188C22×C32—21no—
189C22×C32C22⋊C812yescomplemented
190C22×C32C2×C22⋊C411yescharacteristic
191C2×C3⋊D3—21no—
192C2×C3⋊D3—21no—
193C2×C3⋊D3—24no—
194C2×C3⋊D3—24no—
195C2×C3⋊D3C22⋊C812yescomplemented
196C2×C3⋊D3C4×D412yes—
197C2×C3⋊D3C22×C811yescharacteristic
198C2×C3⋊D3C2×M1611yescharacteristic
199C2×C3⋊D3#13112yes—
200C2×C3×D3—22no—
201C2×C3×D3—41no—
202C2×C3×D3—41no—
203D32—24no—
204D32—42no—
205C32×C4—21no—
206C32×C4#12011yescharacteristic
207C32×C4#12311yescharacteristic
208C3⋊Dic3—21no—
209C3⋊Dic3#12112yes—
210C3×Dic3—22no—
211C3×Dic3—41no—
212C3×Dic3—41no—
213C22×D4—91no—
214C23×C4—91no—
215C23×C4—91no—
216C22×C8—61no—
217C22×C8—92no—
218C22×C8—92no—
219C22×C8—184no—
220C22×C8—181no—
221C4×D4—61no—
222C4×D4—181no—
223C4×D4—181no—
224C4×D4—181no—
225C2×C4⋊C4—91no—
226C2×C22⋊C4—91no—
227C2×C22⋊C4—91no—
228C2×C42—91no—
229C22⋊C8—184no—
230C22⋊C8—181no—
231C22⋊C8—181no—
232C23×C3—21no—
233C23×C3—21no—
234C23×C3—61no—
235C22×D3—21no—
236C22×D3—62no—
237C22×D3—61no—
238C22×D3—62no—
239C22×D3—62no—
240C22×D3—61no—
241C22×D3—61no—
242C22×D3—61no—
243C22×D3—62no—
244C22×D3—62no—
245C22×D3—62no—
246C22×D3—64no—
247C22×D3—64no—
248C22×D3—122no—
249C22×D3—122no—
250C22×D3—122no—
251C22×D3—122no—
252C3×D4—64no—
253C2×C3×C4—22no—
254C2×C3×C4—22no—
255C2×C3×C4—21no—
256C2×C3×C4—21no—
257C2×C3×C4—21no—
258C2×C3×C4—21no—
259C2×C3×C4—41no—
260C2×C3×C4—41no—
261C2×C3×C4—61no—
262C2×C3×C4—61no—
263C2×C3×C4—61no—
264C2×C3×C4—121no—
265C2×C3×C4—121no—
266#67—24no—
267#67—64no—
268C2×Dic3—21no—
269C2×Dic3—21no—
270C2×Dic3—21no—
271C2×Dic3—61no—
272C2×Dic3—61no—
273C2×Dic3—61no—
274C2×Dic3—61no—
275C2×Dic3—64no—
276C2×Dic3—122no—
277C2×Dic3—122no—
278D12—64no—
279C4×D3—41no—
280C4×D3—41no—
281C4×D3—62no—
282C4×D3—62no—
283C4×D3—62no—
284C4×D3—62no—
285C4×D3—62no—
286C4×D3—62no—
287C4×D3—62no—
288C4×D3—64no—
289C4×D3—64no—
290C4×D3—62no—
291C4×D3—122no—
292C4×D3—122no—
293C4×D3—122no—
294C4×D3—121no—
295C4×D3—121no—
296C4×D3—122no—
297C4×D3—122no—
298C3×C8—64no—
299C3⋊C8—24no—
300C2×C32—22no—
301C2×C32#38611yescharacteristic
302C2×C32C2×C22⋊C811yescharacteristic
303C2×C32#40811yescharacteristic
304C3⋊D3—22no—
305C3⋊D3C8×D412yes—
306C3⋊D3#43512yes—
307C3×D3—42no—
308C24—91no—
309C24—91no—
310C2×D4—61no—
311C2×D4—94no—
312C2×D4—181no—
313C2×D4—182no—
314C22×C4—61no—
315C22×C4—91no—
316C22×C4—91no—
317C22×C4—91no—
318C22×C4—91no—
319C22×C4—91no—
320C22×C4—91no—
321C22×C4—91no—
322C22×C4—91no—
323C22×C4—182no—
324C22×C4—182no—
325C22×C4—181no—
326C22×C4—182no—
327C22×C4—182no—
328C22×C4GO+(4,𝔽2)11yescharacteristic
329C2×C8—64no—
330C2×C8—62no—
331C2×C8—94no—
332C2×C8—182no—
333C2×C8—184no—
334C2×C8—182no—
335C2×C8—182no—
336C2×C8—182no—
337C4⋊C4—61no—
338C4⋊C4—181no—
339C22⋊C4—61no—
340C22⋊C4—61no—
341C22⋊C4—181no—
342C22⋊C4—181no—
343C42—61no—
344C42—181no—
345C22×C3—22no—
346C22×C3—22no—
347C22×C3—21no—
348C22×C3—21no—
349C22×C3—22no—
350C22×C3—22no—
351C22×C3—41no—
352C22×C3—41no—
353C22×C3—62no—
354C22×C3—121no—
355C22×C3—121no—
356C2×D3—22no—
357C2×D3—41no—
358C2×D3—41no—
359C2×D3—62no—
360C2×D3—62no—
361C2×D3—61no—
362C2×D3—61no—
363C2×D3—62no—
364C2×D3—62no—
365C2×D3—61no—
366C2×D3—61no—
367C2×D3—62no—
368C2×D3—62no—
369C2×D3—62no—
370C2×D3—68no—
371C2×D3—68no—
372C2×D3—62no—
373C2×D3—124no—
374C2×D3—124no—
375C2×D3—124no—
376C2×D3—121no—
377C2×D3—121no—
378C2×D3—124no—
379C3×C4—22no—
380C3×C4—21no—
381C3×C4—21no—
382C3×C4—21no—
383C3×C4—21no—
384C3×C4—41no—
385C3×C4—62no—
386C3×C4—121no—
387C3×C4—121no—
388Dic3—22no—
389Dic3—41no—
390Dic3—41no—
391Dic3—62no—
392Dic3—62no—
393Dic3—62no—
394Dic3—121no—
395C32#186911yescharacteristic, complemented, Sylow
396C23—61no—
397C23—92no—
398C23—91no—
399C23—91no—
400C23—92no—
401C23—92no—
402C23—182no—
403C23—182no—
404C23—181no—
405C23—182no—
406C23—182no—
407C23D32⋊C411yescharacteristic
408D4—64no—
409D4—184no—
410C2×C4—21no—
411C2×C4—21no—
412C2×C4—61no—
413C2×C4—61no—
414C2×C4—61no—
415C2×C4—92no—
416C2×C4—91no—
417C2×C4—92no—
418C2×C4—92no—
419C2×C4—91no—
420C2×C4—121no—
421C2×C4—121no—
422C2×C4—182no—
423C2×C4—182no—
424C2×C4—181no—
425C2×C4—182no—
426C2×C4—182no—
427C2×C4—182no—
428C2×C4—181no—
429C2×C4—181no—
430C2×C4—362no—
431C2×C4D32⋊C411yescharacteristic
432C2×C4C2×GO+(4,𝔽2)11yescharacteristic
433C8—64no—
434C8—362no—
435C2×C3—24no—
436C2×C3—21no—
437C2×C3—21no—
438C2×C3—21no—
439C2×C3—21no—
440C2×C3—21no—
441C2×C3—21no—
442C2×C3—42no—
443C2×C3—122no—
444D3—42no—
445D3—62no—
446D3—62no—
447D3—62no—
448D3—62no—
449D3—64no—
450D3—122no—
451C22—21no—
452C22—21no—
453C22—62no—
454C22—91no—
455C22—91no—
456C22—92no—
457C22—92no—
458C22—92no—
459C22—121no—
460C22—121no—
461C22—184no—
462C22—184no—
463C22—184no—
464C22—181no—
465C22—181no—
466C22—362no—
467C22D32⋊C812yescomplemented
468C22C2×D32⋊C411yescharacteristic
469C4—21no—
470C4—62no—
471C4—92no—
472C4—121no—
473C4—121no—
474C4—181no—
475C4#6423311yescharacteristic
476C4#6424111yescharacteristic
477C3—21no—
478C3—21no—
479C2—22no—
480C2—92no—
481C2—92no—
482C2—122no—
483C2—182no—
484C2C2×D32⋊C811yescharacteristic
485C2#1059330811yescharacteristic
486C2#1059351411yescharacteristic
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
2. C22×#3710C2semidirect
3. #10593800C2semidirect
4. C2×D32⋊C8C2semidirect
5. #10590043C2semidirect
40. #3811C8semidirect
63. #3710D4semidirect
97. C2×D32C2×C8semidirect
121. #624C2×C8semidirect
189. C22×C32C22⋊C8semidirect
195. C2×C3⋊D3C22⋊C8semidirect
395. C32#1869semidirect
467. C22D32⋊C8semidirect

Maximal subgroups

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

subgroupindex
C4×#38112
C22×#37102
#105938002
C2×D32⋊C82
#105900432
#18699