Finite Group Databasebeta

← #11773305

Subgroups — #11773305

Order 1152.

1454 conjugacy classes of subgroups; only the 140 normal subgroups are shown.

normal subgroups only

This group has 1454 conjugacy classes of subgroups — too many to draw. Only its 140 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×C22⋊#157C212yesmaximal
2C22×C4⋊#157C211yesmaximal, characteristic, complemented
3C2×C22⋊C4×C3⋊D3C211yesmaximal, characteristic
4C2×#63781C211yesmaximal, characteristic, complemented
5C2×#63781C212yesmaximal, complemented
6C24×C3⋊D3C2211yescharacteristic
7C23×#157C2212yes—
8C23×#157C2212yes—
9C22×C4×C3⋊D3C2211yescharacteristic
10C22×C4×C3⋊D3C2211yescharacteristic
11C2×C22⋊#157—24no—
12C2×C22⋊#157—24no—
13C2×C22⋊#157—24no—
14C2×C4⋊#157—22no—
15C2×C4⋊#157—24no—
16C2×C32×C22⋊C4C411yescharacteristic, complemented
17C2×#3757C411yescharacteristic, complemented
18C2×#3752C411yescharacteristic, complemented
19C2×#3752C411yescharacteristic, complemented
20C22⋊C4×C3⋊D3—22no—
21C22⋊C4×C3⋊D3—22no—
22#63781—22no—
23#63781—24no—
24C2×D3×C22⋊C4—62no—
25C24×C32C2×C411yescharacteristic
26C23×C3⋊D3—21no—
27C23×C3⋊D3—21no—
28C23×C3⋊D3—22no—
29C23×C3⋊D3—22no—
30C23×C3⋊D3—21no—
31C23×C3⋊D3—21no—
32C23×C3⋊D3—22no—
33C23×C3⋊D3—22no—
34C23×C3⋊D3C2×C411yescharacteristic
35C23×C3⋊D3D412yes—
36C23×C3⋊D3Q812yes—
37C23×C3⋊D3C2311yescharacteristic
38C22×#157—22no—
39C22×#157—22no—
40C22×#157—22no—
41C22×#157—22no—
42C22×#157—22no—
43C22×#157—22no—
44C22×#157—22no—
45C22×#157—22no—
46C22×#157—22no—
47C22×#157—22no—
48C22×#157—22no—
49C22×#157—22no—
50C22×#157D414yes—
51C22×#157D414yes—
52C22×C32×C4C2×C411yescharacteristic
53C22×C32×C4C2×C411yescharacteristic
54C22×C3⋊Dic3C2×C411yescharacteristic
55C22×C3⋊Dic3C2×C411yescharacteristic
56C2×C4×C3⋊D3—22no—
57C2×C4×C3⋊D3—22no—
58C2×C4×C3⋊D3—21no—
59C2×C4×C3⋊D3—21no—
60C2×C4×C3⋊D3—22no—
61C2×C4×C3⋊D3—22no—
62C2×C4×C3⋊D3D412yes—
63C2×C4×C3⋊D3D412yes—
64C22⋊#157—216no—
65C4⋊#157—28no—
66C32×C22⋊C4—22no—
67#3757—22no—
68#3752—22no—
69#3752—22no—
70#1439—91nomaximal, Sylow
71C24×D3—62no—
72C22×C4×D3—62no—
73C22×C4×D3—62no—
74C2×C3×C22⋊C4—22no—
75C2×#217—62no—
76C2×(C2×D3)⋊C4—62no—
77C2×(C2×D3)⋊C4—62no—
78D3×C22⋊C4—68no—
79D3×C22⋊C4—68no—
80C23×C32—21no—
81C23×C32—21no—
82C23×C32—22no—
83C23×C32—22no—
84C23×C32C4⋊C412yescomplemented
85C23×C32C22×C411yescharacteristic
86C22×C3⋊D3—21no—
87C22×C3⋊D3—21no—
88C22×C3⋊D3—22no—
89C22×C3⋊D3—22no—
90C22×C3⋊D3—22no—
91C22×C3⋊D3—22no—
92C22×C3⋊D3—22no—
93C22×C3⋊D3—22no—
94C22×C3⋊D3—24no—
95C22×C3⋊D3—24no—
96C22×C3⋊D3—24no—
97C22×C3⋊D3—24no—
98C22×C3⋊D3—41no—
99C22×C3⋊D3—42no—
100C22×C3⋊D3—42no—
101C22×C3⋊D3—42no—
102C22×C3⋊D3—41no—
103C22×C3⋊D3—42no—
104C22×C3⋊D3—42no—
105C22×C3⋊D3—41no—
106C22×C3⋊D3—42no—
107C22×C3⋊D3—41no—
108C22×C3⋊D3C4⋊C412yescomplemented
109C22×C3⋊D3C2×D412yes—
110C22×C3⋊D3C2×D411yescharacteristic
111C22×C3⋊D3C2×D412yes—
112C22×C3⋊D3C2×D411yescharacteristic
113C22×C3⋊D3C2×D411yescharacteristic
114C22×C3⋊D3C2×Q811yescharacteristic
115C22×C3⋊D3#4111yescharacteristic
116C22×C3⋊D3#4112yes—
117C22×C3⋊D3#4111yescharacteristic
118C22×C3⋊D3#4112yes—
119C2×#157—24no—
120C2×#157—24no—
121C2×#157—24no—
122C2×#157—24no—
123C2×#157—24no—
124C2×#157—24no—
125C2×#157—42no—
126C2×#157—42no—
127C2×#157—42no—
128C2×#157—42no—
129C2×#157—42no—
130C2×#157—42no—
131C2×#157—42no—
132C2×#157—42no—
133C2×C32×C4—21no—
134C2×C32×C4—22no—
135C2×C32×C4—22no—
136C2×C32×C4C22⋊C412yescomplemented
137C2×C3⋊Dic3—21no—
138C2×C3⋊Dic3—22no—
139C2×C3⋊Dic3—22no—
140C2×C3⋊Dic3C22⋊C412yescomplemented
141C4×C3⋊D3—22no—
142C4×C3⋊D3—24no—
143C4×C3⋊D3—22no—
144C4×C3⋊D3—42no—
145C4×C3⋊D3—41no—
146C4×C3⋊D3—41no—
147C22×C4⋊C4—91no—
148C22×C22⋊C4—91no—
149C22×C22⋊C4—92no—
150C2×#95—91no—
151C2×#95—92no—
152C24×C3—22no—
153C23×D3—62no—
154C23×D3—62no—
155C23×D3—64no—
156C23×D3—64no—
157C23×D3—64no—
158C23×D3—64no—
159C23×D3—64no—
160C23×D3—64no—
161C23×D3—124no—
162C23×D3—124no—
163C23×D3—124no—
164C23×D3—124no—
165C22×C3×C4—22no—
166C22×C3×C4—22no—
167C22×Dic3—62no—
168C22×Dic3—62no—
169C2×C4×D3—64no—
170C2×C4×D3—64no—
171C2×C4×D3—64no—
172C2×C4×D3—64no—
173C2×C4×D3—124no—
174C2×C4×D3—124no—
175C2×C4×D3—124no—
176C2×C4×D3—124no—
177C3×C22⋊C4—28no—
178#217—68no—
179(C2×D3)⋊C4—68no—
180(C2×D3)⋊C4—68no—
181C22×C32—22no—
182C22×C32—24no—
183C22×C32—41no—
184C22×C32—42no—
185C22×C32—41no—
186C22×C32C2×C22⋊C411yescharacteristic
187C22×C32C2×C4⋊C411yescharacteristic
188C22×C32#11711yescharacteristic
189C22×C32C4×D412yes—
190C22×C32C4×D412yes—
191C2×C3⋊D3—22no—
192C2×C3⋊D3—24no—
193C2×C3⋊D3—41no—
194C2×C3⋊D3—42no—
195C2×C3⋊D3—41no—
196C2×C3⋊D3—42no—
197C2×C3⋊D3—42no—
198C2×C3⋊D3—44no—
199C2×C3⋊D3—44no—
200C2×C3⋊D3—42no—
201C2×C3⋊D3—42no—
202C2×C3⋊D3#12011yescharacteristic
203C2×C3⋊D3#12012yes—
204C2×C3⋊D3#12112yes—
205C2×C3⋊D3#12111yescharacteristic
206C2×C3⋊D3#12111yescharacteristic
207C2×C3⋊D3#12112yes—
208C2×C3⋊D3#12112yes—
209C2×C3⋊D3C22⋊Q812yes—
210C2×C3⋊D3C22⋊Q811yescharacteristic
211C2×C3⋊D3C22⋊Q811yescharacteristic
212C2×C3⋊D3C22⋊Q812yes—
213C2×C3⋊D3#12312yes—
214C2×C3⋊D3#12312yes—
215C2×C3⋊D3#12311yescharacteristic
216C2×C3⋊D3#12412yes—
217C2×C3⋊D3#12411yescharacteristic
218C2×C3⋊D3#12612yes—
219C2×C3⋊D3#12711yescharacteristic
220#157—44no—
221#157—44no—
222C32×C4—22no—
223C32×C4—41no—
224C3⋊Dic3—22no—
225C3⋊Dic3—41no—
226C25—91no—
227C23×C4—91no—
228C23×C4—92no—
229C23×C4—92no—
230C23×C4—91no—
231C2×C4⋊C4—182no—
232C2×C4⋊C4—184no—
233C2×C22⋊C4—91no—
234C2×C22⋊C4—91no—
235C2×C22⋊C4—91no—
236C2×C22⋊C4—182no—
237C2×C22⋊C4—182no—
238C2×C22⋊C4—184no—
239C2×C22⋊C4—184no—
240C2×C22⋊C4—184no—
241C2×C22⋊C4#15711yescharacteristic, complemented
242#95—182no—
243#95—184no—
244C23×C3—24no—
245C23×C3—24no—
246C23×C3—22no—
247C23×C3—24no—
248C23×C3—44no—
249C23×C3—44no—
250C22×D3—62no—
251C22×D3—62no—
252C22×D3—64no—
253C22×D3—64no—
254C22×D3—62no—
255C22×D3—62no—
256C22×D3—64no—
257C22×D3—64no—
258C22×D3—62no—
259C22×D3—62no—
260C22×D3—64no—
261C22×D3—64no—
262C22×D3—68no—
263C22×D3—68no—
264C22×D3—68no—
265C22×D3—68no—
266C22×D3—64no—
267C22×D3—128no—
268C22×D3—128no—
269C22×D3—128no—
270C22×D3—128no—
271C22×D3—124no—
272C22×D3—124no—
273C22×D3—124no—
274C22×D3—128no—
275C22×D3—128no—
276C22×D3—124no—
277C22×D3—124no—
278C22×D3—124no—
279C22×D3—128no—
280C22×D3—128no—
281C22×D3—128no—
282C22×D3—128no—
283C2×C3×C4—24no—
284C2×C3×C4—24no—
285C2×C3×C4—44no—
286C2×C3×C4—44no—
287C2×Dic3—64no—
288C2×Dic3—64no—
289C2×Dic3—124no—
290C2×Dic3—124no—
291C4×D3—124no—
292C4×D3—128no—
293C4×D3—124no—
294C4×D3—128no—
295C4×D3—124no—
296C4×D3—124no—
297C2×C32—42no—
298C2×C32#38011yescharacteristic
299C2×C32#38512yes—
300C2×C32#38611yescharacteristic
301C2×C32#38812yes—
302C2×C32#39011yescharacteristic
303C3⋊D3—42no—
304C3⋊D3#39211yescharacteristic
305C3⋊D3#39311yescharacteristic
306C3⋊D3#39412yes—
307C3⋊D3#39612yes—
308C3⋊D3#39711yescharacteristic
309C3⋊D3#39911yescharacteristic
310C24—91no—
311C24—91no—
312C24—92no—
313C24—92no—
314C24—181no—
315C24—181no—
316C24—182no—
317C24—182no—
318C24—181no—
319C24—181no—
320C24—182no—
321C24—182no—
322C24C2×#15711yescharacteristic
323C22×C4—91no—
324C22×C4—92no—
325C22×C4—92no—
326C22×C4—94no—
327C22×C4—94no—
328C22×C4—91no—
329C22×C4—182no—
330C22×C4—182no—
331C22×C4—182no—
332C22×C4—181no—
333C22×C4—181no—
334C22×C4—182no—
335C22×C4—182no—
336C22×C4—182no—
337C22×C4—182no—
338C22×C4—182no—
339C22×C4—182no—
340C22×C4—182no—
341C22×C4—182no—
342C22×C4—182no—
343C22×C4—182no—
344C22×C4—182no—
345C22×C4—182no—
346C22×C4—182no—
347C22×C4C2×#15711yescharacteristic
348C22×C4C2×#15711yescharacteristic
349C4⋊C4—188no—
350C22⋊C4—22no—
351C22⋊C4—182no—
352C22⋊C4—182no—
353C22⋊C4—1816no—
354C22⋊C4—182no—
355C22×C3—28no—
356C22×C3—22no—
357C22×C3—22no—
358C22×C3—24no—
359C22×C3—24no—
360C22×C3—22no—
361C22×C3—48no—
362C22×C3—44no—
363C22×C3—44no—
364C22×C3—48no—
365C2×D3—62no—
366C2×D3—64no—
367C2×D3—64no—
368C2×D3—62no—
369C2×D3—62no—
370C2×D3—62no—
371C2×D3—62no—
372C2×D3—62no—
373C2×D3—62no—
374C2×D3—62no—
375C2×D3—64no—
376C2×D3—64no—
377C2×D3—64no—
378C2×D3—64no—
379C2×D3—64no—
380C2×D3—64no—
381C2×D3—64no—
382C2×D3—64no—
383C2×D3—68no—
384C2×D3—128no—
385C2×D3—128no—
386C2×D3—128no—
387C2×D3—1216no—
388C2×D3—1216no—
389C2×D3—128no—
390C2×D3—128no—
391C2×D3—124no—
392C2×D3—124no—
393C2×D3—128no—
394C3×C4—44no—
395C3×C4—44no—
396Dic3—124no—
397Dic3—124no—
398C32#143911yescharacteristic, complemented, Sylow
399C23—21no—
400C23—22no—
401C23—22no—
402C23—21no—
403C23—91no—
404C23—91no—
405C23—91no—
406C23—91no—
407C23—92no—
408C23—92no—
409C23—92no—
410C23—92no—
411C23—91no—
412C23—91no—
413C23—92no—
414C23—184no—
415C23—182no—
416C23—182no—
417C23—182no—
418C23—182no—
419C23—184no—
420C23—184no—
421C23—184no—
422C23—181no—
423C23—182no—
424C23—182no—
425C23—181no—
426C23—361no—
427C23—362no—
428C23—361no—
429C23—362no—
430C23—361no—
431C23—361no—
432C23—362no—
433C23—362no—
434C23—362no—
435C23—362no—
436C23C4⋊#15712yescomplemented
437C23C22×#15711yescharacteristic
438C2×C4—22no—
439C2×C4—22no—
440C2×C4—21no—
441C2×C4—92no—
442C2×C4—182no—
443C2×C4—184no—
444C2×C4—182no—
445C2×C4—182no—
446C2×C4—184no—
447C2×C4—184no—
448C2×C4—184no—
449C2×C4—184no—
450C2×C4—184no—
451C2×C4—184no—
452C2×C4—182no—
453C2×C4—181no—
454C2×C4—362no—
455C2×C4—361no—
456C2×C4—361no—
457C2×C4—362no—
458C2×C4—362no—
459C2×C4—362no—
460C2×C4—362no—
461C2×C4—362no—
462C2×C4—362no—
463C2×C4—362no—
464C2×C4—362no—
465C2×C4C22⋊#15712yescomplemented
466C2×C3—22no—
467C2×C3—22no—
468C2×C3—22no—
469C2×C3—24no—
470C2×C3—24no—
471C2×C3—48no—
472D3—62no—
473D3—62no—
474D3—64no—
475D3—64no—
476D3—62no—
477D3—62no—
478D3—128no—
479C22—22no—
480C22—24no—
481C22—41no—
482C22—41no—
483C22—42no—
484C22—91no—
485C22—91no—
486C22—91no—
487C22—91no—
488C22—91no—
489C22—92no—
490C22—91no—
491C22—92no—
492C22—91no—
493C22—91no—
494C22—92no—
495C22—92no—
496C22—92no—
497C22—92no—
498C22—92no—
499C22—92no—
500C22—92no—
501C22—92no—
502C22—182no—
503C22—184no—
504C22—362no—
505C22—364no—
506C22—362no—
507C22—362no—
508C22—364no—
509C22—362no—
510C22—361no—
511C22—361no—
512C22—362no—
513C22C2×C4⋊#15711yescharacteristic
514C22#6428611yescharacteristic
515C22D4×#15712yes—
516C22D4×#15712yes—
517C22C2×C22⋊#15711yescharacteristic
518C4—22no—
519C4—41no—
520C4—182no—
521C4—364no—
522C4—364no—
523C4—361no—
524C3—22no—
525C2—42no—
526C2—91no—
527C2—92no—
528C2—91no—
529C2—91no—
530C2—92no—
531C2—91no—
532C2—362no—
533C2#1059374612yes—
534C2#1059374712yes—
535C2#1059381211yescharacteristic
536C2#1059382611yescharacteristic
537C2#1059382711yescharacteristic
subgroupG/H
center, FrattiniC23C22×#157
commutator, socleC23×C32C22×C4
FittingC2×C32×C22⋊C4C4
prime radicalC24×C3⋊D3C22
cosocleC23×C3⋊D3C23
hypercenterC2×C22⋊C4#157
hypocenterC32#1439
LCSG ▷ C23×C32 ▷ C32
UCSC1 ◁ C23 ◁ C2×C22⋊C4

Complements

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

NH ≅ G/Ntype
2. C22×C4⋊#157C2semidirect
4. C2×#63781C2semidirect
5. C2×#63781C2semidirect
16. C2×C32×C22⋊C4C4semidirect
17. C2×#3757C4semidirect
18. C2×#3752C4semidirect
19. C2×#3752C4semidirect
84. C23×C32C4⋊C4semidirect
108. C22×C3⋊D3C4⋊C4semidirect
136. C2×C32×C4C22⋊C4semidirect
140. C2×C3⋊Dic3C22⋊C4semidirect
241. C2×C22⋊C4#157semidirect
398. C32#1439semidirect
436. C23C4⋊#157semidirect
465. C2×C4C22⋊#157semidirect

Maximal subgroups

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

subgroupindex
C22×C22⋊#1572
C22×C4⋊#1572
C2×C22⋊C4×C3⋊D32
C2×#637812
#14399