Finite Group Databasebeta

← C2×#85563 #11715654

Subgroups — C2×#85563 #11715654

Order 768.

1029 conjugacy classes of subgroups; only the 39 normal subgroups are shown.

normal subgroups only

This group has 1029 conjugacy classes of subgroups — too many to draw. Only its 39 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×#5948C211yesmaximal, characteristic
2#85563C212yesmaximal, direct product factor
3C2×#1789—31nomaximal, Sylow
4#5948C411yescharacteristic, complemented
5#5948C2211yescharacteristic
6C2×C22⋊SL(2,𝔽3)C411yescharacteristic, complemented
7C2×GU(2,𝔽3)—42nomaximal
8C2×A4⋊C8—41nomaximal
9C23×#41D311yescharacteristic
10C22×#104—32no—
11C22×#104—32no—
12C2×#407—31no—
13C2×C4×C22⋊C4—31no—
14#1789—32no—
15#1789—34no—
16#1789—32no—
17C22⋊SL(2,𝔽3)C2×C411yescharacteristic, complemented
18C2×#41⋊C3—42no—
19C2×#41⋊C3—81no—
20C2×C4×A4—41no—
21GU(2,𝔽3)—44no—
22A4⋊C8—42no—
23C22×#41—31no—
24C22×#41—31no—
25C22×#41—32no—
26C22×#41—32no—
27C22×#41—32no—
28C22×#41—32no—
29C22×#41—61no—
30C22×#41—61no—
31C22×#41—61no—
32C22×#41—61no—
33C22×#41Dic311yescharacteristic
34C22×#41C2×D311yescharacteristic
35C23×Q8Dic311yescharacteristic
36C23×D4—31no—
37C24×C4—31no—
38C22×M16—31no—
39C22×M16—31no—
40C22×C22⋊C4—61no—
41C22×C42—32no—
42C2×#104—34no—
43C2×#104—34no—
44C2×#104—34no—
45C2×#104—34no—
46C2×#104—62no—
47C2×#104—62no—
48C2×#104—62no—
49C2×#104—62no—
50C2×#104—62no—
51C2×#104—62no—
52C2×#104—62no—
53C2×#104—62no—
54#407—32no—
55#407—32no—
56#407—61no—
57#407—61no—
58C2×C22⋊C8—61no—
59C2×C22⋊C8—61no—
60C4×C22⋊C4—32no—
61C4×C22⋊C4—32no—
62C4×C22⋊C4—61no—
63C4×C22⋊C4—61no—
64C2×#95—61no—
65C22×A4—41no—
66#41⋊C3—42no—
67#41⋊C3—42no—
68#41⋊C3—81no—
69#41⋊C3—81no—
70C2×SL(2,𝔽3)—42no—
71C2×SL(2,𝔽3)—81no—
72C4×A4—41no—
73C4×A4—41no—
74C2×C3⋊C8—161no—
75C25—31no—
76C2×#41—21no—
77C2×#41—32no—
78C2×#41—32no—
79C2×#41—32no—
80C2×#41—32no—
81C2×#41—32no—
82C2×#41—32no—
83C2×#41—32no—
84C2×#41—32no—
85C2×#41—32no—
86C2×#41—32no—
87C2×#41—32no—
88C2×#41—32no—
89C2×#41—32no—
90C2×#41—32no—
91C2×#41—32no—
92C2×#41—61no—
93C2×#41—62no—
94C2×#41—61no—
95C2×#41—61no—
96C2×#41—61no—
97C2×#41—61no—
98C2×#41—62no—
99C2×#41—62no—
100C2×#41—62no—
101C2×#41—62no—
102C2×#41—62no—
103C2×#41—62no—
104C2×#41—62no—
105C2×#41—62no—
106C2×#41—62no—
107C2×#41—62no—
108C2×#41—61no—
109C2×#41—61no—
110C2×#41—61no—
111C2×#41—61no—
112C2×#41—61no—
113C2×#41—62no—
114C2×#41—62no—
115C2×#41—61no—
116C2×#41—62no—
117C2×#41—62no—
118C2×#41—62no—
119C2×#41—61no—
120C2×#41—61no—
121C2×#41—61no—
122C2×#41—61no—
123C2×#41—62no—
124C2×#41—62no—
125C2×#41—62no—
126C2×#41—62no—
127C2×#41—62no—
128C2×#41—62no—
129C2×#41S412yes—
130C22×Q8—31no—
131C22×Q8—32no—
132C22×Q8—32no—
133C22×Q8—61no—
134C22×Q8—61no—
135C22×Q8C2×Dic311yescharacteristic
136C22×D4—32no—
137C22×D4—31no—
138C22×D4—31no—
139C22×D4—32no—
140C22×D4—61no—
141C22×D4—61no—
142C22×D4—61no—
143C22×D4—62no—
144C22×D4—62no—
145C22×D4—62no—
146C22×D4—62no—
147C23×C4—31no—
148C23×C4—31no—
149C23×C4—31no—
150C23×C4—31no—
151C23×C4—31no—
152C23×C4—31no—
153C23×C4—31no—
154C23×C4—31no—
155C23×C4—31no—
156C23×C4—31no—
157C23×C4—31no—
158C23×C4—31no—
159C23×C4—31no—
160C23×C4—62no—
161C23×C4—62no—
162C23×C4—62no—
163C23×C4—62no—
164C23×C4—62no—
165C23×C4S411yescharacteristic
166C2×M16—32no—
167C2×M16—32no—
168C2×M16—61no—
169C2×M16—61no—
170C2×M16—62no—
171C2×M16—62no—
172C2×M16—62no—
173C2×M16—62no—
174C22×C8—61no—
175C22×C8—61no—
176C2×C22⋊C4—62no—
177C2×C22⋊C4—62no—
178C2×C22⋊C4—64no—
179C2×C22⋊C4—64no—
180C2×C42—34no—
181C2×C42—62no—
182C2×C42—62no—
183C2×C42—62no—
184C2×C42—62no—
185C2×C42—62no—
186#104—64no—
187#104—64no—
188#104—64no—
189#104—64no—
190#104—64no—
191#104—64no—
192#104—64no—
193#104—64no—
194C22⋊C8—62no—
195C22⋊C8—62no—
196C22⋊C8—62no—
197C22⋊C8—62no—
198#95—62no—
199#95—62no—
200C2×A4—42no—
201C2×A4—41no—
202C2×C3×C4—161no—
203SL(2,𝔽3)—42no—
204SL(2,𝔽3)—81no—
205C3⋊C8—162no—
206C24—31no—
207C24—31no—
208C24—31no—
209C24—31no—
210C24—31no—
211C24—31no—
212C24—62no—
213C24—62no—
214C24—62no—
215C24—62no—
216C24—122no—
217C24SL(2,ℤ/4)11yescharacteristic
218#41—21no—
219#41—21no—
220#41—32no—
221#41—32no—
222#41—32no—
223#41—32no—
224#41—32no—
225#41—32no—
226#41—32no—
227#41—32no—
228#41—32no—
229#41—32no—
230#41—32no—
231#41—32no—
232#41—32no—
233#41—32no—
234#41—61no—
235#41—62no—
236#41—61no—
237#41—61no—
238#41—62no—
239#41—61no—
240#41—62no—
241#41—62no—
242#41—62no—
243#41—62no—
244#41—62no—
245#41—62no—
246#41—62no—
247#41—62no—
248#41—62no—
249#41—62no—
250#41—62no—
251#41—62no—
252#41—62no—
253#41—62no—
254#41—61no—
255#41—61no—
256#41—62no—
257#41—62no—
258#41—61no—
259#41—61no—
260#41—62no—
261#41—62no—
262#41—62no—
263#41—62no—
264#41—61no—
265#41—61no—
266#41—62no—
267#41—62no—
268#41—62no—
269#41—62no—
270#41—62no—
271#41—61no—
272#41—62no—
273#41—61no—
274#41—61no—
275#41—61no—
276#41SL(2,ℤ/4)12yes—
277#41C2×S412yes—
278C2×Q8—21no—
279C2×Q8—32no—
280C2×Q8—32no—
281C2×Q8—32no—
282C2×Q8—32no—
283C2×Q8—32no—
284C2×Q8—61no—
285C2×Q8—61no—
286C2×Q8—62no—
287C2×Q8—62no—
288C2×Q8—62no—
289C2×Q8—61no—
290C2×Q8—61no—
291C2×Q8—61no—
292C2×Q8—62no—
293C2×Q8SL(2,ℤ/4)12yes—
294C2×D4—32no—
295C2×D4—32no—
296C2×D4—32no—
297C2×D4—32no—
298C2×D4—32no—
299C2×D4—32no—
300C2×D4—62no—
301C2×D4—62no—
302C2×D4—62no—
303C2×D4—61no—
304C2×D4—61no—
305C2×D4—62no—
306C2×D4—62no—
307C2×D4—62no—
308C2×D4—62no—
309C2×D4—62no—
310C2×D4—62no—
311C2×D4—62no—
312C2×D4—62no—
313C2×D4—62no—
314C2×D4—61no—
315C2×D4—62no—
316C2×D4—62no—
317C2×D4—62no—
318C2×D4—62no—
319C2×D4—62no—
320C2×D4—62no—
321C2×D4—62no—
322C2×D4—61no—
323C2×D4—62no—
324C2×D4—62no—
325C2×D4—61no—
326C2×D4—61no—
327C2×D4—62no—
328C22×C4—31no—
329C22×C4—31no—
330C22×C4—32no—
331C22×C4—32no—
332C22×C4—32no—
333C22×C4—31no—
334C22×C4—31no—
335C22×C4—31no—
336C22×C4—32no—
337C22×C4—31no—
338C22×C4—31no—
339C22×C4—31no—
340C22×C4—31no—
341C22×C4—31no—
342C22×C4—31no—
343C22×C4—31no—
344C22×C4—31no—
345C22×C4—31no—
346C22×C4—31no—
347C22×C4—31no—
348C22×C4—31no—
349C22×C4—31no—
350C22×C4—31no—
351C22×C4—31no—
352C22×C4—31no—
353C22×C4—31no—
354C22×C4—31no—
355C22×C4—31no—
356C22×C4—31no—
357C22×C4—31no—
358C22×C4—62no—
359C22×C4—62no—
360C22×C4—62no—
361C22×C4—62no—
362C22×C4—62no—
363C22×C4—62no—
364C22×C4—62no—
365C22×C4—62no—
366C22×C4—62no—
367C22×C4—62no—
368C22×C4—62no—
369C22×C4—62no—
370C22×C4—62no—
371C22×C4—62no—
372C22×C4—62no—
373C22×C4—62no—
374C22×C4—62no—
375C22×C4—62no—
376C22×C4—62no—
377C22×C4—62no—
378C22×C4—62no—
379C22×C4—61no—
380C22×C4—61no—
381C22×C4—61no—
382C22×C4—62no—
383C22×C4—61no—
384C22×C4—64no—
385C22×C4—68no—
386C22×C4—122no—
387C22×C4—122no—
388C22×C4—124no—
389C22×C4SL(2,ℤ/4)11yescharacteristic
390C22×C4C2×S411yescharacteristic
391M16—64no—
392M16—64no—
393M16—122no—
394M16—122no—
395C2×C8—62no—
396C2×C8—62no—
397C2×C8—121no—
398C2×C8—121no—
399C2×C8—121no—
400C2×C8—121no—
401C22⋊C4—68no—
402C22⋊C4—64no—
403C22⋊C4—64no—
404C42—64no—
405C42—64no—
406C42—122no—
407C42—122no—
408C22×C3—161no—
409A4—41no—
410C3×C4—161no—
411C3×C4—161no—
412C23—31no—
413C23—31no—
414C23—32no—
415C23—31no—
416C23—31no—
417C23—32no—
418C23—32no—
419C23—31no—
420C23—31no—
421C23—62no—
422C23—62no—
423C23—62no—
424C23—62no—
425C23—62no—
426C23—62no—
427C23—62no—
428C23—61no—
429C23—61no—
430C23—62no—
431C23—61no—
432C23—62no—
433C23—62no—
434C23—61no—
435C23—61no—
436C23—61no—
437C23—121no—
438C23—121no—
439C23—124no—
440C23—124no—
441C23—124no—
442C23—121no—
443C23—121no—
444C23—124no—
445C23GU(2,𝔽3)12yescomplemented
446C23C2×SL(2,ℤ/4)11yescharacteristic
447Q8—21no—
448Q8—32no—
449Q8—32no—
450Q8—32no—
451Q8—61no—
452Q8—62no—
453Q8—61no—
454Q8—62no—
455Q8—61no—
456Q8C2×SL(2,ℤ/4)12yes—
457D4—32no—
458D4—32no—
459D4—32no—
460D4—32no—
461D4—62no—
462D4—62no—
463D4—62no—
464D4—62no—
465D4—62no—
466D4—62no—
467D4—61no—
468D4—61no—
469D4—62no—
470D4—62no—
471D4—62no—
472D4—62no—
473D4—62no—
474D4—61no—
475D4—61no—
476D4—62no—
477C2×C4—32no—
478C2×C4—32no—
479C2×C4—32no—
480C2×C4—31no—
481C2×C4—31no—
482C2×C4—32no—
483C2×C4—32no—
484C2×C4—31no—
485C2×C4—32no—
486C2×C4—32no—
487C2×C4—31no—
488C2×C4—31no—
489C2×C4—32no—
490C2×C4—32no—
491C2×C4—31no—
492C2×C4—31no—
493C2×C4—32no—
494C2×C4—31no—
495C2×C4—31no—
496C2×C4—31no—
497C2×C4—31no—
498C2×C4—61no—
499C2×C4—62no—
500C2×C4—62no—
501C2×C4—62no—
502C2×C4—62no—
503C2×C4—62no—
504C2×C4—62no—
505C2×C4—62no—
506C2×C4—62no—
507C2×C4—62no—
508C2×C4—62no—
509C2×C4—62no—
510C2×C4—62no—
511C2×C4—61no—
512C2×C4—61no—
513C2×C4—61no—
514C2×C4—61no—
515C2×C4—61no—
516C2×C4—61no—
517C2×C4—61no—
518C2×C4—61no—
519C2×C4—61no—
520C2×C4—61no—
521C2×C4—68no—
522C2×C4—124no—
523C2×C4—124no—
524C2×C4—128no—
525C2×C4—128no—
526C2×C4#102011yescharacteristic
527C8—122no—
528C8—122no—
529C2×C3—162no—
530C2×C3—161no—
531C22—32no—
532C22—31no—
533C22—32no—
534C22—32no—
535C22—31no—
536C22—31no—
537C22—32no—
538C22—32no—
539C22—61no—
540C22—61no—
541C22—61no—
542C22—64no—
543C22—61no—
544C22—61no—
545C22—61no—
546C22—61no—
547C22—61no—
548C22—61no—
549C22—64no—
550C22—61no—
551C22—61no—
552C22—122no—
553C22—122no—
554C22—122no—
555C22—122no—
556C22—124no—
557C22—124no—
558C22—122no—
559C22—122no—
560C22C2×GU(2,𝔽3)11yescharacteristic, complemented
561C22#593611yescharacteristic
562C4—31no—
563C4—31no—
564C4—32no—
565C4—32no—
566C4—61no—
567C4—61no—
568C4—128no—
569C4#593611yescharacteristic
570C4C2×#102011yescharacteristic
571C3—161noSylow
572C2—31no—
573C2—31no—
574C2—32no—
575C2—62no—
576C2—62no—
577C2—121no—
578C2—121no—
579C2#8556312yesdirect product factor
580C2C2×#593611yescharacteristic
subgroupG/H
center, hypercenterC2×C4#1020
commutator, hypocenterC22⋊SL(2,𝔽3)C2×C4
FrattiniC4C2×#1020
FittingC23×#41D3
socleC24SL(2,ℤ/4)
prime radicalC2×#5948C2
cosocle#5948C22
LCSG ▷ C22⋊SL(2,𝔽3)
UCSC1 ◁ C2×C4

Complements

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

NH ≅ G/Ntype
2. #85563C2direct (N × H)
4. #5948C4semidirect
6. C2×C22⋊SL(2,𝔽3)C4semidirect
17. C22⋊SL(2,𝔽3)C2×C4semidirect
445. C23GU(2,𝔽3)semidirect
560. C22C2×GU(2,𝔽3)semidirect
579. C2#85563direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#59482
#855632
C2×#17893
C2×GU(2,𝔽3)4
C2×A4⋊C84