Finite Group Databasebeta

← C2×#89331⋊C3 #72276563

Subgroups — C2×#89331⋊C3 #72276563

Order 2688.

876 conjugacy classes of subgroups; only the 121 normal subgroups are shown.

normal subgroups only

This group has 876 conjugacy classes of subgroups — too many to draw. Only its 121 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×ΓL(1,𝔽8)×#120C211yesmaximal, characteristic
2C2×#6407⋊C3C211yesmaximal, characteristic, complemented
3C2×#6408⋊C3C211yesmaximal, characteristic, complemented
4#89331⋊C3C214yesmaximal, direct product factor
5C2×#89331C311yesmaximal, characteristic, complemented, Hall
6C25×ΓL(1,𝔽8)C411yescharacteristic, complemented
7C22×D4×ΓL(1,𝔽8)—21no—
8C22×D4×ΓL(1,𝔽8)C2211yescharacteristic
9ΓL(1,𝔽8)×#120—21no—
10ΓL(1,𝔽8)×#120—21no—
11ΓL(1,𝔽8)×#120C412yescomplemented
12ΓL(1,𝔽8)×#120C2212yes—
13C2×C22⋊C4×ΓL(1,𝔽8)—21no—
14C2×C22⋊C4×ΓL(1,𝔽8)C411yescharacteristic, complemented
15C2×(C22⋊Dic7)⋊C3—21no—
16C2×#1136⋊C3—21no—
17#6408⋊C3—21no—
18#6408⋊C3C2212yescomplemented
19#6407⋊C3—21no—
20#6407⋊C3C2212yescomplemented
21C2×C7×#120C2×C311yescharacteristic
22C2×#6408C2×C311yescharacteristic, complemented
23C2×#6407C2×C311yescharacteristic, complemented
24#89331C2×C314yescomplemented
25C2×C3×AΣL(1,𝔽16)—71nomaximal, Hall
26C24×ΓL(1,𝔽8)—21no—
27C24×ΓL(1,𝔽8)—21no—
28C24×ΓL(1,𝔽8)—44no—
29C24×ΓL(1,𝔽8)—41no—
30C24×ΓL(1,𝔽8)—41no—
31C24×ΓL(1,𝔽8)C2×C411yescharacteristic
32C24×ΓL(1,𝔽8)C2×C412yescomplemented
33C24×ΓL(1,𝔽8)D411yescharacteristic
34C2×D4×ΓL(1,𝔽8)—22no—
35C2×D4×ΓL(1,𝔽8)—22no—
36C2×D4×ΓL(1,𝔽8)—41no—
37C2×D4×ΓL(1,𝔽8)—41no—
38C2×D4×ΓL(1,𝔽8)—44no—
39C2×D4×ΓL(1,𝔽8)C2×C411yescharacteristic
40C2×D4×ΓL(1,𝔽8)D411yescharacteristic
41C2×D4×ΓL(1,𝔽8)D411yescharacteristic
42C2×D4×ΓL(1,𝔽8)C2311yescharacteristic
43C22×C4×ΓL(1,𝔽8)—21no—
44C22×C4×ΓL(1,𝔽8)D411yescharacteristic
45C22×ΓL(1,𝔽8)⋊C4—41no—
46C22⋊C4×ΓL(1,𝔽8)—21no—
47C22⋊C4×ΓL(1,𝔽8)—22no—
48C22⋊C4×ΓL(1,𝔽8)—22no—
49C22⋊C4×ΓL(1,𝔽8)C2×C412yescomplemented
50(C22⋊Dic7)⋊C3—22no—
51(C22⋊Dic7)⋊C3—41no—
52#1136⋊C3—22no—
53#1136⋊C3—41no—
54C2×ΓL(1,𝔽8)⋊C8—41no—
55C25×C7C3×C411yescharacteristic, complemented
56C22×C7×D4—21no—
57C22×C7×D4C22×C311yescharacteristic
58C7×#120—21no—
59C7×#120—21no—
60C7×#120C3×C412yescomplemented
61C7×#120C22×C312yes—
62C2×C7×C22⋊C4—21no—
63C2×C7×C22⋊C4C3×C411yescharacteristic, complemented
64C2×C22⋊Dic7—21no—
65C2×#1136—21no—
66#6408—21no—
67#6408C22×C312yescomplemented
68#6407—21no—
69#6407C22×C312yescomplemented
70C2×C3×#120—71no—
71C2×C3×#100—71no—
72C2×C3×#99—71no—
73C3×AΣL(1,𝔽16)—74no—
74C23×ΓL(1,𝔽8)—24no—
75C23×ΓL(1,𝔽8)—21no—
76C23×ΓL(1,𝔽8)—21no—
77C23×ΓL(1,𝔽8)—21no—
78C23×ΓL(1,𝔽8)—22no—
79C23×ΓL(1,𝔽8)—21no—
80C23×ΓL(1,𝔽8)—42no—
81C23×ΓL(1,𝔽8)—44no—
82C23×ΓL(1,𝔽8)—42no—
83C23×ΓL(1,𝔽8)—42no—
84C23×ΓL(1,𝔽8)—44no—
85C23×ΓL(1,𝔽8)—42no—
86C23×ΓL(1,𝔽8)—42no—
87C23×ΓL(1,𝔽8)—44no—
88C23×ΓL(1,𝔽8)—48no—
89C23×ΓL(1,𝔽8)—44no—
90C23×ΓL(1,𝔽8)—41no—
91C23×ΓL(1,𝔽8)—41no—
92C23×ΓL(1,𝔽8)C22⋊C411yescharacteristic
93C23×ΓL(1,𝔽8)C22⋊C411yescharacteristic
94C23×ΓL(1,𝔽8)C22⋊C411yescharacteristic
95C23×ΓL(1,𝔽8)C22×C411yescharacteristic
96C23×ΓL(1,𝔽8)C2×D411yescharacteristic
97D4×ΓL(1,𝔽8)—41no—
98D4×ΓL(1,𝔽8)—41no—
99D4×ΓL(1,𝔽8)—41no—
100D4×ΓL(1,𝔽8)—41no—
101D4×ΓL(1,𝔽8)—44no—
102D4×ΓL(1,𝔽8)—44no—
103C2×C4×ΓL(1,𝔽8)—21no—
104C2×C4×ΓL(1,𝔽8)—21no—
105C2×C4×ΓL(1,𝔽8)—22no—
106C2×C4×ΓL(1,𝔽8)—42no—
107C2×C4×ΓL(1,𝔽8)C22⋊C411yescharacteristic
108C2×C4×ΓL(1,𝔽8)C2×D411yescharacteristic
109C2×ΓL(1,𝔽8)⋊C4—42no—
110C2×ΓL(1,𝔽8)⋊C4—82no—
111ΓL(1,𝔽8)⋊C8—42no—
112C2×AΣL(1,𝔽16)—71noSylow
113C24×C7—21no—
114C24×C7—21no—
115C24×C7—44no—
116C24×C7—41no—
117C24×C7—41no—
118C24×C7C2×C3×C411yescharacteristic
119C24×C7C2×C3×C412yescomplemented
120C24×C7C3×D411yescharacteristic
121C2×C7×D4—22no—
122C2×C7×D4—22no—
123C2×C7×D4—41no—
124C2×C7×D4—41no—
125C2×C7×D4—44no—
126C2×C7×D4C2×C3×C411yescharacteristic
127C2×C7×D4C3×D411yescharacteristic
128C2×C7×D4C3×D411yescharacteristic
129C2×C7×D4C23×C311yescharacteristic
130C22×C4×C7—21no—
131C22×C4×C7C3×D411yescharacteristic
132C22×Dic7—41no—
133C7×C22⋊C4—21no—
134C7×C22⋊C4—22no—
135C7×C22⋊C4—22no—
136C7×C22⋊C4C2×C3×C412yescomplemented
137C22⋊Dic7—22no—
138C22⋊Dic7—41no—
139#1136—22no—
140#1136—41no—
141C2×C7⋊C8—41no—
142C25×C3—71no—
143C22×C3×D4—71no—
144C22×C3×D4—141no—
145C2×C3×M16—141no—
146C3×#120—72no—
147C3×#120—72no—
148C3×#120—141no—
149C3×#120—141no—
150C2×C3×C22⋊C4—71no—
151C2×C3×C22⋊C4—141no—
152C2×C3×C22⋊C4—141no—
153C3×#100—72no—
154C3×#100—141no—
155C3×#99—72no—
156C3×#99—141no—
157C22×ΓL(1,𝔽8)—22no—
158C22×ΓL(1,𝔽8)—24no—
159C22×ΓL(1,𝔽8)—21no—
160C22×ΓL(1,𝔽8)—21no—
161C22×ΓL(1,𝔽8)—21no—
162C22×ΓL(1,𝔽8)—21no—
163C22×ΓL(1,𝔽8)—41no—
164C22×ΓL(1,𝔽8)—41no—
165C22×ΓL(1,𝔽8)—41no—
166C22×ΓL(1,𝔽8)—41no—
167C22×ΓL(1,𝔽8)—42no—
168C22×ΓL(1,𝔽8)—44no—
169C22×ΓL(1,𝔽8)—42no—
170C22×ΓL(1,𝔽8)—44no—
171C22×ΓL(1,𝔽8)—44no—
172C22×ΓL(1,𝔽8)—48no—
173C22×ΓL(1,𝔽8)—42no—
174C22×ΓL(1,𝔽8)—44no—
175C22×ΓL(1,𝔽8)—42no—
176C22×ΓL(1,𝔽8)—42no—
177C22×ΓL(1,𝔽8)—81no—
178C22×ΓL(1,𝔽8)#9911yescharacteristic
179C22×ΓL(1,𝔽8)#9911yescharacteristic
180C22×ΓL(1,𝔽8)C2×C22⋊C411yescharacteristic
181C4×ΓL(1,𝔽8)—21no—
182C4×ΓL(1,𝔽8)—21no—
183C4×ΓL(1,𝔽8)—42no—
184ΓL(1,𝔽8)⋊C4—82no—
185C2×#120AGL(1,𝔽7)11yescharacteristic
186C2×#100—71no—
187C2×#99—71no—
188AΣL(1,𝔽16)—74no—
189C23×C7—24no—
190C23×C7—21no—
191C23×C7—21no—
192C23×C7—21no—
193C23×C7—22no—
194C23×C7—21no—
195C23×C7—42no—
196C23×C7—44no—
197C23×C7—42no—
198C23×C7—42no—
199C23×C7—44no—
200C23×C7—42no—
201C23×C7—42no—
202C23×C7—44no—
203C23×C7—48no—
204C23×C7—44no—
205C23×C7—41no—
206C23×C7—41no—
207C23×C7C3×C22⋊C411yescharacteristic
208C23×C7C3×C22⋊C411yescharacteristic
209C23×C7C3×C22⋊C411yescharacteristic
210C23×C7C22×C3×C411yescharacteristic
211C23×C7C2×C3×D411yescharacteristic
212C7×D4—41no—
213C7×D4—41no—
214C7×D4—41no—
215C7×D4—41no—
216C7×D4—44no—
217C7×D4—44no—
218C2×C4×C7—21no—
219C2×C4×C7—21no—
220C2×C4×C7—22no—
221C2×C4×C7—42no—
222C2×C4×C7C3×C22⋊C411yescharacteristic
223C2×C4×C7C2×C3×D411yescharacteristic
224C2×Dic7—42no—
225C2×Dic7—82no—
226C7⋊C8—42no—
227C24×C3—71no—
228C24×C3—72no—
229C24×C3—71no—
230C24×C3—141no—
231C24×C3—141no—
232C24×C3—281no—
233C24×C3—281no—
234C24×C3—284no—
235C2×C3×D4—71no—
236C2×C3×D4—71no—
237C2×C3×D4—71no—
238C2×C3×D4—71no—
239C2×C3×D4—142no—
240C2×C3×D4—142no—
241C2×C3×D4—281no—
242C2×C3×D4—281no—
243C2×C3×D4—284no—
244C22×C3×C4—71no—
245C22×C3×C4—141no—
246C22×C3×C4—281no—
247C3×M16—142no—
248C3×M16—281no—
249C2×C3×C8—281no—
250C3×C22⋊C4—72no—
251C3×C22⋊C4—142no—
252C3×C22⋊C4—142no—
253C3×C22⋊C4—141no—
254C3×C22⋊C4—142no—
255C3×C22⋊C4—281no—
256C2×ΓL(1,𝔽8)—21no—
257C2×ΓL(1,𝔽8)—21no—
258C2×ΓL(1,𝔽8)—41no—
259C2×ΓL(1,𝔽8)—44no—
260C2×ΓL(1,𝔽8)—41no—
261C2×ΓL(1,𝔽8)—41no—
262C2×ΓL(1,𝔽8)—41no—
263C2×ΓL(1,𝔽8)AΣL(1,𝔽16)12yescomplemented
264C2×ΓL(1,𝔽8)C2×#9911yescharacteristic
265C25ΓL(1,𝔽8)⋊C411yescharacteristic, complemented
266C22×D4—21no—
267C22×D4C2×AGL(1,𝔽7)11yescharacteristic
268C2×M16—141no—
269#120—21no—
270#120—21no—
271#120ΓL(1,𝔽8)⋊C412yescomplemented
272#120C2×AGL(1,𝔽7)12yes—
273C2×C22⋊C4—21no—
274C2×C22⋊C4—141no—
275C2×C22⋊C4ΓL(1,𝔽8)⋊C411yescharacteristic, complemented
276#100—72no—
277#100—141no—
278#99—72no—
279#99—141no—
280C22×C7—22no—
281C22×C7—24no—
282C22×C7—21no—
283C22×C7—21no—
284C22×C7—21no—
285C22×C7—21no—
286C22×C7—41no—
287C22×C7—41no—
288C22×C7—41no—
289C22×C7—41no—
290C22×C7—42no—
291C22×C7—44no—
292C22×C7—42no—
293C22×C7—44no—
294C22×C7—44no—
295C22×C7—48no—
296C22×C7—42no—
297C22×C7—42no—
298C22×C7—44no—
299C22×C7—42no—
300C22×C7—81no—
301C22×C7C3×#9911yescharacteristic
302C22×C7C3×#9911yescharacteristic
303C22×C7C2×C3×C22⋊C411yescharacteristic
304C4×C7—21no—
305C4×C7—21no—
306C4×C7—42no—
307Dic7—82no—
308C23×C3—71no—
309C23×C3—71no—
310C23×C3—71no—
311C23×C3—71no—
312C23×C3—71no—
313C23×C3—142no—
314C23×C3—141no—
315C23×C3—141no—
316C23×C3—141no—
317C23×C3—141no—
318C23×C3—144no—
319C23×C3—281no—
320C23×C3—281no—
321C23×C3—282no—
322C23×C3—282no—
323C23×C3—284no—
324C23×C3—282no—
325C23×C3—282no—
326C23×C3—284no—
327C23×C3—282no—
328C23×C3—284no—
329C23×C3—284no—
330C23×C3—288no—
331C3×D4—281no—
332C3×D4—281no—
333C3×D4—281no—
334C3×D4—281no—
335C3×D4—284no—
336C3×D4—284no—
337C2×C3×C4—71no—
338C2×C3×C4—71no—
339C2×C3×C4—141no—
340C2×C3×C4—141no—
341C2×C3×C4—142no—
342C2×C3×C4—282no—
343C2×C3×C4—282no—
344C2×C3×C4—562no—
345C3×C8—282no—
346ΓL(1,𝔽8)C2×AΣL(1,𝔽16)11yescharacteristic, complemented, Hall
347C24—21no—
348C24—21no—
349C24—44no—
350C24—41no—
351C24—41no—
352C24C2×ΓL(1,𝔽8)⋊C411yescharacteristic
353C24C2×ΓL(1,𝔽8)⋊C412yescomplemented
354C24(C22⋊D7)⋊C311yescharacteristic
355C2×D4—22no—
356C2×D4—22no—
357C2×D4—41no—
358C2×D4—41no—
359C2×D4—44no—
360C2×D4C2×ΓL(1,𝔽8)⋊C411yescharacteristic
361C2×D4(C22⋊D7)⋊C311yescharacteristic
362C2×D4(C22⋊D7)⋊C311yescharacteristic
363C2×D4C22×AGL(1,𝔽7)11yescharacteristic
364C22×C4—21no—
365C22×C4—281no—
366C22×C4(C22⋊D7)⋊C311yescharacteristic
367M16—142no—
368M16—281no—
369C2×C8—281no—
370C22⋊C4—21no—
371C22⋊C4—22no—
372C22⋊C4—22no—
373C22⋊C4—142no—
374C22⋊C4—281no—
375C22⋊C4C2×ΓL(1,𝔽8)⋊C412yescomplemented
376C2×C7—21no—
377C2×C7—21no—
378C2×C7—41no—
379C2×C7—44no—
380C2×C7—41no—
381C2×C7—41no—
382C2×C7—41no—
383C2×C7C3×AΣL(1,𝔽16)12yescomplemented
384C2×C7C2×C3×#9911yescharacteristic
385C22×C3—71no—
386C22×C3—71no—
387C22×C3—71no—
388C22×C3—142no—
389C22×C3—141no—
390C22×C3—141no—
391C22×C3—141no—
392C22×C3—141no—
393C22×C3—144no—
394C22×C3—282no—
395C22×C3—282no—
396C22×C3—284no—
397C22×C3—282no—
398C22×C3—281no—
399C22×C3—281no—
400C22×C3—281no—
401C22×C3—281no—
402C22×C3—282no—
403C22×C3—282no—
404C22×C3—284no—
405C22×C3—284no—
406C22×C3—284no—
407C22×C3—288no—
408C22×C3—561no—
409C3×C4—141no—
410C3×C4—141no—
411C3×C4—282no—
412C3×C4—562no—
413C23—24no—
414C23—21no—
415C23—21no—
416C23—21no—
417C23—22no—
418C23—21no—
419C23—42no—
420C23—44no—
421C23—42no—
422C23—42no—
423C23—44no—
424C23—42no—
425C23—42no—
426C23—44no—
427C23—48no—
428C23—44no—
429C23—41no—
430C23—41no—
431C23(C22⋊Dic7)⋊C311yescharacteristic
432C23(C22⋊Dic7)⋊C311yescharacteristic
433C23(C22⋊Dic7)⋊C311yescharacteristic
434C23C22×ΓL(1,𝔽8)⋊C411yescharacteristic
435C23C2×(C22⋊D7)⋊C311yescharacteristic
436D4—41no—
437D4—41no—
438D4—41no—
439D4—41no—
440D4—44no—
441D4—44no—
442C2×C4—21no—
443C2×C4—21no—
444C2×C4—22no—
445C2×C4—42no—
446C2×C4—282no—
447C2×C4—562no—
448C2×C4(C22⋊Dic7)⋊C311yescharacteristic
449C2×C4C2×(C22⋊D7)⋊C311yescharacteristic
450C8—282no—
451C7C2×C3×AΣL(1,𝔽16)11yescharacteristic, complemented, Sylow
452C2×C3—72no—
453C2×C3—71no—
454C2×C3—141no—
455C2×C3—141no—
456C2×C3—281no—
457C2×C3—281no—
458C2×C3—281no—
459C2×C3—284no—
460C2×C3—281no—
461C22—22no—
462C22—24no—
463C22—21no—
464C22—21no—
465C22—21no—
466C22—21no—
467C22—41no—
468C22—41no—
469C22—41no—
470C22—41no—
471C22—42no—
472C22—44no—
473C22—42no—
474C22—44no—
475C22—44no—
476C22—48no—
477C22—42no—
478C22—44no—
479C22—42no—
480C22—42no—
481C22—81no—
482C22#6408⋊C311yescharacteristic
483C22#6408⋊C311yescharacteristic
484C22C2×(C22⋊Dic7)⋊C311yescharacteristic
485C4—21no—
486C4—21no—
487C4—42no—
488C4—562no—
489C3—71noSylow
490C2—21no—
491C2—21no—
492C2—41no—
493C2—44no—
494C2—41no—
495C2—41no—
496C2—41no—
497C2#89331⋊C312yesdirect product factor
498C2C2×#6408⋊C311yescharacteristic
subgroupG/H
centerC22#6408⋊C3
commutatorC23×C7C22×C3×C4
FrattiniC2×D4C22×AGL(1,𝔽7)
FittingC2×C7×#120C2×C3
socleC22×C7C3×#99
prime radicalC2×ΓL(1,𝔽8)×#120C2
cosocleC2×C7×D4C23×C3
hypercenterC2×#120AGL(1,𝔽7)
hypocenterC7C2×C3×AΣL(1,𝔽16)
LCSG ▷ C23×C7 ▷ C22×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C22 ◁ C23 ◁ C22×D4 ◁ C2×#120

Complements

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

NH ≅ G/Ntype
2. C2×#6407⋊C3C2semidirect
3. C2×#6408⋊C3C2semidirect
4. #89331⋊C3C2direct (N × H)
5. C2×#89331C3semidirect
6. C25×ΓL(1,𝔽8)C4semidirect
11. ΓL(1,𝔽8)×#120C4semidirect
14. C2×C22⋊C4×ΓL(1,𝔽8)C4semidirect
18. #6408⋊C3C22semidirect
20. #6407⋊C3C22semidirect
22. C2×#6408C2×C3semidirect
23. C2×#6407C2×C3semidirect
24. #89331C2×C3semidirect
32. C24×ΓL(1,𝔽8)C2×C4semidirect
49. C22⋊C4×ΓL(1,𝔽8)C2×C4semidirect
55. C25×C7C3×C4semidirect
60. C7×#120C3×C4semidirect
63. C2×C7×C22⋊C4C3×C4semidirect
67. #6408C22×C3semidirect
69. #6407C22×C3semidirect
119. C24×C7C2×C3×C4semidirect
136. C7×C22⋊C4C2×C3×C4semidirect
263. C2×ΓL(1,𝔽8)AΣL(1,𝔽16)semidirect
265. C25ΓL(1,𝔽8)⋊C4semidirect
271. #120ΓL(1,𝔽8)⋊C4semidirect
275. C2×C22⋊C4ΓL(1,𝔽8)⋊C4semidirect
346. ΓL(1,𝔽8)C2×AΣL(1,𝔽16)semidirect
353. C24C2×ΓL(1,𝔽8)⋊C4semidirect
375. C22⋊C4C2×ΓL(1,𝔽8)⋊C4semidirect
383. C2×C7C3×AΣL(1,𝔽16)semidirect
451. C7C2×C3×AΣL(1,𝔽16)semidirect
497. C2#89331⋊C3direct (N × H)

Maximal subgroups

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

subgroupindex
C2×ΓL(1,𝔽8)×#1202
C2×#6407⋊C32
C2×#6408⋊C32
#89331⋊C32
C2×#893313
C2×C3×AΣL(1,𝔽16)7