Finite Group Databasebeta

← C2×#89270⋊C3 #72276549

Subgroups — C2×#89270⋊C3 #72276549

Order 2688.

876 conjugacy classes of subgroups; only the 113 normal subgroups are shown.

normal subgroups only

This group has 876 conjugacy classes of subgroups — too many to draw. Only its 113 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×#6500⋊C3C211yesmaximal, characteristic
2C2×ΓL(1,𝔽8)×#100C211yesmaximal, characteristic, complemented
3C2×#6408⋊C3C211yesmaximal, characteristic, complemented
4#89270⋊C3C214yesmaximal, direct product factor
5C2×#89270C311yesmaximal, characteristic, complemented, Hall
6C24×AGL(1,𝔽7)C411yescharacteristic, complemented
7C22×D4×ΓL(1,𝔽8)C2211yescharacteristic
8C22×(C22⋊D7)⋊C3—21no—
9C2×M16×ΓL(1,𝔽8)—21no—
10C2×(C22⋊Dic7)⋊C3—21no—
11C2×(C22⋊Dic7)⋊C3C411yescharacteristic, complemented
12#6500⋊C3—21no—
13#6500⋊C3—21no—
14#6500⋊C3C412yescomplemented
15#6500⋊C3C2212yes—
16C2×(C2×AGL(1,𝔽7))⋊C4—21no—
17ΓL(1,𝔽8)×#100—21no—
18ΓL(1,𝔽8)×#100C2212yescomplemented
19#6408⋊C3—21no—
20#6408⋊C3C2212yescomplemented
21C2×#6500C2×C311yescharacteristic
22C2×C7×#100C2×C311yescharacteristic, complemented
23C2×#6408C2×C311yescharacteristic, complemented
24#89270C2×C314yescomplemented
25C2×C3×AΣL(1,𝔽16)—71nomaximal, Hall
26C24×ΓL(1,𝔽8)C2×C411yescharacteristic
27C24×ΓL(1,𝔽8)D411yescharacteristic
28C23×AGL(1,𝔽7)—21no—
29C23×AGL(1,𝔽7)—21no—
30C23×AGL(1,𝔽7)—44no—
31C23×AGL(1,𝔽7)—41no—
32C23×AGL(1,𝔽7)—41no—
33C23×AGL(1,𝔽7)C2×C412yescomplemented
34C2×D4×ΓL(1,𝔽8)—41no—
35C2×D4×ΓL(1,𝔽8)—41no—
36C2×D4×ΓL(1,𝔽8)C2×C411yescharacteristic
37C2×D4×ΓL(1,𝔽8)D411yescharacteristic
38C2×D4×ΓL(1,𝔽8)D411yescharacteristic
39C2×D4×ΓL(1,𝔽8)C2311yescharacteristic
40C22×C4×ΓL(1,𝔽8)D411yescharacteristic
41C2×(C22⋊D7)⋊C3—22no—
42C2×(C22⋊D7)⋊C3—22no—
43C2×(C22⋊D7)⋊C3—44no—
44C22×ΓL(1,𝔽8)⋊C4—21no—
45C22×ΓL(1,𝔽8)⋊C4—41no—
46M16×ΓL(1,𝔽8)—22no—
47M16×ΓL(1,𝔽8)—41no—
48C2×C8×ΓL(1,𝔽8)—41no—
49(C22⋊Dic7)⋊C3—21no—
50(C22⋊Dic7)⋊C3—22no—
51(C22⋊Dic7)⋊C3—41no—
52(C22⋊Dic7)⋊C3C2×C412yescomplemented
53(C2×AGL(1,𝔽7))⋊C4—22no—
54(C2×AGL(1,𝔽7))⋊C4—22no—
55C24×D7C3×C411yescharacteristic, complemented
56C22×C7×D4C22×C311yescharacteristic
57C22×C22⋊D7—21no—
58C2×C7×M16—21no—
59C2×C22⋊Dic7—21no—
60C2×C22⋊Dic7C3×C411yescharacteristic, complemented
61#6500—21no—
62#6500—21no—
63#6500C3×C412yescomplemented
64#6500C22×C312yes—
65C2×(C2×D7)⋊C4—21no—
66C7×#100—21no—
67C7×#100C22×C312yescomplemented
68#6408—21no—
69#6408C22×C312yescomplemented
70C2×C3×#120—71no—
71C2×C3×#100—71no—
72C2×C3×#99—71no—
73C3×AΣL(1,𝔽16)—74no—
74C23×ΓL(1,𝔽8)—21no—
75C23×ΓL(1,𝔽8)—21no—
76C23×ΓL(1,𝔽8)—21no—
77C23×ΓL(1,𝔽8)—21no—
78C23×ΓL(1,𝔽8)—42no—
79C23×ΓL(1,𝔽8)—42no—
80C23×ΓL(1,𝔽8)C22⋊C411yescharacteristic
81C23×ΓL(1,𝔽8)C22⋊C411yescharacteristic
82C23×ΓL(1,𝔽8)C22⋊C411yescharacteristic
83C23×ΓL(1,𝔽8)C22×C411yescharacteristic
84C23×ΓL(1,𝔽8)C2×D411yescharacteristic
85C22×AGL(1,𝔽7)—24no—
86C22×AGL(1,𝔽7)—22no—
87C22×AGL(1,𝔽7)—44no—
88C22×AGL(1,𝔽7)—44no—
89C22×AGL(1,𝔽7)—42no—
90C22×AGL(1,𝔽7)—42no—
91C22×AGL(1,𝔽7)—48no—
92C22×AGL(1,𝔽7)—44no—
93C22×AGL(1,𝔽7)—41no—
94C22×AGL(1,𝔽7)—41no—
95C22×AGL(1,𝔽7)—42no—
96C22×AGL(1,𝔽7)—44no—
97D4×ΓL(1,𝔽8)—41no—
98D4×ΓL(1,𝔽8)—41no—
99D4×ΓL(1,𝔽8)—41no—
100D4×ΓL(1,𝔽8)—41no—
101C2×C4×ΓL(1,𝔽8)—21no—
102C2×C4×ΓL(1,𝔽8)—21no—
103C2×C4×ΓL(1,𝔽8)C22⋊C411yescharacteristic
104C2×C4×ΓL(1,𝔽8)C2×D411yescharacteristic
105(C22⋊D7)⋊C3—44no—
106(C22⋊D7)⋊C3—44no—
107C2×ΓL(1,𝔽8)⋊C4—22no—
108C2×ΓL(1,𝔽8)⋊C4—42no—
109C2×ΓL(1,𝔽8)⋊C4—42no—
110C2×ΓL(1,𝔽8)⋊C4—82no—
111C8×ΓL(1,𝔽8)—42no—
112C2×AΣL(1,𝔽16)—71noSylow
113C24×C7C2×C3×C411yescharacteristic
114C24×C7C3×D411yescharacteristic
115C23×D7—21no—
116C23×D7—21no—
117C23×D7—44no—
118C23×D7—41no—
119C23×D7—41no—
120C23×D7C2×C3×C412yescomplemented
121C2×C7×D4—41no—
122C2×C7×D4—41no—
123C2×C7×D4C2×C3×C411yescharacteristic
124C2×C7×D4C3×D411yescharacteristic
125C2×C7×D4C3×D411yescharacteristic
126C2×C7×D4C23×C311yescharacteristic
127C22×C4×C7C3×D411yescharacteristic
128C2×C22⋊D7—22no—
129C2×C22⋊D7—22no—
130C2×C22⋊D7—44no—
131C22×Dic7—21no—
132C22×Dic7—41no—
133C7×M16—22no—
134C7×M16—41no—
135C2×C7×C8—41no—
136C22⋊Dic7—21no—
137C22⋊Dic7—22no—
138C22⋊Dic7—41no—
139C22⋊Dic7C2×C3×C412yescomplemented
140(C2×D7)⋊C4—22no—
141(C2×D7)⋊C4—22no—
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)—21no—
159C22×ΓL(1,𝔽8)—21no—
160C22×ΓL(1,𝔽8)—21no—
161C22×ΓL(1,𝔽8)—21no—
162C22×ΓL(1,𝔽8)—41no—
163C22×ΓL(1,𝔽8)—41no—
164C22×ΓL(1,𝔽8)—41no—
165C22×ΓL(1,𝔽8)—41no—
166C22×ΓL(1,𝔽8)—42no—
167C22×ΓL(1,𝔽8)—42no—
168C22×ΓL(1,𝔽8)—42no—
169C22×ΓL(1,𝔽8)—81no—
170C22×ΓL(1,𝔽8)#9911yescharacteristic
171C22×ΓL(1,𝔽8)#9911yescharacteristic
172C22×ΓL(1,𝔽8)C2×C22⋊C411yescharacteristic
173C2×AGL(1,𝔽7)—24no—
174C2×AGL(1,𝔽7)—42no—
175C2×AGL(1,𝔽7)—44no—
176C2×AGL(1,𝔽7)—48no—
177C2×AGL(1,𝔽7)—44no—
178C2×AGL(1,𝔽7)—44no—
179C2×AGL(1,𝔽7)—42no—
180C2×AGL(1,𝔽7)—44no—
181C4×ΓL(1,𝔽8)—21no—
182C4×ΓL(1,𝔽8)—21no—
183ΓL(1,𝔽8)⋊C4—42no—
184ΓL(1,𝔽8)⋊C4—82no—
185C2×#120—71no—
186C2×#100AGL(1,𝔽7)11yescharacteristic, complemented
187C2×#99—71no—
188AΣL(1,𝔽16)—74no—
189C23×C7—21no—
190C23×C7—21no—
191C23×C7—21no—
192C23×C7—21no—
193C23×C7—42no—
194C23×C7—42no—
195C23×C7C3×C22⋊C411yescharacteristic
196C23×C7C3×C22⋊C411yescharacteristic
197C23×C7C3×C22⋊C411yescharacteristic
198C23×C7C22×C3×C411yescharacteristic
199C23×C7C2×C3×D411yescharacteristic
200C22×D7—24no—
201C22×D7—22no—
202C22×D7—44no—
203C22×D7—44no—
204C22×D7—42no—
205C22×D7—42no—
206C22×D7—48no—
207C22×D7—44no—
208C22×D7—41no—
209C22×D7—41no—
210C22×D7—42no—
211C22×D7—44no—
212C7×D4—41no—
213C7×D4—41no—
214C7×D4—41no—
215C7×D4—41no—
216C2×C4×C7—21no—
217C2×C4×C7—21no—
218C2×C4×C7C3×C22⋊C411yescharacteristic
219C2×C4×C7C2×C3×D411yescharacteristic
220C22⋊D7—44no—
221C22⋊D7—44no—
222C2×Dic7—22no—
223C2×Dic7—42no—
224C2×Dic7—42no—
225C2×Dic7—82no—
226C7×C8—42no—
227C24×C3—72no—
228C24×C3—71no—
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)—41no—
260C2×ΓL(1,𝔽8)—41no—
261C2×ΓL(1,𝔽8)—41no—
262C2×ΓL(1,𝔽8)AΣL(1,𝔽16)12yescomplemented
263C2×ΓL(1,𝔽8)C2×#9911yescharacteristic
264AGL(1,𝔽7)—44no—
265C25—71no—
266C22×D4—141no—
267C22×D4C2×AGL(1,𝔽7)11yescharacteristic
268C2×M16—21no—
269#120—72no—
270#120—72no—
271#120—141no—
272#120—141no—
273C2×C22⋊C4—71no—
274C2×C22⋊C4—141no—
275C2×C22⋊C4—141no—
276#100—21no—
277#100C2×AGL(1,𝔽7)12yescomplemented
278#99—72no—
279#99—141no—
280C22×C7—22no—
281C22×C7—21no—
282C22×C7—21no—
283C22×C7—21no—
284C22×C7—21no—
285C22×C7—41no—
286C22×C7—41no—
287C22×C7—41no—
288C22×C7—41no—
289C22×C7—42no—
290C22×C7—42no—
291C22×C7—42no—
292C22×C7—81no—
293C22×C7C3×#9911yescharacteristic
294C22×C7C3×#9911yescharacteristic
295C22×C7C2×C3×C22⋊C411yescharacteristic
296C2×D7—24no—
297C2×D7—42no—
298C2×D7—44no—
299C2×D7—48no—
300C2×D7—44no—
301C2×D7—42no—
302C2×D7—44no—
303C2×D7—44no—
304C4×C7—21no—
305C4×C7—21no—
306Dic7—42no—
307Dic7—82no—
308C23×C3—71no—
309C23×C3—71no—
310C23×C3—71no—
311C23×C3—71no—
312C23×C3—71no—
313C23×C3—141no—
314C23×C3—141no—
315C23×C3—141no—
316C23×C3—141no—
317C23×C3—142no—
318C23×C3—144no—
319C23×C3—282no—
320C23×C3—282no—
321C23×C3—281no—
322C23×C3—281no—
323C23×C3—282no—
324C23×C3—284no—
325C23×C3—282no—
326C23×C3—282no—
327C23×C3—284no—
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—72no—
348C24—141no—
349C24—141no—
350C24—281no—
351C24—281no—
352C24—284no—
353C24C4×AGL(1,𝔽7)11yescharacteristic
354C24C4⋊AGL(1,𝔽7)11yescharacteristic
355C2×D4—41no—
356C2×D4—41no—
357C2×D4—142no—
358C2×D4—142no—
359C2×D4—284no—
360C2×D4C4×AGL(1,𝔽7)11yescharacteristic
361C2×D4C4⋊AGL(1,𝔽7)11yescharacteristic
362C2×D4(C22⋊D7)⋊C311yescharacteristic
363C2×D4C22×AGL(1,𝔽7)11yescharacteristic
364C22×C4—141no—
365C22×C4—281no—
366C22×C4(C22⋊D7)⋊C311yescharacteristic
367M16—22no—
368M16—41no—
369C2×C8—41no—
370C22⋊C4—72no—
371C22⋊C4—141no—
372C22⋊C4—142no—
373C22⋊C4—142no—
374C22⋊C4—142no—
375C22⋊C4—281no—
376C2×C7—21no—
377C2×C7—21no—
378C2×C7—41no—
379C2×C7—41no—
380C2×C7—41no—
381C2×C7—41no—
382C2×C7C3×AΣL(1,𝔽16)12yescomplemented
383C2×C7C2×C3×#9911yescharacteristic
384D7—44no—
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—281no—
397C22×C3—281no—
398C22×C3—281no—
399C22×C3—281no—
400C22×C3—282no—
401C22×C3—284no—
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—21no—
414C23—21no—
415C23—21no—
416C23—21no—
417C23—42no—
418C23—42no—
419C23—144no—
420C23—142no—
421C23—282no—
422C23—282no—
423C23—282no—
424C23—284no—
425C23—284no—
426C23—284no—
427C23—288no—
428C23—284no—
429C23—281no—
430C23—281no—
431C23(C2×AGL(1,𝔽7))⋊C411yescharacteristic
432C23(C2×AGL(1,𝔽7))⋊C411yescharacteristic
433C23(C2×AGL(1,𝔽7))⋊C411yescharacteristic
434C23C2×C4×AGL(1,𝔽7)11yescharacteristic
435C23C2×C4⋊AGL(1,𝔽7)11yescharacteristic
436D4—41no—
437D4—41no—
438D4—41no—
439D4—41no—
440D4—284no—
441D4—284no—
442C2×C4—21no—
443C2×C4—21no—
444C2×C4—142no—
445C2×C4—282no—
446C2×C4—282no—
447C2×C4—562no—
448C2×C4(C2×AGL(1,𝔽7))⋊C411yescharacteristic
449C2×C4C2×(C22⋊D7)⋊C311yescharacteristic
450C8—42no—
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—281no—
460C2×C3—284no—
461C22—22no—
462C22—21no—
463C22—21no—
464C22—21no—
465C22—21no—
466C22—41no—
467C22—41no—
468C22—41no—
469C22—41no—
470C22—42no—
471C22—42no—
472C22—42no—
473C22—81no—
474C22—144no—
475C22—282no—
476C22—284no—
477C22—284no—
478C22—284no—
479C22—288no—
480C22—284no—
481C22—282no—
482C22#6380⋊C311yescharacteristic
483C22#6380⋊C311yescharacteristic
484C22C2×(C2×AGL(1,𝔽7))⋊C411yescharacteristic
485C4—21no—
486C4—21no—
487C4—282no—
488C4—562no—
489C3—71noSylow
490C2—21no—
491C2—21no—
492C2—41no—
493C2—41no—
494C2—41no—
495C2—41no—
496C2—284no—
497C2#89270⋊C312yesdirect product factor
498C2C2×#6380⋊C311yescharacteristic
subgroupG/H
centerC22#6380⋊C3
commutatorC23×C7C22×C3×C4
FrattiniC2×D4C22×AGL(1,𝔽7)
FittingC2×C7×#100C2×C3
socleC22×C7C3×#99
prime radicalC2×#6500⋊C3C2
cosocleC2×C7×D4C23×C3
hypercenterC2×#100AGL(1,𝔽7)
hypocenterC7C2×C3×AΣL(1,𝔽16)
LCSG ▷ C23×C7 ▷ C22×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C22 ◁ C23 ◁ C22×D4 ◁ C2×#100

Complements

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

NH ≅ G/Ntype
2. C2×ΓL(1,𝔽8)×#100C2semidirect
3. C2×#6408⋊C3C2semidirect
4. #89270⋊C3C2direct (N × H)
5. C2×#89270C3semidirect
6. C24×AGL(1,𝔽7)C4semidirect
11. C2×(C22⋊Dic7)⋊C3C4semidirect
14. #6500⋊C3C4semidirect
18. ΓL(1,𝔽8)×#100C22semidirect
20. #6408⋊C3C22semidirect
22. C2×C7×#100C2×C3semidirect
23. C2×#6408C2×C3semidirect
24. #89270C2×C3semidirect
33. C23×AGL(1,𝔽7)C2×C4semidirect
52. (C22⋊Dic7)⋊C3C2×C4semidirect
55. C24×D7C3×C4semidirect
60. C2×C22⋊Dic7C3×C4semidirect
63. #6500C3×C4semidirect
67. C7×#100C22×C3semidirect
69. #6408C22×C3semidirect
120. C23×D7C2×C3×C4semidirect
139. C22⋊Dic7C2×C3×C4semidirect
186. C2×#100AGL(1,𝔽7)semidirect
262. C2×ΓL(1,𝔽8)AΣL(1,𝔽16)semidirect
277. #100C2×AGL(1,𝔽7)semidirect
346. ΓL(1,𝔽8)C2×AΣL(1,𝔽16)semidirect
382. C2×C7C3×AΣL(1,𝔽16)semidirect
451. C7C2×C3×AΣL(1,𝔽16)semidirect
497. C2#89270⋊C3direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#6500⋊C32
C2×ΓL(1,𝔽8)×#1002
C2×#6408⋊C32
#89270⋊C32
C2×#892703
C2×C3×AΣL(1,𝔽16)7