Finite Group Databasebeta

← C2×ΓL(1,𝔽8)×#5934 #72276690

Subgroups — C2×ΓL(1,𝔽8)×#5934 #72276690

Order 8064.

1808 conjugacy classes of subgroups; only the 54 normal subgroups are shown.

normal subgroups only

This group has 1808 conjugacy classes of subgroups — too many to draw. Only its 54 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×C7×#5934C311yesmaximal, characteristic, complemented
2C22×ΓL(1,𝔽8)×S4—43nomaximal
3C2×ΓL(1,𝔽8)×#997C211yesmaximal, characteristic, complemented
4C2×ΓL(1,𝔽8)×B(4,𝔽2)—31nomaximal
5ΓL(1,𝔽8)×#5934C212yesmaximal, direct product factor
6ΓL(1,𝔽8)×#997C2211yescharacteristic, complemented
7C2×C7⋊#997—21no—
8C2×C7×#997C2×C311yescharacteristic, complemented
9C2×ΓL(1,𝔽8)×H(5,𝔽2)D311yescharacteristic, complemented
10C7×#5934C2×C312yescomplemented
11C2×ΓL(1,𝔽8)×#120—33no—
12ΓL(1,𝔽8)×B(4,𝔽2)—36no—
13ΓL(1,𝔽8)×B(4,𝔽2)—32no—
14C2×ΓL(1,𝔽8)×#99—33no—
15C2×C3×#5934—71nomaximal, Hall
16C22×A4×ΓL(1,𝔽8)—43no—
17C2×ΓL(1,𝔽8)×S4—46no—
18C2×ΓL(1,𝔽8)×S4—412no—
19C2×ΓL(1,𝔽8)×SL(2,𝔽3)—81no—
20C2×C7×B(4,𝔽2)—31noHall
21C25×ΓL(1,𝔽8)—31no—
22C22×C7×S4—43no—
23C7⋊#997—21no—
24C7×#997C22×C311yescharacteristic, complemented
25ΓL(1,𝔽8)×H(5,𝔽2)—33no—
26ΓL(1,𝔽8)×H(5,𝔽2)—61no—
27ΓL(1,𝔽8)×H(5,𝔽2)C2×D311yescharacteristic, complemented
28C2×#41×ΓL(1,𝔽8)—61no—
29C22×D4×ΓL(1,𝔽8)—33no—
30C22×D4×ΓL(1,𝔽8)—63no—
31ΓL(1,𝔽8)×#120—36no—
32ΓL(1,𝔽8)×#120—36no—
33ΓL(1,𝔽8)×#120—63no—
34ΓL(1,𝔽8)×#120—63no—
35C2×C22⋊C4×ΓL(1,𝔽8)—33no—
36C2×C22⋊C4×ΓL(1,𝔽8)—63no—
37ΓL(1,𝔽8)×#99—36no—
38ΓL(1,𝔽8)×#99—36no—
39C2×C3×#997—71no—
40C3×#5934—72no—
41C22×D3×ΓL(1,𝔽8)—161no—
42C2×A4×ΓL(1,𝔽8)—46no—
43C2×A4×ΓL(1,𝔽8)—43no—
44ΓL(1,𝔽8)×S4—412no—
45ΓL(1,𝔽8)×SL(2,𝔽3)—81no—
46C2×C7×H(5,𝔽2)C3×D311yescharacteristic, complemented
47C2×C7×#120—33no—
48C7×B(4,𝔽2)—36no—
49C7×B(4,𝔽2)—32no—
50C2×C7×#99—33no—
51C2×#5934ΓL(1,𝔽8)11yescharacteristic, direct product factor
52C2×C3×B(4,𝔽2)—211no—
53C22×C22⋊ΓL(1,𝔽8)—83no—
54C22×C7×A4—43no—
55C24×ΓL(1,𝔽8)—31no—
56C24×ΓL(1,𝔽8)—32no—
57C24×ΓL(1,𝔽8)—63no—
58C24×ΓL(1,𝔽8)—63no—
59C24×ΓL(1,𝔽8)—124no—
60C24×ΓL(1,𝔽8)S413yescomplemented
61C2×C7×S4—46no—
62C2×C7×S4—412no—
63C2×C7⋊SL(2,𝔽3)—82no—
64C2×C7×SL(2,𝔽3)—81no—
65#41×ΓL(1,𝔽8)—61no—
66#41×ΓL(1,𝔽8)—63no—
67#41×ΓL(1,𝔽8)—63no—
68#41×ΓL(1,𝔽8)—61no—
69C2×Q8×ΓL(1,𝔽8)—21no—
70C2×D4×ΓL(1,𝔽8)—33no—
71C2×D4×ΓL(1,𝔽8)—33no—
72C2×D4×ΓL(1,𝔽8)—33no—
73C2×D4×ΓL(1,𝔽8)—33no—
74C2×D4×ΓL(1,𝔽8)—63no—
75C2×D4×ΓL(1,𝔽8)—66no—
76C2×D4×ΓL(1,𝔽8)—66no—
77C2×D4×ΓL(1,𝔽8)—66no—
78C2×D4×ΓL(1,𝔽8)—1212no—
79C22×C4×ΓL(1,𝔽8)—33no—
80C22×C4×ΓL(1,𝔽8)—63no—
81C22⋊C4×ΓL(1,𝔽8)—36no—
82C22⋊C4×ΓL(1,𝔽8)—66no—
83C22⋊C4×ΓL(1,𝔽8)—66no—
84C22⋊C4×ΓL(1,𝔽8)—63no—
85C22×C3×S4—283no—
86C3×#997—71no—
87C22×C3×ΓL(1,𝔽8)—161no—
88C2×D3×ΓL(1,𝔽8)—164no—
89C2×D3×ΓL(1,𝔽8)—162no—
90A4×ΓL(1,𝔽8)—43no—
91C25×C7—31no—
92C7×H(5,𝔽2)—33no—
93C7×H(5,𝔽2)—61no—
94C7×H(5,𝔽2)C2×C3×D311yescharacteristic, complemented
95C2×C7×#41—61no—
96C22×C7×D4—33no—
97C22×C7×D4—63no—
98C7×#120—36no—
99C7×#120—36no—
100C7×#120—63no—
101C7×#120—63no—
102C2×C7×C22⋊C4—33no—
103C2×C7×C22⋊C4—63no—
104C7×#99—36no—
105C7×#99—36no—
106C2×C3×H(5,𝔽2)—71no—
107C2×#997—141no—
108C2×#997C2×ΓL(1,𝔽8)11yescharacteristic, complemented
109#5934C2×ΓL(1,𝔽8)12yesdirect product factor
110C2×C3×#120—213no—
111C3×B(4,𝔽2)—212no—
112C3×B(4,𝔽2)—216no—
113C2×C3×#99—213no—
114C22×D3×C7—161no—
115C2×C22⋊ΓL(1,𝔽8)—86no—
116C2×C22⋊ΓL(1,𝔽8)—83no—
117C2×C7×A4—46no—
118C2×C7×A4—43no—
119C23×ΓL(1,𝔽8)—31no—
120C23×ΓL(1,𝔽8)—33no—
121C23×ΓL(1,𝔽8)—33no—
122C23×ΓL(1,𝔽8)—33no—
123C23×ΓL(1,𝔽8)—46no—
124C23×ΓL(1,𝔽8)—66no—
125C23×ΓL(1,𝔽8)—612no—
126C23×ΓL(1,𝔽8)—123no—
127C23×ΓL(1,𝔽8)—122no—
128C23×ΓL(1,𝔽8)—124no—
129C23×ΓL(1,𝔽8)—126no—
130C23×ΓL(1,𝔽8)—1212no—
131C23×ΓL(1,𝔽8)—121no—
132C23×ΓL(1,𝔽8)C2×S413yescomplemented
133C7×S4—412no—
134C7⋊SL(2,𝔽3)—82no—
135C7×SL(2,𝔽3)—81no—
136Q8×ΓL(1,𝔽8)—21no—
137Q8×ΓL(1,𝔽8)—61no—
138D4×ΓL(1,𝔽8)—63no—
139D4×ΓL(1,𝔽8)—63no—
140D4×ΓL(1,𝔽8)—66no—
141D4×ΓL(1,𝔽8)—1212no—
142D4×ΓL(1,𝔽8)—1212no—
143C2×C4×ΓL(1,𝔽8)—33no—
144C2×C4×ΓL(1,𝔽8)—33no—
145C2×C4×ΓL(1,𝔽8)—63no—
146C2×C4×ΓL(1,𝔽8)—61no—
147C2×C4×ΓL(1,𝔽8)—66no—
148C2×C4×ΓL(1,𝔽8)—126no—
149C22×C3×A4—283no—
150C2×C3×S4—286no—
151C2×C3×S4—2812no—
152C2×C3×SL(2,𝔽3)—561no—
153C2×B(4,𝔽2)—31noSylow
154C2×C3×ΓL(1,𝔽8)—162no—
155C2×C3×ΓL(1,𝔽8)—161no—
156D3×ΓL(1,𝔽8)—164no—
157C24×C7—31no—
158C24×C7—32no—
159C24×C7—63no—
160C24×C7—63no—
161C24×C7—124no—
162C24×C7C3×S413yescomplemented
163C7×#41—61no—
164C7×#41—63no—
165C7×#41—63no—
166C7×#41—61no—
167C2×C7×Q8—21no—
168C2×C7×D4—33no—
169C2×C7×D4—33no—
170C2×C7×D4—33no—
171C2×C7×D4—33no—
172C2×C7×D4—63no—
173C2×C7×D4—66no—
174C2×C7×D4—66no—
175C2×C7×D4—66no—
176C2×C7×D4—1212no—
177C22×C4×C7—33no—
178C22×C4×C7—63no—
179C7×C22⋊C4—36no—
180C7×C22⋊C4—66no—
181C7×C22⋊C4—66no—
182C7×C22⋊C4—63no—
183C25×C3—211no—
184C22×S4—43no—
185C3×H(5,𝔽2)—71no—
186C3×H(5,𝔽2)—213no—
187C3×H(5,𝔽2)—421no—
188C2×C3×#41—421no—
189C22×C3×D4—213no—
190C22×C3×D4—423no—
191#997—141no—
192#997C22×ΓL(1,𝔽8)11yescharacteristic, complemented
193C3×#120—216no—
194C3×#120—216no—
195C3×#120—423no—
196C3×#120—423no—
197C2×C3×C22⋊C4—213no—
198C2×C3×C22⋊C4—423no—
199C3×#99—216no—
200C3×#99—216no—
201C22×C3×C7—161no—
202C2×D3×C7—162no—
203C2×D3×C7—164no—
204C22⋊ΓL(1,𝔽8)—83no—
205C7×A4—43no—
206C22×ΓL(1,𝔽8)—33no—
207C22×ΓL(1,𝔽8)—33no—
208C22×ΓL(1,𝔽8)—43no—
209C22×ΓL(1,𝔽8)—66no—
210C22×ΓL(1,𝔽8)—612no—
211C22×ΓL(1,𝔽8)—123no—
212C22×ΓL(1,𝔽8)—121no—
213C22×ΓL(1,𝔽8)—123no—
214C22×ΓL(1,𝔽8)—126no—
215C22×ΓL(1,𝔽8)—1212no—
216C22×ΓL(1,𝔽8)—124no—
217C22×ΓL(1,𝔽8)—122no—
218C22×ΓL(1,𝔽8)—321no—
219C22×ΓL(1,𝔽8)#102011yescharacteristic
220C4×ΓL(1,𝔽8)—61no—
221C4×ΓL(1,𝔽8)—61no—
222C4×ΓL(1,𝔽8)—126no—
223C22×C3×D3—1121no—
224C2×C3×A4—283no—
225C2×C3×A4—286no—
226C3×S4—2812no—
227C3×SL(2,𝔽3)—561no—
228C2×H(5,𝔽2)D3×ΓL(1,𝔽8)11yescharacteristic, complemented
229C2×#120—33no—
230B(4,𝔽2)—36no—
231B(4,𝔽2)—32no—
232C2×#99—33no—
233C3×ΓL(1,𝔽8)—161noHall
234C23×C7—31no—
235C23×C7—33no—
236C23×C7—33no—
237C23×C7—33no—
238C23×C7—46no—
239C23×C7—66no—
240C23×C7—612no—
241C23×C7—123no—
242C23×C7—122no—
243C23×C7—124no—
244C23×C7—126no—
245C23×C7—1212no—
246C23×C7—121no—
247C23×C7C2×C3×S413yescomplemented
248C7×Q8—21no—
249C7×Q8—61no—
250C7×D4—63no—
251C7×D4—63no—
252C7×D4—66no—
253C7×D4—1212no—
254C7×D4—1212no—
255C2×C4×C7—33no—
256C2×C4×C7—33no—
257C2×C4×C7—63no—
258C2×C4×C7—61no—
259C2×C4×C7—66no—
260C2×C4×C7—126no—
261C24×C3—73no—
262C24×C3—211no—
263C24×C3—212no—
264C24×C3—423no—
265C24×C3—423no—
266C24×C3—844no—
267C22×A4—43no—
268C22×A4—563no—
269C2×S4—46no—
270C2×S4—412no—
271C3×#41—421no—
272C3×#41—423no—
273C3×#41—423no—
274C3×#41—421no—
275C2×C3×Q8—141no—
276C2×C3×D4—213no—
277C2×C3×D4—213no—
278C2×C3×D4—213no—
279C2×C3×D4—213no—
280C2×C3×D4—423no—
281C2×C3×D4—426no—
282C2×C3×D4—426no—
283C2×C3×D4—426no—
284C2×C3×D4—8412no—
285C22×C3×C4—213no—
286C22×C3×C4—423no—
287C2×SL(2,𝔽3)—81no—
288C2×SL(2,𝔽3)—562no—
289C3×C22⋊C4—216no—
290C3×C22⋊C4—426no—
291C3×C22⋊C4—426no—
292C3×C22⋊C4—423no—
293C2×C3×C7—161no—
294C2×C3×C7—162no—
295D3×C7—164no—
296C2×ΓL(1,𝔽8)—63no—
297C2×ΓL(1,𝔽8)—63no—
298C2×ΓL(1,𝔽8)—124no—
299C2×ΓL(1,𝔽8)—321no—
300C2×ΓL(1,𝔽8)—322no—
301C2×ΓL(1,𝔽8)#593412yesdirect product factor
302C2×ΓL(1,𝔽8)C2×#102011yescharacteristic
303C22×C32—1121no—
304C2×C3×D3—1122no—
305C2×C3×D3—1124no—
306C3×A4—283no—
307C25—31no—
308H(5,𝔽2)—33no—
309H(5,𝔽2)—61no—
310H(5,𝔽2)C2×D3×ΓL(1,𝔽8)11yescharacteristic, complemented
311C2×#41—61no—
312C22×D4—33no—
313C22×D4—63no—
314#120—36no—
315#120—36no—
316#120—63no—
317#120—63no—
318C2×C22⋊C4—33no—
319C2×C22⋊C4—63no—
320#99—36no—
321#99—36no—
322C22×C7—33no—
323C22×C7—33no—
324C22×C7—43no—
325C22×C7—66no—
326C22×C7—612no—
327C22×C7—123no—
328C22×C7—121no—
329C22×C7—123no—
330C22×C7—126no—
331C22×C7—1212no—
332C22×C7—122no—
333C22×C7—124no—
334C22×C7C3×#102011yescharacteristic
335C4×C7—61no—
336C4×C7—61no—
337C4×C7—126no—
338C23×C3—73no—
339C23×C3—213no—
340C23×C3—211no—
341C23×C3—213no—
342C23×C3—213no—
343C23×C3—286no—
344C23×C3—426no—
345C23×C3—4212no—
346C23×C3—841no—
347C23×C3—843no—
348C23×C3—842no—
349C23×C3—844no—
350C23×C3—846no—
351C23×C3—8412no—
352C22×D3—161no—
353C2×A4—46no—
354C2×A4—43no—
355C2×A4—566no—
356C2×A4—563no—
357S4—412no—
358C3×Q8—141no—
359C3×Q8—421no—
360C3×D4—423no—
361C3×D4—423no—
362C3×D4—426no—
363C3×D4—8412no—
364C3×D4—8412no—
365C2×C3×C4—213no—
366C2×C3×C4—213no—
367C2×C3×C4—421no—
368C2×C3×C4—423no—
369C2×C3×C4—426no—
370C2×C3×C4—846no—
371SL(2,𝔽3)—81no—
372SL(2,𝔽3)—562no—
373C3×C7—161no—
374ΓL(1,𝔽8)—321no—
375ΓL(1,𝔽8)C2×#593411yescharacteristic, direct product factor
376C2×C32—1121no—
377C2×C32—1122no—
378C3×D3—1124no—
379C24—31no—
380C24—32no—
381C24—63no—
382C24—63no—
383C24—124no—
384C24ΓL(1,𝔽8)×S413yescomplemented
385#41—61no—
386#41—63no—
387#41—63no—
388#41—61no—
389C2×Q8—21no—
390C2×D4—33no—
391C2×D4—33no—
392C2×D4—33no—
393C2×D4—33no—
394C2×D4—63no—
395C2×D4—66no—
396C2×D4—66no—
397C2×D4—66no—
398C2×D4—1212no—
399C22×C4—33no—
400C22×C4—63no—
401C22⋊C4—36no—
402C22⋊C4—66no—
403C22⋊C4—66no—
404C22⋊C4—63no—
405C2×C7—63no—
406C2×C7—63no—
407C2×C7—124no—
408C2×C7C3×#593412yescomplemented
409C2×C7C2×C3×#102011yescharacteristic
410C22×C3—71no—
411C22×C3—161no—
412C22×C3—213no—
413C22×C3—213no—
414C22×C3—283no—
415C22×C3—426no—
416C22×C3—4212no—
417C22×C3—842no—
418C22×C3—844no—
419C22×C3—843no—
420C22×C3—841no—
421C22×C3—843no—
422C22×C3—846no—
423C22×C3—8412no—
424C22×C3—2241no—
425C2×D3—162no—
426C2×D3—164no—
427A4—43no—
428A4—563no—
429C3×C4—421no—
430C3×C4—421no—
431C3×C4—846no—
432C32—1121noSylow
433C23—31no—
434C23—33no—
435C23—33no—
436C23—33no—
437C23—46no—
438C23—66no—
439C23—612no—
440C23—123no—
441C23—122no—
442C23—124no—
443C23—126no—
444C23—1212no—
445C23—121no—
446C23C2×ΓL(1,𝔽8)×S413yescomplemented
447Q8—21no—
448Q8—61no—
449D4—63no—
450D4—63no—
451D4—66no—
452D4—1212no—
453D4—1212no—
454C2×C4—33no—
455C2×C4—33no—
456C2×C4—63no—
457C2×C4—61no—
458C2×C4—66no—
459C2×C4—126no—
460C7C2×C3×#593411yescharacteristic, complemented, Sylow
461C2×C3—71no—
462C2×C3—72no—
463C2×C3—161no—
464C2×C3—162no—
465C2×C3—423no—
466C2×C3—423no—
467C2×C3—844no—
468C2×C3—2241no—
469C2×C3—2242no—
470D3—164no—
471C22—33no—
472C22—33no—
473C22—43no—
474C22—66no—
475C22—612no—
476C22—123no—
477C22—121no—
478C22—123no—
479C22—126no—
480C22—1212no—
481C22—122no—
482C22—124no—
483C22ΓL(1,𝔽8)×#102011yescharacteristic
484C4—61no—
485C4—61no—
486C4—126no—
487C3—71no—
488C3—161no—
489C3—2241no—
490C2—63no—
491C2—63no—
492C2—124no—
493C2ΓL(1,𝔽8)×#593412yesdirect product factor
494C2C2×ΓL(1,𝔽8)×#102011yescharacteristic
subgroupG/H
center, hypercenterC22ΓL(1,𝔽8)×#1020
commutator, cosocle, hypocenterC7×#997C22×C3
FrattiniC2C2×ΓL(1,𝔽8)×#1020
FittingC2×C7×H(5,𝔽2)C3×D3
socleC22×C7C3×#1020
LCSG ▷ C7×#997
UCSC1 ◁ C22

Complements

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

NH ≅ G/Ntype
1. C2×C7×#5934C3semidirect
3. C2×ΓL(1,𝔽8)×#997C2semidirect
5. ΓL(1,𝔽8)×#5934C2direct (N × H)
6. ΓL(1,𝔽8)×#997C22semidirect
8. C2×C7×#997C2×C3semidirect
9. C2×ΓL(1,𝔽8)×H(5,𝔽2)D3semidirect
10. C7×#5934C2×C3semidirect
24. C7×#997C22×C3semidirect
27. ΓL(1,𝔽8)×H(5,𝔽2)C2×D3semidirect
46. C2×C7×H(5,𝔽2)C3×D3semidirect
51. C2×#5934ΓL(1,𝔽8)direct (N × H)
60. C24×ΓL(1,𝔽8)S4semidirect
94. C7×H(5,𝔽2)C2×C3×D3semidirect
108. C2×#997C2×ΓL(1,𝔽8)semidirect
109. #5934C2×ΓL(1,𝔽8)direct (N × H)
132. C23×ΓL(1,𝔽8)C2×S4semidirect
162. C24×C7C3×S4semidirect
192. #997C22×ΓL(1,𝔽8)semidirect
228. C2×H(5,𝔽2)D3×ΓL(1,𝔽8)semidirect
247. C23×C7C2×C3×S4semidirect
301. C2×ΓL(1,𝔽8)#5934direct (N × H)
310. H(5,𝔽2)C2×D3×ΓL(1,𝔽8)semidirect
375. ΓL(1,𝔽8)C2×#5934direct (N × H)
384. C24ΓL(1,𝔽8)×S4semidirect
408. C2×C7C3×#5934semidirect
446. C23C2×ΓL(1,𝔽8)×S4semidirect
460. C7C2×C3×#5934semidirect
493. C2ΓL(1,𝔽8)×#5934direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C7×#59343
C22×ΓL(1,𝔽8)×S44
C2×ΓL(1,𝔽8)×#9972
C2×ΓL(1,𝔽8)×B(4,𝔽2)3
ΓL(1,𝔽8)×#59342
C2×C3×#59347