Finite Group Databasebeta

← #13060208

Subgroups — #13060208

Order 1344.

580 conjugacy classes of subgroups.

The subgroup lattice has 464 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#4248C211yesmaximal, characteristic, complemented
2C2×#66683C211yesmaximal, characteristic, complemented
3#10621422C211yesmaximal, characteristic, complemented
4C7×D4×Dic3C211yesmaximal, characteristic, complemented
5C3×#6502C211yesmaximal, characteristic, complemented
6#10621137C211yesmaximal, characteristic, complemented
7#10621131C211yesmaximal, characteristic, complemented
8(C22⋊D7)⋊Dic3C211yesmaximal, characteristic, complemented
9#10621114C211yesmaximal, characteristic, complemented
10#10621096C211yesmaximal, characteristic, complemented
11#10621088C211yesmaximal, characteristic, complemented
12#10620992C211yesmaximal, characteristic, complemented
13#10620981C211yesmaximal, characteristic, complemented
14#10620915C211yesmaximal, characteristic, complemented
15#10620886C211yesmaximal, characteristic, complemented
16#90243—31nomaximal, Hall
17C2×C3×C7×D4C2211yescharacteristic
18C2×#4268C2211yescharacteristic
19C2×#4268C2211yescharacteristic
20C22×Dic21C2211yescharacteristic
21C22×C7×Dic3C2211yescharacteristic
22C22×C7×Dic3C2211yescharacteristic
23C2×C3×C22⋊D7C2211yescharacteristic
24C2×C3×C22⋊D7C2211yescharacteristic
25C22×C3×Dic7C2211yescharacteristic
26C2×#4248—24no—
27C2×#4248C2211yescharacteristic
28C2×#4248C2211yescharacteristic
29C2×#4248C2211yescharacteristic
30C2×#4248C2211yescharacteristic
31C2×#4246C2211yescharacteristic
32C2×D21⋊C4C2211yescharacteristic
33#66683—24no—
34C2×Dic3×D7C2211yescharacteristic
35#66636C2211yescharacteristic
36#66631C2211yescharacteristic
37#66629C2211yescharacteristic
38C7×#217C2211yescharacteristic
39C7×#217C2211yescharacteristic
40C7×C4⋊Dic3C2211yescharacteristic
41C4×C7×Dic3C2211yescharacteristic
42C3×C22⋊Dic7C2211yescharacteristic
43C3×(C2×D7)⋊C4C2211yescharacteristic
44C3×Dic7⋊C4C2211yescharacteristic
45Dic3⋊Dic7C2211yescharacteristic
46#66577C2211yescharacteristic
47Dic7⋊Dic3C2211yescharacteristic
48#66575C2211yescharacteristic
49#66575C2211yescharacteristic
50(C2×D7)⋊Dic3C2211yescharacteristic
51(C2×D7)⋊Dic3C2211yescharacteristic
52Dic3×Dic7C2211yescharacteristic
53Dic3×Dic7C2211yescharacteristic
54C2×#1157—31no—
55C22×Dic14—31no—
56C4×C7×D4—31no—
57#6502—31no—
58#6502D311yescharacteristic, complemented
59#6493—31no—
60#6487—31no—
61#6459—31no—
62#6459—31no—
63#6448—31no—
64#6448—31no—
65(C22⋊D7)⋊C4—31no—
66(C22⋊D7)⋊C4—31no—
67#6438—31no—
68C4×Dic14—31no—
69#5598—71nomaximal, Hall
70C23×C3×C7C2311yescharacteristic
71C23×C3×C7C2311yescharacteristic
72C22×D21C2311yescharacteristic
73C22×C3×D7C2311yescharacteristic
74C3×C7×D4—22no—
75C2×C3×C4×C7C2311yescharacteristic
76#4268—22no—
77#4268—22no—
78C2×Dic21—22no—
79C2×Dic21C2311yescharacteristic
80C2×Dic21C2311yescharacteristic
81C2×Dic21C2311yescharacteristic
82C2×C7×Dic3—22no—
83C2×C7×Dic3—22no—
84C2×C7×Dic3C2311yescharacteristic
85C2×C7×Dic3C2311yescharacteristic
86C2×C7×Dic3C2311yescharacteristic
87C2×C7×Dic3C2311yescharacteristic
88C3×C22⋊D7—22no—
89C3×C22⋊D7—22no—
90C2×C3×Dic7—22no—
91C2×C3×Dic7C2311yescharacteristic
92C2×C3×Dic7C2311yescharacteristic
93C2×C3×Dic7C2311yescharacteristic
94#4248—22no—
95#4248—22no—
96#4248—22no—
97#4248D414yescomplemented
98#4246—22no—
99D21⋊C4—22no—
100Dic3×D7—22no—
101C2×C7×D4C2×D311yescharacteristic
102C22×C4×C7—31no—
103C22×C4×C7—31no—
104C2×C22⋊D7—31no—
105C2×C22⋊D7—31no—
106C2×C22⋊D7C2×D311yescharacteristic
107C2×C22⋊D7C2×D311yescharacteristic
108C22×Dic7—31no—
109C22×Dic7C2×D311yescharacteristic
110#1157—64no—
111C2×D28—31no—
112C2×C4×D7—31no—
113C2×C4×D7—31no—
114C2×Dic14—31no—
115C2×Dic14—31no—
116C2×Dic14—31no—
117C2×Dic14—31no—
118C2×Dic14—64no—
119C7×C4⋊C4—31no—
120C7×C22⋊C4—31no—
121C7×C22⋊C4—31no—
122C42×C7—31no—
123C22⋊Dic7—31no—
124C22⋊Dic7C2×D311yescharacteristic
125(C2×D7)⋊C4—31no—
126(C2×D7)⋊C4—31no—
127(C2×D7)⋊C4—31no—
128(C2×D7)⋊C4—31no—
129(C2×D7)⋊C4—31no—
130(C2×D7)⋊C4C2×D311yescharacteristic
131C4⋊Dic7—31no—
132Dic7⋊C4—31no—
133Dic7⋊C4—31no—
134Dic7⋊C4—31no—
135Dic7⋊C4C2×D311yescharacteristic
136C4×Dic7—31no—
137C4×Dic7—31no—
138C2×#237—71no—
139C22×Dic6—71no—
140C3×#121—71no—
141#939—71no—
142#939—71no—
143#936—71no—
144D4×Dic3—71no—
145D4×Dic3D711yescharacteristic, complemented
146C4×#67—71no—
147#897—71no—
148#887—71no—
149#885—71no—
150#885—71no—
151#878—71no—
152#878—71no—
153C22×C3×C7—21no—
154C22×C3×C7—21no—
155C22×C3×C7—21no—
156C22×C3×C7—21no—
157C22×C3×C7—21no—
158C22×C3×C7#4112yescomplemented
159C22×C3×C7C2411yescharacteristic
160C2×D21—21no—
161C2×D21—21no—
162C2×D21—21no—
163C2×C3×D7—21no—
164C2×C3×D7—21no—
165C2×C3×D7—21no—
166C3×C4×C7—21no—
167Dic21—21no—
168Dic21—21no—
169Dic21C2×D412yes—
170C3×Dic7—21no—
171C3×Dic7—21no—
172C3×Dic7C2×D412yes—
173C7×Dic3—21no—
174C7×Dic3—21no—
175C7×Dic3—21no—
176C7×Dic3C2×D412yes—
177#548—211noSylow
178C23×C7C22×D311yescharacteristic
179C23×C7C22×D311yescharacteristic
180C22×D7—31no—
181C22×D7C22×D311yescharacteristic
182C7×D4—22no—
183C2×C4×C7—31no—
184C2×C4×C7—31no—
185C2×C4×C7—31no—
186C2×C4×C7—31no—
187C2×C4×C7—62no—
188C2×C4×C7—62no—
189C2×C4×C7C22×D311yescharacteristic
190C22⋊D7—22no—
191C22⋊D7—22no—
192C22⋊D7—62no—
193C22⋊D7—62no—
194C2×Dic7—22no—
195C2×Dic7—31no—
196C2×Dic7—31no—
197C2×Dic7—31no—
198C2×Dic7—62no—
199C2×Dic7C22×D311yescharacteristic
200C2×Dic7C22×D311yescharacteristic
201C2×Dic7C22×D311yescharacteristic
202D28—62no—
203C4×D7—62no—
204C4×D7—62no—
205Dic14—34no—
206Dic14—62no—
207Dic14—62no—
208Dic14—62no—
209C2×C3×D4—71no—
210C2×C3×D4—71no—
211C2×C3×D4C2×D711yescharacteristic
212C22×C3×C4—71no—
213C2×#67—71no—
214C2×#67—71no—
215C2×#67—71no—
216C22×Dic3—71no—
217C22×Dic3—71no—
218C22×Dic3C2×D711yescharacteristic
219C22×Dic3C2×D711yescharacteristic
220#237—144no—
221C2×C4×D3—71no—
222C2×Dic6—71no—
223C2×Dic6—71no—
224C2×Dic6—71no—
225C2×Dic6—71no—
226C2×Dic6—144no—
227C3×C4⋊C4—71no—
228C3×C22⋊C4—71no—
229C3×C22⋊C4—71no—
230#217—71no—
231#217—71no—
232#217—71no—
233#217C2×D711yescharacteristic
234#217C2×D711yescharacteristic
235(C2×D3)⋊C4—71no—
236(C2×D3)⋊C4—71no—
237(C2×D3)⋊C4—71no—
238C4⋊Dic3—71no—
239C4⋊Dic3C2×D711yescharacteristic
240Dic3⋊C4—71no—
241Dic3⋊C4—71no—
242Dic3⋊C4—71no—
243C4×Dic3—71no—
244C4×Dic3—71no—
245C4×Dic3C2×D711yescharacteristic
246C2×C3×C7—22no—
247C2×C3×C7—41no—
248C2×C3×C7C22×D411yescharacteristic
249C2×C3×C7C2×#4111yescharacteristic
250C2×C3×C72−1+411yescharacteristic
251D21—41no—
252C3×D7—41no—
253C2×#41—211no—
254C22×Q8—211no—
255#124—211no—
256#124—211no—
257#124—211no—
258C22⋊Q8—211no—
259C22⋊Q8—211no—
260C22⋊Q8—211no—
261#121—71no—
262#121—211no—
263#121—211no—
264C4×Q8—211no—
265C4×D4—31no—
266C4×D4—211no—
267C4×D4—211no—
268C22×C7—21no—
269C22×C7—21no—
270C22×C7—21no—
271C22×C7—21no—
272C22×C7—21no—
273C22×C7#23712yescomplemented
274C22×C7C23×D311yescharacteristic
275C2×D7—21no—
276C2×D7—21no—
277C2×D7—21no—
278C2×D7—61no—
279C2×D7—61no—
280C2×D7—61no—
281C4×C7—21no—
282C4×C7—32no—
283C4×C7—61no—
284C4×C7—61no—
285C4×C7—61no—
286Dic7—21no—
287Dic7—21no—
288Dic7—32no—
289Dic7—61no—
290Dic7—61no—
291Dic7D3×D412yes—
292C23×C3—71no—
293C23×C3C22×D711yescharacteristic
294C23×C3C22×D711yescharacteristic
295C22×D3—71no—
296C3×D4—22no—
297C3×D4—142no—
298C3×D4—142no—
299C2×C3×C4—71no—
300C2×C3×C4—71no—
301C2×C3×C4—71no—
302C2×C3×C4—142no—
303C2×C3×C4C22×D711yescharacteristic
304#67—142no—
305#67—142no—
306#67—142no—
307C2×Dic3—22no—
308C2×Dic3—22no—
309C2×Dic3—71no—
310C2×Dic3—71no—
311C2×Dic3—71no—
312C2×Dic3—142no—
313C2×Dic3—142no—
314C2×Dic3C22×D711yescharacteristic
315C2×Dic3C22×D711yescharacteristic
316C2×Dic3C22×D711yescharacteristic
317C2×Dic3C22×D711yescharacteristic
318C4×D3—142no—
319Dic6—74no—
320Dic6—142no—
321Dic6—142no—
322Dic6—142no—
323C3×C7#54811yescharacteristic, complemented, Hall
324#41—424no—
325C2×Q8—211no—
326C2×Q8—211no—
327C2×Q8—211no—
328C2×Q8—211no—
329C2×Q8—424no—
330C2×D4—71no—
331C2×D4—71no—
332C2×D4—211no—
333C2×D4—211no—
334C2×D4—211no—
335C2×D4D3×D711yescharacteristic
336C22×C4—31no—
337C22×C4—31no—
338C22×C4—71no—
339C22×C4—211no—
340C22×C4—211no—
341C22×C4—211no—
342C4⋊C4—31no—
343C4⋊C4—71no—
344C4⋊C4—211no—
345C4⋊C4—211no—
346C4⋊C4—211no—
347C4⋊C4—211no—
348C22⋊C4—31no—
349C22⋊C4—31no—
350C22⋊C4—71no—
351C22⋊C4—71no—
352C22⋊C4—211no—
353C22⋊C4—211no—
354C22⋊C4—211no—
355C22⋊C4—211no—
356C22⋊C4—211no—
357C22⋊C4—211no—
358C42—31no—
359C42—211no—
360C42—211no—
361C2×C7—22no—
362C2×C7—41no—
363C2×C7C2×D3×D411yescharacteristic
364C2×C7C2×#23711yescharacteristic
365C2×C7#101011yescharacteristic
366D7—41no—
367D7—121no—
368C22×C3—21no—
369C22×C3—21no—
370C22×C3—21no—
371C22×C3—21no—
372C22×C3—21no—
373C22×C3—141no—
374C22×C3—141no—
375C22×C3—141no—
376C22×C3#115712yescomplemented
377C22×C3C23×D711yescharacteristic
378C2×D3—141no—
379C2×D3—141no—
380C2×D3—141no—
381C3×C4—21no—
382C3×C4—72no—
383C3×C4—141no—
384C3×C4—141no—
385Dic3—21no—
386Dic3—21no—
387Dic3—21no—
388Dic3—72no—
389Dic3—141no—
390Dic3—141no—
391Dic3D4×D712yes—
392C23—71no—
393C23—211no—
394C23C2×D3×D711yescharacteristic
395C23C2×D3×D711yescharacteristic
396Q8—214no—
397Q8—422no—
398Q8—422no—
399Q8—422no—
400D4—22no—
401D4—142no—
402D4—142no—
403D4—422no—
404D4—422no—
405D4—422no—
406C2×C4—31no—
407C2×C4—31no—
408C2×C4—31no—
409C2×C4—31no—
410C2×C4—62no—
411C2×C4—62no—
412C2×C4—71no—
413C2×C4—71no—
414C2×C4—71no—
415C2×C4—142no—
416C2×C4—211no—
417C2×C4—211no—
418C2×C4—211no—
419C2×C4—422no—
420C2×C4—422no—
421C2×C4—422no—
422C2×C4C2×D3×D711yescharacteristic
423C7#559811yescharacteristic, complemented, Sylow
424C2×C3—22no—
425C2×C3—41no—
426C2×C3—281no—
427C2×C3C2×#115711yescharacteristic
428C2×C3C2×D4×D711yescharacteristic
429C2×C3#655411yescharacteristic
430D3—281no—
431C22—21no—
432C22—21no—
433C22—21no—
434C22—21no—
435C22—21no—
436C22—141no—
437C22—141no—
438C22—141no—
439C22—421no—
440C22—421no—
441C22—421no—
442C22#6668312yescomplemented
443C22C22×D3×D711yescharacteristic
444C4—21no—
445C4—32no—
446C4—61no—
447C4—61no—
448C4—61no—
449C4—72no—
450C4—141no—
451C4—141no—
452C4—212no—
453C4—421no—
454C4—421no—
455C3#9024311yescharacteristic, complemented, Sylow
456C2—22no—
457C2—41no—
458C2—281no—
459C2—841no—
460C2#1062156711yescharacteristic
461C2D3×D4×D711yescharacteristic
462C2C2×#6668311yescharacteristic
subgroupG/H
center, FrattiniC22C22×D3×D7
commutator, socle, cosocleC22×C3×C7C24
FittingC2×C3×C7×D4C22
hypercenterC2×D4D3×D7
hypocenterC3×C7#548
LCSG ▷ C22×C3×C7 ▷ C3×C7
UCSC1 ◁ C22 ◁ C2×D4

Complements

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

NH ≅ G/Ntype
1. C22×#4248C2semidirect
2. C2×#66683C2semidirect
3. #10621422C2semidirect
4. C7×D4×Dic3C2semidirect
5. C3×#6502C2semidirect
6. #10621137C2semidirect
7. #10621131C2semidirect
8. (C22⋊D7)⋊Dic3C2semidirect
9. #10621114C2semidirect
10. #10621096C2semidirect
11. #10621088C2semidirect
12. #10620992C2semidirect
13. #10620981C2semidirect
14. #10620915C2semidirect
15. #10620886C2semidirect
58. #6502D3semidirect
97. #4248D4semidirect
145. D4×Dic3D7semidirect
158. C22×C3×C7#41semidirect
273. C22×C7#237semidirect
323. C3×C7#548semidirect
376. C22×C3#1157semidirect
423. C7#5598semidirect
442. C22#66683semidirect
455. C3#90243semidirect

Maximal subgroups

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

subgroupindex
C22×#42482
C2×#666832
#106214222
C7×D4×Dic32
C3×#65022
#106211372
#106211312
(C22⋊D7)⋊Dic32
#106211142
#106210962
#106210882
#106209922
#106209812
#106209152
#106208862
#902433
#55987