Finite Group Databasebeta

← #13060207

Subgroups — #13060207

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×#66686C211yesmaximal, characteristic, complemented
3#10621422C211yesmaximal, characteristic, complemented
4C7×#939C211yesmaximal, characteristic, complemented
5C3×D4×Dic7C211yesmaximal, characteristic, complemented
6#10621136C211yesmaximal, characteristic, complemented
7#10621128C211yesmaximal, characteristic, complemented
8#67⋊Dic7C211yesmaximal, characteristic, complemented
9#10621118C211yesmaximal, characteristic, complemented
10#10621088C211yesmaximal, characteristic, complemented
11#10621087C211yesmaximal, characteristic, complemented
12#10620998C211yesmaximal, characteristic, complemented
13#10620991C211yesmaximal, characteristic, complemented
14#10620912C211yesmaximal, characteristic, complemented
15#10620885C211yesmaximal, characteristic, complemented
16#90289—31nomaximal, Hall
17C2×C3×C7×D4C2211yescharacteristic
18C2×#4268C2211yescharacteristic
19C2×#4268C2211yescharacteristic
20C22×Dic21C2211yescharacteristic
21C2×C7×#67C2211yescharacteristic
22C2×C7×#67C2211yescharacteristic
23C22×C7×Dic3C2211yescharacteristic
24C22×C3×Dic7C2211yescharacteristic
25C22×C3×Dic7C2211yescharacteristic
26C2×#4248—24no—
27C2×#4248C2211yescharacteristic
28C2×#4248C2211yescharacteristic
29C2×#4248C2211yescharacteristic
30C2×#4248C2211yescharacteristic
31C2×#4247C2211yescharacteristic
32C2×D21⋊C4C2211yescharacteristic
33#66686—24no—
34C2×D3×Dic7C2211yescharacteristic
35#66636C2211yescharacteristic
36#66631C2211yescharacteristic
37#66629C2211yescharacteristic
38C7×#217C2211yescharacteristic
39C7×(C2×D3)⋊C4C2211yescharacteristic
40C7×Dic3⋊C4C2211yescharacteristic
41C3×C22⋊Dic7C2211yescharacteristic
42C3×C22⋊Dic7C2211yescharacteristic
43C3×C4⋊Dic7C2211yescharacteristic
44C3×C4×Dic7C2211yescharacteristic
45Dic3⋊Dic7C2211yescharacteristic
46#66577C2211yescharacteristic
47Dic7⋊Dic3C2211yescharacteristic
48#66575C2211yescharacteristic
49#66575C2211yescharacteristic
50(C2×D3)⋊Dic7C2211yescharacteristic
51(C2×D3)⋊Dic7C2211yescharacteristic
52Dic3×Dic7C2211yescharacteristic
53Dic3×Dic7C2211yescharacteristic
54C2×#1159—31no—
55C22×Dic14—31no—
56C7×#121—31no—
57#6502—31no—
58#6502—31no—
59#6499—31no—
60D4×Dic7—31no—
61D4×Dic7D311yescharacteristic, complemented
62C4×C22⋊D7—31no—
63#6460—31no—
64#6450—31no—
65#6448—31no—
66#6448—31no—
67#6441—31no—
68#6441—31no—
69#5552—71nomaximal, Hall
70C23×C3×C7C2311yescharacteristic
71C23×C3×C7C2311yescharacteristic
72C22×D21C2311yescharacteristic
73C22×D3×C7C2311yescharacteristic
74C3×C7×D4—22no—
75C2×C3×C4×C7C2311yescharacteristic
76#4268—22no—
77#4268—22no—
78C2×Dic21—22no—
79C2×Dic21C2311yescharacteristic
80C2×Dic21C2311yescharacteristic
81C2×Dic21C2311yescharacteristic
82C7×#67—22no—
83C7×#67—22no—
84C2×C7×Dic3—22no—
85C2×C7×Dic3C2311yescharacteristic
86C2×C7×Dic3C2311yescharacteristic
87C2×C7×Dic3C2311yescharacteristic
88C2×C3×Dic7—22no—
89C2×C3×Dic7—22no—
90C2×C3×Dic7C2311yescharacteristic
91C2×C3×Dic7C2311yescharacteristic
92C2×C3×Dic7C2311yescharacteristic
93C2×C3×Dic7C2311yescharacteristic
94#4248—22no—
95#4248—22no—
96#4248—22no—
97#4248D414yescomplemented
98#4247—22no—
99D21⋊C4—22no—
100D3×Dic7—22no—
101C2×C7×D4—31no—
102C2×C7×D4—31no—
103C2×C7×D4C2×D311yescharacteristic
104C22×C4×C7—31no—
105C2×C22⋊D7—31no—
106C2×C22⋊D7—31no—
107C2×C22⋊D7—31no—
108C22×Dic7—31no—
109C22×Dic7—31no—
110C22×Dic7C2×D311yescharacteristic
111C22×Dic7C2×D311yescharacteristic
112#1159—64no—
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—
122C22⋊Dic7—31no—
123C22⋊Dic7—31no—
124C22⋊Dic7—31no—
125C22⋊Dic7C2×D311yescharacteristic
126C22⋊Dic7C2×D311yescharacteristic
127(C2×D7)⋊C4—31no—
128(C2×D7)⋊C4—31no—
129(C2×D7)⋊C4—31no—
130C4⋊Dic7—31no—
131C4⋊Dic7C2×D311yescharacteristic
132Dic7⋊C4—31no—
133Dic7⋊C4—31no—
134Dic7⋊C4—31no—
135C4×Dic7—31no—
136C4×Dic7—31no—
137C4×Dic7C2×D311yescharacteristic
138C2×#235—71no—
139C22×Dic6—71no—
140C3×C4×D4—71no—
141#939—71no—
142#939D711yescharacteristic, complemented
143#930—71no—
144#924—71no—
145#896—71no—
146#896—71no—
147#885—71no—
148#885—71no—
149#67⋊C4—71no—
150#67⋊C4—71no—
151#875—71no—
152C4×Dic6—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×D3×C7—21no—
164C2×D3×C7—21no—
165C2×D3×C7—21no—
166C3×C4×C7—21no—
167Dic21—21no—
168Dic21—21no—
169Dic21C2×D412yes—
170C3×Dic7—21no—
171C3×Dic7—21no—
172C3×Dic7—21no—
173C3×Dic7C2×D412yes—
174C7×Dic3—21no—
175C7×Dic3—21no—
176C7×Dic3C2×D412yes—
177#548—211noSylow
178C23×C7—31no—
179C23×C7C22×D311yescharacteristic
180C23×C7C22×D311yescharacteristic
181C22×D7—31no—
182C7×D4—22no—
183C7×D4—62no—
184C7×D4—62no—
185C2×C4×C7—31no—
186C2×C4×C7—31no—
187C2×C4×C7—31no—
188C2×C4×C7—62no—
189C2×C4×C7C22×D311yescharacteristic
190C22⋊D7—62no—
191C22⋊D7—62no—
192C22⋊D7—62no—
193C2×Dic7—22no—
194C2×Dic7—22no—
195C2×Dic7—31no—
196C2×Dic7—31no—
197C2×Dic7—31no—
198C2×Dic7—62no—
199C2×Dic7—62no—
200C2×Dic7C22×D311yescharacteristic
201C2×Dic7C22×D311yescharacteristic
202C2×Dic7C22×D311yescharacteristic
203C2×Dic7C22×D311yescharacteristic
204C4×D7—62no—
205Dic14—34no—
206Dic14—62no—
207Dic14—62no—
208Dic14—62no—
209C2×C3×D4C2×D711yescharacteristic
210C22×C3×C4—71no—
211C22×C3×C4—71no—
212C2×#67—71no—
213C2×#67—71no—
214C2×#67C2×D711yescharacteristic
215C2×#67C2×D711yescharacteristic
216C22×Dic3—71no—
217C22×Dic3C2×D711yescharacteristic
218#235—144no—
219C2×D12—71no—
220C2×C4×D3—71no—
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—
230C3×C42—71no—
231#217—71no—
232#217C2×D711yescharacteristic
233(C2×D3)⋊C4—71no—
234(C2×D3)⋊C4—71no—
235(C2×D3)⋊C4—71no—
236(C2×D3)⋊C4—71no—
237(C2×D3)⋊C4—71no—
238(C2×D3)⋊C4C2×D711yescharacteristic
239C4⋊Dic3—71no—
240Dic3⋊C4—71no—
241Dic3⋊C4—71no—
242Dic3⋊C4—71no—
243Dic3⋊C4C2×D711yescharacteristic
244C4×Dic3—71no—
245C4×Dic3—71no—
246C2×C3×C7—22no—
247C2×C3×C7—41no—
248C2×C3×C7C22×D411yescharacteristic
249C2×C3×C7C2×#4111yescharacteristic
250C2×C3×C72−1+411yescharacteristic
251D21—41no—
252D3×C7—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—31no—
262#121—211no—
263#121—211no—
264C4×Q8—211no—
265C4×D4—71no—
266C4×D4—211no—
267C4×D4—211no—
268C22×C7—21no—
269C22×C7—21no—
270C22×C7—21no—
271C22×C7—21no—
272C22×C7—21no—
273C22×C7—61no—
274C22×C7—61no—
275C22×C7—61no—
276C22×C7#23512yescomplemented
277C22×C7C23×D311yescharacteristic
278C2×D7—61no—
279C2×D7—61no—
280C2×D7—61no—
281C4×C7—21no—
282C4×C7—32no—
283C4×C7—61no—
284C4×C7—61no—
285Dic7—21no—
286Dic7—21no—
287Dic7—21no—
288Dic7—32no—
289Dic7—61no—
290Dic7—61no—
291Dic7D3×D412yes—
292C23×C3C22×D711yescharacteristic
293C23×C3C22×D711yescharacteristic
294C22×D3—71no—
295C22×D3C22×D711yescharacteristic
296C3×D4—22no—
297C2×C3×C4—71no—
298C2×C3×C4—71no—
299C2×C3×C4—71no—
300C2×C3×C4—71no—
301C2×C3×C4—142no—
302C2×C3×C4—142no—
303C2×C3×C4C22×D711yescharacteristic
304#67—22no—
305#67—22no—
306#67—142no—
307#67—142no—
308C2×Dic3—22no—
309C2×Dic3—71no—
310C2×Dic3—71no—
311C2×Dic3—71no—
312C2×Dic3—142no—
313C2×Dic3C22×D711yescharacteristic
314C2×Dic3C22×D711yescharacteristic
315C2×Dic3C22×D711yescharacteristic
316D12—142no—
317C4×D3—142no—
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—31no—
331C2×D4—31no—
332C2×D4—211no—
333C2×D4—211no—
334C2×D4—211no—
335C2×D4D3×D711yescharacteristic
336C22×C4—31no—
337C22×C4—71no—
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—71no—
359C42—211no—
360C42—211no—
361C2×C7—22no—
362C2×C7—41no—
363C2×C7—121no—
364C2×C7C2×#23511yescharacteristic
365C2×C7C2×D3×D411yescharacteristic
366C2×C7#101011yescharacteristic
367D7—121no—
368C22×C3—21no—
369C22×C3—21no—
370C22×C3—21no—
371C22×C3—21no—
372C22×C3—21no—
373C22×C3#115912yescomplemented
374C22×C3C23×D711yescharacteristic
375C2×D3—21no—
376C2×D3—21no—
377C2×D3—21no—
378C2×D3—141no—
379C2×D3—141no—
380C2×D3—141no—
381C3×C4—21no—
382C3×C4—72no—
383C3×C4—141no—
384C3×C4—141no—
385C3×C4—141no—
386Dic3—21no—
387Dic3—21no—
388Dic3—72no—
389Dic3—141no—
390Dic3—141no—
391Dic3D4×D712yes—
392C23—31no—
393C23—211no—
394C23C2×D3×D711yescharacteristic
395C23C2×D3×D711yescharacteristic
396Q8—214no—
397Q8—422no—
398Q8—422no—
399Q8—422no—
400D4—22no—
401D4—62no—
402D4—62no—
403D4—422no—
404D4—422no—
405D4—422no—
406C2×C4—31no—
407C2×C4—31no—
408C2×C4—31no—
409C2×C4—62no—
410C2×C4—71no—
411C2×C4—71no—
412C2×C4—71no—
413C2×C4—71no—
414C2×C4—142no—
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#555211yescharacteristic, complemented, Sylow
424C2×C3—22no—
425C2×C3—41no—
426C2×C3C2×D4×D711yescharacteristic
427C2×C3C2×#115911yescharacteristic
428C2×C3#655411yescharacteristic
429D3—41no—
430D3—281no—
431C22—21no—
432C22—21no—
433C22—21no—
434C22—21no—
435C22—21no—
436C22—61no—
437C22—61no—
438C22—61no—
439C22—421no—
440C22—421no—
441C22—421no—
442C22#6668612yescomplemented
443C22C22×D3×D711yescharacteristic
444C4—21no—
445C4—32no—
446C4—61no—
447C4—61no—
448C4—72no—
449C4—141no—
450C4—141no—
451C4—141no—
452C4—212no—
453C4—421no—
454C4—421no—
455C3#9028911yescharacteristic, complemented, Sylow
456C2—22no—
457C2—41no—
458C2—121no—
459C2—841no—
460C2#1062156711yescharacteristic
461C2D3×D4×D711yescharacteristic
462C2C2×#6668611yescharacteristic
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×#66686C2semidirect
3. #10621422C2semidirect
4. C7×#939C2semidirect
5. C3×D4×Dic7C2semidirect
6. #10621136C2semidirect
7. #10621128C2semidirect
8. #67⋊Dic7C2semidirect
9. #10621118C2semidirect
10. #10621088C2semidirect
11. #10621087C2semidirect
12. #10620998C2semidirect
13. #10620991C2semidirect
14. #10620912C2semidirect
15. #10620885C2semidirect
61. D4×Dic7D3semidirect
97. #4248D4semidirect
142. #939D7semidirect
158. C22×C3×C7#41semidirect
276. C22×C7#235semidirect
323. C3×C7#548semidirect
373. C22×C3#1159semidirect
423. C7#5552semidirect
442. C22#66686semidirect
455. C3#90289semidirect

Maximal subgroups

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

subgroupindex
C22×#42482
C2×#666862
#106214222
C7×#9392
C3×D4×Dic72
#106211362
#106211282
#67⋊Dic72
#106211182
#106210882
#106210872
#106209982
#106209912
#106209122
#106208852
#902893
#55527