Finite Group Databasebeta

← #13059336

Subgroups — #13059336

Order 1344.

496 conjugacy classes of subgroups.

The subgroup lattice has 496 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#10621576C211yesmaximal, characteristic, complemented
2#10621575C211yesmaximal, characteristic, complemented
3#10621401C211yesmaximal, characteristic, complemented
4D3×C7×Q16C211yesmaximal, characteristic, complemented
5C3×#6481C211yesmaximal, characteristic, complemented
6#10621083C211yesmaximal, characteristic, complemented
7#10621082C211yesmaximal, characteristic, complemented
8#10621080C211yesmaximal, characteristic, complemented
9#10621078C211yesmaximal, characteristic, complemented
10#10621073C211yesmaximal, characteristic, complemented
11D7×#216C211yesmaximal, characteristic, complemented
12#10620840C211yesmaximal, characteristic, complemented
13#10620834C211yesmaximal, characteristic, complemented
14#10620829C211yesmaximal, characteristic, complemented
15#10620823C211yesmaximal, characteristic, complemented
16#90457—31nomaximal, Hall
17#66732C2211yescharacteristic
18Q8×D21C2211yescharacteristic
19D3×C7×Q8C2211yescharacteristic
20D3×C7×Q8C2211yescharacteristic
21C3×#1161C2211yescharacteristic
22C3×Q8×D7C2211yescharacteristic
23#66677—21no—
24#66677C2211yescharacteristic
25#66675—21no—
26#66675—21no—
27#66675C2211yescharacteristic
28D21⋊Q8—21no—
29D21⋊Q8C2211yescharacteristic
30#66673—21no—
31#66673C2211yescharacteristic
32#66670—21no—
33#66670C2211yescharacteristic
34#66669—21no—
35#66669C2211yescharacteristic
36D7×Dic6—21no—
37D7×Dic6C2211yescharacteristic
38C3×C7×Q16C2211yescharacteristic
39#66635C2211yescharacteristic
40#66634C2211yescharacteristic
41#66623C2211yescharacteristic
42#66622C2211yescharacteristic
43C7×#216C2211yescharacteristic
44C7×#216C2211yescharacteristic
45C7×Dic12C2211yescharacteristic
46D3×C7×C8C2211yescharacteristic
47C3×#1144C2211yescharacteristic
48C3×#1143C2211yescharacteristic
49C3×#1132C2211yescharacteristic
50C3×#1131C2211yescharacteristic
51#66571C2211yescharacteristic
52#66570C2211yescharacteristic
53#66569C2211yescharacteristic
54#66568C2211yescharacteristic
55#66563C2211yescharacteristic
56#66561C2211yescharacteristic
57#66558C2211yescharacteristic
58D21⋊C8C2211yescharacteristic
59D7×C3⋊C8C2211yescharacteristic
60#6551—31no—
61#6551—31no—
62C2×C7×Q16—31no—
63#6505—31no—
64#6505—31no—
65#6482—31no—
66#6482—31no—
67#6481—31no—
68#6481—31no—
69#6481—31no—
70#6481D311yescharacteristic, complemented
71D7×Q16—31no—
72D7×Q16—31no—
73#6467—31no—
74#6464—31no—
75#5779—71nomaximal, Hall
76C3×C7×Q8C2311yescharacteristic
77C3×C7×Q8C2311yescharacteristic
78D84—21no—
79D84C2311yescharacteristic
80C4×D21—21no—
81C4×D21—21no—
82C4×D21C2311yescharacteristic
83Dic42—21no—
84Dic42C2311yescharacteristic
85C4×D3×C7—21no—
86C4×D3×C7—21no—
87C4×D3×C7C2311yescharacteristic
88C7×Dic6—21no—
89C7×Dic6—21no—
90C7×Dic6C2311yescharacteristic
91C7×Dic6C2311yescharacteristic
92C3×D28—21no—
93C3×D28C2311yescharacteristic
94C3×C4×D7—21no—
95C3×C4×D7—21no—
96C3×C4×D7C2311yescharacteristic
97C3×Dic14—21no—
98C3×Dic14C2311yescharacteristic
99#4248—21no—
100#4248—21no—
101#4248D411yescharacteristic
102#4247—21no—
103#4247D411yescharacteristic
104#4246—21no—
105#4246—21no—
106#4246D411yescharacteristic
107#4245—21no—
108#4245D411yescharacteristic
109D21⋊C4—21no—
110D21⋊C4—21no—
111D21⋊C4—21no—
112Dic3×D7—21no—
113Dic3×D7—21no—
114Dic3×D7—21no—
115C3×C7×C8C2311yescharacteristic
116C3⋊(C7⋊C8)C2311yescharacteristic
117C3×C7⋊C8C2311yescharacteristic
118C7×C3⋊C8C2311yescharacteristic
119C2×C7×Q8—31no—
120C2×C7×Q8—31no—
121#1161—31no—
122#1161—31no—
123#1161—31no—
124#1161—61no—
125#1161—61no—
126#1161C2×D311yescharacteristic
127Q8×D7—31no—
128Q8×D7—31no—
129Q8×D7—31no—
130Q8×D7—61no—
131Q8×D7—61no—
132Q8×D7C2×D311yescharacteristic
133#1157—31no—
134#1157—31no—
135#1157—31no—
136#1157—61no—
137#1157—61no—
138#1157—61no—
139#1157—61no—
140C7×Q16—31no—
141C7×Q16—31no—
142C7×Q16—31no—
143C7×Q16C2×D311yescharacteristic
144C2×C7×C8—31no—
145#1144—31no—
146#1144—31no—
147#1144—31no—
148#1144C2×D311yescharacteristic
149#1143—31no—
150#1143—31no—
151#1143—31no—
152#1143C2×D311yescharacteristic
153#1136—31no—
154Dic28—31no—
155D56—31no—
156#1132—31no—
157#1132C2×D311yescharacteristic
158#1131—31no—
159#1131C2×D311yescharacteristic
160C8×D7—31no—
161C8×D7—31no—
162#1010—71no—
163#1007—71no—
164C3×#137—71no—
165#950—71no—
166C2×#216—71no—
167#918—71no—
168#918—71no—
169D3×Q16—71no—
170D3×Q16D711yescharacteristic, complemented
171#916—71no—
172#916—71no—
173#915—71no—
174#915—71no—
175#909—71no—
176#907—71no—
177C2×D21—21no—
178C2×D21C2×D411yescharacteristic
179C2×D3×C7C2×D411yescharacteristic
180C2×C3×D7—21no—
181C2×C3×D7C2×D411yescharacteristic
182C3×C4×C7—21no—
183C3×C4×C7—21no—
184C3×C4×C7C2411yescharacteristic
185Dic21—21no—
186Dic21C2×D411yescharacteristic
187C3×Dic7—21no—
188C3×Dic7C2×D411yescharacteristic
189C7×Dic3—21no—
190C7×Dic3—21no—
191C7×Dic3C2×D411yescharacteristic
192#578—211noSylow
193C7×Q8—31no—
194C7×Q8—31no—
195C7×Q8—61no—
196C7×Q8—61no—
197C7×Q8C22×D311yescharacteristic
198C7×Q8C22×D311yescharacteristic
199C2×C4×C7—31no—
200C2×C4×C7—61no—
201C2×C4×C7—61no—
202C22⋊D7—31no—
203C22⋊D7—31no—
204C22⋊D7—61no—
205C22⋊D7—61no—
206D28—21no—
207D28—31no—
208D28—31no—
209D28—61no—
210D28—61no—
211D28—61no—
212D28C22×D311yescharacteristic
213C4×D7—21no—
214C4×D7—21no—
215C4×D7—31no—
216C4×D7—61no—
217C4×D7—61no—
218C4×D7—61no—
219C4×D7—61no—
220C4×D7—61no—
221C4×D7—61no—
222C4×D7—61no—
223C4×D7—61no—
224C4×D7C22×D311yescharacteristic
225Dic14—21no—
226Dic14—31no—
227Dic14—31no—
228Dic14—61no—
229Dic14—61no—
230Dic14—61no—
231Dic14C22×D311yescharacteristic
232C7×C8—31no—
233C7×C8C22×D311yescharacteristic
234C7⋊C8—31no—
235C7⋊C8C22×D311yescharacteristic
236C3×#41—71no—
237C2×C3×Q8—71no—
238#239—71no—
239#239—71no—
240#239—141no—
241D3×Q8—71no—
242D3×Q8—71no—
243D3×Q8—141no—
244D3×Q8C2×D711yescharacteristic
245D3×Q8C2×D711yescharacteristic
246#237—71no—
247#237—71no—
248#237—141no—
249#237—141no—
250#235—71no—
251#235—71no—
252#235—141no—
253#235—141no—
254#235—141no—
255C2×Dic6—71no—
256C2×Dic6—141no—
257C3×Q16—71no—
258C3×Q16C2×D711yescharacteristic
259C3×SD16—71no—
260C3×SD16—71no—
261C3×M16—71no—
262#216—71no—
263#216—71no—
264#216—71no—
265#216C2×D711yescharacteristic
266#216C2×D711yescharacteristic
267#215—71no—
268#214—71no—
269#213—71no—
270C2×C3⋊C8—71no—
271Dic12—71no—
272Dic12C2×D711yescharacteristic
273#204—71no—
274#204—71no—
275ΓL(1,𝔽25)—71no—
276ΓL(1,𝔽25)—71no—
277D3×C8—71no—
278D3×C8C2×D711yescharacteristic
279C2×C3×C7C22×D411yescharacteristic
280D21—21no—
281D21—41no—
282C3×D7—21no—
283C3×D7—41no—
284D3×C7—21no—
2852−1+4—211no—
2862−1+4—211no—
287#137—71no—
288#137—211no—
289#137—211no—
290#137—211no—
291#137—211no—
292#137—211no—
293#135—211no—
294#135—211no—
295#135—211no—
296C2×Q16—31no—
297C2×Q16—211no—
298C2×Q16—211no—
299#131—211no—
300C22×C7—31no—
301C2×D7—21no—
302C2×D7—31no—
303C2×D7—61no—
304C2×D7D3×D411yescharacteristic
305C4×C7—21no—
306C4×C7—21no—
307C4×C7—31no—
308C4×C7—61no—
309C4×C7—61no—
310C4×C7C23×D311yescharacteristic
311Dic7—21no—
312Dic7—31no—
313Dic7—61no—
314Dic7D3×D411yescharacteristic
315C3×Q8—71no—
316C3×Q8—141no—
317C3×Q8C22×D711yescharacteristic
318C3×Q8C22×D711yescharacteristic
319C3×D4—71no—
320C3×D4—141no—
321C2×C3×C4—71no—
322C2×C3×C4—141no—
323C2×C3×C4—141no—
324#67—71no—
325#67—71no—
326#67—141no—
327#67—141no—
328#67—141no—
329C2×Dic3—141no—
330C2×Dic3—141no—
331C2×Dic3—141no—
332D12—71no—
333D12—71no—
334D12—141no—
335D12—141no—
336C4×D3—21no—
337C4×D3—21no—
338C4×D3—71no—
339C4×D3—141no—
340C4×D3—141no—
341C4×D3—141no—
342C4×D3—141no—
343C4×D3—141no—
344C4×D3C22×D711yescharacteristic
345Dic6—21no—
346Dic6—21no—
347Dic6—71no—
348Dic6—71no—
349Dic6—141no—
350Dic6—141no—
351Dic6—141no—
352Dic6C22×D711yescharacteristic
353Dic6C22×D711yescharacteristic
354C3×C8—71no—
355C3×C8C22×D711yescharacteristic
356C3⋊C8—71no—
357C3⋊C8C22×D711yescharacteristic
358C3×C7#57811yescharacteristic, complemented, Hall
359#41—71no—
360#41—211no—
361#41—211no—
362#41—211no—
363#41—211no—
364#41—211no—
365#41—211no—
366#41—421no—
367#41—421no—
368#41—421no—
369#41—421no—
370#41—421no—
371#41—421no—
372C2×Q8—31no—
373C2×Q8—31no—
374C2×Q8—71no—
375C2×Q8—211no—
376C2×Q8—211no—
377C2×Q8—211no—
378C2×Q8—421no—
379C2×Q8—421no—
380Q16—31no—
381Q16—31no—
382Q16—31no—
383Q16—71no—
384Q16—211no—
385Q16—211no—
386Q16—211no—
387Q16—211no—
388Q16D3×D711yescharacteristic
389SD16—71no—
390SD16—71no—
391SD16—211no—
392SD16—211no—
393SD16—211no—
394SD16—211no—
395D8—211no—
396M16—71no—
397M16—211no—
398M16—211no—
399C2×C8—31no—
400C2×C8—211no—
401C2×C8—211no—
402C2×C7—61no—
403C2×C7C2×D3×D411yescharacteristic
404D7—21no—
405D7—41no—
406D7—61no—
407D7—121no—
408C22×C3—71no—
409C22×C3—141no—
410C2×D3—71no—
411C2×D3—141no—
412C2×D3D4×D711yescharacteristic
413C3×C4—21no—
414C3×C4—21no—
415C3×C4—71no—
416C3×C4—141no—
417C3×C4C23×D711yescharacteristic
418Dic3—21no—
419Dic3—21no—
420Dic3—71no—
421Dic3—141no—
422Dic3D4×D711yescharacteristic
423Q8—31no—
424Q8—31no—
425Q8—61no—
426Q8—61no—
427Q8—71no—
428Q8—141no—
429Q8—211no—
430Q8—211no—
431Q8—421no—
432Q8—421no—
433Q8—421no—
434Q8C2×D3×D711yescharacteristic
435Q8C2×D3×D711yescharacteristic
436D4—71no—
437D4—141no—
438D4—211no—
439D4—211no—
440D4—211no—
441D4—211no—
442D4—421no—
443D4—421no—
444D4—421no—
445D4—421no—
446D4—421no—
447C2×C4—31no—
448C2×C4—61no—
449C2×C4—61no—
450C2×C4—71no—
451C2×C4—141no—
452C2×C4—141no—
453C2×C4—211no—
454C2×C4—421no—
455C2×C4—421no—
456C2×C4—421no—
457C2×C4—421no—
458C2×C4—421no—
459C2×C4—421no—
460C2×C4—421no—
461C2×C4—421no—
462C8—31no—
463C8—71no—
464C8—211no—
465C8C2×D3×D711yescharacteristic
466C7#577911yescharacteristic, complemented, Sylow
467C2×C3—141no—
468C2×C3—281no—
469C2×C3C2×D4×D711yescharacteristic
470D3—21no—
471D3—141no—
472D3—281no—
473C22—31no—
474C22—71no—
475C22—141no—
476C22—211no—
477C22—421no—
478C4—21no—
479C4—21no—
480C4—31no—
481C4—61no—
482C4—61no—
483C4—71no—
484C4—141no—
485C4—211no—
486C4—421no—
487C4C22×D3×D711yescharacteristic
488C3#9045711yescharacteristic, complemented, Sylow
489C2—61no—
490C2—141no—
491C2—281no—
492C2—421no—
493C2—841no—
494C2D3×D4×D711yescharacteristic
subgroupG/H
centerC2D3×D4×D7
commutator, cosocleC3×C4×C7C24
FrattiniC4C22×D3×D7
FittingC3×C7×Q16C22
socleC2×C3×C7C22×D4
hypercenterQ16D3×D7
hypocenterC3×C7#578
LCSG ▷ C3×C4×C7 ▷ C2×C3×C7 ▷ C3×C7
UCSC1 ◁ C2 ◁ C4 ◁ Q16

Complements

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

NH ≅ G/Ntype
1. #10621576C2semidirect
2. #10621575C2semidirect
3. #10621401C2semidirect
4. D3×C7×Q16C2semidirect
5. C3×#6481C2semidirect
6. #10621083C2semidirect
7. #10621082C2semidirect
8. #10621080C2semidirect
9. #10621078C2semidirect
10. #10621073C2semidirect
11. D7×#216C2semidirect
12. #10620840C2semidirect
13. #10620834C2semidirect
14. #10620829C2semidirect
15. #10620823C2semidirect
70. #6481D3semidirect
170. D3×Q16D7semidirect
358. C3×C7#578semidirect
466. C7#5779semidirect
488. C3#90457semidirect

Maximal subgroups

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

subgroupindex
#106215762
#106215752
#106214012
D3×C7×Q162
C3×#64812
#106210832
#106210822
#106210802
#106210782
#106210732
D7×#2162
#106208402
#106208342
#106208292
#106208232
#904573
#57797