Finite Group Databasebeta

← #13332580

Subgroups — #13332580

Order 1792.

1138 conjugacy classes of subgroups.

The subgroup lattice has 492 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C7×D4×Q8C211yesmaximal, characteristic
2#11740668C211yesmaximal, characteristic, complemented
3#11739876C211yesmaximal, characteristic, complemented
4#11739759C211yesmaximal, characteristic
5#11729671C214yesmaximal, complemented
6#11729669C214yesmaximal, complemented
7#11729527C211yesmaximal, characteristic, complemented
8#11729375C211yesmaximal, characteristic, complemented
9#11729352C211yesmaximal, characteristic
10C23×C7×Q8—21no—
11C7×D4×Q8—22no—
12C7×D4×Q8—22no—
13C7×D4×Q8—22no—
14C7×D4×Q8C2214yes—
15C2×C7×C4⋊Q8—21no—
16C2×C7×C4⋊Q8C2211yescharacteristic
17C2×C7×C22⋊Q8—21no—
18C2×C7×C22⋊Q8—22no—
19C2×C4×C7×Q8C2211yescharacteristic
20C2×C4×C7×D4—21no—
21C2×C4×C7×D4C2211yescharacteristic
22#90161C2211yescharacteristic
23#89997—21no—
24#89990—21no—
25#89866—22no—
26#89865C2212yescomplemented
27#89851C2212yescomplemented
28#89794—22no—
29#89793C2212yescomplemented
30#89792C2212yescomplemented
31#89791C2212yes—
32C4×#1142—22no—
33#89780C2212yes—
34#89743—21no—
35#89743—22no—
36#89736—21no—
37#89697C2211yescharacteristic
38#89697C2211yescharacteristic
39#89696C2211yescharacteristic
40#89342C2214yescomplemented
41Q8⋊(C7⋊C8)C2214yescomplemented
42D4⋊(C7⋊C8)C2214yes—
43#18777—71nomaximal, Sylow
44C22×C7×Q8—22no—
45C22×C7×Q8—24no—
46C22×C7×Q8—24no—
47C22×C7×Q8—21no—
48C22×C7×Q8—44no—
49C22×C7×Q8—44no—
50C22×C7×Q8D411yescharacteristic
51C22×C7×D4D411yescharacteristic
52C23×C4×C7—22no—
53C23×C4×C7—21no—
54C7×C4⋊Q8—24no—
55C7×C4⋊Q8—24no—
56C7×C4⋊Q8—21no—
57C7×C4⋊Q8—21no—
58C7×C4⋊Q8D412yes—
59C7×C4⋊Q8C2312yes—
60C7×C22⋊Q8—24no—
61C7×C22⋊Q8—28no—
62C7×C22⋊Q8—24no—
63C7×C22⋊Q8—28no—
64C4×C7×Q8—21no—
65C4×C7×Q8—21no—
66C4×C7×Q8D412yes—
67C4×C7×Q8C2312yes—
68C4×C7×D4—24no—
69C4×C7×D4—24no—
70C4×C7×D4—21no—
71C4×C7×D4—21no—
72C4×C7×D4D412yes—
73C4×C7×D4C2312yes—
74C2×C7×C4⋊C4—22no—
75C2×C7×C4⋊C4—21no—
76C2×C7×C4⋊C4—21no—
77C2×C7×C4⋊C4D411yescharacteristic
78C2×C7×C4⋊C4D411yescharacteristic
79C2×C7×C4⋊C4D412yes—
80C2×C7×C22⋊C4—22no—
81C2×C7×C22⋊C4—21no—
82C2×C42×C7—21no—
83C2×C42×C7C2311yescharacteristic
84C2×#1142—22no—
85#6489—21no—
86#6487—21no—
87#6487—21no—
88C2×#1136—21no—
89#6433—21no—
90#6432C2311yescharacteristic
91#6432C2311yescharacteristic
92C4×Dic14—21no—
93C4×Dic14—21no—
94#6409—22no—
95#6409—22no—
96#6406—22no—
97C22⋊(C7⋊C8)—21no—
98C22⋊(C7⋊C8)—21no—
99#6384—24no—
100#6384—22no—
101#6382—24no—
102#6382—22no—
103#6382—22no—
104#6381—22no—
105C4⋊(C7⋊C8)—21no—
106C4⋊(C7⋊C8)—21no—
107C4⋊(C7⋊C8)C2312yes—
108C4⋊(C7⋊C8)C2312yes—
109C4×C7⋊C8—21no—
110C4×C7⋊C8C2312yes—
111C2×D4×Q8D711yescharacteristic
112#3229—71no—
113#3101—71no—
114#3078—71no—
115#1651—74no—
116#1637—74no—
117#1546—71no—
118#1486—71no—
119#1479—71no—
120C24×C7—21no—
121C2×C7×Q8—28no—
122C2×C7×Q8—21no—
123C2×C7×Q8—22no—
124C2×C7×Q8—22no—
125C2×C7×Q8—22no—
126C2×C7×Q8—48no—
127C2×C7×Q8—48no—
128C2×C7×Q8—416no—
129C2×C7×Q8—48no—
130C2×C7×Q8—42no—
131C2×C7×Q8—42no—
132C2×C7×Q8SD1614yescomplemented
133C2×C7×Q8C2×D412yes—
134C2×C7×D4—24no—
135C2×C7×D4—21no—
136C2×C7×D4C2×D412yes—
137C22×C4×C7—21no—
138C22×C4×C7—28no—
139C22×C4×C7—21no—
140C22×C4×C7—22no—
141C22×C4×C7—21no—
142C22×C4×C7—22no—
143C22×C4×C7—22no—
144C22×C4×C7—48no—
145C22×C4×C7—42no—
146C22×C4×C7—42no—
147C22×C4×C7C2×D411yescharacteristic
148C22×C4×C7C2×D411yescharacteristic
149C22×C4×C7C2×D411yescharacteristic
150C2×Dic14—21no—
151C2×Dic14—21no—
152C7×C4⋊C4—28no—
153C7×C4⋊C4—22no—
154C7×C4⋊C4—22no—
155C7×C4⋊C4—21no—
156C7×C4⋊C4—21no—
157C7×C4⋊C4—22no—
158C7×C4⋊C4—22no—
159C7×C4⋊C4—22no—
160C7×C4⋊C4C2×D412yes—
161C7×C4⋊C4C2×D412yes—
162C7×C4⋊C4C2×D414yes—
163C7×C22⋊C4—28no—
164C7×C22⋊C4—22no—
165C7×C22⋊C4—22no—
166C42×C7—22no—
167C42×C7—22no—
168C42×C7C2×D411yescharacteristic
169C42×C7C2×D411yescharacteristic
170C42×C7C2×D411yescharacteristic
171C42×C7C2411yescharacteristic
172C22⋊Dic7—41no—
173#1142—44no—
174C4⋊Dic7—21no—
175C4⋊Dic7—21no—
176C4⋊Dic7—21no—
177C4⋊Dic7—21no—
178Dic7⋊C4—41no—
179Dic7⋊C4—41no—
180C4×Dic7—41no—
181#1136—42no—
182C2×C7⋊C8—22no—
183C2×C7⋊C8—22no—
184C2×C7⋊C8—22no—
185C23×Q8—21no—
186D4×Q8—22no—
187D4×Q8—22no—
188D4×Q8—22no—
189D4×Q8C2×D714yes—
190#533—71no—
191C2×C4⋊Q8—21no—
192C2×C4⋊Q8C2×D711yescharacteristic
193C2×C22⋊Q8—21no—
194C2×C22⋊Q8—22no—
195C2×C4×Q8C2×D711yescharacteristic
196C2×C4×D4—21no—
197C2×C4×D4C2×D711yescharacteristic
198#498—72no—
199#487—72no—
200#483—141no—
201#483—142no—
202#479—142no—
203#477—72no—
204#476—72no—
205#475—72no—
206#475—142no—
207#461—72no—
208#450—141no—
209C4×SD16—142no—
210#431—71no—
211#431—71no—
212#428—141no—
213#423—71no—
214#419—141no—
215#332—74no—
216Q8⋊C8—74no—
217D4⋊C8—74no—
218C23×C7—24no—
219C23×C7—22no—
220C23×C7—42no—
221C23×C7—42no—
222C23×C7#12011yescharacteristic
223C7×Q8—22no—
224C7×Q8—24no—
225C7×Q8—416no—
226C7×Q8—42no—
227C7×Q8—42no—
228C7×Q8—44no—
229C7×Q8C2×SD1614yes—
230C7×D4—24no—
231C7×D4—22no—
232C7×D4#13714yes—
233C2×C4×C7—22no—
234C2×C4×C7—24no—
235C2×C4×C7—22no—
236C2×C4×C7—22no—
237C2×C4×C7—24no—
238C2×C4×C7—22no—
239C2×C4×C7—41no—
240C2×C4×C7—416no—
241C2×C4×C7—42no—
242C2×C4×C7—44no—
243C2×C4×C7—42no—
244C2×C4×C7—42no—
245C2×C4×C7—42no—
246C2×C4×C7—44no—
247C2×C4×C7#12011yescharacteristic
248C2×C4×C7#12011yescharacteristic
249C2×C4×C7#12011yescharacteristic
250C2×C4×C7C2×SD1612yes—
251C2×C4×C7AGL(1,ℤ/8)12yes—
252C2×C4×C7C22×D411yescharacteristic
253C2×C4×C7C22×D411yescharacteristic
254C2×C4×C7C22×D411yescharacteristic
255C2×Dic7—41no—
256C2×Dic7—41no—
257Dic14—42no—
258Dic14—42no—
259C7⋊C8—42no—
260C7⋊C8—42no—
261C7⋊C8—42no—
262C22×Q8—22no—
263C22×Q8—24no—
264C22×Q8—24no—
265C22×Q8—21no—
266C22×Q8—44no—
267C22×Q8—44no—
268C22×Q8C22⋊D711yescharacteristic
269C22×D4C22⋊D711yescharacteristic
270C23×C4—22no—
271C23×C4—21no—
272C2×SD16—142no—
273C2×M16—141no—
274C4⋊Q8—24no—
275C4⋊Q8—24no—
276C4⋊Q8—21no—
277C4⋊Q8—21no—
278C4⋊Q8—71no—
279C4⋊Q8—71no—
280C4⋊Q8C22⋊D712yes—
281C4⋊Q8C22×D712yes—
282#125—141no—
283C22⋊Q8—24no—
284C22⋊Q8—28no—
285C22⋊Q8—24no—
286C22⋊Q8—28no—
287C22⋊Q8—141no—
288C22⋊Q8—141no—
289C4×Q8—21no—
290C4×Q8—21no—
291C4×Q8—141no—
292C4×Q8—141no—
293C4×Q8C22⋊D712yes—
294C4×Q8C22×D712yes—
295C4×D4—24no—
296C4×D4—24no—
297C4×D4—21no—
298C4×D4—21no—
299C4×D4C22⋊D712yes—
300C4×D4C22×D712yes—
301#117—141no—
302C2×C4⋊C4—22no—
303C2×C4⋊C4—21no—
304C2×C4⋊C4—21no—
305C2×C4⋊C4C22⋊D711yescharacteristic
306C2×C4⋊C4C22⋊D711yescharacteristic
307C2×C4⋊C4C22⋊D712yes—
308C2×C22⋊C4—22no—
309C2×C22⋊C4—21no—
310C2×C42—21no—
311C2×C42C22×D711yescharacteristic
312#107—142no—
313#106—142no—
314#106—142no—
315#106—144no—
316C4⋊C8—72no—
317C4⋊C8—72no—
318C4⋊C8—141no—
319C4⋊C8—141no—
320#103—142no—
321#103—142no—
322#103—142no—
323#103—144no—
324#102—142no—
325C22⋊C8—141no—
326C22⋊C8—141no—
327C4×C8—72no—
328C4×C8—141no—
329C22×C7—21no—
330C22×C7—21no—
331C22×C7—24no—
332C22×C7—48no—
333C22×C7—42no—
334C22×C7—42no—
335C22×C7#45012yes—
336C22×C7C2×#12011yescharacteristic
337C4×C7—24no—
338C4×C7—21no—
339C4×C7—21no—
340C4×C7—21no—
341C4×C7—44no—
342C4×C7—42no—
343C4×C7—42no—
344C4×C7#45012yes—
345C4×C7C22×SD1611yescharacteristic
346C4×C7C2×AGL(1,ℤ/8)11yescharacteristic
347C4×C7C2×#13712yes—
348Dic7—81no—
349Dic7—81no—
350C24—21no—
351C2×Q8—28no—
352C2×Q8—22no—
353C2×Q8—21no—
354C2×Q8—22no—
355C2×Q8—22no—
356C2×Q8—48no—
357C2×Q8—48no—
358C2×Q8—416no—
359C2×Q8—48no—
360C2×Q8—42no—
361C2×Q8—42no—
362C2×Q8—141no—
363C2×Q8—141no—
364C2×Q8#114214yescomplemented
365C2×Q8C2×C22⋊D712yes—
366C2×D4—24no—
367C2×D4—21no—
368C2×D4C2×C22⋊D712yes—
369C22×C4—21no—
370C22×C4—28no—
371C22×C4—21no—
372C22×C4—22no—
373C22×C4—21no—
374C22×C4—22no—
375C22×C4—22no—
376C22×C4—48no—
377C22×C4—42no—
378C22×C4—42no—
379C22×C4C2×C22⋊D711yescharacteristic
380C22×C4C2×C22⋊D711yescharacteristic
381C22×C4C2×C22⋊D711yescharacteristic
382SD16—284no—
383M16—282no—
384C2×C8—142no—
385C2×C8—142no—
386C2×C8—142no—
387C4⋊C4—28no—
388C4⋊C4—22no—
389C4⋊C4—22no—
390C4⋊C4—21no—
391C4⋊C4—21no—
392C4⋊C4—22no—
393C4⋊C4—22no—
394C4⋊C4—22no—
395C4⋊C4—141no—
396C4⋊C4—141no—
397C4⋊C4—141no—
398C4⋊C4—141no—
399C4⋊C4—281no—
400C4⋊C4—281no—
401C4⋊C4C2×C22⋊D712yes—
402C4⋊C4C2×C22⋊D714yes—
403C4⋊C4C2×C22⋊D712yes—
404C22⋊C4—28no—
405C22⋊C4—22no—
406C22⋊C4—22no—
407C22⋊C4—281no—
408C42—22no—
409C42—22no—
410C42—281no—
411C42C2×C22⋊D711yescharacteristic
412C42C2×C22⋊D711yescharacteristic
413C42C2×C22⋊D711yescharacteristic
414C42C23×D711yescharacteristic
415C2×C7—22no—
416C2×C7—44no—
417C2×C7C2×#45011yescharacteristic
418C2×C7#300411yescharacteristic
419C2×C7#302011yescharacteristic
420C23—24no—
421C23—22no—
422C23—42no—
423C23—42no—
424C23#651611yescharacteristic
425Q8—22no—
426Q8—24no—
427Q8—416no—
428Q8—42no—
429Q8—42no—
430Q8—44no—
431Q8—282no—
432Q8—282no—
433Q8C2×#114214yes—
434D4—24no—
435D4—22no—
436D4#650514yes—
437C2×C4—24no—
438C2×C4—22no—
439C2×C4—22no—
440C2×C4—22no—
441C2×C4—24no—
442C2×C4—22no—
443C2×C4—41no—
444C2×C4—416no—
445C2×C4—42no—
446C2×C4—44no—
447C2×C4—42no—
448C2×C4—42no—
449C2×C4—42no—
450C2×C4—44no—
451C2×C4—281no—
452C2×C4—281no—
453C2×C4#649512yes—
454C2×C4C2×#114212yes—
455C2×C4#651611yescharacteristic
456C2×C4#651611yescharacteristic
457C2×C4#651611yescharacteristic
458C2×C4C22×C22⋊D711yescharacteristic
459C2×C4C22×C22⋊D711yescharacteristic
460C2×C4C22×C22⋊D711yescharacteristic
461C8—282no—
462C8—282no—
463C8—282no—
464C7#1877711yescharacteristic, complemented, Sylow
465C22—21no—
466C22—21no—
467C22—24no—
468C22—48no—
469C22—42no—
470C22—42no—
471C22#8999012yes—
472C22C2×#651611yescharacteristic
473C4—24no—
474C4—21no—
475C4—21no—
476C4—21no—
477C4—44no—
478C4—42no—
479C4—42no—
480C4—561no—
481C4—561no—
482C4#8999012yes—
483C4C2×#649511yescharacteristic
484C4C22×#114211yescharacteristic
485C4C2×#650512yes—
486C2—22no—
487C2—44no—
488C2C2×#8999011yescharacteristic
489C2#1174150411yescharacteristic
490C2#1174156111yescharacteristic
subgroupG/H
centerC22C2×#6516
commutator, cosocleC42×C7C24
FrattiniC42C23×D7
FittingC2×C7×D4×Q8C2
socleC22×C7C2×#120
prime radicalC22×C7×D4D4
hypercenterC2×D4×Q8D7
hypocenterC7#18777
LCSG ▷ C42×C7 ▷ C22×C7 ▷ C7
UCSC1 ◁ C22 ◁ C2×C42 ◁ C2×D4×Q8

Complements

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

NH ≅ G/Ntype
2. #11740668C2semidirect
3. #11739876C2semidirect
5. #11729671C2semidirect
6. #11729669C2semidirect
7. #11729527C2semidirect
8. #11729375C2semidirect
26. #89865C22semidirect
27. #89851C22semidirect
29. #89793C22semidirect
30. #89792C22semidirect
40. #89342C22semidirect
41. Q8⋊(C7⋊C8)C22semidirect
132. C2×C7×Q8SD16semidirect
364. C2×Q8#1142semidirect
464. C7#18777semidirect

Maximal subgroups

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

subgroupindex
C2×C7×D4×Q82
#117406682
#117398762
#117397592
#117296712
#117296692
#117295272
#117293752
#117293522
#187777