Finite Group Databasebeta

← #34259358

Subgroups — #34259358

Order 20224.

592 conjugacy classes of subgroups.

The subgroup lattice has 568 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23773478C211yesmaximal, characteristic
2#23772401C211yesmaximal, characteristic
3Q8×#11929607C211yesmaximal, characteristic, complemented
4#23762299C211yesmaximal, characteristic, complemented
5#23762298C211yesmaximal, characteristic, complemented
6#23762279C211yesmaximal, characteristic, complemented
7#23762270C211yesmaximal, characteristic, complemented
8#23762250C211yesmaximal, characteristic, complemented
9#23762236C211yesmaximal, characteristic, complemented
10#23762216C211yesmaximal, characteristic, complemented
11#23762187C211yesmaximal, characteristic, complemented
12#23761862C211yesmaximal, characteristic
13#23761793C211yesmaximal, characteristic, complemented
14#23761694C211yesmaximal, characteristic, complemented
15C8×SD16×C79C211yesmaximal, characteristic, complemented
16#19099095C2211yescharacteristic
17C4×Q8×D79C2211yescharacteristic
18#19098895C411yescharacteristic
19#19098875C411yescharacteristic
20#19098873C411yescharacteristic
21#19098872C411yescharacteristic
22#19098784C411yescharacteristic
23#19098783C411yescharacteristic
24#19098720C411yescharacteristic
25#19098700C411yescharacteristic
26#19098693—21no—
27#19098693C2211yescharacteristic
28#19098693C2211yescharacteristic
29#19098690—21no—
30#19098690C2211yescharacteristic
31Q8×C79⋊C8C2211yescharacteristic
32#19098687C2211yescharacteristic
33C4×Dic316C2211yescharacteristic
34C4×#11929614C2211yescharacteristic
35C4×#11929612C2211yescharacteristic
36#19098641C2211yescharacteristic
37#19098638—21no—
38#19098634—21no—
39#19098621—21no—
40#19098615C2211yescharacteristic
41#19098614C2211yescharacteristic
42#19098614C2211yescharacteristic
43#19098585—21no—
44#19098585C2211yescharacteristic
45#19098518—21no—
46C4×#11929607—21no—
47C4×#11929607C2211yescharacteristic
48#19098316C2211yescharacteristic
49#19098315C2211yescharacteristic
50#19098313C2211yescharacteristic
51#19098295C2211yescharacteristic
52#19098295C2211yescharacteristic
53Q8⋊(C79⋊C8)C2211yescharacteristic
54#19098293C2211yescharacteristic
55#19098293C2211yescharacteristic
56D4⋊(C79⋊C8)C2211yescharacteristic
57#19098285C2211yescharacteristic
58#19098285C2211yescharacteristic
59#19098284C2211yescharacteristic
60C8×Q8×C79C2211yescharacteristic
61C4×SD16×C79C2211yescharacteristic
62C8×D4×C79C2211yescharacteristic
63#334×C79C2211yescharacteristic
64Q8⋊C8×C79C2211yescharacteristic
65D4⋊C8×C79C2211yescharacteristic
66C82×C79C2211yescharacteristic
67C2×Q8×D79C2×C411yescharacteristic
68C2×Dic316C2×C411yescharacteristic
69C2×#11929614C2×C411yescharacteristic
70C2×#11929612C2×C411yescharacteristic
71C2×#11929608—21no—
72C2×#11929607—21no—
73C2×#11929607D411yescharacteristic
74C2×#11929607D411yescharacteristic
75#14579154—21no—
76#14579152C2×C411yescharacteristic
77#14579148C2×C411yescharacteristic
78#14579143—21no—
79#14579141C2×C411yescharacteristic
80#14579132C2×C411yescharacteristic
81Q8×Dic79C2×C411yescharacteristic
82C4×Dic158—21no—
83C4×Dic158C2311yescharacteristic
84C4×Dic158C2311yescharacteristic
85#14579126—21no—
86#14579126C2×C411yescharacteristic
87C4×C22⋊D79—21no—
88#14579120—21no—
89C4⋊C4×D79—21no—
90C4⋊C4×D79C2×C411yescharacteristic
91C42×D79—21no—
92C42×D79C2311yescharacteristic
93#14579095C2×C411yescharacteristic
94#14579094C2×C411yescharacteristic
95#14579092C2×C411yescharacteristic
96C4⋊(C79⋊C8)—21no—
97C4⋊(C79⋊C8)C2311yescharacteristic
98C4⋊(C79⋊C8)C2311yescharacteristic
99Dic79⋊C8—21no—
100Dic79⋊C8—21no—
101Dic79⋊C8—21no—
102Dic79⋊C8—21no—
103Dic79⋊C8—21no—
104Dic79⋊C8—21no—
105Dic79⋊C8C2×C411yescharacteristic
106Dic79⋊C8C2×C411yescharacteristic
107#14579086C2×C411yescharacteristic
108#14579086C2×C411yescharacteristic
109#14579085C2×C411yescharacteristic
110#14579084C2×C411yescharacteristic
111#14579084C2×C411yescharacteristic
112#14579082C2×C411yescharacteristic
113C22⋊(C79⋊C8)—21no—
114(C2×D79)⋊C8—21no—
115(C2×D79)⋊C8—21no—
116(C2×D79)⋊C8C2×C411yescharacteristic
117(C2×D79)⋊C8C2×C411yescharacteristic
118#14579071—21no—
119#14579071—21no—
120#14579071—21no—
121#14579071D411yescharacteristic
122#14579071D411yescharacteristic
123C4×C79⋊C8—21no—
124C4×C79⋊C8C2311yescharacteristic
125C4×C79⋊C8C2311yescharacteristic
126C4×C79⋊C8C2311yescharacteristic
127C8×Dic79—21no—
128C8×Dic79—21no—
129C2×SD16×C79C2×C411yescharacteristic
130C22×C8×C79—21no—
131C4×Q8×C79C2311yescharacteristic
132C4×D4×C79C2311yescharacteristic
133#106×C79C2×C411yescharacteristic
134C4⋊C8×C79—21no—
135C4⋊C8×C79C2311yescharacteristic
136C4⋊C8×C79C2311yescharacteristic
137#103×C79C2×C411yescharacteristic
138#102×C79C2×C411yescharacteristic
139C22⋊C8×C79—21no—
140C4×C8×C79—21no—
141C4×C8×C79C2311yescharacteristic
142C4×C8×C79C2311yescharacteristic
143C4×C8×C79C2311yescharacteristic
144C2×Dic158—21no—
145C2×Dic158C22×C411yescharacteristic
146C2×Dic158C22×C411yescharacteristic
147Q8×D79—22no—
148Q8×D79—41no—
149C2×C22⋊D79—21no—
150C2×C4×D79—21no—
151C2×C4×D79—21no—
152C2×C4×D79C22×C411yescharacteristic
153C2×C4×D79C2×D411yescharacteristic
154C2×C4×D79#4111yescharacteristic
155Dic316—41no—
156#11929614—22no—
157#11929612—41no—
158#11929608—42no—
159#11929607—22no—
160#11929607—41no—
161#11929607—42no—
162C2×C79⋊C8—21no—
163C2×C79⋊C8—21no—
164C2×C79⋊C8—21no—
165C2×C79⋊C8C22×C411yescharacteristic
166C2×C79⋊C8C22×C411yescharacteristic
167C2×C79⋊C8C2×D411yescharacteristic
168C2×C79⋊C8C2×D411yescharacteristic
169C4⋊Dic79—21no—
170C4⋊Dic79—21no—
171C4⋊Dic79—21no—
172C4⋊Dic79C22×C411yescharacteristic
173C4⋊Dic79C22×C411yescharacteristic
174Dic79⋊C4—21no—
175Dic79⋊C4—21no—
176Dic79⋊C4—21no—
177Dic79⋊C4—21no—
178Dic79⋊C4—21no—
179C22⋊Dic79—21no—
180C22⋊Dic79—21no—
181C22⋊Dic79—21no—
182(C2×D79)⋊C4—21no—
183C4×Dic79—21no—
184C4×Dic79—21no—
185C4×Dic79—21no—
186C4×Dic79—21no—
187C4×Dic79C22×C411yescharacteristic
188C4×Dic79C2×D411yescharacteristic
189C4×Dic79#4111yescharacteristic
190C2×Q8×C79C22×C411yescharacteristic
191C2×D4×C79C22×C411yescharacteristic
192C22×C4×C79—21no—
193SD16×C79—22no—
194C2×C8×C79—21no—
195C2×C8×C79—21no—
196C2×C8×C79—21no—
197C2×C8×C79—42no—
198C2×C8×C79C22×C411yescharacteristic
199C2×C8×C79C22×C411yescharacteristic
200C2×C8×C79C2×D411yescharacteristic
201C2×C8×C79C2×D411yescharacteristic
202C4⋊C4×C79—21no—
203C4⋊C4×C79C22×C411yescharacteristic
204C4⋊C4×C79C22×C411yescharacteristic
205C22⋊C4×C79—21no—
206C42×C79—21no—
207C42×C79C2411yescharacteristic
208C23×C79—21no—
209C22×D79C4×D411yescharacteristic
210Q8×C79—21no—
211Q8×C79#13112yes—
212D4×C79—21no—
213D4×C79—21no—
214C2×C4×C79—21no—
215C2×C4×C79—21no—
216C2×C4×C79—21no—
217C2×C4×C79—41no—
218C2×C4×C79—41no—
219C2×C4×C79C23×C411yescharacteristic
220C2×C4×C79C22×D411yescharacteristic
221C2×C4×C79C2×#4111yescharacteristic
222C22⋊D79—81no—
223C2×Dic79—21no—
224C2×Dic79—21no—
225C2×Dic79—21no—
226C2×Dic79—21no—
227C2×Dic79C4×D411yescharacteristic
228C2×Dic79C4×D411yescharacteristic
229C2×Dic79C4×D411yescharacteristic
230C4×D79—21no—
231C4×D79—21no—
232C4×D79—21no—
233C4×D79—21no—
234C4×D79—41no—
235C4×D79—42no—
236C4×D79—81no—
237Dic158—21no—
238Dic158—21no—
239Dic158—21no—
240Dic158—21no—
241Dic158—42no—
242C8×C79—21no—
243C8×C79—21no—
244C8×C79—21no—
245C8×C79—41no—
246C8×C79—41no—
247C8×C79—41no—
248C8×C79#13712yes—
249C79⋊C8—21no—
250C79⋊C8—21no—
251C79⋊C8—21no—
252C79⋊C8—21no—
253C79⋊C8—41no—
254C79⋊C8—41no—
255C79⋊C8—41no—
256C22×C79—41no—
257C22×C79—41no—
258C22×C79—41no—
259C22×C79C2×C4×D411yescharacteristic
260C2×D79—21no—
261C2×D79—21no—
262C2×D79—21no—
263C4×C79—22no—
264C4×C79—21no—
265C4×C79—41no—
266C4×C79—41no—
267C4×C79C2×#13111yescharacteristic
268C4×C79#56811yescharacteristic
269C4×C79C2×#13711yescharacteristic
270C4×C79#57511yescharacteristic
271Dic79—21no—
272Dic79—21no—
273Dic79—21no—
274Dic79—41no—
275Dic79—41no—
276Dic79—41no—
277Dic79—41no—
278#18010—791nomaximal, Sylow
279C2×C79—81no—
280C2×C79#293311yescharacteristic
281C2×C79C4×#13711yescharacteristic
282C2×C79#295711yescharacteristic
283D79—41no—
284#3108—791no—
285#2969—791no—
286Q8×M16—791no—
287#1592—791no—
288#1591—791no—
289#1588—791no—
290#1587—791no—
291#1584—791no—
292#1582—791no—
293#1579—791no—
294C8×SD16D7911yescharacteristic, complemented
295#1574—791no—
296#1495—791no—
297#1479—791no—
298#1453—791no—
299C79#1801011yescharacteristic, complemented, Sylow
300#529—791no—
301C2×C4×Q8—791no—
302#483—791no—
303#480—791no—
304#479—791no—
305#479—791no—
306#464—791no—
307#464—791no—
308#451—791no—
309#448—791no—
310#446—791no—
311#446—791no—
312#446—791no—
313#446—1581no—
314#446—1581no—
315C8×Q8—791no—
316C8×Q8—791no—
317C8×Q8C2×D7911yescharacteristic
318C4×Q16—791no—
319C4×Q16—791no—
320C4×SD16—791no—
321C4×SD16C2×D7911yescharacteristic
322#436—791no—
323#436—1581no—
324#435—1581no—
325C8×D4C2×D7911yescharacteristic
326#433—1581no—
327#431—791no—
328#431—791no—
329#431—791no—
330#423—791no—
331#423—1581no—
332#405—1581no—
333C4×M16—791no—
334C4×M16—1581no—
335#335—791no—
336#335—791no—
337#334—791no—
338#334C2×D7911yescharacteristic
339Q8⋊C8—791no—
340Q8⋊C8—791no—
341Q8⋊C8—791no—
342Q8⋊C8—791no—
343Q8⋊C8—791no—
344Q8⋊C8C2×D7911yescharacteristic
345D4⋊C8—791no—
346D4⋊C8C2×D7911yescharacteristic
347#322—791no—
348#322—791no—
349#322—791no—
350C82C2×D7911yescharacteristic
351C22×Q8—791no—
352C2×Q16—791no—
353C2×Q16—791no—
354C2×SD16—791no—
355C2×SD16C4×D7911yescharacteristic
356C2×M16—791no—
357C2×M16—791no—
358C2×M16—1581no—
359C2×M16—1581no—
360C22×C8—21no—
361#126—1581no—
362#125—791no—
363#124—791no—
364#123—1581no—
365C22⋊Q8—791no—
366#121—791no—
367C4×Q8—791no—
368C4×Q8—791no—
369C4×Q8—791no—
370C4×Q8—791no—
371C4×Q8—1581no—
372C4×Q8—1581no—
373C4×Q8C22×D7911yescharacteristic
374C4×D4—1581no—
375C4×D4C22×D7911yescharacteristic
376#117—1581no—
377C2×C4⋊C4—791no—
378C2×C4⋊C4—1581no—
379C2×C42—791no—
380C2×C42—1581no—
381#107—791no—
382#107—791no—
383#106—791no—
384#106C4×D7911yescharacteristic
385C4⋊C8—21no—
386C4⋊C8—791no—
387C4⋊C8—791no—
388C4⋊C8—791no—
389C4⋊C8—791no—
390C4⋊C8—1581no—
391C4⋊C8—1581no—
392C4⋊C8—1581no—
393C4⋊C8—1581no—
394C4⋊C8—1581no—
395C4⋊C8—1581no—
396C4⋊C8—1581no—
397C4⋊C8C22×D7911yescharacteristic
398C4⋊C8C22×D7911yescharacteristic
399#103—791no—
400#103—791no—
401#103—791no—
402#103—791no—
403#103—791no—
404#103C4×D7911yescharacteristic
405#102—791no—
406#102C4×D7911yescharacteristic
407C22⋊C8—21no—
408C22⋊C8—791no—
409C22⋊C8—791no—
410C22⋊C8—1581no—
411C22⋊C8—1581no—
412C22⋊C8—1581no—
413#97—791no—
414#97—791no—
415#97—1581no—
416#97—1581no—
417#97—1581no—
418C4×C8—21no—
419C4×C8—791no—
420C4×C8—791no—
421C4×C8—791no—
422C4×C8—1581no—
423C4×C8—1581no—
424C4×C8—1581no—
425C4×C8C22×D7911yescharacteristic
426C4×C8C22×D7911yescharacteristic
427C4×C8C22×D7911yescharacteristic
428C2×Q8—3161no—
429C2×Q8—791no—
430C2×Q8—791no—
431C2×Q8—1581no—
432C2×Q8—1582no—
433C2×Q8C2×C4×D7911yescharacteristic
434C2×D4—1581no—
435C2×D4C2×C4×D7911yescharacteristic
436C22×C4—21no—
437C22×C4—791no—
438C22×C4—791no—
439C22×C4—791no—
440C22×C4—1581no—
441C22×C4—1581no—
442Q16—3161no—
443Q16—1582no—
444SD16—22no—
445SD16—3161no—
446M16—3161no—
447M16—3162no—
448M16—3162no—
449M16—1582no—
450C2×C8—21no—
451C2×C8—21no—
452C2×C8—21no—
453C2×C8—42no—
454C2×C8—791no—
455C2×C8—791no—
456C2×C8—791no—
457C2×C8—791no—
458C2×C8—1581no—
459C2×C8—1581no—
460C2×C8—1581no—
461C2×C8C2×C4×D7911yescharacteristic
462C2×C8C2×C4×D7911yescharacteristic
463C2×C8D4×D7911yescharacteristic
464C2×C8D4×D7911yescharacteristic
465C4⋊C4—21no—
466C4⋊C4—791no—
467C4⋊C4—791no—
468C4⋊C4—1581no—
469C4⋊C4—1581no—
470C4⋊C4—1581no—
471C4⋊C4—1581no—
472C4⋊C4—1581no—
473C4⋊C4—1581no—
474C4⋊C4—1581no—
475C4⋊C4—1581no—
476C4⋊C4C2×C4×D7911yescharacteristic
477C4⋊C4C2×C4×D7911yescharacteristic
478C22⋊C4—21no—
479C22⋊C4—1581no—
480C22⋊C4—1581no—
481C22⋊C4—1581no—
482C22⋊C4—1581no—
483C42—21no—
484C42—791no—
485C42—791no—
486C42—791no—
487C42—1581no—
488C42—1581no—
489C42—1581no—
490C42—1581no—
491C42C23×D7911yescharacteristic
492C23—21no—
493C23—791no—
494Q8—21no—
495Q8—3162no—
496Q8—1581no—
497Q8—1581no—
498Q8—1581no—
499Q8—1581no—
500Q8#1457916712yes—
501D4—21no—
502D4—21no—
503D4—6321no—
504C2×C4—21no—
505C2×C4—21no—
506C2×C4—21no—
507C2×C4—41no—
508C2×C4—41no—
509C2×C4—3162no—
510C2×C4—3161no—
511C2×C4—791no—
512C2×C4—791no—
513C2×C4—791no—
514C2×C4—6321no—
515C2×C4—1581no—
516C2×C4—1581no—
517C2×C4—1581no—
518C2×C4—1581no—
519C2×C4—1581no—
520C2×C4—1581no—
521C2×C4—1581no—
522C2×C4—1581no—
523C2×C4C22×C4×D7911yescharacteristic
524C2×C4C2×D4×D7911yescharacteristic
525C2×C4#41×D7911yescharacteristic
526C8—21no—
527C8—21no—
528C8—21no—
529C8—41no—
530C8—41no—
531C8—41no—
532C8—3161no—
533C8—3161no—
534C8—3161no—
535C8—1581no—
536C8—1581no—
537C8—1581no—
538C8—1581no—
539C8#1457919012yes—
540C22—41no—
541C22—41no—
542C22—41no—
543C22—1581no—
544C22—1581no—
545C22—1581no—
546C22C4×D4×D7911yescharacteristic
547C4—21no—
548C4—22no—
549C4—41no—
550C4—41no—
551C4—3161no—
552C4—3161no—
553C4—3161no—
554C4—3161no—
555C4—1581no—
556C4—1581no—
557C4—1581no—
558C4C2×#1457916711yescharacteristic
559C4#1909932811yescharacteristic
560C4C2×#1457919011yescharacteristic
561C4#1909936011yescharacteristic
562C2—81no—
563C2—3161no—
564C2#2377212011yescharacteristic
565C2C4×#1457919011yescharacteristic
566C2#2377231211yescharacteristic
subgroupG/H
centerC2×C4C2×D4×D79
commutatorC2×C4×C79C23×C4
FrattiniC42C23×D79
FittingC8×SD16×C79C2
socleC22×C79C2×C4×D4
prime radical#14579132C2×C4
cosocleC42×C79C24
hypercenterC8×SD16D79
hypocenterC79#18010
LCSG ▷ C2×C4×C79 ▷ C2×C79 ▷ C79
UCSC1 ◁ C2×C4 ◁ C4×C8 ◁ C8×SD16

Complements

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

NH ≅ G/Ntype
3. Q8×#11929607C2semidirect
4. #23762299C2semidirect
5. #23762298C2semidirect
6. #23762279C2semidirect
7. #23762270C2semidirect
8. #23762250C2semidirect
9. #23762236C2semidirect
10. #23762216C2semidirect
11. #23762187C2semidirect
13. #23761793C2semidirect
14. #23761694C2semidirect
15. C8×SD16×C79C2semidirect
294. C8×SD16D79semidirect
299. C79#18010semidirect

Maximal subgroups

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

subgroupindex
#237734782
#237724012
Q8×#119296072
#237622992
#237622982
#237622792
#237622702
#237622502
#237622362
#237622162
#237621872
#237618622
#237617932
#237616942
C8×SD16×C792
#1801079