Finite Group Databasebeta

← #34016089

Subgroups — #34016089

Order 18688.

984 conjugacy classes of subgroups.

The subgroup lattice has 540 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22626786C211yesmaximal, characteristic, complemented
2#22626781C211yesmaximal, characteristic, complemented
3#22625213C212yesmaximal, complemented
4#22625211C212yesmaximal, complemented
5#22625203C212yesmaximal, complemented
6#22625197C212yesmaximal, complemented
7#22625166C212yesmaximal, complemented
8#22625160C212yesmaximal, complemented
9C2×#17989183C211yesmaximal, characteristic
10#17989617—24no—
11#17989615—24no—
12#17989607C2212yes—
13#17989603C2212yes—
14#17989558C2212yes—
15#17989555C2212yes—
16#17989555C2212yes—
17#17989555C2212yes—
18#17989433C2214yes—
19#17989433C2214yes—
20#17989432C2214yes—
21#17989430C2214yes—
22#17989183—24no—
23#17989183—24no—
24C2×#14556586C411yescharacteristic
25C2×#14556584C2211yescharacteristic
26C2×#14556581C411yescharacteristic
27C2×#14556581C2211yescharacteristic
28C2×#14556576C411yescharacteristic
29C2×#14556574C411yescharacteristic
30C2×#14556573C411yescharacteristic
31C2×#14556573C2211yescharacteristic
32C2×#14556567C411yescharacteristic
33C2×#14556567C2211yescharacteristic
34C2×#14556564C411yescharacteristic
35C2×#14556553C2211yescharacteristic
36C2×C4×C22⋊D73C2211yescharacteristic
37C2×C4⋊C4×D73C2211yescharacteristic
38C2×C22⋊Q8×C73C411yescharacteristic
39C2×#11924038—22no—
40C2×#11924036—22no—
41C2×#11924036—22no—
42C2×#11924036—22no—
43#14556658—22no—
44C4⋊#10596833—22no—
45#14556648—22no—
46#14556648—22no—
47#14556648—22no—
48#14556648—22no—
49C22⋊#10596833C2314yes—
50C22⋊#10596833C2314yes—
51#14556646—22no—
52C4×#10596833—22no—
53C22×D292C2×C411yescharacteristic
54C22×C22⋊D73C2×C411yescharacteristic
55C22×C22⋊D73C2×C411yescharacteristic
56C22×C4×D73C2×C411yescharacteristic
57C22×C4×D73C2×C411yescharacteristic
58C22×C4×D73C2311yescharacteristic
59C22×C4×D73C2311yescharacteristic
60#14556586—24no—
61#14556584—22no—
62#14556584—22no—
63#14556581—22no—
64#14556581—22no—
65#14556581—24no—
66#14556576—24no—
67#14556574—24no—
68#14556573—22no—
69#14556573—22no—
70#14556573—24no—
71#14556567—22no—
72#14556567—22no—
73#14556567—24no—
74#14556564—24no—
75#14556553D414yes—
76#14556553D414yes—
77C4×C22⋊D73—22no—
78C4×C22⋊D73—22no—
79C2×C4⋊Dic73C2×C411yescharacteristic
80C2×C4⋊Dic73C2×C411yescharacteristic
81C2×C4⋊Dic73C2311yescharacteristic
82C2×#11924007C2×C411yescharacteristic
83C2×#11924007C2×C411yescharacteristic
84C2×#11924007C2×C411yescharacteristic
85C2×#11924007C2×C411yescharacteristic
86C2×#11924007C2×C411yescharacteristic
87C2×#11924007C2×C411yescharacteristic
88C2×#11924007C2311yescharacteristic
89C2×#11924007C2311yescharacteristic
90C4⋊C4×D73—22no—
91C4⋊C4×D73—22no—
92C2×C22⋊Dic73C2×C411yescharacteristic
93C2×C22⋊Dic73C2311yescharacteristic
94C2×#11924005C2×C411yescharacteristic
95C2×#11924005C2×C411yescharacteristic
96C2×#11924005C2×C411yescharacteristic
97C2×#11924005C2×C411yescharacteristic
98C2×#11924005C2×C411yescharacteristic
99C2×#11924005C2×C411yescharacteristic
100C2×C4×Dic73C2×C411yescharacteristic
101C2×C4×Dic73C2311yescharacteristic
102C22×Q8×C73C2×C411yescharacteristic
103C23×C4×C73C2×C411yescharacteristic
104C22⋊Q8×C73—24no—
105C2×C4⋊C4×C73C2×C411yescharacteristic
106C2×C4⋊C4×C73C2×C411yescharacteristic
107C2×C4⋊C4×C73C2×C411yescharacteristic
108C2×C22⋊C4×C73C2×C411yescharacteristic
109C2×C22⋊C4×C73C2×C411yescharacteristic
110#11924038—82no—
111#11924036—82no—
112#11924036—82no—
113#11924036—82no—
114C2×#10596833—24no—
115C2×#10596833—24no—
116C23×D73C22×C411yescharacteristic
117C23×D73C22×C411yescharacteristic
118C2×D292—24no—
119C2×D292C22⋊C414yes—
120C2×C22⋊D73—22no—
121C2×C22⋊D73—24no—
122C2×C22⋊D73—22no—
123C2×C22⋊D73—81no—
124C22×Dic73C22×C411yescharacteristic
125C22×Dic73C22×C411yescharacteristic
126C22×Dic73C22×C411yescharacteristic
127C22×Dic73C22×C411yescharacteristic
128C22×Dic73C22×C411yescharacteristic
129C22×Dic73C2411yescharacteristic
130C2×C4×D73—22no—
131C2×C4×D73—24no—
132C2×C4×D73—22no—
133C2×C4×D73—81no—
134C2×C4×D73—81no—
135C2×C4×D73—81no—
136C2×C4×D73C2×D412yes—
137C2×C4×D73C2×D412yes—
138C2×C4×D73C2×D412yes—
139C2×C4×D73C2×D412yes—
140C4⋊Dic73—22no—
141C4⋊Dic73—22no—
142C4⋊Dic73—21no—
143C4⋊Dic73—21no—
144#11924007—22no—
145#11924007—22no—
146#11924007—22no—
147#11924007—22no—
148#11924007—22no—
149#11924007—22no—
150#11924007—21no—
151#11924007—21no—
152#11924007—21no—
153#11924007—21no—
154C22⋊Dic73—22no—
155C22⋊Dic73—21no—
156C22⋊Dic73—21no—
157#11924005—22no—
158#11924005—22no—
159#11924005—22no—
160#11924005—22no—
161#11924005C22⋊C414yes—
162#11924005C22⋊C414yes—
163C4×Dic73—22no—
164C4×Dic73C2×D412yes—
165C4×Dic73C2×D412yes—
166C24×C73C22×C411yescharacteristic
167C2×Q8×C73—22no—
168C2×Q8×C73—81no—
169C22×C4×C73—22no—
170C22×C4×C73—81no—
171C22×C4×C73C22×C411yescharacteristic
172C22×C4×C73C22×C411yescharacteristic
173C22×C4×C73C22×C411yescharacteristic
174C22×C4×C73C22×C411yescharacteristic
175C22×C4×C73C22×C411yescharacteristic
176C22×C4×C73C22×C411yescharacteristic
177C4⋊C4×C73—22no—
178C4⋊C4×C73—22no—
179C4⋊C4×C73C22⋊C414yes—
180C22⋊C4×C73—22no—
181C22⋊C4×C73—22no—
182C23×C73—21no—
183C23×C73—22no—
184C23×C73—41no—
185C23×C73—41no—
186C23×C73C23×C411yescharacteristic
187C22×D73—22no—
188C22×D73—41no—
189C22×D73—41no—
190C22×D73—41no—
191C22×D73—41no—
192C22×D73C2×C22⋊C412yes—
193C22×D73C2×C22⋊C412yes—
194C22×D73#13114yes—
195Q8×C73—82no—
196C2×C4×C73—22no—
197C2×C4×C73—21no—
198C2×C4×C73—21no—
199C2×C4×C73—21no—
200C2×C4×C73—22no—
201C2×C4×C73—41no—
202C2×C4×C73—41no—
203C2×C4×C73—41no—
204C2×C4×C73—41no—
205C2×C4×C73—82no—
206C2×C4×C73C2×C22⋊C412yes—
207C2×C4×C73C2×C22⋊C412yes—
208C2×C4×C73C2×C22⋊C412yes—
209C22⋊D73—44no—
210C22⋊D73—82no—
211C2×Dic73—21no—
212C2×Dic73—21no—
213C2×Dic73—21no—
214C2×Dic73—21no—
215C2×Dic73—22no—
216C2×Dic73—22no—
217C2×Dic73—41no—
218C2×Dic73—41no—
219C2×Dic73—41no—
220C2×Dic73—41no—
221C2×Dic73C2×C22⋊C412yes—
222C2×Dic73C22×D411yescharacteristic
223C2×Dic73C22×D411yescharacteristic
224D292—28no—
225C4×D73—42no—
226C4×D73—42no—
227C4×D73—42no—
228C4×D73—42no—
229C4×D73—44no—
230C4×D73—82no—
231#10596833—44no—
232#10596833—44no—
233C22×C73—21no—
234C22×C73—21no—
235C22×C73—41no—
236C22×C73—41no—
237C22×C73—41no—
238C22×C73—82no—
239C22×C73C22×C22⋊C411yescharacteristic
240C22×C73C2×#13112yes—
241C2×D73—22no—
242C2×D73—22no—
243C2×D73—22no—
244C2×D73—22no—
245C2×D73—22no—
246C2×D73—44no—
247C2×D73—82no—
248C2×D73#40814yes—
249C4×C73—22no—
250C4×C73—22no—
251C4×C73—22no—
252C4×C73—41no—
253C4×C73—41no—
254C4×C73—41no—
255Dic73—21no—
256Dic73—21no—
257Dic73—22no—
258Dic73—41no—
259Dic73—41no—
260Dic73—41no—
261Dic73—41no—
262#17387—731nomaximal, Sylow
263C2×C73—22no—
264C2×C73—81no—
265C2×C73C2×#40811yescharacteristic
266C2×C73#288311yescharacteristic
267C2×C73#288911yescharacteristic
268D73—24no—
269D73—42no—
270C2×#539—731no—
271#2973—731no—
272#2972—731no—
273#1473—732no—
274#1472—732no—
275#1470—732no—
276#1468—732no—
277#1459—732no—
278C2×#324—732no—
279C73#1738711yescharacteristic, complemented, Sylow
280#539—1464no—
281#539—1464no—
282C2×#126—731no—
283C2×#126—731no—
284C2×#125—731no—
285C2×#125—731no—
286C2×#123—731no—
287C2×#123—731no—
288C2×#123—731no—
289C2×#123—731no—
290C2×C22⋊Q8—731no—
291C2×C22⋊Q8—731no—
292C2×C22⋊Q8#6440811yescharacteristic
293C2×#121—731no—
294C2×C4×D4—731no—
295C2×C4×D4—731no—
296C22×C4⋊C4—731no—
297#435—1464no—
298C8×D4—1464no—
299#432—732no—
300#431—732no—
301#407—732no—
302C2×C22⋊C8—732no—
303C2×C22⋊C8—732no—
304C2×C22⋊C8—732no—
305#324—734no—
306#324—734no—
307#324—734no—
308#323—734no—
309C22×Q8C2×#6440811yescharacteristic
310C22×D4—731no—
311C22×D4—731no—
312C22×D4—731no—
313C23×C4—731no—
314C23×C4—731no—
315C23×C4—731no—
316C23×C4—731no—
317C23×C4C2×#6440811yescharacteristic
318C2×M16—1462no—
319C22×C8—1462no—
320C22×C8—1462no—
321C22×C8—1462no—
322#126—1462no—
323#126—1462no—
324#126—1464no—
325#125—1462no—
326#125—1462no—
327#125—1464no—
328#123—1464no—
329#123—1462no—
330#123—1462no—
331#123—1464no—
332#123—1464no—
333C22⋊Q8—24no—
334C22⋊Q8—1464no—
335C22⋊Q8—1462no—
336C22⋊Q8—1462no—
337#121—1464no—
338C4×D4—734no—
339C4×D4—734no—
340C4×D4—1462no—
341C4×D4—1462no—
342C2×C4⋊C4—731no—
343C2×C4⋊C4—731no—
344C2×C4⋊C4—731no—
345C2×C4⋊C4—731no—
346C2×C4⋊C4—731no—
347C2×C4⋊C4—731no—
348C2×C4⋊C4—731no—
349C2×C4⋊C4—731no—
350C2×C4⋊C4—731no—
351C2×C4⋊C4—731no—
352C2×C4⋊C4—731no—
353C2×C4⋊C4—1462no—
354C2×C4⋊C4—1462no—
355C2×C4⋊C4C2×#6440811yescharacteristic
356C2×C4⋊C4C2×#6440811yescharacteristic
357C2×C4⋊C4C2×#6440811yescharacteristic
358C2×C22⋊C4—731no—
359C2×C22⋊C4—731no—
360C2×C22⋊C4—731no—
361C2×C22⋊C4—731no—
362C2×C22⋊C4—731no—
363C2×C22⋊C4—731no—
364C2×C22⋊C4—731no—
365C2×C22⋊C4—731no—
366C2×C22⋊C4C2×#6440811yescharacteristic
367C2×C22⋊C4C2×#6440811yescharacteristic
368C2×C42—731no—
369C2×C42—731no—
370C4⋊C8—1462no—
371C4⋊C8—1462no—
372C22⋊C8—734no—
373C22⋊C8—734no—
374C22⋊C8—1462no—
375C22⋊C8—1462no—
376C22⋊C8—1462no—
377C22⋊C8—1462no—
378#97—1462no—
379C4×C8—1462no—
380C24—731no—
381C24—731no—
382C24C22×#6440811yescharacteristic
383C2×Q8—22no—
384C2×Q8—81no—
385C2×D4—5841no—
386C2×D4—734no—
387C2×D4—1464no—
388C2×D4—1462no—
389C2×D4—1464no—
390C2×D4—1462no—
391C22×C4—22no—
392C22×C4—81no—
393C22×C4—5841no—
394C22×C4—5841no—
395C22×C4—5841no—
396C22×C4—731no—
397C22×C4—731no—
398C22×C4—732no—
399C22×C4—732no—
400C22×C4—732no—
401C22×C4—732no—
402C22×C4—731no—
403C22×C4—731no—
404C22×C4—731no—
405C22×C4—731no—
406C22×C4—1462no—
407C22×C4—1464no—
408C22×C4—1462no—
409C22×C4C22×#6440811yescharacteristic
410C22×C4C22×#6440811yescharacteristic
411C22×C4C22×#6440811yescharacteristic
412C22×C4C22×#6440811yescharacteristic
413C22×C4C22×#6440811yescharacteristic
414C22×C4C22×#6440811yescharacteristic
415M16—5842no—
416C2×C8—5842no—
417C2×C8—5842no—
418C2×C8—5842no—
419C2×C8—1464no—
420C2×C8—1464no—
421C4⋊C4—22no—
422C4⋊C4—22no—
423C4⋊C4—1461no—
424C4⋊C4—1461no—
425C4⋊C4—1461no—
426C4⋊C4—1461no—
427C4⋊C4—1462no—
428C4⋊C4—1462no—
429C4⋊C4—1461no—
430C4⋊C4—1461no—
431C4⋊C4—1462no—
432C4⋊C4—1462no—
433C4⋊C4—1462no—
434C4⋊C4—1462no—
435C4⋊C4—1462no—
436C4⋊C4—1462no—
437C4⋊C4C22⋊#6440814yes—
438C22⋊C4—22no—
439C22⋊C4—22no—
440C22⋊C4—734no—
441C22⋊C4—734no—
442C22⋊C4—1462no—
443C22⋊C4—1462no—
444C22⋊C4—1462no—
445C22⋊C4—1462no—
446C22⋊C4—1461no—
447C22⋊C4—1461no—
448C22⋊C4—1462no—
449C42—732no—
450C42—732no—
451C42—1462no—
452C23—21no—
453C23—22no—
454C23—41no—
455C23—41no—
456C23—2921no—
457C23—2921no—
458C23—2921no—
459C23—2921no—
460C23—734no—
461C23—732no—
462C23—732no—
463C23—1462no—
464C23C23×#6440811yescharacteristic
465Q8—82no—
466D4—2924no—
467D4—5842no—
468D4—1468no—
469C2×C4—22no—
470C2×C4—21no—
471C2×C4—21no—
472C2×C4—21no—
473C2×C4—22no—
474C2×C4—41no—
475C2×C4—41no—
476C2×C4—41no—
477C2×C4—41no—
478C2×C4—82no—
479C2×C4—2922no—
480C2×C4—2922no—
481C2×C4—2922no—
482C2×C4—2922no—
483C2×C4—2924no—
484C2×C4—2921no—
485C2×C4—2921no—
486C2×C4—2921no—
487C2×C4—2921no—
488C2×C4—5842no—
489C2×C4—732no—
490C2×C4—731no—
491C2×C4—731no—
492C2×C4—1461no—
493C2×C4—1461no—
494C2×C4—1461no—
495C2×C4—1461no—
496C2×C4—1462no—
497C2×C4—1462no—
498C2×C4C2×C22⋊#6440812yes—
499C2×C4C2×C22⋊#6440812yes—
500C2×C4C2×C22⋊#6440812yes—
501C8—2924no—
502C8—2924no—
503C22—21no—
504C22—21no—
505C22—41no—
506C22—41no—
507C22—41no—
508C22—82no—
509C22—2924no—
510C22—5842no—
511C22—734no—
512C22—1462no—
513C22—1462no—
514C22—1462no—
515C22—1462no—
516C22—1462no—
517C22C22×C22⋊#6440811yescharacteristic
518C22D73⋊#13112yes—
519C4—22no—
520C4—22no—
521C4—22no—
522C4—41no—
523C4—41no—
524C4—41no—
525C4—2921no—
526C4—2921no—
527C4—2921no—
528C4—2921no—
529C4—1461no—
530C4—1461no—
531C4—1462no—
532C2—22no—
533C2—81no—
534C2—2922no—
535C2—1464no—
536C2D73⋊#40811yescharacteristic
537C2#2262653011yescharacteristic
538C2#2262654411yescharacteristic
subgroupG/H
centerC22C22×C22⋊#64408
commutatorC23×C73C23×C4
FrattiniC23C23×#64408
FittingC2×C22⋊Q8×C73C4
socleC22×C73C22×C22⋊C4
prime radicalC2×#14556564C4
cosocleC22×Dic73C24
hypercenterC2×C22⋊Q8#64408
hypocenterC73#17387
LCSG ▷ C23×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C22 ◁ C2×C4⋊C4 ◁ C2×C22⋊Q8

Complements

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

NH ≅ G/Ntype
1. #22626786C2semidirect
2. #22626781C2semidirect
3. #22625213C2semidirect
4. #22625211C2semidirect
5. #22625203C2semidirect
6. #22625197C2semidirect
7. #22625166C2semidirect
8. #22625160C2semidirect
279. C73#17387semidirect

Maximal subgroups

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

subgroupindex
#226267862
#226267812
#226252132
#226252112
#226252032
#226251972
#226251662
#226251602
C2×#179891832
#1738773