Finite Group Databasebeta

← #34018271

Subgroups — #34018271

Order 18688.

734 conjugacy classes of subgroups.

The subgroup lattice has 500 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22626735C211yesmaximal, characteristic, complemented
2#41×#10596833C211yesmaximal, characteristic, complemented
3#22625528C212yesmaximal, complemented
4#22625525C212yesmaximal, complemented
5#22625522C212yesmaximal, complemented
6#22625521C212yesmaximal, complemented
7#22625480C211yesmaximal, characteristic, complemented
8#22625224C211yesmaximal, characteristic, complemented
9#22625222C211yesmaximal, characteristic, complemented
10C2×#17989428C211yesmaximal, characteristic, complemented
11#22618699C211yesmaximal, characteristic
12#17989636C2212yes—
13Q8×#10596833C2212yes—
14#17989615C2212yes—
15#17989614—22no—
16D4×#10596833—22no—
17D4×#10596833C2212yes—
18#17989605—21no—
19#17989603—21no—
20#17989603C2211yescharacteristic
21#17989603C2211yescharacteristic
22C4×#11924036—21no—
23C2×C4×#10596833—21no—
24C2×C4×#10596833C2211yescharacteristic
25C2×C4×#10596833C2211yescharacteristic
26#17989451C2212yes—
27#17989449C2212yes—
28#17989437C2212yes—
29Q8⋊#10596833C2212yes—
30#17989435C2212yes—
31D4⋊#10596833C2212yes—
32#17989428C2212yes—
33#17989428C2212yes—
34C2×#14556611C411yescharacteristic
35#41×Dic73C2211yescharacteristic
36#17989055C2211yescharacteristic
37#17988679C411yescharacteristic
38#17988679C2211yescharacteristic
39#17988678C411yescharacteristic
40#17988678C2211yescharacteristic
41#17988672C411yescharacteristic
42#17988672C2211yescharacteristic
43#17988667C411yescharacteristic
44#17988667C2211yescharacteristic
45#17988623C411yescharacteristic
46#17988584C411yescharacteristic
47C2×C8×Dic73C2211yescharacteristic
48#416×C73C411yescharacteristic
49C2×#11924036—22no—
50C22×#10596833—22no—
51C22×#10596833—21no—
52C22×#10596833D412yes—
53#14556658—22no—
54C4⋊#10596833—22no—
55C4⋊#10596833C2312yes—
56C4⋊#10596833C2312yes—
57#14556648—22no—
58C22⋊#10596833—22no—
59C22⋊#10596833—21no—
60C22⋊#10596833—21no—
61C4×#10596833—22no—
62C4×#10596833—22no—
63C4×#10596833—21no—
64C4×#10596833—21no—
65C4×#10596833—21no—
66C4×#10596833D411yescharacteristic
67C4×#10596833D411yescharacteristic
68C4×#10596833C2312yes—
69C4×#10596833C2312yes—
70C2×#11924028C2×C411yescharacteristic
71#14556611C818yescomplemented
72C2×#11924018C2×C411yescharacteristic
73C2×#11924016C2×C411yescharacteristic
74C2×#11924015C2×C411yescharacteristic
75C2×#11924013C2×C411yescharacteristic
76C22×#10596831C2×C411yescharacteristic
77Q8×Dic73C2×C411yescharacteristic
78Q8×Dic73C2311yescharacteristic
79#14556556C2×C411yescharacteristic
80#14556556C2311yescharacteristic
81D4×Dic73—21no—
82D4×Dic73—21no—
83D4×Dic73C2×C411yescharacteristic
84D4×Dic73C2311yescharacteristic
85#14556553C2×C411yescharacteristic
86#14556553C2311yescharacteristic
87#14556551—21no—
88#14556551—21no—
89#14556550—21no—
90#14556550C2×C411yescharacteristic
91#14556548—21no—
92C2×C4×Dic73—21no—
93C2×C4×Dic73C2311yescharacteristic
94C42×D73—21no—
95#14556524C2×C411yescharacteristic
96#14556524C2×C411yescharacteristic
97#14556522C2×C411yescharacteristic
98#14556522C2×C411yescharacteristic
99#14556516C2×C411yescharacteristic
100#14556516C2×C411yescharacteristic
101#14556513C2×C411yescharacteristic
102#14556513C2×C411yescharacteristic
103C8×Dic73C2×C411yescharacteristic
104C8×Dic73C2×C411yescharacteristic
105C8×Dic73C2311yescharacteristic
106C8×Dic73C2311yescharacteristic
107C2×#41×C73C2×C411yescharacteristic
108C22×C8×C73C2×C411yescharacteristic
109#117×C73C2×C411yescharacteristic
110#103×C73C2×C411yescharacteristic
111#103×C73C2×C411yescharacteristic
112#102×C73C2×C411yescharacteristic
113#102×C73C2×C411yescharacteristic
114#11924036—44no—
115C2×#10596833—22no—
116C2×#10596833—22no—
117C2×#10596833—22no—
118C2×#10596833—42no—
119C2×#10596833—44no—
120C2×#10596833—42no—
121C2×#10596833C2×D412yes—
122C2×#10596833C2×D412yes—
123#11924028—22no—
124#11924028C2×C814yescomplemented
125C2×Dic146C22×C411yescharacteristic
126C2×D292C22×C411yescharacteristic
127C2×C22⋊D73—21no—
128C22×Dic73—21no—
129C22×Dic73—21no—
130C22×Dic73C2×D411yescharacteristic
131C22×Dic73#4111yescharacteristic
132C2×C4×D73—21no—
133C2×C4×D73—21no—
134C2×C4×D73—21no—
135#11924018C2×C814yescomplemented
136#11924016C2×C814yescomplemented
137#11924015C2×C814yescomplemented
138#11924013C2×C814yescomplemented
139C2×#10596831C2×C814yescomplemented
140C2×#10596831C22×C411yescharacteristic
141C2×#10596831C22×C411yescharacteristic
142C4⋊Dic73—21no—
143C4⋊Dic73—21no—
144C4⋊Dic73C22×C411yescharacteristic
145C4⋊Dic73C22×C411yescharacteristic
146#11924007—21no—
147#11924007—21no—
148C22⋊Dic73—21no—
149C22⋊Dic73—21no—
150C22⋊Dic73—21no—
151#11924005—21no—
152#11924005—21no—
153C4×Dic73—21no—
154C4×Dic73—21no—
155C4×Dic73—21no—
156C4×Dic73—21no—
157C4×Dic73—21no—
158C4×Dic73C22×C411yescharacteristic
159C4×Dic73C2×D411yescharacteristic
160C4×Dic73#4111yescharacteristic
161C4×Dic73C2411yescharacteristic
162#41×C73—22no—
163#41×C73C2×C814yescomplemented
164C2×Q8×C73C22×C411yescharacteristic
165C2×D4×C73—21no—
166C2×D4×C73C22×C411yescharacteristic
167C22×C4×C73—21no—
168C22×C4×C73C22×C411yescharacteristic
169C2×C8×C73—22no—
170C2×C8×C73C22×C411yescharacteristic
171C2×C8×C73C22×C411yescharacteristic
172C4⋊C4×C73C22×C411yescharacteristic
173C4⋊C4×C73C22×C411yescharacteristic
174C22⋊C4×C73—21no—
175C42×C73—21no—
176C23×C73—21no—
177C23×C73C4×D411yescharacteristic
178C22×D73—21no—
179Q8×C73—21no—
180Q8×C73C22×C812yes—
181D4×C73—24no—
182D4×C73—21no—
183D4×C73C22×C812yes—
184C2×C4×C73—21no—
185C2×C4×C73—24no—
186C2×C4×C73—21no—
187C2×C4×C73—21no—
188C2×C4×C73C4×D411yescharacteristic
189C2×C4×C73C22×C812yes—
190C2×C4×C73#13112yes—
191C2×C4×C73C23×C411yescharacteristic
192C22⋊D73—24no—
193C2×Dic73—21no—
194C2×Dic73—22no—
195C2×Dic73—22no—
196C2×Dic73—21no—
197C2×Dic73—21no—
198C2×Dic73—42no—
199C2×Dic73—42no—
200C2×Dic73C4×D411yescharacteristic
201C2×Dic73C4×D411yescharacteristic
202C2×Dic73C22×D411yescharacteristic
203C2×Dic73C2×#4111yescharacteristic
204D292—21no—
205D292C22×C812yes—
206C4×D73—24no—
207C4×D73—42no—
208C4×D73—42no—
209Dic146—21no—
210Dic146C22×C812yes—
211#10596833—22no—
212#10596833—22no—
213#10596833—42no—
214#10596833—42no—
215#10596833—42no—
216C8×C73—21no—
217C8×C73—21no—
218#10596831C22×C812yes—
219#10596831C22×C812yes—
220C22×C73—21no—
221C22×C73—21no—
222C22×C73—22no—
223C22×C73—41no—
224C22×C73—41no—
225C22×C73C8×D412yes—
226C22×C73C2×C4×D411yescharacteristic
227C2×D73—22no—
228C2×D73—41no—
229C2×D73—41no—
230C4×C73—22no—
231C4×C73—41no—
232C4×C73—41no—
233C4×C73C8×D412yes—
234C4×C73C23×C811yescharacteristic
235C4×C73C2×#13111yescharacteristic
236Dic73—22no—
237Dic73—21no—
238Dic73—21no—
239Dic73—41no—
240Dic73—41no—
241Dic73#44412yes—
242Dic73#57512yes—
243#17849—731nomaximal, Sylow
244C2×C73—22no—
245C2×C73—42no—
246C2×C73C2×C8×D411yescharacteristic
247C2×C73#294311yescharacteristic
248C2×C73C2×#44411yescharacteristic
249D73—42no—
250C8×#41—731no—
251C8×#41—731no—
252C4×#135—731no—
253#1580—732no—
254#1579—732no—
255SD16⋊C8—732no—
256SD16⋊C8—732no—
257#1564—731no—
258#1476—731no—
259#1476—731no—
260C2×#322—731no—
261C73#1784911yescharacteristic, complemented, Sylow
262C2×#135—731no—
263C4×#41—731no—
264C4×#41—731no—
265C8×Q8—732no—
266C8×Q8—732no—
267C4×Q16—731no—
268C4×Q16—731no—
269C4×SD16—731no—
270C4×SD16—731no—
271C4×SD16—731no—
272C4×SD16—731no—
273C4×D8—731no—
274C4×D8—731no—
275C8×D4—732no—
276C8×D4—732no—
277C8×D4—1462no—
278C8×D4—1462no—
279#431—731no—
280#431—731no—
281#431—1461no—
282#431—1461no—
283#427—731no—
284#416—731no—
285#416#6440811yescharacteristic
286C2×C4×C8—731no—
287C2×C4×C8—731no—
288C2×C4×C8—731no—
289C2×C4×C8—1461no—
290C2×C4×C8—1461no—
291#335—732no—
292#334—732no—
293Q8⋊C8—732no—
294Q8⋊C8—732no—
295D4⋊C8—732no—
296D4⋊C8—732no—
297#322—732no—
298#322—732no—
299C2×#41—731no—
300C2×#41C2×#6440811yescharacteristic
301#135—738no—
302C2×Q16—731no—
303C2×SD16—731no—
304C2×SD16—731no—
305C2×D8—731no—
306C22×C8—732no—
307C22×C8—731no—
308C22×C8—1462no—
309C22×C8—1462no—
310C22×C8—1461no—
311C22×C8C2×#6440811yescharacteristic
312C4×Q8—731no—
313C4×Q8—731no—
314C4×Q8—731no—
315C4×Q8—731no—
316C4×D4—731no—
317C4×D4—731no—
318C4×D4—731no—
319C4×D4—731no—
320C4×D4—1461no—
321C4×D4—1461no—
322C4×D4—1461no—
323C4×D4—1461no—
324#117—731no—
325#117—1461no—
326#117—1461no—
327#117C2×#6440811yescharacteristic
328C2×C42—731no—
329C2×C42—1461no—
330C2×C42—1461no—
331#107—731no—
332#107—731no—
333#106—731no—
334#106—731no—
335C4⋊C8—732no—
336C4⋊C8—732no—
337C4⋊C8—1462no—
338C4⋊C8—1462no—
339#103—731no—
340#103—731no—
341#103C2×#6440811yescharacteristic
342#103C2×#6440811yescharacteristic
343#102—731no—
344#102—731no—
345#102C2×#6440811yescharacteristic
346#102C2×#6440811yescharacteristic
347C22⋊C8—1462no—
348C22⋊C8—1462no—
349C22⋊C8—1461no—
350C22⋊C8—1461no—
351C4×C8—732no—
352C4×C8—731no—
353C4×C8—731no—
354C4×C8—731no—
355C4×C8—731no—
356C4×C8—731no—
357C4×C8—731no—
358C4×C8—732no—
359C4×C8—1461no—
360C4×C8—1461no—
361C4×C8—1461no—
362C4×C8—1462no—
363C4×C8—1462no—
364#41—22no—
365#41—734no—
366#41—1462no—
367#41#1192403614yescomplemented
368C2×Q8—731no—
369C2×Q8C22×#6440811yescharacteristic
370C2×D4—21no—
371C2×D4—731no—
372C2×D4—1461no—
373C2×D4C22×#6440811yescharacteristic
374C22×C4—21no—
375C22×C4—731no—
376C22×C4—731no—
377C22×C4—1461no—
378C22×C4—1461no—
379C22×C4—1461no—
380C22×C4—1461no—
381C22×C4—1461no—
382C22×C4C22×#6440811yescharacteristic
383Q16—734no—
384SD16—734no—
385SD16—734no—
386D8—734no—
387C2×C8—22no—
388C2×C8—2922no—
389C2×C8—2924no—
390C2×C8—2924no—
391C2×C8—2922no—
392C2×C8—734no—
393C2×C8—731no—
394C2×C8—731no—
395C2×C8—732no—
396C2×C8—732no—
397C2×C8—1462no—
398C2×C8—1462no—
399C2×C8—1462no—
400C2×C8C22×#6440811yescharacteristic
401C2×C8C22×#6440811yescharacteristic
402C4⋊C4—731no—
403C4⋊C4—731no—
404C4⋊C4—1461no—
405C4⋊C4—1461no—
406C4⋊C4—1461no—
407C4⋊C4—1461no—
408C4⋊C4C22×#6440811yescharacteristic
409C4⋊C4C22×#6440811yescharacteristic
410C22⋊C4—21no—
411C22⋊C4—1461no—
412C22⋊C4—1461no—
413C22⋊C4—1461no—
414C22⋊C4—1461no—
415C22⋊C4—1461no—
416C42—21no—
417C42—731no—
418C42—731no—
419C42—731no—
420C42—731no—
421C42—1461no—
422C42—1461no—
423C42—1461no—
424C42—1461no—
425C42—1461no—
426C23—21no—
427C23—1461no—
428C23D4×#6440811yescharacteristic
429Q8—21no—
430Q8—732no—
431Q8—1461no—
432Q8C2×#1192403612yes—
433D4—24no—
434D4—21no—
435D4—732no—
436D4—1464no—
437D4—1461no—
438D4C2×#1192403612yes—
439C2×C4—21no—
440C2×C4—24no—
441C2×C4—21no—
442C2×C4—21no—
443C2×C4—2922no—
444C2×C4—2922no—
445C2×C4—2922no—
446C2×C4—2922no—
447C2×C4—731no—
448C2×C4—731no—
449C2×C4—731no—
450C2×C4—731no—
451C2×C4—1461no—
452C2×C4—1461no—
453C2×C4—1462no—
454C2×C4—1462no—
455C2×C4—1461no—
456C2×C4—1464no—
457C2×C4D4×#6440811yescharacteristic
458C2×C4C2×#1192403612yes—
459C2×C4#1455667812yes—
460C2×C4C23×#6440811yescharacteristic
461C8—21no—
462C8—21no—
463C8—2922no—
464C8—2922no—
465C8—2922no—
466C8—732no—
467C8—732no—
468C8—1462no—
469C8—1462no—
470C22—21no—
471C22—21no—
472C22—22no—
473C22—41no—
474C22—41no—
475C22—2921no—
476C22—2921no—
477C22—1462no—
478C22#1798961412yes—
479C22C2×D4×#6440811yescharacteristic
480C4—22no—
481C4—41no—
482C4—41no—
483C4—2921no—
484C4—2921no—
485C4—732no—
486C4—732no—
487C4—1462no—
488C4—1461no—
489C4—1461no—
490C4#1798961412yes—
491C4C22×#1192403611yescharacteristic
492C4C2×#1455667811yescharacteristic
493C2—22no—
494C2—42no—
495C2—2922no—
496C2C2×#1798961411yescharacteristic
497C2#2262668511yescharacteristic
498C2D73⋊#44411yescharacteristic
subgroupG/H
centerC2×C4D4×#64408
commutatorC4×C73C23×C8
FrattiniC2×C4C23×#64408
Fitting#416×C73C4
socleC22×C73C2×C4×D4
prime radicalC2×#14556611C4
cosocleC4×Dic73C24
hypercenter#416#64408
hypocenterC73#17849
LCSG ▷ C4×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C2×C4 ◁ C22×C4 ◁ #416

Complements

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

NH ≅ G/Ntype
1. #22626735C2semidirect
2. #41×#10596833C2semidirect
3. #22625528C2semidirect
4. #22625525C2semidirect
5. #22625522C2semidirect
6. #22625521C2semidirect
7. #22625480C2semidirect
8. #22625224C2semidirect
9. #22625222C2semidirect
10. C2×#17989428C2semidirect
71. #14556611C8semidirect
124. #11924028C2×C8semidirect
135. #11924018C2×C8semidirect
136. #11924016C2×C8semidirect
137. #11924015C2×C8semidirect
138. #11924013C2×C8semidirect
139. C2×#10596831C2×C8semidirect
163. #41×C73C2×C8semidirect
261. C73#17849semidirect
367. #41#11924036semidirect

Maximal subgroups

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

subgroupindex
#226267352
#41×#105968332
#226255282
#226255252
#226255222
#226255212
#226254802
#226252242
#226252222
C2×#179894282
#226186992
#1784973