Finite Group Databasebeta

← #33240637

Subgroups — #33240637

Order 18688.

598 conjugacy classes of subgroups.

The subgroup lattice has 484 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22620444C211yesmaximal, characteristic, complemented
2C4×D8×D73C211yesmaximal, characteristic
3#22612089C211yesmaximal, characteristic, complemented
4#22612088C211yesmaximal, characteristic, complemented
5#22612085C211yesmaximal, characteristic, complemented
6#22612077C211yesmaximal, characteristic, complemented
7#22612076C211yesmaximal, characteristic, complemented
8#22612067C211yesmaximal, characteristic, complemented
9#22612066C211yesmaximal, characteristic, complemented
10#22611718C211yesmaximal, characteristic, complemented
11#22611716C211yesmaximal, characteristic, complemented
12#22611685C211yesmaximal, characteristic, complemented
13#22611645C211yesmaximal, characteristic
14#22611645C211yesmaximal, characteristic
15C73×#2228C211yesmaximal, characteristic, complemented
16C2×D8×D73C2211yescharacteristic
17#17989143—21no—
18C4×D4×D73—21no—
19C2×#14556530C2211yescharacteristic
20C2×#14556530C2211yescharacteristic
21#17988991C2211yescharacteristic, complemented
22#17988991C2211yescharacteristic, complemented
23#17988989C2211yescharacteristic, complemented
24#17988986—21no—
25#17988984—21no—
26C4×D584C2211yescharacteristic
27D8×Dic73C2211yescharacteristic
28#17988669C2211yescharacteristic
29C4×#11924013—21no—
30#17988667—21no—
31#17988624C2211yescharacteristic
32#17988624C2211yescharacteristic
33#17988622—21no—
34#107×D73C2211yescharacteristic
35#102×D73—21no—
36C4×C8×D73C2211yescharacteristic
37#17988415C2211yescharacteristic, complemented
38#17988414C2211yescharacteristic, complemented
39#17988414C2211yescharacteristic, complemented
40#17988413C2211yescharacteristic, complemented
41#17988412C2211yescharacteristic, complemented
42#17988412C2211yescharacteristic, complemented
43C4⋊#11924003C2211yescharacteristic, complemented
44#17988405C2211yescharacteristic, complemented
45#17988405C2211yescharacteristic, complemented
46#17988389C2211yescharacteristic
47#17988389C2211yescharacteristic
48#17988388C2211yescharacteristic
49#17988388C2211yescharacteristic
50#17988387C2211yescharacteristic
51#17988387C2211yescharacteristic
52#499×C73C2211yescharacteristic, complemented
53C4×D8×C73C2211yescharacteristic
54#367×C73C2211yescharacteristic, complemented
55#366×C73C2211yescharacteristic, complemented
56C4⋊C16×C73C2211yescharacteristic, complemented
57#357×C73C2211yescharacteristic
58#357×C73C2211yescharacteristic
59C2×D4×D73—21no—
60C22×C4×D73—41no—
61C2×D584C2311yescharacteristic
62C2×#11924013—21no—
63D8×D73—22no—
64D8×D73Q814yescomplemented
65C2×C8×D73D411yescharacteristic
66C2×C8×D73C2311yescharacteristic
67#14556590—21no—
68#14556590—21no—
69#14556583—21no—
70#14556569—41no—
71#14556569—41no—
72#14556556—41no—
73#14556556—41no—
74C4×D292—21no—
75D4×Dic73—21no—
76#14556553—21no—
77C4×C22⋊D73—41no—
78#14556551—41no—
79#14556549—21no—
80#14556549—21no—
81C4⋊C4×D73—21no—
82C22⋊C4×D73—41no—
83C42×D73D411yescharacteristic
84#14556530—22no—
85#14556530—22no—
86C2×#11924003C2311yescharacteristic
87C2×#11924003C2311yescharacteristic
88#14556525C2311yescharacteristic
89#14556525C2311yescharacteristic
90#14556525C2311yescharacteristic
91#14556524—21no—
92#14556524—21no—
93#14556524—21no—
94#14556523—21no—
95#14556522—21no—
96#14556522—21no—
97#14556513—21no—
98#14556512—21no—
99#14556511—21no—
100C4×#10596831D411yescharacteristic
101C8×Dic73D411yescharacteristic
102C8×Dic73C2311yescharacteristic
103C2×D8×C73C2311yescharacteristic
104#125×C73—21no—
105C4×D4×C73—21no—
106C2×C16×C73C2311yescharacteristic
107C2×C16×C73C2311yescharacteristic
108#107×C73C2311yescharacteristic
109#107×C73C2311yescharacteristic
110#107×C73C2311yescharacteristic
111#106×C73—21no—
112#102×C73—21no—
113C4×C8×C73C2311yescharacteristic
114C23×D73—41no—
115C2×Dic146—41no—
116C2×Dic146—41no—
117C2×D292—21no—
118C2×C22⋊D73—41no—
119D4×D73—24no—
120D4×D73—42no—
121C22×Dic73—41no—
122C2×C4×D73—41no—
123C2×C4×D73—41no—
124C2×C4×D73—41no—
125C2×C4×D73—84no—
126C2×C4×D73D811yescharacteristic
127C2×C4×D73D811yescharacteristic
128C2×C4×D73C2×D411yescharacteristic
129D584—21no—
130D584C2×Q812yes—
131#11924013—24no—
132C2×#10596831C2×D411yescharacteristic
133C2×#10596831C2×D411yescharacteristic
134C8×D73—22no—
135C8×D73C2×Q812yes—
136C8×D73#4112yes—
137C4⋊Dic73—21no—
138C4⋊Dic73—21no—
139C4⋊Dic73—21no—
140#11924007—41no—
141#11924007—41no—
142#11924007—41no—
143#11924007—41no—
144#11924007—41no—
145C22⋊Dic73—41no—
146#11924005—41no—
147#11924005—41no—
148#11924005—41no—
149#11924005—41no—
150C4×Dic73—41no—
151C4×Dic73—41no—
152C4×Dic73D811yescharacteristic
153C4×Dic73D811yescharacteristic
154C4×Dic73C2×D411yescharacteristic
155#11924003—21no—
156#11924003—21no—
157C2×D4×C73—21no—
158C22×C4×C73—41no—
159D8×C73—21no—
160D8×C73C2×Q812yes—
161C2×C8×C73C2×D411yescharacteristic
162C2×C8×C73C2411yescharacteristic
163C4⋊C4×C73—21no—
164C4⋊C4×C73—21no—
165C4⋊C4×C73—21no—
166C4⋊C4×C73—41no—
167C4⋊C4×C73—41no—
168C22⋊C4×C73—41no—
169C42×C73C2×D411yescharacteristic
170C16×C73—21no—
171C16×C73—21no—
172C23×C73—41no—
173C22×D73—41no—
174C22×D73—44no—
175C22×D73—82no—
176C22×D73—82no—
177C22×D73C2×D811yescharacteristic
178D4×C73—22no—
179D4×C73—41no—
180C2×C4×C73—41no—
181C2×C4×C73—41no—
182C2×C4×C73—41no—
183C2×C4×C73—82no—
184C2×C4×C73C2×D811yescharacteristic
185C2×C4×C73C2×D811yescharacteristic
186C2×C4×C73C22×D411yescharacteristic
187C22⋊D73—44no—
188C2×Dic73—41no—
189C2×Dic73—41no—
190C2×Dic73—41no—
191C2×Dic73—82no—
192C2×Dic73C2×D811yescharacteristic
193C2×Dic73C2×D811yescharacteristic
194C2×Dic73C2×D811yescharacteristic
195D292—22no—
196D292—41no—
197C4×D73—22no—
198C4×D73—41no—
199C4×D73—82no—
200C4×D73—82no—
201C4×D73—82no—
202C4×D73C22⋊Q812yes—
203C4×D73#13712yes—
204Dic146—82no—
205Dic146—82no—
206C8×C73—21no—
207C8×C73C22×Q811yescharacteristic
208C8×C73C2×#4111yescharacteristic
209#10596831—21no—
210#10596831C22⋊Q812yes—
211C22×C73—42no—
212C22×C73—81no—
213C22×C73—81no—
214C22×C73C22×D811yescharacteristic
215C2×D73—21no—
216C2×D73—21no—
217C2×D73—42no—
218C2×D73—88no—
219C2×D73—81no—
220C2×D73—81no—
221C2×D73#47412yes—
222C4×C73—21no—
223C4×C73—21no—
224C4×C73—81no—
225C4×C73—81no—
226C4×C73—81no—
227C4×C73C2×C22⋊Q811yescharacteristic
228C4×C73C2×#13711yescharacteristic
229Dic73—21no—
230Dic73—81no—
231Dic73—81no—
232Dic73—81no—
233Dic73#47412yes—
234Dic73#50912yes—
235#33627—731nomaximal, Sylow
236C2×C73—82no—
237C2×C73C2×#47411yescharacteristic
238C2×C73C2×#50911yescharacteristic
239C2×C73#341411yescharacteristic
240D73—22no—
241D73—82no—
242#3160—731no—
243C2×C4×D8—731no—
244#2228—731no—
245#2228—731no—
246#2228—731no—
247#2228D7311yescharacteristic, complemented
248#2226—731no—
249#2226—731no—
250#2224—731no—
251#2224—731no—
252#2159—731no—
253#2159—731no—
254#2150—731no—
255#2140—731no—
256#2140—731no—
257C73#3362711yescharacteristic, complemented, Sylow
258C22×D8—731no—
259#533—1461no—
260C2×C4×D4—1461no—
261C2×M32—731no—
262C2×M32—731no—
263#500—731no—
264#500—731no—
265#499—731no—
266#499—1461no—
267#499C2×D7311yescharacteristic, complemented
268#498—1461no—
269C4×D8—731no—
270C4×D8—731no—
271C4×D8—731no—
272C4×D8—1461no—
273C4×D8—1461no—
274C4×D8C2×D7311yescharacteristic
275#427—731no—
276#427—731no—
277#427—1461no—
278C2×#107—731no—
279C2×#102—1461no—
280C2×C4×C8—731no—
281#367—731no—
282#367—731no—
283#367—731no—
284#367C2×D7311yescharacteristic, complemented
285#366—731no—
286#366—731no—
287#366—731no—
288#366C2×D7311yescharacteristic, complemented
289C4⋊C16—731no—
290C4⋊C16—731no—
291C4⋊C16—731no—
292C4⋊C16C2×D7311yescharacteristic, complemented
293#357—731no—
294#357—731no—
295#357—731no—
296#357—731no—
297#357—731no—
298#357—731no—
299#357C2×D7311yescharacteristic
300#357C2×D7311yescharacteristic
301C22×D4—1461no—
302C23×C4—2921no—
303C2×D8—734no—
304C2×D8—731no—
305C2×D8—1462no—
306C2×D8—1461no—
307C2×D8C22×D7311yescharacteristic
308C22×C8—731no—
309C22×C8—731no—
310C4⋊Q8—1461no—
311C4⋊Q8—1461no—
312#125—21no—
313#125—1461no—
314C22⋊Q8—2921no—
315C22⋊Q8—2921no—
316C4×Q8—2921no—
317C4×Q8—2921no—
318C4×D4—21no—
319C4×D4—2921no—
320C4×D4—2921no—
321C4×D4—1461no—
322C4×D4—1461no—
323C4×D4—1461no—
324#117—1461no—
325#117—1461no—
326C2×C4⋊C4—1461no—
327C2×C22⋊C4—2921no—
328C2×C42—731no—
329M32—1462no—
330M32—1462no—
331C2×C16—731no—
332C2×C16—731no—
333C2×C16C22×D7311yescharacteristic
334C2×C16C22×D7311yescharacteristic
335#107—731no—
336#107—731no—
337#107—731no—
338#107—1461no—
339#107—1461no—
340#107—1461no—
341#107C22×D7311yescharacteristic
342#107C22×D7311yescharacteristic
343#107C22×D7311yescharacteristic
344#106—21no—
345#106—1461no—
346#106—1461no—
347#106—1461no—
348#102—21no—
349#102—1461no—
350#102—1461no—
351#102—1461no—
352C4×C8—731no—
353C4×C8—731no—
354C4×C8—731no—
355C4×C8C22×D7311yescharacteristic
356C24—2921no—
357C2×Q8—2921no—
358C2×Q8—2921no—
359C2×D4—21no—
360C2×D4—2922no—
361C2×D4—2921no—
362C2×D4—1464no—
363C2×D4—1461no—
364C22×C4—41no—
365C22×C4—2921no—
366C22×C4—2921no—
367C22×C4—2921no—
368C22×C4—2921no—
369C22×C4—5844no—
370C22×C4—731no—
371C22×C4—731no—
372C22×C4—731no—
373D8—21no—
374D8—732no—
375D8—1464no—
376D8—1461no—
377D8Q8×D7312yes—
378C2×C8—731no—
379C2×C8—732no—
380C2×C8—732no—
381C2×C8—731no—
382C2×C8—1462no—
383C2×C8D4×D7311yescharacteristic
384C2×C8C23×D7311yescharacteristic
385C4⋊C4—21no—
386C4⋊C4—21no—
387C4⋊C4—21no—
388C4⋊C4—41no—
389C4⋊C4—41no—
390C4⋊C4—2921no—
391C4⋊C4—2921no—
392C4⋊C4—2921no—
393C4⋊C4—2921no—
394C4⋊C4—2921no—
395C4⋊C4—1461no—
396C4⋊C4—1461no—
397C4⋊C4—1461no—
398C22⋊C4—41no—
399C22⋊C4—2921no—
400C22⋊C4—2921no—
401C22⋊C4—2921no—
402C22⋊C4—2921no—
403C22⋊C4—2921no—
404C42—2921no—
405C42—2921no—
406C42—731no—
407C42—731no—
408C42—731no—
409C42D4×D7311yescharacteristic
410C16—21no—
411C16—21no—
412C16—1461no—
413C16—1461no—
414C23—41no—
415C23—2924no—
416C23—2921no—
417C23—5842no—
418C23—5842no—
419C23—731no—
420Q8—5842no—
421Q8—5842no—
422D4—22no—
423D4—41no—
424D4—2924no—
425D4—2921no—
426D4—1462no—
427C2×C4—41no—
428C2×C4—41no—
429C2×C4—41no—
430C2×C4—82no—
431C2×C4—2921no—
432C2×C4—2921no—
433C2×C4—2921no—
434C2×C4—2921no—
435C2×C4—5842no—
436C2×C4—5842no—
437C2×C4—5842no—
438C2×C4—5842no—
439C2×C4—731no—
440C2×C4—731no—
441C2×C4—732no—
442C2×C4—732no—
443C2×C4—731no—
444C2×C4—1462no—
445C2×C4D8×D7311yescharacteristic
446C2×C4D8×D7311yescharacteristic
447C2×C4C2×D4×D7311yescharacteristic
448C8—21no—
449C8—732no—
450C8—1461no—
451C8C2×Q8×D7311yescharacteristic
452C8#41×D7311yescharacteristic
453C22—42no—
454C22—81no—
455C22—81no—
456C22—2922no—
457C22—5848no—
458C22—5841no—
459C22—5841no—
460C22—732no—
461C22—1461no—
462C22—1461no—
463C22C2×D8×D7311yescharacteristic
464C4—21no—
465C4—21no—
466C4—81no—
467C4—81no—
468C4—81no—
469C4—5841no—
470C4—5841no—
471C4—5841no—
472C4—732no—
473C4—732no—
474C4—1461no—
475C4C22⋊Q8×D7311yescharacteristic
476C4#137×D7311yescharacteristic
477C2—82no—
478C2—5842no—
479C2—1462no—
480C2#474×D7311yescharacteristic
481C2#509×D7311yescharacteristic
482C2#2262306411yescharacteristic
subgroupG/H
centerC22C2×D8×D73
commutator, cosocleC2×C8×C73C24
FrattiniC2×C8C23×D73
FittingC73×#2228C2
socleC22×C73C22×D8
prime radicalC2×D8×D73C22
hypercenter#2228D73
hypocenterC73#33627
LCSG ▷ C2×C8×C73 ▷ C4×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C22 ◁ C42 ◁ C4×C8 ◁ #2228

Complements

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

NH ≅ G/Ntype
1. #22620444C2semidirect
3. #22612089C2semidirect
4. #22612088C2semidirect
5. #22612085C2semidirect
6. #22612077C2semidirect
7. #22612076C2semidirect
8. #22612067C2semidirect
9. #22612066C2semidirect
10. #22611718C2semidirect
11. #22611716C2semidirect
12. #22611685C2semidirect
15. C73×#2228C2semidirect
21. #17988991C22semidirect
22. #17988991C22semidirect
23. #17988989C22semidirect
37. #17988415C22semidirect
38. #17988414C22semidirect
39. #17988414C22semidirect
40. #17988413C22semidirect
41. #17988412C22semidirect
42. #17988412C22semidirect
43. C4⋊#11924003C22semidirect
44. #17988405C22semidirect
45. #17988405C22semidirect
52. #499×C73C22semidirect
54. #367×C73C22semidirect
55. #366×C73C22semidirect
56. C4⋊C16×C73C22semidirect
64. D8×D73Q8semidirect
247. #2228D73semidirect
257. C73#33627semidirect
267. #499C2×D73semidirect
284. #367C2×D73semidirect
288. #366C2×D73semidirect
292. C4⋊C16C2×D73semidirect

Maximal subgroups

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

subgroupindex
#226204442
C4×D8×D732
#226120892
#226120882
#226120852
#226120772
#226120762
#226120672
#226120662
#226117182
#226117162
#226116852
#226116452
C73×#22282
#3362773