Finite Group Databasebeta

← #33238899

Subgroups — #33238899

Order 18688.

1124 conjugacy classes of subgroups.

The subgroup lattice has 448 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#17989401C211yesmaximal, characteristic
2C2×#17989003C211yesmaximal, characteristic, complemented
3#22618528C211yesmaximal, characteristic, complemented
4#22611650C211yesmaximal, characteristic, complemented
5#22611649C212yesmaximal, complemented
6#22611643C211yesmaximal, characteristic, complemented
7#22611640C212yesmaximal, complemented
8#22611635C212yesmaximal, complemented
9C2×#17988392C211yesmaximal, characteristic, complemented
10#358×D73C212yesmaximal, complemented
11C73×#2138C211yesmaximal, characteristic, complemented
12C2×#14556638—21no—
13#17989401—24no—
14#17989401C418yescomplemented
15C2×#14556621—21no—
16C2×#14556620—22no—
17C2×#14556610C2212yes—
18C22×Dic292C2211yescharacteristic
19C2×Q16×D73C2212yes—
20C2×#14556597C2211yescharacteristic
21#17989067—21no—
22#17989003—24no—
23C2×#14556531C2211yescharacteristic
24C2×#14556530C2212yes—
25C2×C16×D73C2212yes—
26#17988630—21no—
27#17988626—22no—
28#17988624C2212yes—
29C2×#14556525C2211yescharacteristic
30#107×D73C2212yes—
31#17988392C2212yes—
32#17988392C2212yes—
33#17988391C2212yes—
34#17988390C2212yes—
35#17988390C2212yes—
36#17988388C2212yes—
37C2×#135×C73C2211yescharacteristic
38C22×C16×C73C2211yescharacteristic
39#427×C73C2211yescharacteristic
40#358×C73C2211yescharacteristic
41#358×C73C2211yescharacteristic
42#357×C73C2211yescharacteristic
43#357×C73C2211yescharacteristic
44#14556638—28no—
45#14556638—44no—
46C2×#11924028—41no—
47C2×#11924028D411yescharacteristic
48C2×#11924027—22no—
49C2×#11924027—42no—
50C22×Dic146—21no—
51C22×Dic146—41no—
52C2×Q8×D73—22no—
53#14556621—28no—
54#14556620—216no—
55#14556610—24no—
56#14556610C2×C418yescomplemented
57C2×Dic292—21no—
58C2×Dic292—22no—
59C2×Dic292C2×C414yescomplemented
60C2×Dic292C2312yes—
61C2×#11924018—22no—
62Q16×D73—24no—
63Q16×D73C2×C418yescomplemented
64C2×#11924016—22no—
65#14556597C2×C414yes—
66#14556597D414yes—
67C2×#11924012D411yescharacteristic
68C2×#11924011D412yes—
69C2×C8×D73C2312yes—
70C4×Dic146—41no—
71C4×D292—41no—
72#14556551—42no—
73#14556549—22no—
74C2×C4⋊Dic73—21no—
75C4⋊C4×D73—22no—
76#14556531—22no—
77#14556530—24no—
78C2×#11924003C2312yes—
79C16×D73—24no—
80#14556525—21no—
81#14556525C2312yes—
82#14556524—22no—
83#14556522—22no—
84C2×#41×C73—21no—
85#135×C73—22no—
86#135×C73C2×C414yescomplemented
87C2×Q16×C73C2311yescharacteristic
88C2×SD16×C73—21no—
89C2×D8×C73C2311yescharacteristic
90C22×C8×C73C2311yescharacteristic
91#117×C73—21no—
92C2×C16×C73—22no—
93C2×C16×C73C2311yescharacteristic
94C2×C16×C73C2311yescharacteristic
95#107×C73C2311yescharacteristic
96#107×C73C2311yescharacteristic
97#106×C73—21no—
98#11924028—48no—
99#11924028C22⋊C414yes—
100#11924028SD1614yes—
101#11924027—28no—
102#11924027—416no—
103#11924027—44no—
104C2×Dic146—24no—
105C2×Dic146—22no—
106C2×Dic146—41no—
107C2×Dic146—42no—
108C2×Dic146—42no—
109C2×Dic146—48no—
110C2×Dic146D811yescharacteristic
111Q8×D73—28no—
112Q8×D73—44no—
113C2×D292D811yescharacteristic
114C2×C22⋊D73—42no—
115C2×C22⋊D73D812yes—
116C22×Dic73—41no—
117C22×Dic73—42no—
118C22×Dic73—41no—
119C2×C4×D73—42no—
120C2×C4×D73—42no—
121C2×C4×D73C2×D412yes—
122Dic292—24no—
123Dic292—22no—
124Dic292C22×C414yes—
125#11924018—28no—
126#11924016—28no—
127#11924012C22⋊C414yes—
128#11924011C22⋊C418yes—
129C2×#10596831C2×D412yes—
130C8×D73C22×C414yes—
131C8×D73C2×D414yes—
132C4⋊Dic73—22no—
133C4⋊Dic73—41no—
134#11924007—42no—
135#11924005—42no—
136C4×Dic73—42no—
137#11924003—22no—
138#41×C73—24no—
139#41×C73—42no—
140C2×Q8×C73—21no—
141C2×D4×C73—21no—
142C2×D4×C73—41no—
143C22×C4×C73—41no—
144C22×C4×C73C2×D411yescharacteristic
145Q16×C73—21no—
146Q16×C73C22×C412yes—
147SD16×C73—24no—
148D8×C73—21no—
149D8×C73C22×C412yes—
150C2×C8×C73C22×C412yes—
151C2×C8×C73C2×D411yescharacteristic
152C2×C8×C73C2×D412yes—
153C2×C8×C73C2411yescharacteristic
154C4⋊C4×C73—21no—
155C4⋊C4×C73—21no—
156C22⋊C4×C73—41no—
157C42×C73—41no—
158C16×C73—21no—
159C16×C73—21no—
160C23×C73—41no—
161C23×C73C2×D811yescharacteristic
162C22×D73C2×D812yes—
163Q8×C73—22no—
164Q8×C73—41no—
165D4×C73—22no—
166D4×C73—44no—
167D4×C73—41no—
168C2×C4×C73—41no—
169C2×C4×C73—44no—
170C2×C4×C73—41no—
171C2×C4×C73—41no—
172C2×C4×C73C2×C22⋊C412yes—
173C2×C4×C73C2×D811yescharacteristic
174C2×C4×C73C2×SD1612yes—
175C2×C4×C73C22×D411yescharacteristic
176C22⋊D73—48no—
177C22⋊D73#10218yes—
178C2×Dic73—42no—
179C2×Dic73—44no—
180C2×Dic73—48no—
181C2×Dic73—42no—
182C2×Dic73—161no—
183C2×Dic73C2×D812yes—
184D292#10214yes—
185C4×D73—48no—
186C4×D73—162no—
187C4×D73C2×C22⋊C414yes—
188C4×D73C2×SD1614yes—
189Dic146—24no—
190Dic146—42no—
191Dic146—44no—
192Dic146—48no—
193Dic146#10214yes—
194C8×C73C2×C22⋊C412yes—
195C8×C73C23×C411yescharacteristic
196C8×C73C22×D411yescharacteristic
197#10596831C2×C22⋊C414yes—
198C22×C73—21no—
199C22×C73—21no—
200C22×C73—42no—
201C22×C73—81no—
202C22×C73—81no—
203C22×C73C2×#10212yes—
204C22×C73C22×D811yescharacteristic
205C2×D73—22no—
206C2×D73—22no—
207C2×D73C2×#10214yes—
208C4×C73—42no—
209C4×C73—81no—
210C4×C73—81no—
211C4×C73C2×#10212yes—
212C4×C73C22×C22⋊C411yescharacteristic
213C4×C73C22×SD1611yescharacteristic
214Dic73—44no—
215Dic73—82no—
216Dic73C2×#10214yes—
217#33382—731nomaximal, Sylow
218C2×C73—22no—
219C2×C73—82no—
220C2×C73C22×#10211yescharacteristic
221C2×C73#341611yescharacteristic
222C2×C73#341711yescharacteristic
223D73—24no—
224C2×#578—731no—
225C2×#504—731no—
226#2913—731no—
227#2141—731no—
228#2141—732no—
229#2139—732no—
230#2139—731no—
231#2138—732no—
232#2138D7311yescharacteristic, complemented
233C2×#358—731no—
234C2×#358—732no—
235C73#3338211yescharacteristic, complemented, Sylow
236C2×2−1+4—1461no—
237#578—738no—
238#578—1464no—
239C2×#137—1462no—
240C2×#137—1461no—
241C2×#135—732no—
242C2×#135C2×D7311yescharacteristic
243C22×Q16—732no—
244C22×Q16—731no—
245C2×#131—731no—
246#520—1461no—
247#504—1464no—
248C2×M32—731no—
249C2×M32—732no—
250C22×C16—732no—
251C22×C16C2×D7311yescharacteristic
252#428—1462no—
253#428—1461no—
254#427—732no—
255#427C2×D7311yescharacteristic
256C2×#107—731no—
257C2×#107—732no—
258#358—732no—
259#358—732no—
260#358—732no—
261#358—732no—
262#358—732no—
263#358C2×D7311yescharacteristic
264#358C2×D7311yescharacteristic
265#357—732no—
266#357C2×D7311yescharacteristic
267#357C2×D7311yescharacteristic
2682−1+4—2924no—
2692−1+4—1468no—
270C2×#41—21no—
271C2×#41—2921no—
272C2×#41—2922no—
273C2×#41—731no—
274C2×#41—1462no—
275C22×Q8—2921no—
276C22×Q8—1462no—
277C22×Q8—1461no—
278#137—14616no—
279#137—1468no—
280#135—22no—
281#135—738no—
282#135—1464no—
283#135C4×D7314yescomplemented
284C2×Q16—738no—
285C2×Q16—734no—
286C2×Q16—732no—
287C2×Q16—1464no—
288C2×Q16—1462no—
289C2×Q16—1461no—
290C2×Q16—1462no—
291C2×Q16C22×D7311yescharacteristic
292C2×SD16—21no—
293C2×SD16—1462no—
294C2×D8C22×D7311yescharacteristic
295#131—734no—
296#131—734no—
297C2×M16—731no—
298C2×M16—732no—
299C22×C8—732no—
300C22×C8C22×D7311yescharacteristic
301C4×Q8—2921no—
302C4×D4—2921no—
303C4×D4—2922no—
304#117—21no—
305#117—1462no—
306C2×C4⋊C4—1462no—
307C2×C4⋊C4—1461no—
308M32—1464no—
309M32—1462no—
310C2×C16—22no—
311C2×C16—732no—
312C2×C16—1464no—
313C2×C16C22×D7311yescharacteristic
314C2×C16C22×D7311yescharacteristic
315#107—732no—
316#107—1462no—
317#107—1461no—
318#107C22×D7311yescharacteristic
319#107C22×D7311yescharacteristic
320#106—21no—
321#106—1462no—
322#41—24no—
323#41—42no—
324#41—2928no—
325#41—29216no—
326#41—2924no—
327#41—734no—
328#41—734no—
329#41—1468no—
330C2×Q8—21no—
331C2×Q8—2928no—
332C2×Q8—2922no—
333C2×Q8—2924no—
334C2×Q8—2921no—
335C2×Q8—2922no—
336C2×Q8—731no—
337C2×Q8—1468no—
338C2×Q8—1464no—
339C2×Q8—1462no—
340C2×D4—21no—
341C2×D4—41no—
342C2×D4—2922no—
343C2×D4—732no—
344C2×D4—731no—
345C22×C4—41no—
346C22×C4—2922no—
347C22×C4—2922no—
348C22×C4—2921no—
349C22×C4—2922no—
350C22×C4—2921no—
351C22×C4—732no—
352C22×C4D4×D7311yescharacteristic
353Q16—21no—
354Q16—734no—
355Q16—1468no—
356Q16—1464no—
357Q16—1462no—
358Q16C2×C4×D7312yes—
359SD16—24no—
360SD16—1468no—
361D8—21no—
362D8C2×C4×D7312yes—
363M16—734no—
364M16—738no—
365C2×C8—732no—
366C2×C8—734no—
367C2×C8—734no—
368C2×C8C2×C4×D7312yes—
369C2×C8D4×D7311yescharacteristic
370C2×C8D4×D7312yes—
371C2×C8C23×D7311yescharacteristic
372C4⋊C4—21no—
373C4⋊C4—21no—
374C4⋊C4—2922no—
375C4⋊C4—2921no—
376C4⋊C4—1462no—
377C22⋊C4—41no—
378C22⋊C4—2922no—
379C42—41no—
380C42—2922no—
381C16—21no—
382C16—21no—
383C16—1462no—
384C23—41no—
385C23—732no—
386C23D8×D7311yescharacteristic
387Q8—22no—
388Q8—41no—
389Q8—2928no—
390Q8—2924no—
391Q8—2922no—
392Q8—734no—
393Q8—1464no—
394D4—22no—
395D4—44no—
396D4—41no—
397D4—2928no—
398D4—738no—
399D4—734no—
400C2×C4—41no—
401C2×C4—44no—
402C2×C4—41no—
403C2×C4—41no—
404C2×C4—2928no—
405C2×C4—2924no—
406C2×C4—2922no—
407C2×C4—2928no—
408C2×C4—2922no—
409C2×C4—732no—
410C2×C4—734no—
411C2×C4—734no—
412C2×C4—11682no—
413C2×C4—11681no—
414C2×C4C22⋊C4×D7312yes—
415C2×C4D8×D7311yescharacteristic
416C2×C4SD16×D7312yes—
417C2×C4C2×D4×D7311yescharacteristic
418C8—734no—
419C8C22⋊C4×D7312yes—
420C8C22×C4×D7311yescharacteristic
421C8C2×D4×D7311yescharacteristic
422C22—21no—
423C22—21no—
424C22—42no—
425C22—81no—
426C22—81no—
427C22—734no—
428C22—1462no—
429C22—1462no—
430C22#102×D7312yes—
431C22C2×D8×D7311yescharacteristic
432C4—42no—
433C4—81no—
434C4—81no—
435C4—2924no—
436C4—5842no—
437C4—734no—
438C4#102×D7312yes—
439C4C2×C22⋊C4×D7311yescharacteristic
440C4C2×SD16×D7311yescharacteristic
441C2—22no—
442C2—82no—
443C2—1464no—
444C2C2×#102×D7311yescharacteristic
445C2#2262307411yescharacteristic
446C2#2262308111yescharacteristic
subgroupG/H
centerC22C2×D8×D73
commutatorC8×C73C23×C4
FrattiniC2×C8C23×D73
FittingC73×#2138C2
socleC22×C73C22×D8
prime radicalC2×#17989401C2
cosocleC2×C8×C73C24
hypercenter#2138D73
hypocenterC73#33382
LCSG ▷ C8×C73 ▷ C4×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22×C8 ◁ #2138

Complements

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

NH ≅ G/Ntype
2. C2×#17989003C2semidirect
3. #22618528C2semidirect
4. #22611650C2semidirect
5. #22611649C2semidirect
6. #22611643C2semidirect
7. #22611640C2semidirect
8. #22611635C2semidirect
9. C2×#17988392C2semidirect
10. #358×D73C2semidirect
11. C73×#2138C2semidirect
14. #17989401C4semidirect
56. #14556610C2×C4semidirect
59. C2×Dic292C2×C4semidirect
63. Q16×D73C2×C4semidirect
86. #135×C73C2×C4semidirect
232. #2138D73semidirect
235. C73#33382semidirect
283. #135C4×D73semidirect

Maximal subgroups

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

subgroupindex
C2×#179894012
C2×#179890032
#226185282
#226116502
#226116492
#226116432
#226116402
#226116352
C2×#179883922
#358×D732
C73×#21382
#3338273