Finite Group Databasebeta

← #33102541

Subgroups — #33102541

Order 18688.

1114 conjugacy classes of subgroups.

The subgroup lattice has 528 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#17989294C211yesmaximal, characteristic
2#22612810C212yesmaximal, complemented
3#22610915C212yesmaximal, complemented
4#22610914C214yesmaximal, complemented
5#22609917C212yesmaximal
6#99×Dic73C212yesmaximal
7C4×#14556506C211yesmaximal, characteristic
8C73×#1799C211yesmaximal, characteristic
9#17989294—22no—
10#17989294—24no—
11#17989294—22no—
12C2×#14556591C411yescharacteristic, complemented
13C2×#14556582C412yescomplemented
14C2×#14556570C411yescharacteristic, complemented
15C2×#14556570C2211yescharacteristic
16C2×#14556568C412yescomplemented
17C2×#14556568C2212yes—
18C2×C4×Dic146C411yescharacteristic, complemented
19C2×D4×Dic73C2212yes—
20C22×C4⋊Dic73C411yescharacteristic, complemented
21C22×C4⋊Dic73C2211yescharacteristic
22C2×#14556506C2211yescharacteristic
23C2×#14556506C2211yescharacteristic
24#17988483—22no—
25#17988472—24no—
26#17988445C2214yes—
27C4⋊C4×Dic73—22no—
28C22⋊C4×Dic73C2214yes—
29C4×C22⋊Dic73C2212yes—
30#17988354C2212yes—
31#17988353C2214yes—
32#17988353C2214yes—
33C2×C4×D4×C73C2211yescharacteristic
34C2×#99×C73C2212yes—
35#380×C73C2212yes—
36#342×C73C2212yes—
37C22×Dic146C2×C411yescharacteristic
38C22×Dic146C2×C412yes—
39C23×Dic73C2×C412yes—
40C23×Dic73C2312yes—
41#14556591—22no—
42#14556591—22no—
43#14556582—24no—
44#14556582—24no—
45#14556570—22no—
46#14556570—22no—
47#14556570—22no—
48#14556570—22no—
49#14556568—24no—
50#14556568—24no—
51#14556568—24no—
52#14556568—24no—
53C4×Dic146—22no—
54C4×Dic146—21no—
55C4×Dic146—21no—
56D4×Dic73—24no—
57D4×Dic73—24no—
58C2×C4⋊Dic73—22no—
59C2×C4⋊Dic73—21no—
60C2×C4⋊Dic73—21no—
61C2×C4⋊Dic73C2×C411yescharacteristic
62C2×C4⋊Dic73C2×C412yes—
63C2×C4⋊Dic73C2×C412yes—
64C2×C4⋊Dic73C2×C411yescharacteristic
65C2×C4⋊Dic73C2×C411yescharacteristic
66C2×C4⋊Dic73C2×C412yes—
67C2×C4⋊Dic73D412yes—
68C2×C4⋊Dic73D414yes—
69C2×C4⋊Dic73D412yes—
70C2×#11924007—24no—
71C2×#11924007C2×C412yes—
72C2×#11924007C2×C412yes—
73C2×#11924007C2×C412yes—
74C2×C22⋊Dic73C2×C412yes—
75C2×C22⋊Dic73C2312yes—
76C2×C22⋊Dic73C2312yes—
77C2×C22⋊Dic73C2312yes—
78C2×C4×Dic73—24no—
79C2×C4×Dic73—22no—
80C2×C4×Dic73C2×C412yes—
81#14556506—22no—
82#14556506Q814yescomplemented
83#14556498—24no—
84#14556498—24no—
85#14556498—22no—
86C22×D4×C73C2311yescharacteristic
87C23×C4×C73C2×C411yescharacteristic
88C23×C4×C73C2311yescharacteristic
89C4×D4×C73—22no—
90C4×D4×C73—22no—
91C2×C4⋊C4×C73—22no—
92C2×C4⋊C4×C73C2×C411yescharacteristic
93C2×C22⋊C4×C73C2×C411yescharacteristic
94C2×C22⋊C4×C73C2314yes—
95C2×C22⋊C4×C73C2311yescharacteristic
96C2×C42×C73C2×C411yescharacteristic
97#99×C73—24no—
98#95×C73—22no—
99C2×Dic146—21no—
100C2×Dic146—21no—
101C2×Dic146—24no—
102C2×Dic146—42no—
103C2×Dic146—82no—
104C22×Dic73—22no—
105C22×Dic73—22no—
106C22×Dic73—22no—
107C22×Dic73—22no—
108C22×Dic73—82no—
109C22×Dic73—82no—
110C22×Dic73C22×C412yes—
111C22×Dic73C22×C412yes—
112C22×Dic73C22×C412yes—
113C22×Dic73C22×C412yes—
114C22×Dic73C2×D414yes—
115C22×Dic73C2×D414yes—
116C4⋊Dic73—21no—
117C4⋊Dic73—21no—
118C4⋊Dic73—22no—
119C4⋊Dic73—22no—
120C4⋊Dic73—24no—
121C4⋊Dic73C22⋊C412yes—
122C4⋊Dic73C22⋊C411yescharacteristic
123C4⋊Dic73C22⋊C411yescharacteristic
124C4⋊Dic73C22⋊C414yes—
125C4⋊Dic73C22⋊C414yes—
126C4⋊Dic73C22⋊C412yes—
127C4⋊Dic73C22⋊C412yes—
128#11924007—24no—
129#11924007—22no—
130#11924007—22no—
131#11924007—24no—
132#11924007—84no—
133C22⋊Dic73—24no—
134C22⋊Dic73—22no—
135C22⋊Dic73—24no—
136C22⋊Dic73—22no—
137C22⋊Dic73—22no—
138C22⋊Dic73C2×Q814yes—
139C4×Dic73—22no—
140C4×Dic73—22no—
141C4×Dic73—44no—
142C4×Dic73—84no—
143C24×C73C22×C411yescharacteristic
144C24×C73C2411yescharacteristic
145C2×D4×C73—22no—
146C2×D4×C73—22no—
147C2×D4×C73C2×Q812yes—
148C2×D4×C73#4112yes—
149C22×C4×C73—21no—
150C22×C4×C73—21no—
151C22×C4×C73—24no—
152C22×C4×C73—42no—
153C22×C4×C73—42no—
154C22×C4×C73C22×C411yescharacteristic
155C22×C4×C73C22×C411yescharacteristic
156C22×C4×C73C22×C411yescharacteristic
157C22×C4×C73C22×C411yescharacteristic
158C22×C4×C73C22×C411yescharacteristic
159C22×C4×C73C2×D412yes—
160C22×C4×C73C2×D412yes—
161C4⋊C4×C73—22no—
162C4⋊C4×C73—44no—
163C22⋊C4×C73—22no—
164C22⋊C4×C73—22no—
165C22⋊C4×C73—24no—
166C22⋊C4×C73—24no—
167C42×C73—21no—
168C42×C73—21no—
169C23×C73—21no—
170C23×C73—21no—
171C23×C73—41no—
172C23×C73—41no—
173C23×C73—41no—
174C23×C73—41no—
175C23×C73C4×Q812yes—
176C23×C73C23×C411yescharacteristic
177C23×C73C22×D412yes—
178C23×C73C22×Q811yescharacteristic
179C23×C73C2×#4111yescharacteristic
180C23×C732−1+412yes—
181D4×C73—24no—
182D4×C73—24no—
183C2×C4×C73—21no—
184C2×C4×C73—21no—
185C2×C4×C73—24no—
186C2×C4×C73—22no—
187C2×C4×C73—21no—
188C2×C4×C73—21no—
189C2×C4×C73—41no—
190C2×C4×C73—42no—
191C2×C4×C73—42no—
192C2×C4×C73—42no—
193C2×C4×C73—41no—
194C2×C4×C73—41no—
195C2×C4×C73—44no—
196C2×C4×C73—44no—
197C2×C4×C73—44no—
198C2×C4×C73C2×C22⋊C411yescharacteristic
199C2×C4×C73C2×C22⋊C412yes—
200C2×C4×C73C2×C22⋊C411yescharacteristic
201C2×C4×C73C4×Q812yes—
202C2×Dic73—22no—
203C2×Dic73—22no—
204C2×Dic73—24no—
205C2×Dic73—44no—
206C2×Dic73—44no—
207C2×Dic73—42no—
208C2×Dic73—42no—
209C2×Dic73—42no—
210C2×Dic73—42no—
211C2×Dic73—42no—
212C2×Dic73—42no—
213C2×Dic73—42no—
214C2×Dic73—42no—
215C2×Dic73—42no—
216C2×Dic73—82no—
217C2×Dic73—82no—
218C2×Dic73C2×C22⋊C412yes—
219C2×Dic73C2×C22⋊C412yes—
220C2×Dic73C2×C22⋊C414yes—
221Dic146—42no—
222Dic146—42no—
223Dic146—84no—
224C22×C73—22no—
225C22×C73—21no—
226C22×C73—22no—
227C22×C73—21no—
228C22×C73—22no—
229C22×C73—41no—
230C22×C73—41no—
231C22×C73—42no—
232C22×C73—42no—
233C22×C73—42no—
234C22×C73—41no—
235C22×C73—41no—
236C22×C73C22×C22⋊C411yescharacteristic
237C22×C73C2×C4×Q811yescharacteristic
238C22×C73#51911yescharacteristic
239C22×C73#54812yes—
240C22×C73D4×Q812yes—
241C4×C73—22no—
242C4×C73—21no—
243C4×C73—22no—
244C4×C73—41no—
245C4×C73—41no—
246C4×C73—84no—
247C4×C73#41012yes—
248Dic73—24no—
249Dic73—22no—
250Dic73—22no—
251Dic73—44no—
252Dic73—42no—
253Dic73—42no—
254Dic73—82no—
255#22513—731nomaximal, Sylow
256C2×C73—22no—
257C2×C73—22no—
258C2×C73—21no—
259C2×C73—21no—
260C2×C73—42no—
261C2×C73Q8×C22⋊C411yescharacteristic
262C2×C73C2×#41011yescharacteristic
263C2×C73#288311yescharacteristic
264C2×#554—731no—
265#2345—732no—
266#1996—732no—
267#1996—734no—
268#1799—732no—
269#1799D7311yescharacteristic
270C4×#99—732no—
271C4×#99—731no—
272C73#2251311yescharacteristic, complemented, Sylow
273#554—1462no—
274#554—1464no—
275#554—1462no—
276C2×C4⋊Q8—731no—
277C2×#125—732no—
278C2×C22⋊Q8—731no—
279C2×C22⋊Q8—731no—
280C2×C22⋊Q8—732no—
281C2×C22⋊Q8—732no—
282C2×C4×Q8—731no—
283C2×C4×D4—732no—
284C2×C4×D4C2×D7311yescharacteristic
285C22×C4⋊C4—731no—
286C22×C4⋊C4—731no—
287C2×#99—731no—
288C2×#99—731no—
289C2×#99C2×D7312yes—
290#389—1462no—
291#387—1464no—
292#380—734no—
293#380C2×D7312yes—
294C4×C4⋊C4—1462no—
295C4×C22⋊C4—732no—
296C4×C22⋊C4—734no—
297#342—734no—
298#342—734no—
299#342—732no—
300#342C2×D7312yes—
301C22×Q8—731no—
302C22×Q8—732no—
303C22×D4C22×D7311yescharacteristic
304C23×C4—732no—
305C23×C4—732no—
306C23×C4C4×D7311yescharacteristic
307C23×C4C22×D7311yescharacteristic
308C4⋊Q8—1462no—
309C4⋊Q8—1462no—
310#125—1464no—
311#125—1464no—
312C22⋊Q8—1464no—
313C22⋊Q8—1464no—
314C22⋊Q8—1464no—
315C22⋊Q8—1464no—
316C22⋊Q8—1462no—
317C22⋊Q8—1462no—
318C22⋊Q8—1462no—
319C22⋊Q8—1462no—
320C4×Q8—1462no—
321C4×Q8—1461no—
322C4×Q8—1461no—
323C4×D4—22no—
324C4×D4—22no—
325C4×D4—1464no—
326C4×D4—1464no—
327C2×C4⋊C4—22no—
328C2×C4⋊C4—731no—
329C2×C4⋊C4—732no—
330C2×C4⋊C4—731no—
331C2×C4⋊C4—732no—
332C2×C4⋊C4—734no—
333C2×C4⋊C4—732no—
334C2×C4⋊C4—732no—
335C2×C4⋊C4—731no—
336C2×C4⋊C4—732no—
337C2×C4⋊C4—732no—
338C2×C4⋊C4—732no—
339C2×C4⋊C4—732no—
340C2×C4⋊C4—1464no—
341C2×C4⋊C4—1462no—
342C2×C4⋊C4—1461no—
343C2×C4⋊C4—1461no—
344C2×C4⋊C4C4×D7311yescharacteristic
345C2×C22⋊C4—732no—
346C2×C22⋊C4—732no—
347C2×C22⋊C4—732no—
348C2×C22⋊C4—732no—
349C2×C22⋊C4C4×D7311yescharacteristic
350C2×C22⋊C4C22×D7314yes—
351C2×C22⋊C4C22×D7311yescharacteristic
352C2×C42—732no—
353C2×C42—1462no—
354C2×C42—1464no—
355C2×C42C4×D7311yescharacteristic
356#99—24no—
357#99—734no—
358#99—1462no—
359#95—22no—
360#95—1464no—
361#95—1464no—
362#95—1462no—
363C24C2×C4×D7311yescharacteristic
364C24C23×D7311yescharacteristic
365C2×Q8—2922no—
366C2×Q8—5842no—
367C2×Q8—1464no—
368C2×Q8—1461no—
369C2×Q8—1461no—
370C2×D4—22no—
371C2×D4—22no—
372C2×D4Q8×D7312yes—
373C2×D4#1192402712yes—
374C22×C4—21no—
375C22×C4—21no—
376C22×C4—24no—
377C22×C4—42no—
378C22×C4—42no—
379C22×C4—5842no—
380C22×C4—5842no—
381C22×C4—732no—
382C22×C4—732no—
383C22×C4—734no—
384C22×C4—734no—
385C22×C4—732no—
386C22×C4—732no—
387C22×C4—1462no—
388C22×C4—1462no—
389C22×C4—1462no—
390C22×C4—1462no—
391C22×C4C2×C4×D7311yescharacteristic
392C22×C4C2×C4×D7311yescharacteristic
393C22×C4C2×C4×D7311yescharacteristic
394C22×C4C2×C4×D7311yescharacteristic
395C22×C4C2×C4×D7311yescharacteristic
396C22×C4D4×D7312yes—
397C22×C4D4×D7312yes—
398C4⋊C4—22no—
399C4⋊C4—44no—
400C4⋊C4—5844no—
401C4⋊C4—732no—
402C4⋊C4—731no—
403C4⋊C4—731no—
404C4⋊C4—734no—
405C4⋊C4—734no—
406C4⋊C4—732no—
407C4⋊C4—732no—
408C4⋊C4—1462no—
409C4⋊C4—1462no—
410C4⋊C4—1464no—
411C4⋊C4—1462no—
412C4⋊C4—1462no—
413C4⋊C4—1464no—
414C4⋊C4—1464no—
415C4⋊C4—1461no—
416C4⋊C4—1461no—
417C22⋊C4—22no—
418C22⋊C4—22no—
419C22⋊C4—24no—
420C22⋊C4—24no—
421C22⋊C4—734no—
422C22⋊C4—1462no—
423C22⋊C4—1462no—
424C22⋊C4—1464no—
425C22⋊C4—1464no—
426C22⋊C4—1462no—
427C42—21no—
428C42—21no—
429C42—2924no—
430C42—5844no—
431C42—1462no—
432C42—1462no—
433C23—21no—
434C23—21no—
435C23—41no—
436C23—41no—
437C23—41no—
438C23—41no—
439C23#1455655612yes—
440C23C22×C4×D7311yescharacteristic
441C23C2×D4×D7312yes—
442C23C2×Q8×D7311yescharacteristic
443C23C2×#1192402711yescharacteristic
444C23#1455663812yes—
445Q8—2922no—
446Q8—2922no—
447Q8—5844no—
448D4—24no—
449D4—24no—
450C2×C4—21no—
451C2×C4—21no—
452C2×C4—24no—
453C2×C4—22no—
454C2×C4—21no—
455C2×C4—21no—
456C2×C4—41no—
457C2×C4—42no—
458C2×C4—42no—
459C2×C4—42no—
460C2×C4—41no—
461C2×C4—41no—
462C2×C4—44no—
463C2×C4—44no—
464C2×C4—44no—
465C2×C4—2922no—
466C2×C4—2922no—
467C2×C4—2922no—
468C2×C4—2922no—
469C2×C4—2922no—
470C2×C4—2922no—
471C2×C4—2922no—
472C2×C4—2924no—
473C2×C4—2924no—
474C2×C4—2922no—
475C2×C4—2922no—
476C2×C4—5842no—
477C2×C4—5842no—
478C2×C4—732no—
479C2×C4—732no—
480C2×C4—734no—
481C2×C4—1464no—
482C2×C4—1462no—
483C2×C4—1462no—
484C2×C4C22⋊C4×D7311yescharacteristic
485C2×C4C22⋊C4×D7312yes—
486C2×C4C22⋊C4×D7311yescharacteristic
487C2×C4#1455655612yes—
488C22—22no—
489C22—21no—
490C22—22no—
491C22—21no—
492C22—22no—
493C22—41no—
494C22—41no—
495C22—42no—
496C22—42no—
497C22—42no—
498C22—41no—
499C22—41no—
500C22C2×C22⋊C4×D7311yescharacteristic
501C22C2×#1455655611yescharacteristic
502C22#1798906411yescharacteristic
503C22#1798924312yes—
504C22#1798925012yes—
505C4—21no—
506C4—22no—
507C4—22no—
508C4—41no—
509C4—41no—
510C4—84no—
511C4—2924no—
512C4—2922no—
513C4—2922no—
514C4—5842no—
515C4—1462no—
516C4—1462no—
517C4—1464no—
518C4#1798855912yes—
519C2—22no—
520C2—22no—
521C2—21no—
522C2—21no—
523C2—42no—
524C2#2261276311yescharacteristic
525C2C2×#1798855911yescharacteristic
526C2#2261839011yescharacteristic
subgroupG/H
centerC22C2×C22⋊C4×D73
commutatorC23×C73C23×C4
FrattiniC24C23×D73
FittingC73×#1799C2
socleC22×C73C22×C22⋊C4
prime radicalC22×D4×C73C23
cosocleC24×C73C24
hypercenter#1799D73
hypocenterC73#22513
LCSG ▷ C23×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1799

Complements

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

NH ≅ G/Ntype
2. #22612810C2semidirect
3. #22610915C2semidirect
4. #22610914C2semidirect
12. C2×#14556591C4semidirect
13. C2×#14556582C4semidirect
14. C2×#14556570C4semidirect
16. C2×#14556568C4semidirect
18. C2×C4×Dic146C4semidirect
20. C22×C4⋊Dic73C4semidirect
82. #14556506Q8semidirect
272. C73#22513semidirect

Maximal subgroups

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

subgroupindex
C2×#179892942
#226128102
#226109152
#226109142
#226099172
#99×Dic732
C4×#145565062
C73×#17992
#2251373