Finite Group Databasebeta

← #33102537

Subgroups — #33102537

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#22612806C212yesmaximal, complemented
3#22610916C212yesmaximal, complemented
4#22610914C214yesmaximal, complemented
5#22609919C211yesmaximal, characteristic
6#22609917C212yesmaximal
7#99×Dic73C212yesmaximal
8C4×#99×C73C211yesmaximal, 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×#14556506C2212yes—
23#17988485—22no—
24#17988470—24no—
25#17988447C2212yes—
26#17988445C2214yes—
27C4×C4⋊Dic73—22no—
28C22⋊C4×Dic73C2214yes—
29#17988354C2212yes—
30#17988353C2214yes—
31#17988353C2214yes—
32C2×C4×D4×C73C2211yescharacteristic
33C2×#99×C73C2211yescharacteristic
34C2×#99×C73C2211yescharacteristic
35C4×C22⋊C4×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⋊Dic73—22no—
62C2×C4⋊Dic73C2×C411yescharacteristic
63C2×C4⋊Dic73C2×C412yes—
64C2×C4⋊Dic73C2×C412yes—
65C2×C4⋊Dic73C2×C411yescharacteristic
66C2×C4⋊Dic73C2×C411yescharacteristic
67C2×C4⋊Dic73C2×C412yes—
68C2×C4⋊Dic73D412yes—
69C2×C4⋊Dic73D414yes—
70C2×C4⋊Dic73D412yes—
71C2×#11924007—24no—
72C2×#11924007C2×C412yes—
73C2×#11924007C2×C412yes—
74C2×#11924007C2×C412yes—
75C2×C22⋊Dic73C2×C412yes—
76C2×C22⋊Dic73C2314yes—
77C2×C22⋊Dic73C2312yes—
78C2×C4×Dic73—24no—
79C2×C4×Dic73C2×C412yes—
80#14556506—24no—
81#14556498—24no—
82#14556498—24no—
83#14556498—22no—
84C22×D4×C73C2311yescharacteristic
85C23×C4×C73C2×C411yescharacteristic
86C23×C4×C73C2311yescharacteristic
87C4×D4×C73—22no—
88C4×D4×C73—22no—
89C2×C4⋊C4×C73C2×C411yescharacteristic
90C2×C22⋊C4×C73C2×C411yescharacteristic
91C2×C22⋊C4×C73C2312yes—
92C2×C22⋊C4×C73C2311yescharacteristic
93C2×C22⋊C4×C73C2312yes—
94C2×C42×C73—22no—
95C2×C42×C73C2×C411yescharacteristic
96#99×C73—22no—
97#99×C73Q814yescomplemented
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—24no—
107C22×Dic73—82no—
108C22×Dic73—82no—
109C22×Dic73C22×C412yes—
110C22×Dic73C22×C412yes—
111C22×Dic73C22×C412yes—
112C22×Dic73C22×C412yes—
113C22×Dic73C2×D414yes—
114C22×Dic73C2×D414yes—
115C4⋊Dic73—21no—
116C4⋊Dic73—21no—
117C4⋊Dic73—22no—
118C4⋊Dic73—22no—
119C4⋊Dic73—24no—
120C4⋊Dic73—44no—
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—24no—
135C22⋊Dic73—24no—
136C22⋊Dic73—24no—
137C4×Dic73—22no—
138C4×Dic73—22no—
139C4×Dic73—84no—
140C24×C73C22×C411yescharacteristic
141C24×C73C2411yescharacteristic
142C2×D4×C73—22no—
143C2×D4×C73—22no—
144C2×D4×C73C2×Q812yes—
145C2×D4×C73#4112yes—
146C22×C4×C73—22no—
147C22×C4×C73—21no—
148C22×C4×C73—21no—
149C22×C4×C73—22no—
150C22×C4×C73—42no—
151C22×C4×C73—42no—
152C22×C4×C73C22×C411yescharacteristic
153C22×C4×C73C22×C411yescharacteristic
154C22×C4×C73C22×C411yescharacteristic
155C22×C4×C73C22×C411yescharacteristic
156C22×C4×C73C22×C411yescharacteristic
157C22×C4×C73C2×D412yes—
158C22×C4×C73C2×D412yes—
159C4⋊C4×C73—22no—
160C22⋊C4×C73—22no—
161C22⋊C4×C73—22no—
162C22⋊C4×C73—22no—
163C22⋊C4×C73—22no—
164C22⋊C4×C73—22no—
165C22⋊C4×C73C2×Q814yes—
166C42×C73—21no—
167C42×C73—21no—
168C42×C73—44no—
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—24no—
190C2×C4×C73—41no—
191C2×C4×C73—42no—
192C2×C4×C73—42no—
193C2×C4×C73—42no—
194C2×C4×C73—42no—
195C2×C4×C73—42no—
196C2×C4×C73—41no—
197C2×C4×C73—41no—
198C2×C4×C73—42no—
199C2×C4×C73—42no—
200C2×C4×C73—42no—
201C2×C4×C73C2×C22⋊C411yescharacteristic
202C2×C4×C73C2×C22⋊C412yes—
203C2×C4×C73C2×C22⋊C411yescharacteristic
204C2×C4×C73C4×Q812yes—
205C2×Dic73—22no—
206C2×Dic73—22no—
207C2×Dic73—44no—
208C2×Dic73—44no—
209C2×Dic73—42no—
210C2×Dic73—42no—
211C2×Dic73—42no—
212C2×Dic73—42no—
213C2×Dic73—44no—
214C2×Dic73—44no—
215C2×Dic73—44no—
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—44no—
245C4×C73—41no—
246C4×C73—41no—
247C4×C73—82no—
248C4×C73#41012yes—
249Dic73—24no—
250Dic73—22no—
251Dic73—22no—
252Dic73—42no—
253Dic73—42no—
254Dic73—84no—
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#1799—731no—
270C4×#99—732no—
271C4×#99D7311yescharacteristic
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—732no—
288C2×#99C2×D7311yescharacteristic
289C2×#99C2×D7311yescharacteristic
290#389—1462no—
291#387—1464no—
292#380—732no—
293#380—734no—
294C4×C4⋊C4—1462no—
295C4×C22⋊C4—734no—
296C4×C22⋊C4C2×D7312yes—
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—731no—
328C2×C4⋊C4—732no—
329C2×C4⋊C4—731no—
330C2×C4⋊C4—732no—
331C2×C4⋊C4—734no—
332C2×C4⋊C4—732no—
333C2×C4⋊C4—732no—
334C2×C4⋊C4—731no—
335C2×C4⋊C4—732no—
336C2×C4⋊C4—732no—
337C2×C4⋊C4—732no—
338C2×C4⋊C4—732no—
339C2×C4⋊C4—1462no—
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—734no—
348C2×C22⋊C4C4×D7311yescharacteristic
349C2×C22⋊C4C22×D7312yes—
350C2×C22⋊C4C22×D7311yescharacteristic
351C2×C22⋊C4C22×D7312yes—
352C2×C42—22no—
353C2×C42—732no—
354C2×C42—1464no—
355C2×C42C4×D7311yescharacteristic
356#99—22no—
357#99—1464no—
358#99Dic14614yescomplemented
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×D4C2×Dic14612yes—
373C2×D4#1192402812yes—
374C22×C4—22no—
375C22×C4—21no—
376C22×C4—21no—
377C22×C4—22no—
378C22×C4—42no—
379C22×C4—42no—
380C22×C4—5842no—
381C22×C4—5842no—
382C22×C4—732no—
383C22×C4—732no—
384C22×C4—734no—
385C22×C4—734no—
386C22×C4—732no—
387C22×C4—732no—
388C22×C4—1464no—
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—2924no—
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—22no—
420C22⋊C4—22no—
421C22⋊C4—22no—
422C22⋊C4—1464no—
423C22⋊C4—1464no—
424C22⋊C4—1464no—
425C22⋊C4—1464no—
426C22⋊C4C2×Dic14614yes—
427C42—21no—
428C42—21no—
429C42—44no—
430C42—5844no—
431C42—1462no—
432C42—1462no—
433C23—21no—
434C23—21no—
435C23—41no—
436C23—41no—
437C23—41no—
438C23—41no—
439C23C4×Dic14612yes—
440C23C22×C4×D7311yescharacteristic
441C23C2×D4×D7312yes—
442C23C22×Dic14611yescharacteristic
443C23C2×#1192402811yescharacteristic
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—24no—
455C2×C4—21no—
456C2×C4—21no—
457C2×C4—41no—
458C2×C4—42no—
459C2×C4—42no—
460C2×C4—42no—
461C2×C4—42no—
462C2×C4—42no—
463C2×C4—41no—
464C2×C4—41no—
465C2×C4—42no—
466C2×C4—42no—
467C2×C4—42no—
468C2×C4—2922no—
469C2×C4—2922no—
470C2×C4—2924no—
471C2×C4—2924no—
472C2×C4—2924no—
473C2×C4—2924no—
474C2×C4—2924no—
475C2×C4—2922no—
476C2×C4—2922no—
477C2×C4—5842no—
478C2×C4—5842no—
479C2×C4—732no—
480C2×C4—732no—
481C2×C4—734no—
482C2×C4—1462no—
483C2×C4—1462no—
484C2×C4C22⋊C4×D7311yescharacteristic
485C2×C4C22⋊C4×D7312yes—
486C2×C4C22⋊C4×D7311yescharacteristic
487C2×C4C4×Dic14612yes—
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×C4×Dic14611yescharacteristic
502C22#1798906411yescharacteristic
503C22#1798924612yes—
504C22D4×Dic14612yes—
505C4—21no—
506C4—22no—
507C4—22no—
508C4—44no—
509C4—41no—
510C4—41no—
511C4—82no—
512C4—2922no—
513C4—2922no—
514C4—5844no—
515C4—1462no—
516C4—1462no—
517C4—1464no—
518C4#1798855912yes—
519C2—22no—
520C2—22no—
521C2—21no—
522C2—21no—
523C2—42no—
524C2C22⋊C4×Dic14611yescharacteristic
525C2C2×#1798855911yescharacteristic
526C2#2261839011yescharacteristic
subgroupG/H
centerC22C2×C22⋊C4×D73
commutatorC23×C73C23×C4
FrattiniC24C23×D73
FittingC4×#99×C73C2
socleC22×C73C22×C22⋊C4
prime radicalC22×D4×C73C23
cosocleC24×C73C24
hypercenterC4×#99D73
hypocenterC73#22513
LCSG ▷ C23×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C22 ◁ C23×C4 ◁ C4×#99

Complements

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

NH ≅ G/Ntype
2. #22612806C2semidirect
3. #22610916C2semidirect
4. #22610914C2semidirect
12. C2×#14556591C4semidirect
13. C2×#14556582C4semidirect
14. C2×#14556570C4semidirect
16. C2×#14556568C4semidirect
18. C2×C4×Dic146C4semidirect
20. C22×C4⋊Dic73C4semidirect
97. #99×C73Q8semidirect
272. C73#22513semidirect
358. #99Dic146semidirect

Maximal subgroups

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

subgroupindex
C2×#179892942
#226128062
#226109162
#226109142
#226099192
#226099172
#99×Dic732
C4×#99×C732
#2251373