Finite Group Databasebeta

← #33089206

Subgroups — #33089206

Order 18688.

800 conjugacy classes of subgroups.

The subgroup lattice has 532 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C4×#14556613C211yesmaximal, characteristic
2#22618576C211yesmaximal, characteristic, complemented
3C4×#14556599C211yesmaximal, characteristic, complemented
4#22610578C212yesmaximal, complemented
5#22610529C212yesmaximal, complemented
6#22610259C212yesmaximal
7#22610203C212yesmaximal, complemented
8C4×#14556517C211yesmaximal, characteristic, complemented
9C4×#14556517C211yesmaximal, characteristic, complemented
10#22609676C211yesmaximal, characteristic, complemented
11C4×#108×C73C211yesmaximal, characteristic, complemented
12C2×#14556613C2211yescharacteristic
13C2×#14556599C2212yes—
14C4×#11924028C2211yescharacteristic
15C4×#11924028C2211yescharacteristic
16C4×Dic292C411yescharacteristic, complemented
17C4×Dic292C411yescharacteristic, complemented
18C4×#11924017C411yescharacteristic, complemented
19C4×#11924017C411yescharacteristic, complemented
20C4×#11924017C411yescharacteristic, complemented
21C4×#11924017C411yescharacteristic, complemented
22C4×D584C411yescharacteristic, complemented
23C4×D584C411yescharacteristic, complemented
24#17988649—22no—
25#17988638—21no—
26#17988638—21no—
27#17988625C2211yescharacteristic
28#17988615C2212yes—
29C2×#14556517C2212yes—
30C2×#14556517C2212yes—
31#17988584C2211yescharacteristic
32#17988584C2211yescharacteristic
33#17988555C2212yes—
34#17988545C2212yes—
35#17988544—22no—
36C4×#11924011—21no—
37C2×C4×#10596831C2211yescharacteristic
38C4×C8×D73—21no—
39C42×Dic73C2211yescharacteristic
40#17988348C2212yes—
41#17988348C2212yes—
42#17988341C2212yes—
43#17988341C2212yes—
44C2×#108×C73C2212yes—
45C4×M16×C73C2211yescharacteristic
46C4×M16×C73C2211yescharacteristic
47C2×C4×C8×C73C2211yescharacteristic
48#341×C73C2212yes—
49C2×#11924028C2311yescharacteristic
50C2×#11924028C2311yescharacteristic
51#14556613—24no—
52C2×Dic292C2×C411yescharacteristic
53C2×#11924017C2×C411yescharacteristic
54C2×#11924017C2×C411yescharacteristic
55C2×D584C2×C411yescharacteristic
56#14556599—24no—
57#14556599—24no—
58C2×#11924011—22no—
59C22×#10596831C2312yes—
60C2×C8×D73—22no—
61C4×Dic146C2×C411yescharacteristic
62C4×Dic146C2×C411yescharacteristic
63C4×Dic146C2×C411yescharacteristic
64C4×Dic146C2×C411yescharacteristic
65C4×D292C2×C411yescharacteristic
66C4×D292C2×C411yescharacteristic
67C4×D292C2×C411yescharacteristic
68C4×D292C2×C411yescharacteristic
69C4×C22⋊D73—21no—
70C4×C22⋊D73—21no—
71C4×C22⋊D73—21no—
72C4×C22⋊D73—21no—
73#14556550C2311yescharacteristic
74#14556550C2311yescharacteristic
75#14556548—21no—
76#14556548—21no—
77C2×C4×Dic73—21no—
78C2×C4×Dic73—21no—
79C2×C4×Dic73C2312yes—
80C42×D73—21no—
81C42×D73—21no—
82#14556525C2×C411yescharacteristic
83#14556525C2×C411yescharacteristic
84#14556523C2×C411yescharacteristic
85#14556523C2×C411yescharacteristic
86#14556520—22no—
87#14556520—22no—
88#14556517—24no—
89#14556517—24no—
90#14556516C2×C411yescharacteristic
91#14556516C2×C411yescharacteristic
92#14556516C2×C411yescharacteristic
93#14556516C2×C411yescharacteristic
94#14556513C2×C411yescharacteristic
95#14556513C2×C411yescharacteristic
96#14556513C2×C411yescharacteristic
97#14556513C2×C411yescharacteristic
98#14556503—22no—
99#14556503—22no—
100#14556501—22no—
101C4×#10596831D411yescharacteristic
102C4×#10596831D411yescharacteristic
103C4×#10596831D412yes—
104C8×Dic73—22no—
105C2×M16×C73C2312yes—
106C2×M16×C73C2312yes—
107C22×C8×C73C2311yescharacteristic
108C22×C8×C73C2311yescharacteristic
109C2×C42×C73C2311yescharacteristic
110#108×C73—24no—
111#97×C73—21no—
112#97×C73—21no—
113C4×C8×C73—21no—
114C4×C8×C73—21no—
115C4×C8×C73C2×C411yescharacteristic
116C4×C8×C73C2×C411yescharacteristic
117C4×C8×C73C2×C411yescharacteristic
118C4×C8×C73C2×C411yescharacteristic
119#11924028—22no—
120#11924028—22no—
121#11924028—22no—
122#11924028#4114yes—
123C2×Dic146C22×C411yescharacteristic
124C2×Dic146C22×C411yescharacteristic
125C2×D292C22×C411yescharacteristic
126C2×D292C22×C411yescharacteristic
127C2×C22⋊D73—21no—
128C2×C22⋊D73—21no—
129C22×Dic73—22no—
130C2×C4×D73—21no—
131C2×C4×D73—21no—
132C2×C4×D73—21no—
133C2×C4×D73—21no—
134C2×C4×D73—21no—
135C2×C4×D73—21no—
136Dic292—22no—
137#11924017—22no—
138#11924017—22no—
139D584—22no—
140#11924011—28no—
141C2×#10596831—22no—
142C2×#10596831—22no—
143C2×#10596831C2×D412yes—
144C2×#10596831C2×D412yes—
145C8×D73—28no—
146C4⋊Dic73C22×C411yescharacteristic
147C4⋊Dic73C22×C411yescharacteristic
148C4⋊Dic73C22×C411yescharacteristic
149C4⋊Dic73C22×C411yescharacteristic
150#11924007—21no—
151#11924007—21no—
152#11924007—21no—
153#11924007—21no—
154C22⋊Dic73—21no—
155C22⋊Dic73—21no—
156#11924005—21no—
157#11924005—21no—
158#11924005—21no—
159#11924005—21no—
160C4×Dic73—24no—
161C4×Dic73—21no—
162C4×Dic73—21no—
163C4×Dic73—21no—
164C4×Dic73—21no—
165C4×Dic73—21no—
166C4×Dic73—21no—
167C4×Dic73—22no—
168C4×Dic73—22no—
169C4×Dic73—42no—
170C22×C4×C73#4111yescharacteristic
171C22×C4×C73#4111yescharacteristic
172C22×C4×C73C2411yescharacteristic
173M16×C73—22no—
174M16×C73—22no—
175M16×C73—22no—
176M16×C73#4114yes—
177C2×C8×C73—22no—
178C2×C8×C73—22no—
179C2×C8×C73—22no—
180C2×C8×C73—22no—
181C2×C8×C73C22×C411yescharacteristic
182C2×C8×C73C22×C411yescharacteristic
183C2×C8×C73C22×C411yescharacteristic
184C2×C8×C73C22×C411yescharacteristic
185C42×C73C22×C411yescharacteristic
186C42×C73C22×C411yescharacteristic
187C42×C73C2×D411yescharacteristic
188C42×C73C2×D411yescharacteristic
189C23×C73C2×#4111yescharacteristic
190C22×D73—21no—
191C22×D73—21no—
192C2×C4×C73—22no—
193C2×C4×C73—22no—
194C2×C4×C73C4×D411yescharacteristic
195C2×C4×C73C4×D411yescharacteristic
196C2×C4×C73C4×D411yescharacteristic
197C2×C4×C73C4×D411yescharacteristic
198C2×C4×C73C23×C411yescharacteristic
199C2×C4×C73C22×D411yescharacteristic
200C2×C4×C73C2×#4112yes—
201C2×C4×C73C2×#4112yes—
202C22⋊D73—24no—
203C22⋊D73—42no—
204C2×Dic73—21no—
205C2×Dic73—24no—
206C2×Dic73—21no—
207C2×Dic73—21no—
208C2×Dic73—21no—
209C2×Dic73—21no—
210C2×Dic73—21no—
211C2×Dic73—42no—
212C2×Dic73—42no—
213D292—21no—
214D292—21no—
215D292—21no—
216D292#11712yes—
217C4×D73—24no—
218C4×D73—42no—
219C4×D73—42no—
220C4×D73—42no—
221C4×D73—42no—
222C4×D73—42no—
223Dic146—21no—
224Dic146—21no—
225Dic146—21no—
226Dic146#11712yes—
227C8×C73—21no—
228C8×C73—21no—
229C8×C73—21no—
230C8×C73—21no—
231C8×C73—24no—
232C8×C73—42no—
233#10596831—22no—
234#10596831—22no—
235C22×C73—21no—
236C22×C73—21no—
237C22×C73C2×C4×D411yescharacteristic
238C22×C73#52912yes—
239C2×D73—22no—
240C2×D73—41no—
241C2×D73—41no—
242C2×D73—41no—
243C2×D73—41no—
244C2×D73—41no—
245C4×C73—21no—
246C4×C73—21no—
247C4×C73#44312yes—
248C4×C73#44412yes—
249C4×C73C2×#11711yescharacteristic
250C4×C73C4×#4111yescharacteristic
251C4×C73#52512yes—
252Dic73—22no—
253Dic73—44no—
254Dic73—41no—
255Dic73—41no—
256Dic73—41no—
257Dic73—41no—
258Dic73—41no—
259#21137—731nomaximal, Sylow
260C2×C73—22no—
261C2×C73#229011yescharacteristic
262C2×C73C2×#44311yescharacteristic
263C2×C73C2×#44411yescharacteristic
264D73—42no—
265D73—81no—
266C4×#135—731no—
267#2922—731no—
268C4×#131—731no—
269#1929—732no—
270#1922—732no—
271#1876—732no—
272#1866—732no—
273C4×#108D7311yescharacteristic, complemented
274C4×#104—731no—
275C4×#104—731no—
276#1744—731no—
277C73#2113711yescharacteristic, complemented, Sylow
278C2×#135—731no—
279C2×#131—732no—
280C4×#41—731no—
281C4×#41—731no—
282C4×Q16—731no—
283C4×Q16—731no—
284C4×SD16—731no—
285C4×SD16—731no—
286C4×SD16—731no—
287C4×SD16—731no—
288C4×D8—731no—
289C4×D8—731no—
290#433—1462no—
291#431—1461no—
292#431—1461no—
293C2×#108C2×D7312yes—
294#427—731no—
295#424—732no—
296C2×#104—732no—
297C2×#104—732no—
298#416—731no—
299#416—731no—
300#408—732no—
301#405—732no—
302#405—1462no—
303C4×M16—1461no—
304C4×M16C2×D7311yescharacteristic
305C4×M16C2×D7311yescharacteristic
306C2×C4×C8—731no—
307C2×C4×C8—1461no—
308C2×C4×C8C2×D7311yescharacteristic
309C43—731no—
310#341C2×D7312yes—
311#339—732no—
312#339—732no—
313#336—732no—
314#336—732no—
315C2×#41—731no—
316C2×#41—731no—
317#135—1464no—
318C2×Q16—731no—
319C2×SD16—731no—
320C2×SD16—731no—
321C2×D8—731no—
322#131—1464no—
323#131—1464no—
324C2×M16—1462no—
325C2×M16C22×D7312yes—
326C2×M16C22×D7312yes—
327C22×C8—732no—
328C22×C8—1462no—
329C22×C8C22×D7311yescharacteristic
330C22×C8C22×D7311yescharacteristic
331C4×Q8—731no—
332C4×Q8—731no—
333C4×Q8—731no—
334C4×Q8—731no—
335C4×D4—731no—
336C4×D4—731no—
337C4×D4—731no—
338C4×D4—731no—
339C4×D4—1461no—
340C4×D4—1461no—
341C4×D4—1461no—
342C4×D4—1461no—
343#117—731no—
344#117—731no—
345#117—1461no—
346#117—1461no—
347C2×C42—732no—
348C2×C42—1461no—
349C2×C42—1461no—
350C2×C42—1461no—
351C2×C42—1461no—
352C2×C42C22×D7311yescharacteristic
353#108—24no—
354#107—731no—
355#107—731no—
356#106—731no—
357#106—731no—
358C4⋊C8—1462no—
359C4⋊C8—1462no—
360#104—1464no—
361#104—1464no—
362#103—731no—
363#103—731no—
364#103—731no—
365#103—731no—
366#102—731no—
367#102—731no—
368#102—731no—
369#102—731no—
370C22⋊C8—1462no—
371C22⋊C8—1462no—
372#97—21no—
373#97—21no—
374#97—1462no—
375C4×C8—21no—
376C4×C8—21no—
377C4×C8—732no—
378C4×C8—731no—
379C4×C8—731no—
380C4×C8—1462no—
381C4×C8C4×D7311yescharacteristic
382C4×C8C4×D7311yescharacteristic
383C4×C8C4×D7311yescharacteristic
384C4×C8C4×D7311yescharacteristic
385#41—734no—
386#41—1462no—
387#41—1462no—
388#41—1462no—
389C2×Q8—731no—
390C2×Q8—731no—
391C2×D4—731no—
392C2×D4—731no—
393C2×D4—1461no—
394C2×D4—1461no—
395C22×C4—1462no—
396C22×C4—1461no—
397C22×C4—1461no—
398C22×C4—1461no—
399C22×C4—1461no—
400C22×C4—1461no—
401C22×C4—1461no—
402C22×C4#1192402911yescharacteristic
403C22×C4#1192402911yescharacteristic
404C22×C4C23×D7311yescharacteristic
405Q16—1462no—
406SD16—1462no—
407SD16—1462no—
408D8—1462no—
409M16—22no—
410M16—22no—
411M16—22no—
412M16—1468no—
413M16#1192402814yes—
414C2×C8—22no—
415C2×C8—22no—
416C2×C8—22no—
417C2×C8—22no—
418C2×C8—732no—
419C2×C8—732no—
420C2×C8—1468no—
421C2×C8—1462no—
422C2×C8—1462no—
423C2×C8C2×C4×D7311yescharacteristic
424C2×C8C2×C4×D7311yescharacteristic
425C2×C8C2×C4×D7311yescharacteristic
426C2×C8C2×C4×D7311yescharacteristic
427C4⋊C4—731no—
428C4⋊C4—731no—
429C4⋊C4—731no—
430C4⋊C4—731no—
431C4⋊C4—1461no—
432C4⋊C4—1461no—
433C4⋊C4—1461no—
434C4⋊C4—1461no—
435C22⋊C4—1461no—
436C22⋊C4—1461no—
437C22⋊C4—1461no—
438C22⋊C4—1461no—
439C22⋊C4—1461no—
440C22⋊C4—1461no—
441C42—2922no—
442C42—1464no—
443C42—1462no—
444C42—1462no—
445C42—1461no—
446C42—1461no—
447C42—1461no—
448C42—1461no—
449C42—1461no—
450C42—1461no—
451C42C2×C4×D7311yescharacteristic
452C42C2×C4×D7311yescharacteristic
453C42D4×D7311yescharacteristic
454C42D4×D7311yescharacteristic
455C23—1461no—
456C23—1461no—
457C23C2×#1192402911yescharacteristic
458Q8—732no—
459Q8—1461no—
460Q8—1461no—
461Q8—1461no—
462D4—2922no—
463D4—732no—
464D4—1464no—
465D4—1461no—
466D4—1461no—
467D4—1461no—
468C2×C4—22no—
469C2×C4—22no—
470C2×C4—2922no—
471C2×C4—2922no—
472C2×C4—2922no—
473C2×C4—2922no—
474C2×C4—2922no—
475C2×C4—2922no—
476C2×C4—2922no—
477C2×C4—1464no—
478C2×C4—1461no—
479C2×C4—1461no—
480C2×C4—1461no—
481C2×C4—1461no—
482C2×C4—1461no—
483C2×C4—1461no—
484C2×C4—1464no—
485C2×C4#1455655311yescharacteristic
486C2×C4#1455655311yescharacteristic
487C2×C4#1455655311yescharacteristic
488C2×C4#1455655311yescharacteristic
489C2×C4C22×C4×D7311yescharacteristic
490C2×C4C2×D4×D7311yescharacteristic
491C2×C4#41×D7312yes—
492C2×C4C2×#1192402812yes—
493C8—21no—
494C8—21no—
495C8—21no—
496C8—21no—
497C8—24no—
498C8—42no—
499C8—1462no—
500C8—1462no—
501C22—21no—
502C22—21no—
503C22—2921no—
504C22—2921no—
505C22—2921no—
506C22—2921no—
507C22—2921no—
508C22—1462no—
509C22C2×#1455655311yescharacteristic
510C22#1798912412yes—
511C4—21no—
512C4—21no—
513C4—2924no—
514C4—2921no—
515C4—2921no—
516C4—2921no—
517C4—2921no—
518C4—2921no—
519C4—1462no—
520C4#1798870312yes—
521C4#1798870812yes—
522C4#117×D7311yescharacteristic
523C4C4×#1192402811yescharacteristic
524C4#1798909612yes—
525C2—22no—
526C2—2922no—
527C2—5841no—
528C2#2261237211yescharacteristic
529C2C2×#1798870311yescharacteristic
530C2C2×#1798870811yescharacteristic
subgroupG/H
centerC42D4×D73
commutatorC2×C4×C73C23×C4
FrattiniC22×C4C23×D73
FittingC4×#108×C73C2
socleC22×C73C2×C4×D4
prime radicalC2×#14556613C22
cosocleC22×C4×C73C24
hypercenterC4×#108D73
hypocenterC73#21137
LCSG ▷ C2×C4×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C42 ◁ C2×C42 ◁ C4×#108

Complements

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

NH ≅ G/Ntype
2. #22618576C2semidirect
3. C4×#14556599C2semidirect
4. #22610578C2semidirect
5. #22610529C2semidirect
7. #22610203C2semidirect
8. C4×#14556517C2semidirect
9. C4×#14556517C2semidirect
10. #22609676C2semidirect
11. C4×#108×C73C2semidirect
16. C4×Dic292C4semidirect
17. C4×Dic292C4semidirect
18. C4×#11924017C4semidirect
19. C4×#11924017C4semidirect
20. C4×#11924017C4semidirect
21. C4×#11924017C4semidirect
22. C4×D584C4semidirect
23. C4×D584C4semidirect
273. C4×#108D73semidirect
277. C73#21137semidirect

Maximal subgroups

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

subgroupindex
C4×#145566132
#226185762
C4×#145565992
#226105782
#226105292
#226102592
#226102032
C4×#145565172
#226096762
C4×#108×C732
#2113773