Finite Group Databasebeta

← #33064675

Subgroups — #33064675

Order 18688.

976 conjugacy classes of subgroups.

The subgroup lattice has 482 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22620856C211yesmaximal, characteristic
2#22619962C211yesmaximal, characteristic
3#22619403C211yesmaximal, characteristic
4#22609152C214yesmaximal, complemented
5#22609009C214yesmaximal, complemented
6#22608612C211yesmaximal, characteristic, complemented
7#22608360C211yesmaximal, characteristic, complemented
8#22608326C211yesmaximal, characteristic, complemented
9C2×Q82×C73C211yesmaximal, characteristic, complemented
10C2×#14556622—21no—
11#17989137C2211yescharacteristic
12#17988961C2212yes—
13#17988937C2212yes—
14#17988905—21no—
15#17988872C2212yes—
16#17988809—22no—
17#17988797C2212yes—
18#17988789—22no—
19#17988787C2212yes—
20#17988787C2212yes—
21#17988726—21no—
22#17988726—22no—
23C4×#11924015—22no—
24#17988640C2211yescharacteristic
25#17988640C2211yescharacteristic
26#17988611C2211yescharacteristic
27#17988586—21no—
28#17988336C2214yes—
29Q8⋊#10596831C2214yes—
30Q8⋊#10596831C2214yes—
31Q82×C73—22no—
32Q82×C73—22no—
33Q82×C73—22no—
34Q82×C73C2214yes—
35C2×C4⋊Q8×C73—22no—
36C2×C4⋊Q8×C73—21no—
37C2×C4⋊Q8×C73—21no—
38C2×C4⋊Q8×C73C2211yescharacteristic
39C2×C4×Q8×C73—21no—
40C2×C4×Q8×C73—21no—
41C2×C4×Q8×C73C2211yescharacteristic
42C2×C4×Q8×C73C2211yescharacteristic
43C2×#11924028—21no—
44#14556622—44no—
45C2×#11924018—22no—
46C2×#11924015—22no—
47C2×#11924015—22no—
48C2×#11924015—24no—
49C2×#11924012—21no—
50#14556591C2311yescharacteristic
51#14556588C2311yescharacteristic
52#14556579—21no—
53#14556565—21no—
54#14556565—21no—
55C4×D292—21no—
56C4×D292—21no—
57#14556522—22no—
58C4⋊#10596831—21no—
59C4⋊#10596831—21no—
60C4⋊#10596831C2312yes—
61C4⋊#10596831C2312yes—
62#14556515—22no—
63#14556515—24no—
64#14556514—22no—
65#14556511—22no—
66#14556511—22no—
67C22⋊#10596831—21no—
68C22⋊#10596831—21no—
69C4×#10596831—21no—
70C4×#10596831C2312yes—
71C22×Q8×C73—22no—
72C22×Q8×C73D411yescharacteristic
73C22×Q8×C73D411yescharacteristic
74C22×Q8×C73D412yes—
75C4⋊Q8×C73—28no—
76C4⋊Q8×C73—24no—
77C4⋊Q8×C73—24no—
78C4⋊Q8×C73—28no—
79C4⋊Q8×C73—24no—
80C4⋊Q8×C73—24no—
81C4⋊Q8×C73—21no—
82C4⋊Q8×C73—21no—
83C4⋊Q8×C73D412yes—
84C4⋊Q8×C73C2312yes—
85C4×Q8×C73—24no—
86C4×Q8×C73—24no—
87C4×Q8×C73—24no—
88C4×Q8×C73—24no—
89C4×Q8×C73—21no—
90C4×Q8×C73—21no—
91C4×Q8×C73—21no—
92C4×Q8×C73—21no—
93C4×Q8×C73D412yes—
94C4×Q8×C73D412yes—
95C4×Q8×C73C2312yes—
96C4×Q8×C73C2312yes—
97C2×C4⋊C4×C73—22no—
98C2×C4⋊C4×C73—22no—
99C2×C4⋊C4×C73—21no—
100C2×C4⋊C4×C73—21no—
101C2×C4⋊C4×C73—21no—
102C2×C4⋊C4×C73—21no—
103C2×C4⋊C4×C73D411yescharacteristic
104C2×C4⋊C4×C73D411yescharacteristic
105C2×C42×C73—22no—
106C2×C42×C73—21no—
107C2×C42×C73—21no—
108C2×C42×C73C2311yescharacteristic
109#11924028—42no—
110#11924028—42no—
111C2×Dic146—21no—
112C2×D292—21no—
113C2×D292—21no—
114C2×D292—21no—
115C2×C22⋊D73—41no—
116C2×C4×D73—41no—
117#11924018—44no—
118#11924015—44no—
119#11924015—44no—
120#11924015—84no—
121#11924012—42no—
122C2×#10596831—22no—
123C2×#10596831—22no—
124C2×#10596831—22no—
125C4⋊Dic73—21no—
126C4⋊Dic73—21no—
127#11924005—41no—
128#11924005—41no—
129C2×Q8×C73—24no—
130C2×Q8×C73—28no—
131C2×Q8×C73—21no—
132C2×Q8×C73—21no—
133C2×Q8×C73—22no—
134C2×Q8×C73—22no—
135C2×Q8×C73—82no—
136C2×Q8×C73—84no—
137C2×Q8×C73SD1614yescomplemented
138C2×Q8×C73C2×D412yes—
139C2×Q8×C73C2×D412yes—
140C2×Q8×C73C2×D414yes—
141C22×C4×C73—21no—
142C22×C4×C73—21no—
143C22×C4×C73—22no—
144C22×C4×C73—21no—
145C22×C4×C73—21no—
146C22×C4×C73C2×D411yescharacteristic
147C22×C4×C73C2×D411yescharacteristic
148C22×C4×C73C2×D411yescharacteristic
149C4⋊C4×C73—28no—
150C4⋊C4×C73—28no—
151C4⋊C4×C73—22no—
152C4⋊C4×C73—22no—
153C4⋊C4×C73—22no—
154C4⋊C4×C73—22no—
155C4⋊C4×C73—21no—
156C4⋊C4×C73—21no—
157C4⋊C4×C73—22no—
158C4⋊C4×C73—22no—
159C4⋊C4×C73—22no—
160C4⋊C4×C73—22no—
161C4⋊C4×C73C2×D412yes—
162C4⋊C4×C73C2×D412yes—
163C42×C73—28no—
164C42×C73—22no—
165C42×C73—22no—
166C42×C73—22no—
167C42×C73—22no—
168C42×C73C2×D411yescharacteristic
169C42×C73C2×D411yescharacteristic
170C42×C73C2×D411yescharacteristic
171C42×C73C2411yescharacteristic
172C23×C73#12011yescharacteristic
173C22×D73—41no—
174Q8×C73—24no—
175Q8×C73—22no—
176Q8×C73—22no—
177Q8×C73—24no—
178Q8×C73—44no—
179Q8×C73—82no—
180Q8×C73—88no—
181Q8×C73C2×SD1614yes—
182Q8×C73#13714yescomplemented
183C2×C4×C73—24no—
184C2×C4×C73—22no—
185C2×C4×C73—22no—
186C2×C4×C73—24no—
187C2×C4×C73—22no—
188C2×C4×C73—24no—
189C2×C4×C73—22no—
190C2×C4×C73—22no—
191C2×C4×C73—41no—
192C2×C4×C73—44no—
193C2×C4×C73—42no—
194C2×C4×C73—42no—
195C2×C4×C73#12011yescharacteristic
196C2×C4×C73#12011yescharacteristic
197C2×C4×C73#12011yescharacteristic
198C2×C4×C73C2×SD1612yes—
199C2×C4×C73#13712yes—
200C2×C4×C73C22×D411yescharacteristic
201C2×C4×C73C22×D411yescharacteristic
202C2×C4×C73C22×D411yescharacteristic
203C22⋊D73—82no—
204C2×Dic73—41no—
205D292—42no—
206D292—41no—
207D292—42no—
208D292—41no—
209C4×D73—82no—
210Dic146—41no—
211Dic146—41no—
212#10596831—42no—
213#10596831—42no—
214#10596831—42no—
215C22×C73—21no—
216C22×C73—21no—
217C22×C73#44812yes—
218C22×C73C2×#12011yescharacteristic
219C2×D73—81no—
220C2×D73—81no—
221C2×D73—81no—
222C4×C73—24no—
223C4×C73—24no—
224C4×C73—21no—
225C4×C73—21no—
226C4×C73—21no—
227C4×C73—44no—
228C4×C73—42no—
229C4×C73—42no—
230C4×C73#44812yes—
231C4×C73C22×SD1611yescharacteristic
232C4×C73C2×#13711yescharacteristic
233C4×C73C2×#13712yes—
234Dic73—81no—
235#18768—731nomaximal, Sylow
236C2×C73—22no—
237C2×C73C2×#44811yescharacteristic
238C2×C73#300411yescharacteristic
239C2×C73#301911yescharacteristic
240D73—161no—
241C2×Q82D7311yescharacteristic, complemented
242#3206—731no—
243#3103—731no—
244#3037—731no—
245#1645—734no—
246#1623—734no—
247#1548—731no—
248#1487—731no—
249#1479—731no—
250C73#1876811yescharacteristic, complemented, Sylow
251C2×#137—1461no—
252Q82—22no—
253Q82—22no—
254Q82—22no—
255Q82C2×D7314yes—
256#532—731no—
257C2×C4⋊Q8—22no—
258C2×C4⋊Q8—21no—
259C2×C4⋊Q8—21no—
260C2×C4⋊Q8C2×D7311yescharacteristic
261C2×C4×Q8—21no—
262C2×C4×Q8—21no—
263C2×C4×Q8C2×D7311yescharacteristic
264C2×C4×Q8C2×D7311yescharacteristic
265#492—732no—
266#487—732no—
267#481—1461no—
268#475—732no—
269#464—1462no—
270#462—732no—
271#461—1462no—
272#460—732no—
273#460—732no—
274#448—1461no—
275#448—1462no—
276C4×SD16—1462no—
277#431—731no—
278#431—731no—
279#423—731no—
280#417—1461no—
281#333—734no—
282Q8⋊C8—734no—
283Q8⋊C8—734no—
284C2×#41—1461no—
285C22×Q8—22no—
286C22×Q8C22⋊D7311yescharacteristic
287C22×Q8C22⋊D7311yescharacteristic
288C22×Q8C22⋊D7312yes—
289#137—2924no—
290C2×Q16—1462no—
291C2×SD16—1462no—
292C2×SD16—1462no—
293C2×SD16—1464no—
294C2×M16—1461no—
295C4⋊Q8—28no—
296C4⋊Q8—24no—
297C4⋊Q8—24no—
298C4⋊Q8—28no—
299C4⋊Q8—24no—
300C4⋊Q8—24no—
301C4⋊Q8—21no—
302C4⋊Q8—21no—
303C4⋊Q8—731no—
304C4⋊Q8C22⋊D7312yes—
305C4⋊Q8C22×D7312yes—
306#127—731no—
307#124—1461no—
308#121—1461no—
309#121—1461no—
310C4×Q8—24no—
311C4×Q8—24no—
312C4×Q8—24no—
313C4×Q8—24no—
314C4×Q8—21no—
315C4×Q8—21no—
316C4×Q8—21no—
317C4×Q8—21no—
318C4×Q8C22⋊D7312yes—
319C4×Q8C22⋊D7312yes—
320C4×Q8C22×D7312yes—
321C4×Q8C22×D7312yes—
322C4×D4—1461no—
323C4×D4—1461no—
324C2×C4⋊C4—22no—
325C2×C4⋊C4—22no—
326C2×C4⋊C4—21no—
327C2×C4⋊C4—21no—
328C2×C4⋊C4—21no—
329C2×C4⋊C4—21no—
330C2×C4⋊C4C22⋊D7311yescharacteristic
331C2×C4⋊C4C22⋊D7311yescharacteristic
332C2×C42—22no—
333C2×C42—21no—
334C2×C42—21no—
335C2×C42C22×D7311yescharacteristic
336#106—1462no—
337C4⋊C8—732no—
338C4⋊C8—732no—
339C4⋊C8—1461no—
340C4⋊C8—1461no—
341#103—1462no—
342#103—1464no—
343#103—1462no—
344#102—1462no—
345#102—1462no—
346C22⋊C8—1461no—
347C22⋊C8—1461no—
348C4×C8—732no—
349C4×C8—1461no—
350#41—2922no—
351#41—2922no—
352C2×Q8—24no—
353C2×Q8—28no—
354C2×Q8—21no—
355C2×Q8—21no—
356C2×Q8—22no—
357C2×Q8—22no—
358C2×Q8—82no—
359C2×Q8—84no—
360C2×Q8—1461no—
361C2×Q8#1192401514yescomplemented
362C2×Q8C2×C22⋊D7312yes—
363C2×Q8C2×C22⋊D7312yes—
364C2×Q8C2×C22⋊D7314yes—
365C2×D4—2921no—
366C2×D4—1461no—
367C2×D4—1461no—
368C2×D4—1461no—
369C22×C4—21no—
370C22×C4—21no—
371C22×C4—22no—
372C22×C4—21no—
373C22×C4—21no—
374C22×C4—2921no—
375C22×C4C2×C22⋊D7311yescharacteristic
376C22×C4C2×C22⋊D7311yescharacteristic
377C22×C4C2×C22⋊D7311yescharacteristic
378Q16—2924no—
379SD16—2924no—
380SD16—2924no—
381SD16—5844no—
382M16—2922no—
383C2×C8—1462no—
384C2×C8—1462no—
385C2×C8—1462no—
386C4⋊C4—28no—
387C4⋊C4—28no—
388C4⋊C4—22no—
389C4⋊C4—22no—
390C4⋊C4—22no—
391C4⋊C4—22no—
392C4⋊C4—21no—
393C4⋊C4—21no—
394C4⋊C4—22no—
395C4⋊C4—22no—
396C4⋊C4—22no—
397C4⋊C4—22no—
398C4⋊C4—1461no—
399C4⋊C4—1461no—
400C4⋊C4C2×C22⋊D7312yes—
401C4⋊C4C2×C22⋊D7312yes—
402C22⋊C4—2921no—
403C22⋊C4—2921no—
404C42—28no—
405C42—22no—
406C42—22no—
407C42—22no—
408C42—22no—
409C42C2×C22⋊D7311yescharacteristic
410C42C2×C22⋊D7311yescharacteristic
411C42C2×C22⋊D7311yescharacteristic
412C42C23×D7311yescharacteristic
413C23—2921no—
414C23#1455656011yescharacteristic
415Q8—24no—
416Q8—22no—
417Q8—22no—
418Q8—24no—
419Q8—44no—
420Q8—82no—
421Q8—88no—
422Q8—2921no—
423Q8—2921no—
424Q8C2×#1192401514yes—
425Q8#1455662214yescomplemented
426D4—2922no—
427D4—2922no—
428D4—2921no—
429D4—2921no—
430D4—5842no—
431C2×C4—24no—
432C2×C4—24no—
433C2×C4—22no—
434C2×C4—22no—
435C2×C4—22no—
436C2×C4—24no—
437C2×C4—22no—
438C2×C4—22no—
439C2×C4—41no—
440C2×C4—44no—
441C2×C4—42no—
442C2×C4—42no—
443C2×C4—2921no—
444C2×C4—5842no—
445C2×C4#1455656011yescharacteristic
446C2×C4#1455656011yescharacteristic
447C2×C4#1455656011yescharacteristic
448C2×C4C2×#1192401512yes—
449C2×C4#1455662212yes—
450C2×C4C22×C22⋊D7311yescharacteristic
451C2×C4C22×C22⋊D7311yescharacteristic
452C2×C4C22×C22⋊D7311yescharacteristic
453C8—2922no—
454C8—2922no—
455C8—2922no—
456C22—21no—
457C22—21no—
458C22—5841no—
459C22—5841no—
460C22—5841no—
461C22#1798872612yes—
462C22C2×#1455656011yescharacteristic
463C4—24no—
464C4—21no—
465C4—21no—
466C4—21no—
467C4—24no—
468C4—44no—
469C4—42no—
470C4—42no—
471C4—5841no—
472C4#1798872612yes—
473C4C22×#1192401511yescharacteristic
474C4C2×#1455662212yes—
475C4C2×#1455662211yescharacteristic
476C2—22no—
477C2—11681no—
478C2C2×#1798872611yescharacteristic
479C2#2261916511yescharacteristic
480C2#2261928811yescharacteristic
subgroupG/H
centerC22C2×#14556560
commutator, cosocleC42×C73C24
FrattiniC42C23×D73
FittingC2×Q82×C73C2
socleC22×C73C2×#120
prime radical#17989137C22
hypercenterC2×Q82D73
hypocenterC73#18768
LCSG ▷ C42×C73 ▷ C22×C73 ▷ C73
UCSC1 ◁ C22 ◁ C2×C42 ◁ C2×Q82

Complements

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

NH ≅ G/Ntype
4. #22609152C2semidirect
5. #22609009C2semidirect
6. #22608612C2semidirect
7. #22608360C2semidirect
8. #22608326C2semidirect
9. C2×Q82×C73C2semidirect
137. C2×Q8×C73SD16semidirect
182. Q8×C73#137semidirect
241. C2×Q82D73semidirect
250. C73#18768semidirect
361. C2×Q8#11924015semidirect
425. Q8#14556622semidirect

Maximal subgroups

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

subgroupindex
#226208562
#226199622
#226194032
#226091522
#226090092
#226086122
#226083602
#226083262
C2×Q82×C732
#1876873