Finite Group Databasebeta

← #34021174

Subgroups — #34021174

Order 18688.

1166 conjugacy classes of subgroups.

The subgroup lattice has 480 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22626611C212yesmaximal
2#22625833C212yesmaximal
3#22625798C212yesmaximal, complemented
4#22625797C212yesmaximal, complemented
5#22625765C212yesmaximal
6#22625684C212yesmaximal, complemented
7C4×#14556652C212yesmaximal, complemented
8C2×#17989064C211yesmaximal, characteristic
9#17989611—28no—
10#17989590C2214yes—
11C2×#14556652C2214yes—
12C2×#14556652C2214yes—
13#17989557C2214yes—
14C4×#11924037C2214yes—
15#17989467C2214yes—
16#17989467C2214yes—
17C23×Dic146C2211yescharacteristic
18#17989064—28no—
19C2×C4×Dic146C411yescharacteristic
20C2×C4×Dic146C411yescharacteristic
21C2×C4×Dic146C2211yescharacteristic
22C2×C4×Dic146C2211yescharacteristic
23C2×#14556556C411yescharacteristic
24C2×#14556556C411yescharacteristic
25C2×#14556556C2211yescharacteristic
26C2×#14556556C2211yescharacteristic
27C2×#14556550C411yescharacteristic
28C2×#14556547C411yescharacteristic
29C2×#14556547C411yescharacteristic
30C2×#14556547C2211yescharacteristic
31C2×#14556547C2211yescharacteristic
32C2×#117×C73C411yescharacteristic
33C2×#11924037C2318yes—
34#14556658—24no—
35C4⋊#10596833—24no—
36#14556652—28no—
37#14556652—28no—
38C22⋊#10596833—28no—
39#14556646—24no—
40C4×#10596833—24no—
41C22×Dic146—24no—
42C22×Dic146—24no—
43C22×Dic146—24no—
44C22×Dic146C2×C411yescharacteristic
45C22×Dic146C2×C411yescharacteristic
46C22×Dic146C2311yescharacteristic
47C22×Dic146C2311yescharacteristic
48C23×Dic73C2×C411yescharacteristic
49C23×Dic73C2311yescharacteristic
50C4×Dic146—24no—
51C4×Dic146—24no—
52C4×Dic146—24no—
53C4×Dic146—24no—
54#14556556—24no—
55#14556556—24no—
56#14556556—24no—
57#14556556—24no—
58#14556550—24no—
59#14556547—24no—
60#14556547—24no—
61#14556547—24no—
62#14556547D418yes—
63C2×C4⋊Dic73C2×C411yescharacteristic
64C2×C4⋊Dic73C2×C411yescharacteristic
65C2×#11924007C2×C411yescharacteristic
66C2×#11924007C2×C411yescharacteristic
67C2×#11924007C2×C411yescharacteristic
68C2×#11924007C2×C411yescharacteristic
69C2×#11924007C2×C411yescharacteristic
70C2×#11924007C2×C411yescharacteristic
71C2×#11924007C2311yescharacteristic
72C2×#11924007C2311yescharacteristic
73C2×C22⋊Dic73C2×C411yescharacteristic
74C2×C22⋊Dic73C2×C411yescharacteristic
75C2×C4×Dic73C2×C411yescharacteristic
76C2×C4×Dic73C2×C411yescharacteristic
77C2×C4×Dic73C2×C411yescharacteristic
78C2×C4×Dic73C2×C411yescharacteristic
79C2×C4×Dic73C2×C411yescharacteristic
80C2×C4×Dic73C2×C411yescharacteristic
81C2×C4×Dic73C2×C411yescharacteristic
82C2×C4×Dic73C2×C411yescharacteristic
83C2×C4×Dic73C2311yescharacteristic
84C2×C4×Dic73C2311yescharacteristic
85C23×C4×C73C2×C411yescharacteristic
86#117×C73—24no—
87C2×C4⋊C4×C73C2×C411yescharacteristic
88C2×C4⋊C4×C73C2×C411yescharacteristic
89C2×C22⋊C4×C73C2×C411yescharacteristic
90C2×C22⋊C4×C73C2×C411yescharacteristic
91C2×C42×C73C2×C411yescharacteristic
92C2×C42×C73C2×C411yescharacteristic
93#11924037—28no—
94#11924037—28no—
95C2×#10596833—28no—
96C2×Dic146—232no—
97C2×Dic146—22no—
98C2×Dic146—22no—
99C2×Dic146—22no—
100C2×Dic146—22no—
101C2×Dic146—22no—
102C2×Dic146—22no—
103C2×Dic146—22no—
104C2×Dic146—22no—
105C2×Dic146—22no—
106C2×Dic146—22no—
107C2×Dic146#4114yes—
108C2×Dic146#4114yes—
109C22×Dic73—22no—
110C22×Dic73—22no—
111C22×Dic73—22no—
112C22×Dic73—22no—
113C22×Dic73—22no—
114C22×Dic73C22×C411yescharacteristic
115C22×Dic73C22×C411yescharacteristic
116C22×Dic73C22×C411yescharacteristic
117C22×Dic73C22×C411yescharacteristic
118C22×Dic73C22×C411yescharacteristic
119C22×Dic73C22×C411yescharacteristic
120C22×Dic73C22×C411yescharacteristic
121C22×Dic73C2×D414yes—
122C22×Dic73C2411yescharacteristic
123C4⋊Dic73—22no—
124C4⋊Dic73—22no—
125#11924007—22no—
126#11924007—22no—
127#11924007—22no—
128#11924007—22no—
129#11924007—22no—
130#11924007—22no—
131#11924007C22⋊C414yes—
132#11924007C22⋊C414yes—
133C22⋊Dic73—22no—
134C22⋊Dic73C22⋊C414yes—
135C4×Dic73—22no—
136C4×Dic73—22no—
137C4×Dic73—22no—
138C4×Dic73—22no—
139C4×Dic73—22no—
140C4×Dic73—22no—
141C4×Dic73—22no—
142C4×Dic73—22no—
143C4×Dic73C2×D414yes—
144C4×Dic73C2×D414yes—
145C24×C73C22×C411yescharacteristic
146C22×C4×C73—22no—
147C22×C4×C73—22no—
148C22×C4×C73—22no—
149C22×C4×C73C22×C411yescharacteristic
150C22×C4×C73C22×C411yescharacteristic
151C22×C4×C73C22×C411yescharacteristic
152C22×C4×C73C22×C411yescharacteristic
153C22×C4×C73C22×C411yescharacteristic
154C22×C4×C73C22×C411yescharacteristic
155C4⋊C4×C73—22no—
156C4⋊C4×C73—22no—
157C22⋊C4×C73—22no—
158C22⋊C4×C73C22⋊C414yes—
159C42×C73—22no—
160C42×C73—22no—
161C23×C73—21no—
162C23×C73—21no—
163C23×C73—41no—
164C23×C73—41no—
165C23×C73C2×C22⋊C412yes—
166C23×C73C23×C411yescharacteristic
167C2×C4×C73—28no—
168C2×C4×C73—21no—
169C2×C4×C73—21no—
170C2×C4×C73—21no—
171C2×C4×C73—21no—
172C2×C4×C73—21no—
173C2×C4×C73—21no—
174C2×C4×C73—41no—
175C2×C4×C73—41no—
176C2×C4×C73—41no—
177C2×C4×C73—41no—
178C2×C4×C73C2×C22⋊C412yes—
179C2×C4×C73C2×C22⋊C412yes—
180C2×C4×C73#11712yes—
181C2×C4×C73#11712yes—
182C2×Dic73—28no—
183C2×Dic73—21no—
184C2×Dic73—21no—
185C2×Dic73—21no—
186C2×Dic73—21no—
187C2×Dic73—21no—
188C2×Dic73—21no—
189C2×Dic73—28no—
190C2×Dic73—21no—
191C2×Dic73—21no—
192C2×Dic73—41no—
193C2×Dic73—41no—
194C2×Dic73—41no—
195C2×Dic73—41no—
196C2×Dic73C2×C22⋊C412yes—
197C2×Dic73C2×C22⋊C412yes—
198C2×Dic73C2×C22⋊C412yes—
199C2×Dic73#11712yes—
200C2×Dic73#11712yes—
201C2×Dic73C22×D412yes—
202C2×Dic73C2×#4112yes—
203C2×Dic73C2×#4112yes—
204Dic146—28no—
205Dic146—28no—
206Dic146—28no—
207Dic146—28no—
208#10596833—48no—
209C22×C73—21no—
210C22×C73—21no—
211C22×C73—22no—
212C22×C73—22no—
213C22×C73—22no—
214C22×C73—42no—
215C22×C73—42no—
216C22×C73C22×C22⋊C411yescharacteristic
217C22×C73C2×#11711yescharacteristic
218C22×C73C2×#11711yescharacteristic
219C4×C73—22no—
220C4×C73—22no—
221C4×C73—22no—
222C4×C73—22no—
223C4×C73—41no—
224C4×C73—41no—
225Dic73—22no—
226Dic73—22no—
227Dic73—22no—
228Dic73—22no—
229Dic73—22no—
230Dic73—41no—
231Dic73—41no—
232Dic73#52514yes—
233#20730—731nomaximal, Sylow
234C2×C73—21no—
235C2×C73—21no—
236C2×C73—42no—
237C2×C73#228011yescharacteristic
238C2×C73#288911yescharacteristic
239C2×C73#288911yescharacteristic
240C2×#519—731no—
241#2921—732no—
242#1822—732no—
243#1801—732no—
244#1801—732no—
245#1784—732no—
246C4×#101—732no—
247C4×#101—732no—
248C73#2073011yescharacteristic, complemented, Sylow
249C23×Q8—731no—
250#519—1468no—
251C2×C4×Q8—731no—
252C2×C4×Q8—731no—
253C2×C4×Q8—731no—
254C2×C4×Q8—731no—
255C2×C4×Q8—731no—
256C2×C4×Q8—731no—
257C2×C4×Q8—731no—
258C2×C4×Q8—731no—
259C2×#117—731no—
260C2×#117—731no—
261C2×#117—731no—
262C2×#117—731no—
263C2×#117—731no—
264C2×#117#6440811yescharacteristic
265#433—1468no—
266#423—734no—
267C2×#101—734no—
268C2×#101—734no—
269#407—734no—
270C4×M16—734no—
271#343—734no—
272#343—734no—
273C22×Q8—731no—
274C22×Q8—731no—
275C22×Q8—731no—
276C22×Q8—731no—
277C22×Q8—1464no—
278C22×Q8—1464no—
279C22×Q8—1464no—
280C23×C4—731no—
281C23×C4—731no—
282C23×C4C2×#6440811yescharacteristic
283C2×M16—738no—
284C4×Q8—1464no—
285C4×Q8—1464no—
286C4×Q8—1464no—
287C4×Q8—1464no—
288C4×Q8—1464no—
289C4×Q8—1464no—
290C4×Q8—1464no—
291C4×Q8—1464no—
292#117—24no—
293#117—738no—
294#117—1464no—
295#117—1464no—
296#117—1464no—
297#117—1464no—
298C2×C4⋊C4—731no—
299C2×C4⋊C4—731no—
300C2×C4⋊C4—731no—
301C2×C4⋊C4—731no—
302C2×C4⋊C4—731no—
303C2×C4⋊C4—731no—
304C2×C4⋊C4—731no—
305C2×C4⋊C4—731no—
306C2×C4⋊C4—731no—
307C2×C4⋊C4—731no—
308C2×C4⋊C4C2×#6440811yescharacteristic
309C2×C4⋊C4C2×#6440811yescharacteristic
310C2×C22⋊C4—731no—
311C2×C22⋊C4—731no—
312C2×C22⋊C4C2×#6440811yescharacteristic
313C2×C22⋊C4C2×#6440811yescharacteristic
314C2×C42—731no—
315C2×C42—731no—
316C2×C42—731no—
317C2×C42—731no—
318C2×C42—731no—
319C2×C42—731no—
320C2×C42—731no—
321C2×C42—731no—
322C2×C42—731no—
323C2×C42—731no—
324C2×C42C2×#6440811yescharacteristic
325C2×C42C2×#6440811yescharacteristic
326C4⋊C8—1464no—
327C4⋊C8—1464no—
328#101—1468no—
329#101—1468no—
330C22⋊C8—1468no—
331#97—1464no—
332C4×C8—1464no—
333C24C22×#6440811yescharacteristic
334C2×Q8—734no—
335C2×Q8—734no—
336C2×Q8—1462no—
337C2×Q8—1462no—
338C2×Q8—1462no—
339C2×Q8—1462no—
340C2×Q8—1462no—
341C2×Q8—1462no—
342C2×Q8—1462no—
343C2×Q8—1462no—
344C2×Q8—14632no—
345C2×Q8—1462no—
346C2×Q8—1462no—
347C22×C4—22no—
348C22×C4—22no—
349C22×C4—22no—
350C22×C4—731no—
351C22×C4—731no—
352C22×C4—731no—
353C22×C4—731no—
354C22×C4—731no—
355C22×C4—731no—
356C22×C4—731no—
357C22×C4—731no—
358C22×C4—734no—
359C22×C4—1462no—
360C22×C4—1462no—
361C22×C4—1462no—
362C22×C4—1462no—
363C22×C4—1462no—
364C22×C4C22×#6440811yescharacteristic
365C22×C4C22×#6440811yescharacteristic
366C22×C4C22×#6440811yescharacteristic
367C22×C4C22×#6440811yescharacteristic
368C22×C4C22×#6440811yescharacteristic
369C22×C4C22×#6440811yescharacteristic
370M16—1468no—
371M16—1468no—
372C2×C8—1468no—
373C4⋊C4—22no—
374C4⋊C4—22no—
375C4⋊C4—734no—
376C4⋊C4—734no—
377C4⋊C4—1462no—
378C4⋊C4—1462no—
379C4⋊C4—1462no—
380C4⋊C4—1462no—
381C4⋊C4—1462no—
382C4⋊C4—1462no—
383C4⋊C4—1462no—
384C4⋊C4—1462no—
385C22⋊C4—22no—
386C22⋊C4—734no—
387C22⋊C4—1462no—
388C22⋊C4C22⋊#6440814yes—
389C42—22no—
390C42—22no—
391C42—734no—
392C42—734no—
393C42—1462no—
394C42—1462no—
395C42—1462no—
396C42—1462no—
397C42—1462no—
398C42—1462no—
399C42—1462no—
400C42—1462no—
401C23—21no—
402C23—21no—
403C23—41no—
404C23—41no—
405C23C2×C22⋊#6440812yes—
406C23C23×#6440811yescharacteristic
407Q8—1468no—
408Q8—1468no—
409Q8—1468no—
410Q8—1468no—
411C2×C4—28no—
412C2×C4—21no—
413C2×C4—21no—
414C2×C4—21no—
415C2×C4—21no—
416C2×C4—21no—
417C2×C4—21no—
418C2×C4—41no—
419C2×C4—41no—
420C2×C4—41no—
421C2×C4—41no—
422C2×C4—2921no—
423C2×C4—2921no—
424C2×C4—2921no—
425C2×C4—2921no—
426C2×C4—732no—
427C2×C4—732no—
428C2×C4—732no—
429C2×C4—732no—
430C2×C4—732no—
431C2×C4—732no—
432C2×C4—732no—
433C2×C4—732no—
434C2×C4—1461no—
435C2×C4—1461no—
436C2×C4—1468no—
437C2×C4—1468no—
438C2×C4—1461no—
439C2×C4—1461no—
440C2×C4—1461no—
441C2×C4—1461no—
442C2×C4—1461no—
443C2×C4—1461no—
444C2×C4C2×C22⋊#6440812yes—
445C2×C4C2×C22⋊#6440812yes—
446C2×C4#1455667012yes—
447C2×C4#1455667012yes—
448C8—2928no—
449C22—21no—
450C22—21no—
451C22—22no—
452C22—22no—
453C22—22no—
454C22—42no—
455C22—42no—
456C22C22×C22⋊#6440811yescharacteristic
457C22C2×#1455667011yescharacteristic
458C22C2×#1455667011yescharacteristic
459C4—22no—
460C4—22no—
461C4—22no—
462C4—22no—
463C4—41no—
464C4—41no—
465C4—2921no—
466C4—2921no—
467C4—734no—
468C4—1462no—
469C4—1462no—
470C4—1462no—
471C4—1462no—
472C4—1462no—
473C2—21no—
474C2—21no—
475C2—42no—
476C2#2262637511yescharacteristic
477C2#2262654311yescharacteristic
478C2#2262654311yescharacteristic
subgroupG/H
centerC22C22×C22⋊#64408
commutatorC23×C73C23×C4
FrattiniC23C23×#64408
FittingC2×#117×C73C4
socleC22×C73C22×C22⋊C4
prime radicalC24×C73C22×C4
cosocleC22×Dic73C24
hypercenterC2×#117#64408
hypocenterC73#20730
LCSG ▷ C23×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C22 ◁ C2×C22⋊C4 ◁ C2×#117

Complements

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

NH ≅ G/Ntype
3. #22625798C2semidirect
4. #22625797C2semidirect
6. #22625684C2semidirect
7. C4×#14556652C2semidirect
248. C73#20730semidirect

Maximal subgroups

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

subgroupindex
#226266112
#226258332
#226257982
#226257972
#226257652
#226256842
C4×#145566522
C2×#179890642
#2073073