Finite Group Databasebeta

← #34021175

Subgroups — #34021175

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#22626613C212yesmaximal
2#22625832C212yesmaximal
3#22625798C212yesmaximal, complemented
4#22625797C212yesmaximal, complemented
5#22625766C212yesmaximal
6#22625684C212yesmaximal, complemented
7C4×#14556652C212yesmaximal, complemented
8C2×#17989063C211yesmaximal, characteristic
9#17989612—28no—
10#17989590C2214yes—
11C2×#14556652C2214yes—
12C2×#14556652C2214yes—
13#17989559C2214yes—
14C4×#11924037C2214yes—
15#17989467C2214yes—
16#17989467C2214yes—
17C22×Q8×D73C2211yescharacteristic
18#17989063—28no—
19C2×Q8×Dic73C411yescharacteristic
20C2×C4×Dic146C411yescharacteristic
21C2×C4×Dic146C2211yescharacteristic
22C2×C4×Dic146C2211yescharacteristic
23C2×#14556556C411yescharacteristic
24C2×#14556556C2211yescharacteristic
25C2×#14556556C2211yescharacteristic
26C2×#14556549C411yescharacteristic
27C2×#14556549C411yescharacteristic
28C2×#14556549C2211yescharacteristic
29C2×#14556548C411yescharacteristic
30C2×#14556548C411yescharacteristic
31C2×#14556548C2211yescharacteristic
32C2×C4×Q8×C73C411yescharacteristic
33C2×#11924037C2318yes—
34#14556658—24no—
35C4⋊#10596833—24no—
36#14556652—28no—
37#14556652—28no—
38#14556648—28no—
39#14556646—24no—
40C4×#10596833—24no—
41C22×Dic146C2×C411yescharacteristic
42C22×Dic146C2311yescharacteristic
43C22×Dic146C2311yescharacteristic
44C2×Q8×D73—24no—
45C2×Q8×D73—24no—
46C2×Q8×D73—24no—
47C22×C4×D73C2×C411yescharacteristic
48C22×C4×D73C2×C411yescharacteristic
49C22×C4×D73C2311yescharacteristic
50Q8×Dic73—24no—
51C4×Dic146—24no—
52C4×Dic146—24no—
53C4×Dic146—24no—
54#14556556—24no—
55#14556556—24no—
56#14556556—24no—
57#14556549—24no—
58#14556549—24no—
59#14556549D418yes—
60#14556548—24no—
61#14556548—24no—
62#14556548—24no—
63C2×C4⋊Dic73C2×C411yescharacteristic
64C2×C4⋊Dic73C2×C411yescharacteristic
65C2×C4⋊Dic73C2×C411yescharacteristic
66C2×#11924007C2×C411yescharacteristic
67C2×#11924007C2×C411yescharacteristic
68C2×#11924007C2×C411yescharacteristic
69C2×#11924007C2×C411yescharacteristic
70C2×#11924007C2311yescharacteristic
71C2×#11924007C2311yescharacteristic
72C2×#11924005C2×C411yescharacteristic
73C2×#11924005C2×C411yescharacteristic
74C2×#11924005C2×C411yescharacteristic
75C2×#11924005C2×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
85C22×Q8×C73C2×C411yescharacteristic
86C4×Q8×C73—24no—
87C2×C4⋊C4×C73C2×C411yescharacteristic
88C2×C4⋊C4×C73C2×C411yescharacteristic
89C2×C4⋊C4×C73C2×C411yescharacteristic
90C2×C42×C73C2×C411yescharacteristic
91C2×C42×C73C2×C411yescharacteristic
92C2×C42×C73C2×C411yescharacteristic
93#11924037—28no—
94#11924037—28no—
95C2×#10596833—28no—
96C23×D73C22×C411yescharacteristic
97C2×Dic146—22no—
98C2×Dic146—22no—
99C2×Dic146—22no—
100C2×Dic146—22no—
101C2×Dic146—22no—
102C2×Dic146—22no—
103C2×Dic146—22no—
104C2×Dic146#4114yes—
105C2×Dic146#4114yes—
106Q8×D73—232no—
107C22×Dic73C22×C411yescharacteristic
108C22×Dic73C22×C411yescharacteristic
109C22×Dic73C22×C411yescharacteristic
110C22×Dic73C22×C411yescharacteristic
111C22×Dic73C22×C411yescharacteristic
112C22×Dic73C22×C411yescharacteristic
113C22×Dic73C2411yescharacteristic
114C2×C4×D73—22no—
115C2×C4×D73—22no—
116C2×C4×D73—22no—
117C2×C4×D73—22no—
118C2×C4×D73—22no—
119C2×C4×D73—22no—
120C2×C4×D73—22no—
121C2×C4×D73—22no—
122C2×C4×D73C2×D414yes—
123C4⋊Dic73—22no—
124C4⋊Dic73—22no—
125C4⋊Dic73C22⋊C414yes—
126#11924007—22no—
127#11924007—22no—
128#11924007—22no—
129#11924007—22no—
130#11924007—22no—
131#11924007—22no—
132#11924005—22no—
133#11924005—22no—
134#11924005C22⋊C414yes—
135#11924005C22⋊C414yes—
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—
145C2×Q8×C73—22no—
146C2×Q8×C73—22no—
147C2×Q8×C73—22no—
148C22×C4×C73C22×C411yescharacteristic
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—
157C4⋊C4×C73C22⋊C414yes—
158C42×C73—22no—
159C42×C73—22no—
160C42×C73—22no—
161C23×C73C23×C411yescharacteristic
162C22×D73—21no—
163C22×D73—21no—
164C22×D73—41no—
165C22×D73—41no—
166C22×D73C2×C22⋊C412yes—
167Q8×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—21no—
175C2×C4×C73—21no—
176C2×C4×C73—41no—
177C2×C4×C73—41no—
178C2×C4×C73—41no—
179C2×C4×C73—41no—
180C2×C4×C73C2×C22⋊C412yes—
181C2×C4×C73C2×C22⋊C412yes—
182C2×C4×C73C2×C22⋊C412yes—
183C2×C4×C73#11712yes—
184C2×C4×C73#11712yes—
185C2×Dic73—21no—
186C2×Dic73—21no—
187C2×Dic73—21no—
188C2×Dic73—21no—
189C2×Dic73—21no—
190C2×Dic73—21no—
191C2×Dic73—41no—
192C2×Dic73—41no—
193C2×Dic73—41no—
194C2×Dic73—41no—
195C2×Dic73C2×C22⋊C412yes—
196C2×Dic73C2×C22⋊C412yes—
197C2×Dic73#11712yes—
198C2×Dic73#11712yes—
199C2×Dic73C22×D412yes—
200C2×Dic73C2×#4112yes—
201C2×Dic73C2×#4112yes—
202C4×D73—28no—
203C4×D73—28no—
204C4×D73—28no—
205Dic146—28no—
206Dic146—28no—
207Dic146—28no—
208#10596833—48no—
209C22×C73—21no—
210C22×C73—21no—
211C22×C73C22×C22⋊C411yescharacteristic
212C22×C73C2×#11711yescharacteristic
213C22×C73C2×#11711yescharacteristic
214C2×D73—22no—
215C2×D73—22no—
216C2×D73—22no—
217C2×D73—42no—
218C2×D73—42no—
219C4×C73—22no—
220C4×C73—22no—
221C4×C73—22no—
222C4×C73—22no—
223C4×C73—22no—
224C4×C73—41no—
225C4×C73—41no—
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#228011yescharacteristic
237C2×C73#288911yescharacteristic
238C2×C73#288911yescharacteristic
239D73—42no—
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#6440811yescharacteristic
259C2×#117—731no—
260C2×#117—731no—
261C2×#117—731no—
262C2×#117—731no—
263C2×#117—731no—
264C2×#117—731no—
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—1464no—
277C22×Q8—1464no—
278C22×Q8—1464no—
279C22×Q8C2×#6440811yescharacteristic
280C23×C4—731no—
281C23×C4—731no—
282C23×C4—731no—
283C2×M16—738no—
284C4×Q8—24no—
285C4×Q8—1464no—
286C4×Q8—1464no—
287C4×Q8—1464no—
288C4×Q8—1464no—
289C4×Q8—1464no—
290C4×Q8—1464no—
291C4×Q8—1464no—
292#117—738no—
293#117—1464no—
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⋊C4C2×#6440811yescharacteristic
308C2×C4⋊C4C2×#6440811yescharacteristic
309C2×C4⋊C4C2×#6440811yescharacteristic
310C2×C22⋊C4—731no—
311C2×C22⋊C4—731no—
312C2×C22⋊C4—731no—
313C2×C22⋊C4—731no—
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×C42C2×#6440811yescharacteristic
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—
333C24—731no—
334C2×Q8—22no—
335C2×Q8—22no—
336C2×Q8—22no—
337C2×Q8—734no—
338C2×Q8—734no—
339C2×Q8—1462no—
340C2×Q8—1462no—
341C2×Q8—1462no—
342C2×Q8—1462no—
343C2×Q8—1462no—
344C2×Q8—1462no—
345C2×Q8—14632no—
346C2×Q8—1462no—
347C22×C4—731no—
348C22×C4—731no—
349C22×C4—731no—
350C22×C4—731no—
351C22×C4—731no—
352C22×C4—731no—
353C22×C4—731no—
354C22×C4—734no—
355C22×C4—1462no—
356C22×C4—1462no—
357C22×C4—1462no—
358C22×C4—1462no—
359C22×C4—1462no—
360C22×C4—1462no—
361C22×C4—1462no—
362C22×C4—1462no—
363C22×C4C22×#6440811yescharacteristic
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—1462no—
377C4⋊C4—1462no—
378C4⋊C4—1462no—
379C4⋊C4—1462no—
380C4⋊C4—1462no—
381C4⋊C4—1462no—
382C4⋊C4—1462no—
383C4⋊C4—1462no—
384C4⋊C4C22⋊#6440814yes—
385C22⋊C4—734no—
386C22⋊C4—734no—
387C22⋊C4—1462no—
388C22⋊C4—1462no—
389C42—22no—
390C42—22no—
391C42—22no—
392C42—734no—
393C42—734no—
394C42—1462no—
395C42—1462no—
396C42—1462no—
397C42—1462no—
398C42—1462no—
399C42—1462no—
400C42—1462no—
401C23—2921no—
402C23—2921no—
403C23—732no—
404C23—1461no—
405C23—1461no—
406C23C23×#6440811yescharacteristic
407Q8—28no—
408Q8—1468no—
409Q8—1468no—
410Q8—1468no—
411C2×C4—21no—
412C2×C4—21no—
413C2×C4—21no—
414C2×C4—21no—
415C2×C4—21no—
416C2×C4—21no—
417C2×C4—21no—
418C2×C4—21no—
419C2×C4—41no—
420C2×C4—41no—
421C2×C4—41no—
422C2×C4—41no—
423C2×C4—2921no—
424C2×C4—2921no—
425C2×C4—2921no—
426C2×C4—2921no—
427C2×C4—732no—
428C2×C4—732no—
429C2×C4—732no—
430C2×C4—732no—
431C2×C4—732no—
432C2×C4—732no—
433C2×C4—732no—
434C2×C4—1468no—
435C2×C4—1468no—
436C2×C4—1468no—
437C2×C4—1461no—
438C2×C4—1461no—
439C2×C4—1461no—
440C2×C4—1461no—
441C2×C4—1461no—
442C2×C4—1461no—
443C2×C4C2×C22⋊#6440812yes—
444C2×C4C2×C22⋊#6440812yes—
445C2×C4C2×C22⋊#6440812yes—
446C2×C4#1455667012yes—
447C2×C4#1455667012yes—
448C8—2928no—
449C22—21no—
450C22—21no—
451C22—2922no—
452C22—2922no—
453C22—1462no—
454C22—1462no—
455C22—1462no—
456C22C22×C22⋊#6440811yescharacteristic
457C22C2×#1455667011yescharacteristic
458C22C2×#1455667011yescharacteristic
459C4—22no—
460C4—22no—
461C4—22no—
462C4—22no—
463C4—22no—
464C4—41no—
465C4—41no—
466C4—2921no—
467C4—2921no—
468C4—734no—
469C4—1462no—
470C4—1462no—
471C4—1462no—
472C4—1462no—
473C2—21no—
474C2—21no—
475C2—2922no—
476C2#2262637511yescharacteristic
477C2#2262654311yescharacteristic
478C2#2262654411yescharacteristic
subgroupG/H
centerC22C22×C22⋊#64408
commutatorC23×C73C23×C4
FrattiniC23C23×#64408
FittingC2×C4×Q8×C73C4
socleC22×C73C22×C22⋊C4
prime radicalC23×D73C22×C4
cosocleC22×Dic73C24
hypercenterC2×C4×Q8#64408
hypocenterC73#20730
LCSG ▷ C23×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C22 ◁ C2×C4⋊C4 ◁ C2×C4×Q8

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
#226266132
#226258322
#226257982
#226257972
#226257662
#226256842
C4×#145566522
C2×#179890632
#2073073