Finite Group Databasebeta

← #27139980

Subgroups — #27139980

Order 12416.

780 conjugacy classes of subgroups.

The subgroup lattice has 514 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#13095490C211yesmaximal, characteristic, complemented
2C22×#13095488C211yesmaximal, characteristic
3C22×#13095488C211yesmaximal, characteristic
4C22×#13095488C211yesmaximal, characteristic
5C2×#15690717C211yesmaximal, characteristic, complemented
6C2×#15690717C211yesmaximal, characteristic, complemented
7C2×#95×C97C211yesmaximal, characteristic, complemented
8C24×D97C2211yescharacteristic
9C23×Dic97C2211yescharacteristic
10C23×Dic97C2211yescharacteristic
11C23×Dic97C2211yescharacteristic
12C2×#13095490—22no—
13C2×#13095490—22no—
14C2×#13095490—22no—
15C2×#13095488—22no—
16C2×#13095488—22no—
17C2×#13095488—22no—
18C2×#13095488—22no—
19C2×#13095488—22no—
20C2×#13095488—22no—
21C2×#13095488—22no—
22C2×#13095488—22no—
23C2×#13095488C414yescomplemented
24#15690717—22no—
25#15690717—22no—
26C23×C4×C97C2211yescharacteristic
27C23×C4×C97C2211yescharacteristic
28C23×C4×C97C2211yescharacteristic
29#95×C97—22no—
30C23×D97—21no—
31C23×D97—21no—
32C23×D97—21no—
33C23×D97—21no—
34C23×D97—21no—
35C23×D97—21no—
36C23×D97—21no—
37C23×D97—21no—
38C23×D97—21no—
39C23×D97—21no—
40C23×D97—21no—
41C23×D97—21no—
42C23×D97C2×C412yes—
43C23×D97D412yes—
44C23×D97Q812yes—
45C22×Dic97—21no—
46C22×Dic97—21no—
47C22×Dic97—21no—
48C22×Dic97—21no—
49C22×Dic97—21no—
50C22×Dic97—21no—
51C22×Dic97—21no—
52C22×Dic97—21no—
53C22×Dic97—21no—
54C22×Dic97—21no—
55C22×Dic97—21no—
56C22×Dic97—21no—
57C22×Dic97—21no—
58C22×Dic97—21no—
59C22×Dic97—21no—
60C22×Dic97—21no—
61C22×Dic97C2×C412yes—
62C22×Dic97D412yes—
63C22×Dic97D412yes—
64C22×Dic97D412yes—
65C22×Dic97D412yes—
66#13095490—28no—
67#13095488—28no—
68#13095488—28no—
69#13095488—28no—
70C24×C97C2311yescharacteristic
71C22×C4×C97—21no—
72C22×C4×C97—21no—
73C22×C4×C97—21no—
74C22×C4×C97—21no—
75C22×C4×C97—21no—
76C22×C4×C97—21no—
77C22×C4×C97—21no—
78C22×C4×C97—21no—
79C22×C4×C97—21no—
80C22×C4×C97—21no—
81C22×C4×C97—21no—
82C22×C4×C97—21no—
83C22×C4×C97—21no—
84C22×C4×C97—21no—
85C22×C4×C97—21no—
86C22×C4×C97—21no—
87C22×C4×C97—21no—
88C22×C4×C97—21no—
89C22×C4×C97C2×C412yes—
90C22×C4×C97D412yes—
91C22×C4×C97D412yes—
92C23×C97C22×C411yescharacteristic
93C23×C97C2×D411yescharacteristic
94C23×C97C2×D411yescharacteristic
95C23×C97C2×D411yescharacteristic
96C23×C97C2×D411yescharacteristic
97C23×C97C2×D411yescharacteristic
98C23×C97C2×D411yescharacteristic
99C23×C97C2×D411yescharacteristic
100C23×C97C2×Q811yescharacteristic
101C23×C97#4111yescharacteristic
102C23×C97#4111yescharacteristic
103C23×C97#4111yescharacteristic
104C23×C97#4111yescharacteristic
105C23×C97#4111yescharacteristic
106C23×C97#4111yescharacteristic
107C22×D97—22no—
108C22×D97—22no—
109C22×D97—22no—
110C22×D97—22no—
111C22×D97—22no—
112C22×D97—22no—
113C22×D97—22no—
114C22×D97—22no—
115C22×D97—22no—
116C22×D97—22no—
117C22×D97—22no—
118C22×D97—22no—
119C22×D97—22no—
120C22×D97—22no—
121C22×D97—22no—
122C22×D97—22no—
123C22×D97—22no—
124C22×D97—22no—
125C22×D97—41no—
126C22×D97—41no—
127C22×D97—41no—
128C22×D97—41no—
129C22×D97—41no—
130C22×D97—41no—
131C22×D97—41no—
132C22×D97—41no—
133C22×D97—41no—
134C22×D97—41no—
135C22×D97—41no—
136C22×D97—41no—
137C22×D97—41no—
138C22×D97—41no—
139C22×D97—41no—
140C22×D97—41no—
141C22×D97C4⋊C414yescomplemented
142C2×C4×C97—22no—
143C2×C4×C97—22no—
144C2×C4×C97—22no—
145C2×C4×C97—22no—
146C2×C4×C97—22no—
147C2×C4×C97—22no—
148C2×C4×C97—22no—
149C2×C4×C97—22no—
150C2×C4×C97—22no—
151C2×C4×C97—41no—
152C2×C4×C97—41no—
153C2×C4×C97—41no—
154C2×C4×C97—41no—
155C2×C4×C97—41no—
156C2×C4×C97—41no—
157C2×C4×C97—41no—
158C2×C4×C97—41no—
159C2×C4×C97—41no—
160C2×C4×C97—41no—
161C2×C4×C97—41no—
162C2×C4×C97—41no—
163C2×Dic97—22no—
164C2×Dic97—22no—
165C2×Dic97—22no—
166C2×Dic97—22no—
167C2×Dic97—22no—
168C2×Dic97—22no—
169C2×Dic97—22no—
170C2×Dic97—22no—
171C2×Dic97—22no—
172C2×Dic97—22no—
173C2×Dic97—22no—
174C2×Dic97—22no—
175C2×Dic97—41no—
176C2×Dic97—41no—
177C2×Dic97—41no—
178C2×Dic97—41no—
179C2×Dic97—41no—
180C2×Dic97—41no—
181C2×Dic97—41no—
182C2×Dic97—41no—
183C2×Dic97C22⋊C414yescomplemented
184C22×C97C2×C22⋊C411yescharacteristic
185C22×C97C2×C4⋊C411yescharacteristic
186C22×C97#11711yescharacteristic
187C22×C97C4×D411yescharacteristic
188C22×C97C4×D411yescharacteristic
189C22×C97C4×D411yescharacteristic
190C22×C97C4×D411yescharacteristic
191C22×C97#12011yescharacteristic
192C22×C97#12011yescharacteristic
193C22×C97#12011yescharacteristic
194C22×C97#12111yescharacteristic
195C22×C97#12111yescharacteristic
196C22×C97#12111yescharacteristic
197C22×C97#12111yescharacteristic
198C22×C97#12111yescharacteristic
199C22×C97#12111yescharacteristic
200C22×C97#12111yescharacteristic
201C22×C97#12111yescharacteristic
202C22×C97C22⋊Q811yescharacteristic
203C22×C97C22⋊Q811yescharacteristic
204C22×C97C22⋊Q811yescharacteristic
205C22×C97C22⋊Q811yescharacteristic
206C22×C97C22⋊Q811yescharacteristic
207C22×C97C22⋊Q811yescharacteristic
208C22×C97#12311yescharacteristic
209C22×C97#12311yescharacteristic
210C22×C97#12311yescharacteristic
211C22×C97#12311yescharacteristic
212C22×C97#12311yescharacteristic
213C22×C97#12411yescharacteristic
214C22×C97#12411yescharacteristic
215C22×C97#12411yescharacteristic
216C22×C97#12611yescharacteristic
217C22×C97#12611yescharacteristic
218C22×C97#12711yescharacteristic
219C2×D97—24no—
220C2×D97—24no—
221C2×D97—24no—
222C2×D97—42no—
223C2×D97—42no—
224C2×D97—42no—
225C2×D97—42no—
226C2×D97—42no—
227C2×D97—42no—
228C2×D97—42no—
229C2×D97—42no—
230C2×D97—42no—
231C2×D97—42no—
232C2×D97—42no—
233C2×D97—42no—
234C4×C97—42no—
235C4×C97—42no—
236C4×C97—42no—
237Dic97—24no—
238Dic97—42no—
239Dic97—42no—
240C2×C97#38011yescharacteristic
241C2×C97#38511yescharacteristic
242C2×C97#38511yescharacteristic
243C2×C97#38611yescharacteristic
244C2×C97#38811yescharacteristic
245C2×C97#38811yescharacteristic
246C2×C97#39011yescharacteristic
247C2×C97#39211yescharacteristic
248C2×C97#39311yescharacteristic
249C2×C97#39411yescharacteristic
250C2×C97#39411yescharacteristic
251C2×C97#39611yescharacteristic
252C2×C97#39611yescharacteristic
253C2×C97#39711yescharacteristic
254C2×C97#39911yescharacteristic
255D97—44no—
256#1439—971nomaximal, Sylow
257C97#143911yescharacteristic, complemented, Sylow
258C22×C4⋊C4—971no—
259C22×C22⋊C4—971no—
260C22×C22⋊C4—971no—
261C22×C22⋊C4—971no—
262C2×#95—971no—
263C2×#95—971no—
264C2×#95D9711yescharacteristic, complemented
265C25—971no—
266C23×C4—971no—
267C23×C4—971no—
268C23×C4—971no—
269C23×C4C2×D9711yescharacteristic
270C23×C4C2×D9711yescharacteristic
271C23×C4C2×D9711yescharacteristic
272C2×C4⋊C4—1942no—
273C2×C4⋊C4—1942no—
274C2×C4⋊C4—1942no—
275C2×C22⋊C4—974no—
276C2×C22⋊C4—1942no—
277C2×C22⋊C4—1942no—
278C2×C22⋊C4—1942no—
279C2×C22⋊C4—1942no—
280C2×C22⋊C4—1942no—
281C2×C22⋊C4—1942no—
282C2×C22⋊C4—1942no—
283C2×C22⋊C4—1942no—
284#95—22no—
285#95—1942no—
286#95—1942no—
287C24—972no—
288C24—972no—
289C24—972no—
290C24—1941no—
291C24—1941no—
292C24—1941no—
293C24—1941no—
294C24—1941no—
295C24—1941no—
296C24—1941no—
297C24—1941no—
298C24—1941no—
299C24—1941no—
300C24—1941no—
301C24—1941no—
302C24C22×D9711yescharacteristic
303C22×C4—21no—
304C22×C4—21no—
305C22×C4—21no—
306C22×C4—21no—
307C22×C4—21no—
308C22×C4—21no—
309C22×C4—21no—
310C22×C4—21no—
311C22×C4—21no—
312C22×C4—21no—
313C22×C4—21no—
314C22×C4—21no—
315C22×C4—21no—
316C22×C4—21no—
317C22×C4—21no—
318C22×C4—21no—
319C22×C4—21no—
320C22×C4—21no—
321C22×C4—972no—
322C22×C4—972no—
323C22×C4—972no—
324C22×C4—972no—
325C22×C4—972no—
326C22×C4—1941no—
327C22×C4—1941no—
328C22×C4—1941no—
329C22×C4—1941no—
330C22×C4—1941no—
331C22×C4—1941no—
332C22×C4—1941no—
333C22×C4—1941no—
334C22×C4—1941no—
335C22×C4—1941no—
336C22×C4—1941no—
337C22×C4—1941no—
338C22×C4—1941no—
339C22×C4—1941no—
340C22×C4—1941no—
341C22×C4—1941no—
342C22×C4C4×D9712yes—
343C22×C4D38812yes—
344C22×C4C22⋊D9712yes—
345C4⋊C4—1948no—
346C22⋊C4—1948no—
347C22⋊C4—1948no—
348C22⋊C4—1948no—
349C23—974no—
350C23—3881no—
351C23—3881no—
352C23—3881no—
353C23—3881no—
354C23—3881no—
355C23—3881no—
356C23—3881no—
357C23—3881no—
358C23—3881no—
359C23—3881no—
360C23—3881no—
361C23—3881no—
362C23—3881no—
363C23—3881no—
364C23—3881no—
365C23—3881no—
366C23—1942no—
367C23—1942no—
368C23—1942no—
369C23—1942no—
370C23—1942no—
371C23—1942no—
372C23—1942no—
373C23—1942no—
374C23—1942no—
375C23—1942no—
376C23—1942no—
377C23—1942no—
378C23—1942no—
379C23—1942no—
380C23—1942no—
381C23—1942no—
382C23—1942no—
383C23—1942no—
384C23C2×C4×D9711yescharacteristic
385C23D4×D9711yescharacteristic
386C23D4×D9711yescharacteristic
387C23D4×D9711yescharacteristic
388C23D4×D9711yescharacteristic
389C23D4×D9711yescharacteristic
390C23C2×C22⋊D9711yescharacteristic
391C23C2×D38811yescharacteristic
392C23Q8×D9711yescharacteristic
393C23#1309551011yescharacteristic
394C23#1309551111yescharacteristic
395C23#1309551111yescharacteristic
396C23#1309551111yescharacteristic
397C23#1309551111yescharacteristic
398C23#1309551211yescharacteristic
399C2×C4—22no—
400C2×C4—22no—
401C2×C4—22no—
402C2×C4—22no—
403C2×C4—22no—
404C2×C4—22no—
405C2×C4—22no—
406C2×C4—22no—
407C2×C4—22no—
408C2×C4—41no—
409C2×C4—41no—
410C2×C4—41no—
411C2×C4—41no—
412C2×C4—41no—
413C2×C4—41no—
414C2×C4—41no—
415C2×C4—41no—
416C2×C4—41no—
417C2×C4—41no—
418C2×C4—41no—
419C2×C4—41no—
420C2×C4—974no—
421C2×C4—3881no—
422C2×C4—3881no—
423C2×C4—3881no—
424C2×C4—3881no—
425C2×C4—3881no—
426C2×C4—3881no—
427C2×C4—3881no—
428C2×C4—3881no—
429C2×C4—1942no—
430C2×C4—1942no—
431C2×C4—1942no—
432C2×C4—1942no—
433C2×C4—1942no—
434C2×C4—1942no—
435C2×C4—1942no—
436C2×C4—1942no—
437C2×C4—1942no—
438C2×C4—1942no—
439C2×C4—1942no—
440C2×C4—1942no—
441C22—3882no—
442C22—3882no—
443C22—3882no—
444C22—3882no—
445C22—3882no—
446C22—3882no—
447C22—3882no—
448C22—3882no—
449C22—3882no—
450C22—3882no—
451C22—3882no—
452C22—3882no—
453C22—1944no—
454C22—1944no—
455C22—1944no—
456C22C22⋊C4×D9711yescharacteristic
457C22C4⋊C4×D9711yescharacteristic
458C22#1569076711yescharacteristic
459C22#1569077011yescharacteristic
460C22C4×C22⋊D9711yescharacteristic
461C22#1569077211yescharacteristic
462C22C4×D38811yescharacteristic
463C22#1569077811yescharacteristic
464C22#1569078011yescharacteristic
465C22#1569078011yescharacteristic
466C22#1569078211yescharacteristic
467C22#1569078311yescharacteristic
468C22#1569078311yescharacteristic
469C22#1569078311yescharacteristic
470C22#1569078311yescharacteristic
471C22#1569078311yescharacteristic
472C22#1569078411yescharacteristic
473C22#1569078511yescharacteristic
474C22#1569078611yescharacteristic
475C22#1569078811yescharacteristic
476C22#1569078811yescharacteristic
477C22#1569078811yescharacteristic
478C22#1569078811yescharacteristic
479C22#1569079011yescharacteristic
480C22#1569079211yescharacteristic
481C22#1569079311yescharacteristic
482C22#1569079511yescharacteristic
483C22#1569079511yescharacteristic
484C22#1569079511yescharacteristic
485C22#1569079611yescharacteristic
486C22#1569079611yescharacteristic
487C22#1569079811yescharacteristic
488C22#1569080411yescharacteristic
489C22#1569080511yescharacteristic
490C22#1569080611yescharacteristic
491C4—42no—
492C4—42no—
493C4—42no—
494C4—3882no—
495C4—3882no—
496C4—1944no—
497C2—3884no—
498C2#2026059411yescharacteristic
499C2#2026060611yescharacteristic
500C2#2026060911yescharacteristic
501C2#2026061311yescharacteristic
502C2#2026062411yescharacteristic
503C2#2026062611yescharacteristic
504C2#2026063511yescharacteristic
505C2#2026064411yescharacteristic
506C2#2026064811yescharacteristic
507C2#2026065011yescharacteristic
508C2#2026065311yescharacteristic
509C2#2026065711yescharacteristic
510C2#2026065911yescharacteristic
511C2#2026066411yescharacteristic
512C2#2026067011yescharacteristic
subgroupG/H
center, FrattiniC24C22×D97
commutatorC23×C97C22×C4
FittingC2×#95×C97C2
socle, cosocleC24×C97C23
prime radicalC24×D97C22
hypercenterC2×#95D97
hypocenterC97#1439
LCSG ▷ C23×C97 ▷ C97
UCSC1 ◁ C24 ◁ C2×#95

Complements

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

NH ≅ G/Ntype
1. C22×#13095490C2semidirect
5. C2×#15690717C2semidirect
6. C2×#15690717C2semidirect
7. C2×#95×C97C2semidirect
23. C2×#13095488C4semidirect
141. C22×D97C4⋊C4semidirect
183. C2×Dic97C22⋊C4semidirect
257. C97#1439semidirect
264. C2×#95D97semidirect

Maximal subgroups

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

subgroupindex
C22×#130954902
C22×#130954882
C2×#156907172
C2×#95×C972
#143997