Finite Group Databasebeta

← #32980757

Subgroups — #32980757

Order 18688.

660 conjugacy classes of subgroups.

The subgroup lattice has 464 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#17988787C211yesmaximal, characteristic
2Q8×C4⋊Dic73C211yesmaximal, characteristic, complemented
3#22610857C211yesmaximal, characteristic
4#22610737C211yesmaximal, characteristic
5#22607815C211yesmaximal, characteristic, complemented
6#22607771C211yesmaximal, characteristic, complemented
7C2×Q8⋊C8×C73C211yesmaximal, characteristic, complemented
8C22×#11924015—21no—
9C2×#14556588C2211yescharacteristic
10C2×Q8×Dic73—21no—
11#17988787—22no—
12#17988787C414yescomplemented
13C2×C4⋊#10596831C2211yescharacteristic
14C2×#14556513—22no—
15C2×#14556513—21no—
16C2×#14556513—21no—
17C2×#14556511—21no—
18#17988486—22no—
19#17988486—21no—
20#17988457—21no—
21#17988457C2211yescharacteristic
22C4×C4⋊Dic73—21no—
23C4×C4⋊Dic73C2211yescharacteristic
24C2×C4×Q8×C73C2211yescharacteristic
25C2×C4⋊C8×C73C2211yescharacteristic
26C2×C4×C8×C73C2211yescharacteristic
27Q8⋊C8×C73—22no—
28C22×D292—21no—
29C22×D292—21no—
30C22×D292—21no—
31C2×#11924015—24no—
32C2×#11924015—42no—
33C2×#11924015—42no—
34C22×#10596831—21no—
35#14556588—21no—
36#14556588—21no—
37#14556588—21no—
38#14556588C2×C412yes—
39Q8×Dic73—42no—
40Q8×Dic73—42no—
41C2×C4⋊Dic73—22no—
42C2×C4⋊Dic73—22no—
43C2×C4⋊Dic73—21no—
44C2×C4⋊Dic73—21no—
45C2×C4⋊Dic73D411yescharacteristic
46C2×C4⋊Dic73D411yescharacteristic
47C2×C4⋊Dic73D412yes—
48C2×C4×Dic73—22no—
49C2×C4×Dic73—21no—
50C4⋊#10596831—21no—
51C4⋊#10596831C2×C412yes—
52#14556513—44no—
53#14556513—42no—
54#14556513—42no—
55#14556511—24no—
56#14556498—22no—
57#14556498—21no—
58C22×Q8×C73D411yescharacteristic
59C22×C8×C73—21no—
60C22×C8×C73—21no—
61C4×Q8×C73—21no—
62C4×Q8×C73—21no—
63C4×Q8×C73C2×C412yes—
64C4×Q8×C73D412yes—
65C2×C4⋊C4×C73—21no—
66C2×C4⋊C4×C73D411yescharacteristic
67C2×C42×C73—21no—
68C2×C42×C73C2311yescharacteristic
69C4⋊C8×C73—21no—
70C4⋊C8×C73—21no—
71C4×C8×C73—21no—
72C4×C8×C73—21no—
73C23×D73—41no—
74C2×D292—22no—
75C2×D292—22no—
76C2×D292—22no—
77C2×D292—42no—
78C2×D292—42no—
79C2×D292—41no—
80C2×D292—41no—
81C2×D292—41no—
82C2×D292—41no—
83C2×D292—41no—
84C2×D292—41no—
85C2×D292—41no—
86C2×D292—41no—
87C2×D292—41no—
88C2×D292—41no—
89C2×D292—41no—
90C22×Dic73—21no—
91C22×Dic73—21no—
92C22×Dic73—22no—
93#11924015—48no—
94C2×#10596831—22no—
95C2×#10596831—41no—
96C2×#10596831—41no—
97C4⋊Dic73—21no—
98C4⋊Dic73—21no—
99C4⋊Dic73—21no—
100C4⋊Dic73—44no—
101C4⋊Dic73—41no—
102C4⋊Dic73—41no—
103C4⋊Dic73—41no—
104C4⋊Dic73—41no—
105C4⋊Dic73—42no—
106C4⋊Dic73—82no—
107C4⋊Dic73SD1612yescomplemented
108C4×Dic73—44no—
109C4×Dic73—41no—
110C4×Dic73—41no—
111C2×Q8×C73—21no—
112C2×Q8×C73—21no—
113C2×Q8×C73—21no—
114C2×Q8×C73C22⋊C412yes—
115C2×Q8×C73D812yes—
116C2×Q8×C73SD1612yes—
117C22×C4×C73—21no—
118C22×C4×C73—21no—
119C22×C4×C73C2×D411yescharacteristic
120C22×C4×C73C2×D411yescharacteristic
121C22×C4×C73C2×D411yescharacteristic
122C2×C8×C73—22no—
123C2×C8×C73—41no—
124C2×C8×C73—41no—
125C2×C8×C73—41no—
126C2×C8×C73—41no—
127C2×C8×C73—41no—
128C4⋊C4×C73—22no—
129C4⋊C4×C73—22no—
130C4⋊C4×C73—21no—
131C4⋊C4×C73C22⋊C412yes—
132C42×C73—22no—
133C42×C73—22no—
134C42×C73C22×C411yescharacteristic
135C42×C73C2×D411yescharacteristic
136C42×C73#4111yescharacteristic
137C42×C73#4111yescharacteristic
138C23×C73#12011yescharacteristic
139C22×D73—42no—
140C22×D73—81no—
141C22×D73—81no—
142C22×D73—81no—
143C22×D73—81no—
144C22×D73—81no—
145C22×D73—81no—
146Q8×C73—22no—
147Q8×C73—22no—
148Q8×C73—22no—
149Q8×C73#10214yescomplemented
150C2×C4×C73—22no—
151C2×C4×C73—22no—
152C2×C4×C73—22no—
153C2×C4×C73—21no—
154C2×C4×C73—21no—
155C2×C4×C73—22no—
156C2×C4×C73—41no—
157C2×C4×C73—41no—
158C2×C4×C73C2×C22⋊C411yescharacteristic
159C2×C4×C73C4×D411yescharacteristic
160C2×C4×C73C4×D411yescharacteristic
161C2×C4×C73#12111yescharacteristic
162C2×C4×C73#12111yescharacteristic
163C2×C4×C73#12311yescharacteristic
164C2×C4×C73C2×D811yescharacteristic
165C2×C4×C73C2×SD1611yescharacteristic
166C2×C4×C73C2×SD1611yescharacteristic
167C2×C4×C73#13511yescharacteristic
168C2×C4×C73AGL(1,ℤ/8)11yescharacteristic
169C2×C4×C73AGL(1,ℤ/8)11yescharacteristic
170C2×C4×C73#13711yescharacteristic
171C2×C4×C73#13711yescharacteristic
172C2×Dic73—22no—
173C2×Dic73—41no—
174C2×Dic73—41no—
175C2×Dic73—42no—
176C2×Dic73—41no—
177C2×Dic73—41no—
178C2×Dic73—42no—
179C2×Dic73—42no—
180C2×Dic73—41no—
181D292—44no—
182D292—42no—
183D292—42no—
184D292—42no—
185D292—42no—
186D292—82no—
187D292—81no—
188D292—81no—
189D292—81no—
190D292—81no—
191C8×C73—42no—
192C8×C73—81no—
193#10596831—42no—
194C22×C73#38611yescharacteristic
195C22×C73#44711yescharacteristic
196C22×C73#44811yescharacteristic
197C22×C73#44811yescharacteristic
198C22×C73#44911yescharacteristic
199C22×C73#45311yescharacteristic
200C22×C73#45411yescharacteristic
201C2×D73—82no—
202C2×D73—82no—
203C2×D73—82no—
204C2×D73—161no—
205C2×D73—161no—
206C2×D73—161no—
207C2×D73—161no—
208C4×C73—24no—
209C4×C73—21no—
210C4×C73—21no—
211C4×C73—42no—
212C4×C73C2×#10211yescharacteristic
213C4×C73#41711yescharacteristic
214C4×C73C4×SD1611yescharacteristic
215C4×C73#44011yescharacteristic
216C4×C73#46011yescharacteristic
217C4×C73#46411yescharacteristic
218C4×C73#48011yescharacteristic
219C4×C73#48111yescharacteristic
220Dic73—42no—
221Dic73—81no—
222Dic73—82no—
223#8979—731nomaximal, Sylow
224C2×C73#162111yescharacteristic
225C2×C73#162311yescharacteristic
226C2×C73#162611yescharacteristic
227C2×C73#162711yescharacteristic
228C2×C73#187411yescharacteristic
229C2×C73#188011yescharacteristic
230C2×C73#188311yescharacteristic
231D73—162no—
232C2×#460—731no—
233Q8×C4⋊C4—731no—
234#1982—731no—
235#1957—731no—
236C2×Q8⋊C8D7311yescharacteristic, complemented
237#1296—731no—
238#1279—731no—
239C73#897911yescharacteristic, complemented, Sylow
240C22×SD16—1461no—
241C2×#127—731no—
242C2×C4×Q8—1461no—
243C2×C4×Q8C2×D7311yescharacteristic
244#460—734no—
245#460—1462no—
246C2×C4⋊C8—731no—
247C2×C4⋊C8C2×D7311yescharacteristic
248C2×#102—1462no—
249C2×#102—1461no—
250C2×#102—1461no—
251C2×#102—1461no—
252C2×C4×C8C2×D7311yescharacteristic
253#389—1462no—
254#389—1461no—
255#384—731no—
256#384—1461no—
257C4×C4⋊C4—731no—
258C4×C4⋊C4—1461no—
259Q8⋊C8—22no—
260C22×Q8D29211yescharacteristic
261C22×D4—1461no—
262C22×D4—1461no—
263C22×D4—1461no—
264C2×SD16—2922no—
265C2×SD16—2922no—
266C2×SD16—1464no—
267C22×C8—21no—
268C22×C8—21no—
269C22×C8—1461no—
270#127—732no—
271#127—1461no—
272#127—1461no—
273#127—1461no—
274C4×Q8—21no—
275C4×Q8—21no—
276C4×Q8—2922no—
277C4×Q8—2922no—
278C4×Q8C4×D7312yes—
279C4×Q8C22⋊D7312yes—
280C2×C4⋊C4—21no—
281C2×C4⋊C4—732no—
282C2×C4⋊C4—731no—
283C2×C4⋊C4—731no—
284C2×C4⋊C4—1462no—
285C2×C4⋊C4—1461no—
286C2×C4⋊C4—1461no—
287C2×C4⋊C4—1462no—
288C2×C4⋊C4D29211yescharacteristic
289C2×C42—21no—
290C2×C42—1462no—
291C2×C42—1461no—
292C2×C42C22×D7311yescharacteristic
293C4⋊C8—21no—
294C4⋊C8—21no—
295C4⋊C8—732no—
296C4⋊C8—1461no—
297#102—2924no—
298#102—2922no—
299#102—2922no—
300#102—1464no—
301C4×C8—21no—
302C4×C8—21no—
303#95—1462no—
304#95—1461no—
305C24—2921no—
306C2×Q8—21no—
307C2×Q8—21no—
308C2×Q8—21no—
309C2×Q8#1192400512yes—
310C2×Q8D58412yes—
311C2×Q8#1192401712yes—
312C2×D4—2922no—
313C2×D4—2922no—
314C2×D4—2921no—
315C2×D4—2921no—
316C2×D4—2921no—
317C2×D4—2921no—
318C2×D4—2921no—
319C2×D4—2921no—
320C2×D4—2921no—
321C2×D4—2921no—
322C2×D4—2921no—
323C2×D4—2921no—
324C2×D4—2921no—
325C2×D4—1462no—
326C2×D4—1462no—
327C2×D4—1462no—
328C22×C4—21no—
329C22×C4—21no—
330C22×C4—1462no—
331C22×C4—1461no—
332C22×C4—1461no—
333C22×C4D4×D7311yescharacteristic
334C22×C4D4×D7311yescharacteristic
335C22×C4C2×D29211yescharacteristic
336SD16—2928no—
337C2×C8—22no—
338C2×C8—41no—
339C2×C8—41no—
340C2×C8—41no—
341C2×C8—41no—
342C2×C8—41no—
343C2×C8—2921no—
344C2×C8—2921no—
345C2×C8—1462no—
346C4⋊C4—22no—
347C4⋊C4—22no—
348C4⋊C4—21no—
349C4⋊C4—2922no—
350C4⋊C4—2924no—
351C4⋊C4—2921no—
352C4⋊C4—2921no—
353C4⋊C4—2921no—
354C4⋊C4—2921no—
355C4⋊C4—5842no—
356C4⋊C4—732no—
357C4⋊C4—1461no—
358C4⋊C4—1461no—
359C4⋊C4—1461no—
360C4⋊C4#1192400512yes—
361C42—22no—
362C42—22no—
363C42—2924no—
364C42—2921no—
365C42—2921no—
366C42C2×C4×D7311yescharacteristic
367C42C2×C22⋊D7311yescharacteristic
368C42#1192402711yescharacteristic
369C42#1192402711yescharacteristic
370C23—2922no—
371C23—5841no—
372C23—5841no—
373C23—5841no—
374C23—5841no—
375C23—5841no—
376C23—5841no—
377C23#1455656111yescharacteristic
378Q8—22no—
379Q8—22no—
380Q8—22no—
381Q8#1455651314yescomplemented
382D4—2924no—
383D4—2922no—
384D4—2922no—
385D4—2922no—
386D4—2922no—
387D4—5842no—
388D4—5841no—
389D4—5841no—
390D4—5841no—
391D4—5841no—
392C2×C4—22no—
393C2×C4—22no—
394C2×C4—22no—
395C2×C4—21no—
396C2×C4—21no—
397C2×C4—22no—
398C2×C4—41no—
399C2×C4—41no—
400C2×C4—2921no—
401C2×C4—2921no—
402C2×C4—2922no—
403C2×C4—2921no—
404C2×C4—2922no—
405C2×C4—2921no—
406C2×C4—2922no—
407C2×C4—2921no—
408C2×C4—1462no—
409C2×C4C2×#1192400511yescharacteristic
410C2×C4#1455655111yescharacteristic
411C2×C4#1455655111yescharacteristic
412C2×C4#1455656311yescharacteristic
413C2×C4#1455656311yescharacteristic
414C2×C4#1455657511yescharacteristic
415C2×C4C2×D58411yescharacteristic
416C2×C4SD16×D7311yescharacteristic
417C2×C4C2×#1192401711yescharacteristic
418C2×C4#1455661411yescharacteristic
419C2×C4#1455661811yescharacteristic
420C2×C4#1455661911yescharacteristic
421C2×C4#1455662311yescharacteristic
422C2×C4#1455662411yescharacteristic
423C8—42no—
424C8—81no—
425C8—2922no—
426C22—5842no—
427C22—5842no—
428C22—5842no—
429C22—11681no—
430C22—11681no—
431C22—11681no—
432C22—11681no—
433C22#1798846411yescharacteristic
434C22#1798872311yescharacteristic
435C22#1798872711yescharacteristic
436C22#1798872811yescharacteristic
437C22#1798873511yescharacteristic
438C22#1798875611yescharacteristic
439C22#1798876111yescharacteristic
440C4—24no—
441C4—21no—
442C4—21no—
443C4—42no—
444C4—2922no—
445C4—5842no—
446C4—5841no—
447C4C2×#1455651311yescharacteristic
448C4#1798858911yescharacteristic
449C4#1798867211yescharacteristic
450C4#1798869011yescharacteristic
451C4#1798878511yescharacteristic
452C4#1798881111yescharacteristic
453C4#1798890111yescharacteristic
454C4#1798890611yescharacteristic
455C2—11682no—
456C2#2260899811yescharacteristic
457C2#2260901111yescharacteristic
458C2#2260903011yescharacteristic
459C2#2260903611yescharacteristic
460C2#2261024611yescharacteristic
461C2#2261028411yescharacteristic
462C2#2261030511yescharacteristic
subgroupG/H
centerC23#14556561
commutatorC42×C73C22×C4
FrattiniC2×C42C22×D73
FittingC2×Q8⋊C8×C73C2
socleC23×C73#120
prime radicalC2×#14556588C22
cosocleC2×C42×C73C23
hypercenterC2×Q8⋊C8D73
hypocenterC73#8979
LCSG ▷ C42×C73 ▷ C22×C73 ▷ C73
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×Q8⋊C8

Complements

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

NH ≅ G/Ntype
2. Q8×C4⋊Dic73C2semidirect
5. #22607815C2semidirect
6. #22607771C2semidirect
7. C2×Q8⋊C8×C73C2semidirect
12. #17988787C4semidirect
107. C4⋊Dic73SD16semidirect
149. Q8×C73#102semidirect
236. C2×Q8⋊C8D73semidirect
239. C73#8979semidirect
381. Q8#14556513semidirect

Maximal subgroups

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

subgroupindex
C2×#179887872
Q8×C4⋊Dic732
#226108572
#226107372
#226078152
#226077712
C2×Q8⋊C8×C732
#897973