Finite Group Databasebeta

← #32989525

Subgroups — #32989525

Order 18688.

530 conjugacy classes of subgroups.

The subgroup lattice has 444 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22614849C211yesmaximal, characteristic
2#22610059C211yesmaximal, characteristic, complemented
3#22609936C211yesmaximal, characteristic, complemented
4#22609932C211yesmaximal, characteristic
5C2×#17988316C211yesmaximal, characteristic
6#22607815C211yesmaximal, characteristic, complemented
7C2×Q8⋊C8×C73C211yesmaximal, characteristic, complemented
8C2×#14556567—21no—
9C2×C4×D292C2211yescharacteristic
10C2×C4⋊#10596831C2211yescharacteristic
11C2×#14556515—21no—
12C2×#14556513—21no—
13#17988509—22no—
14#17988489—21no—
15#17988485—21no—
16#17988457C2211yescharacteristic
17#17988457C2211yescharacteristic
18#17988444—21no—
19#17988350—21no—
20#17988350—22no—
21#17988349—21no—
22#17988316—22no—
23C2×C4×Q8×C73C2211yescharacteristic
24C2×C4⋊C8×C73C2211yescharacteristic
25C2×C4×C8×C73C2211yescharacteristic
26Q8⋊C8×C73—22no—
27C22×D292D411yescharacteristic
28C22×C4×D73—21no—
29C22×#10596831—21no—
30#14556567—42no—
31#14556567—42no—
32C4×D292—21no—
33C4×D292—21no—
34C4×D292—21no—
35C4×D292—21no—
36C2×C4⋊Dic73—21no—
37C2×C4⋊Dic73—21no—
38C2×C4⋊Dic73—21no—
39C2×C4⋊Dic73—21no—
40C2×C4⋊Dic73—21no—
41C2×C4⋊Dic73D411yescharacteristic
42C2×C4⋊Dic73D412yes—
43C2×#11924007—22no—
44C2×#11924005—22no—
45C2×#11924005—22no—
46C2×#11924005—21no—
47C4⋊#10596831—21no—
48C4⋊#10596831—21no—
49#14556515—42no—
50#14556513—42no—
51#14556498—22no—
52C22×Q8×C73D411yescharacteristic
53C22×C8×C73—21no—
54C22×C8×C73—21no—
55C4×Q8×C73—21no—
56C4×Q8×C73—21no—
57C4×Q8×C73—21no—
58C4×Q8×C73—21no—
59C2×C4⋊C4×C73—21no—
60C2×C4⋊C4×C73D411yescharacteristic
61C2×C42×C73—21no—
62C2×C42×C73C2311yescharacteristic
63C4⋊C8×C73—21no—
64C4⋊C8×C73—21no—
65C4×C8×C73—21no—
66C4×C8×C73D412yes—
67C23×D73—21no—
68C2×D292—21no—
69C2×D292—21no—
70C2×D292—21no—
71C2×D292—21no—
72C2×D292—21no—
73C2×D292Q1612yes—
74C22×Dic73—21no—
75C22×Dic73—22no—
76C22×Dic73—41no—
77C2×C4×D73—41no—
78C2×C4×D73—41no—
79C2×C4×D73—41no—
80C2×C4×D73—41no—
81C2×C4×D73—41no—
82C2×C4×D73—41no—
83C2×#10596831—41no—
84C2×#10596831—41no—
85C2×#10596831—41no—
86C4⋊Dic73—21no—
87C4⋊Dic73—21no—
88C4⋊Dic73—41no—
89C4⋊Dic73—41no—
90C4⋊Dic73—41no—
91C4⋊Dic73—41no—
92C4⋊Dic73—42no—
93C4⋊Dic73—42no—
94C4⋊Dic73—41no—
95C4⋊Dic73—41no—
96C4⋊Dic73—41no—
97C4⋊Dic73—41no—
98#11924007—44no—
99#11924005—44no—
100#11924005—41no—
101#11924005—41no—
102#11924005—82no—
103C2×Q8×C73—21no—
104C2×Q8×C73—21no—
105C2×Q8×C73—21no—
106C2×Q8×C73—21no—
107C2×Q8×C73—21no—
108C2×Q8×C73SD1612yes—
109C22×C4×C73—21no—
110C22×C4×C73—21no—
111C22×C4×C73C2×D411yescharacteristic
112C22×C4×C73C2×D411yescharacteristic
113C22×C4×C73C2×D411yescharacteristic
114C2×C8×C73—22no—
115C2×C8×C73—41no—
116C2×C8×C73—41no—
117C2×C8×C73—41no—
118C2×C8×C73—41no—
119C2×C8×C73—41no—
120C4⋊C4×C73—21no—
121C4⋊C4×C73—21no—
122C4⋊C4×C73—41no—
123C4⋊C4×C73—41no—
124C42×C73—41no—
125C42×C73—41no—
126C42×C73C2×D411yescharacteristic
127C42×C73#4111yescharacteristic
128C42×C73#4111yescharacteristic
129C42×C73#4111yescharacteristic
130C23×C73#12011yescharacteristic
131C22×D73—22no—
132C22×D73—41no—
133C22×D73—41no—
134C22×D73—41no—
135C22×D73—41no—
136C22×D73—41no—
137C22×D73—41no—
138Q8×C73—22no—
139Q8×C73—22no—
140Q8×C73—41no—
141Q8×C73—41no—
142C2×C4×C73—21no—
143C2×C4×C73—21no—
144C2×C4×C73—22no—
145C2×C4×C73—41no—
146C2×C4×C73—41no—
147C2×C4×C73—41no—
148C2×C4×C73—41no—
149C2×C4×C73—41no—
150C2×C4×C73#12111yescharacteristic
151C2×C4×C73#12111yescharacteristic
152C2×C4×C73#12111yescharacteristic
153C2×C4×C73#12311yescharacteristic
154C2×C4×C73#12311yescharacteristic
155C2×C4×C73#12411yescharacteristic
156C2×C4×C73C2×SD1611yescharacteristic
157C2×C4×C73C2×Q1611yescharacteristic
158C2×C4×C73#13511yescharacteristic
159C2×C4×C73#13511yescharacteristic
160C2×C4×C73AGL(1,ℤ/8)11yescharacteristic
161C2×C4×C73AGL(1,ℤ/8)11yescharacteristic
162C2×C4×C73AGL(1,ℤ/8)11yescharacteristic
163C2×C4×C73#13711yescharacteristic
164C2×Dic73—41no—
165C2×Dic73—42no—
166C2×Dic73—41no—
167C2×Dic73—42no—
168C2×Dic73—41no—
169C2×Dic73—42no—
170C2×Dic73—81no—
171C2×Dic73—81no—
172C2×Dic73—81no—
173D292—22no—
174D292—22no—
175D292—41no—
176D292—41no—
177C4×D73—81no—
178C4×D73—81no—
179C4×D73—81no—
180C4×D73—81no—
181C8×C73—42no—
182C8×C73—81no—
183#10596831—81no—
184C22×C73#39411yescharacteristic
185C22×C73#44911yescharacteristic
186C22×C73#44911yescharacteristic
187C22×C73#45011yescharacteristic
188C22×C73#45111yescharacteristic
189C22×C73#45411yescharacteristic
190C22×C73#45511yescharacteristic
191C2×D73—42no—
192C2×D73—42no—
193C2×D73—42no—
194C2×D73—81no—
195C2×D73—81no—
196C2×D73—81no—
197C2×D73—81no—
198C4×C73—21no—
199C4×C73—21no—
200C4×C73—42no—
201C4×C73—81no—
202C4×C73#46511yescharacteristic
203C4×C73#46711yescharacteristic
204C4×C73#46811yescharacteristic
205C4×C73#46911yescharacteristic
206C4×C73#48111yescharacteristic
207C4×C73#48211yescharacteristic
208C4×C73#48411yescharacteristic
209C4×C73#48511yescharacteristic
210Dic73—82no—
211Dic73—81no—
212Dic73—161no—
213#10436—731nomaximal, Sylow
214C2×C73#162811yescharacteristic
215C2×C73#163711yescharacteristic
216C2×C73#164211yescharacteristic
217C2×C73#164411yescharacteristic
218C2×C73#203411yescharacteristic
219C2×C73#203611yescharacteristic
220C2×C73#204011yescharacteristic
221D73—82no—
222#2538—731no—
223#1834—731no—
224#1803—731no—
225#1802—731no—
226C2×Q8⋊C8D7311yescharacteristic, complemented
227C2×D4⋊C8—731no—
228#1296—731no—
229C73#1043611yescharacteristic, complemented, Sylow
230C2×C22⋊Q8—1461no—
231C2×C4×Q8C2×D7311yescharacteristic
232C2×C4×D4—731no—
233C2×C4⋊C8—731no—
234C2×C4⋊C8C2×D7311yescharacteristic
235C2×#103—1461no—
236C2×#102—1461no—
237C2×C4×C8C2×D7311yescharacteristic
238#396—1462no—
239#390—1461no—
240#389—1461no—
241#384—731no—
242#384—731no—
243#380—1461no—
244#340—1461no—
245#340—1462no—
246#340—1461no—
247Q8⋊C8—22no—
248D4⋊C8—1462no—
249C22×Q8D29211yescharacteristic
250C22×D4—731no—
251C23×C4—1461no—
252C22×C8—21no—
253C22×C8—21no—
254C22×C8—1461no—
255C22⋊Q8—2922no—
256C22⋊Q8—2922no—
257C4×Q8—21no—
258C4×Q8—21no—
259C4×Q8—21no—
260C4×Q8—21no—
261C4×D4—1461no—
262C4×D4—1461no—
263C4×D4—1461no—
264C4×D4—1461no—
265C2×C4⋊C4—21no—
266C2×C4⋊C4—732no—
267C2×C4⋊C4—731no—
268C2×C4⋊C4—1462no—
269C2×C4⋊C4—1461no—
270C2×C4⋊C4—1461no—
271C2×C4⋊C4—1461no—
272C2×C4⋊C4—1461no—
273C2×C4⋊C4—1461no—
274C2×C4⋊C4D29211yescharacteristic
275C2×C22⋊C4—1462no—
276C2×C22⋊C4—1461no—
277C2×C22⋊C4—1462no—
278C2×C42—21no—
279C2×C42C22×D7311yescharacteristic
280C4⋊C8—21no—
281C4⋊C8—21no—
282C4⋊C8—1461no—
283C4⋊C8—1461no—
284#103—2922no—
285#102—2922no—
286C4×C8—21no—
287C4×C8C22⋊D7312yes—
288#95—1462no—
289C24—1461no—
290C2×Q8—21no—
291C2×Q8—21no—
292C2×Q8—21no—
293C2×Q8—21no—
294C2×Q8—21no—
295C2×Q8#1192401712yes—
296C2×D4—732no—
297C2×D4—1461no—
298C2×D4—1461no—
299C2×D4—1461no—
300C2×D4—1461no—
301C2×D4—1461no—
302C22×C4—21no—
303C22×C4—21no—
304C22×C4—2921no—
305C22×C4—2921no—
306C22×C4—2921no—
307C22×C4—2921no—
308C22×C4—2921no—
309C22×C4—2921no—
310C22×C4—2921no—
311C22×C4—1461no—
312C22×C4—1462no—
313C22×C4D4×D7311yescharacteristic
314C22×C4D4×D7311yescharacteristic
315C22×C4C2×D29211yescharacteristic
316C2×C8—22no—
317C2×C8—41no—
318C2×C8—41no—
319C2×C8—41no—
320C2×C8—41no—
321C2×C8—41no—
322C2×C8—2921no—
323C2×C8—2921no—
324C2×C8—2921no—
325C4⋊C4—21no—
326C4⋊C4—21no—
327C4⋊C4—41no—
328C4⋊C4—41no—
329C4⋊C4—2922no—
330C4⋊C4—2924no—
331C4⋊C4—2922no—
332C4⋊C4—2921no—
333C4⋊C4—2921no—
334C4⋊C4—2921no—
335C4⋊C4—2921no—
336C4⋊C4—2921no—
337C4⋊C4—2921no—
338C4⋊C4—2921no—
339C4⋊C4—2921no—
340C4⋊C4—1461no—
341C4⋊C4—1461no—
342C22⋊C4—2924no—
343C22⋊C4—2921no—
344C22⋊C4—2921no—
345C22⋊C4—5842no—
346C42—41no—
347C42—41no—
348C42C2×C22⋊D7311yescharacteristic
349C42#1192402711yescharacteristic
350C42#1192402711yescharacteristic
351C42#1192402811yescharacteristic
352C23—2921no—
353C23—2921no—
354C23—2921no—
355C23—2921no—
356C23—2921no—
357C23—2921no—
358C23—1462no—
359C23#1455656111yescharacteristic
360Q8—22no—
361Q8—22no—
362Q8—41no—
363Q8—41no—
364D4—2921no—
365D4—2921no—
366D4—1462no—
367D4—1462no—
368C2×C4—21no—
369C2×C4—21no—
370C2×C4—22no—
371C2×C4—41no—
372C2×C4—41no—
373C2×C4—41no—
374C2×C4—41no—
375C2×C4—41no—
376C2×C4—2921no—
377C2×C4—2922no—
378C2×C4—2922no—
379C2×C4—2921no—
380C2×C4—2921no—
381C2×C4—2922no—
382C2×C4—5841no—
383C2×C4—5841no—
384C2×C4—5841no—
385C2×C4—5841no—
386C2×C4—5841no—
387C2×C4—5841no—
388C2×C4—5841no—
389C2×C4#1455656211yescharacteristic
390C2×C4#1455656311yescharacteristic
391C2×C4#1455656511yescharacteristic
392C2×C4#1455657311yescharacteristic
393C2×C4#1455657511yescharacteristic
394C2×C4#1455657711yescharacteristic
395C2×C4C2×#1192401711yescharacteristic
396C2×C4Q16×D7311yescharacteristic
397C2×C4#1455661311yescharacteristic
398C2×C4#1455661411yescharacteristic
399C2×C4#1455661811yescharacteristic
400C2×C4#1455661911yescharacteristic
401C2×C4#1455661911yescharacteristic
402C2×C4#1455662011yescharacteristic
403C8—42no—
404C8—81no—
405C8—5841no—
406C22—2922no—
407C22—2922no—
408C22—2922no—
409C22—5841no—
410C22—5841no—
411C22—5841no—
412C22—5841no—
413C22#1798850311yescharacteristic
414C22#1798873411yescharacteristic
415C22#1798873511yescharacteristic
416C22#1798874011yescharacteristic
417C22#1798874411yescharacteristic
418C22#1798876111yescharacteristic
419C22#1798876611yescharacteristic
420C4—21no—
421C4—21no—
422C4—42no—
423C4—81no—
424C4—5841no—
425C4—5842no—
426C4—11681no—
427C4#1798881611yescharacteristic
428C4#1798882511yescharacteristic
429C4#1798883011yescharacteristic
430C4#1798884011yescharacteristic
431C4#1798890611yescharacteristic
432C4#1798891211yescharacteristic
433C4#1798892111yescharacteristic
434C4#1798892711yescharacteristic
435C2—5842no—
436C2#2260904411yescharacteristic
437C2#2260909811yescharacteristic
438C2#2260913411yescharacteristic
439C2#2260914711yescharacteristic
440C2#2261112111yescharacteristic
441C2#2261113311yescharacteristic
442C2#2261115911yescharacteristic
subgroupG/H
centerC23#14556561
commutator, cosocleC2×C42×C73C23
FrattiniC2×C42C22×D73
FittingC2×Q8⋊C8×C73C2
socleC23×C73#120
prime radicalC22×D292D4
hypercenterC2×Q8⋊C8D73
hypocenterC73#10436
LCSG ▷ C2×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. #22610059C2semidirect
3. #22609936C2semidirect
6. #22607815C2semidirect
7. C2×Q8⋊C8×C73C2semidirect
226. C2×Q8⋊C8D73semidirect
229. C73#10436semidirect

Maximal subgroups

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

subgroupindex
#226148492
#226100592
#226099362
#226099322
C2×#179883162
#226078152
C2×Q8⋊C8×C732
#1043673