Finite Group Databasebeta

← #32982256

Subgroups — #32982256

Order 18688.

604 conjugacy classes of subgroups.

The subgroup lattice has 398 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22619962C211yesmaximal, characteristic
2#22610759C211yesmaximal, characteristic
3#22608425C211yesmaximal, characteristic
4#22608360C211yesmaximal, characteristic, complemented
5#22607834C211yesmaximal, characteristic, complemented
6#22607764C211yesmaximal, characteristic, complemented
7C42×Q8×C73C211yesmaximal, characteristic, complemented
8#17989137C2211yescharacteristic
9#17988905—21no—
10#17988872C412yescomplemented
11#17988787C412yescomplemented
12#17988726—21no—
13C4×#11924015—22no—
14#17988640C2211yescharacteristic
15#17988640C2211yescharacteristic
16#17988611C2211yescharacteristic
17C2×#14556519—21no—
18C2×#14556519—21no—
19#17988320—22no—
20Q8⋊#10596831—22no—
21C2×C4×Q8×C73—21no—
22C2×C4×Q8×C73C2211yescharacteristic
23C4×C4⋊C4×C73—21no—
24C4×C4⋊C4×C73—21no—
25C4×C4⋊C4×C73—22no—
26C4×C4⋊C4×C73C2211yescharacteristic
27C43×C73—21no—
28C43×C73C2211yescharacteristic
29C2×#11924028—21no—
30C2×#11924015—22no—
31C2×#11924012—21no—
32#14556591C2×C411yescharacteristic
33#14556588C2×C411yescharacteristic
34#14556579—21no—
35#14556565—21no—
36#14556565—21no—
37C4×D292—21no—
38C4×D292—21no—
39#14556522—22no—
40C4⋊#10596831—21no—
41C4⋊#10596831—21no—
42C4⋊#10596831—21no—
43C4⋊#10596831C2×C412yes—
44#14556519—42no—
45#14556519—42no—
46#14556515—22no—
47#14556511—22no—
48C22⋊#10596831—21no—
49C22⋊#10596831—21no—
50C4×#10596831—21no—
51C4×#10596831—21no—
52C22×Q8×C73D411yescharacteristic
53C4×Q8×C73—28no—
54C4×Q8×C73—21no—
55C4×Q8×C73—21no—
56C4×Q8×C73C2×C412yes—
57C4×Q8×C73D412yes—
58C2×C4⋊C4×C73—22no—
59C2×C4⋊C4×C73—21no—
60C2×C4⋊C4×C73—21no—
61C2×C4⋊C4×C73D411yescharacteristic
62C2×C42×C73—22no—
63C2×C42×C73—21no—
64C2×C42×C73—21no—
65C2×C42×C73—22no—
66C2×C42×C73—21no—
67C2×C42×C73—21no—
68C2×C42×C73D411yescharacteristic
69C2×C42×C73D411yescharacteristic
70C2×C42×C73C2311yescharacteristic
71#95×C73—22no—
72#95×C73D412yes—
73#11924028—41no—
74#11924028—41no—
75#11924028—41no—
76#11924028—41no—
77C2×Dic146—21no—
78C2×D292—21no—
79C2×D292—21no—
80C2×D292—21no—
81C2×C22⋊D73—41no—
82C2×C4×D73—41no—
83#11924015—44no—
84#11924012—41no—
85#11924012—41no—
86C2×#10596831—22no—
87C2×#10596831—41no—
88C2×#10596831—41no—
89C4⋊Dic73—21no—
90C4⋊Dic73—21no—
91#11924005—41no—
92#11924005—41no—
93C2×Q8×C73—24no—
94C2×Q8×C73—21no—
95C2×Q8×C73C22⋊C412yes—
96C22×C4×C73—21no—
97C22×C4×C73—21no—
98C22×C4×C73—21no—
99C22×C4×C73—21no—
100C22×C4×C73—22no—
101C22×C4×C73C2×D411yescharacteristic
102C22×C4×C73C2×D411yescharacteristic
103C22×C4×C73C2×D411yescharacteristic
104C4⋊C4×C73—28no—
105C4⋊C4×C73—24no—
106C4⋊C4×C73—22no—
107C4⋊C4×C73—22no—
108C4⋊C4×C73—21no—
109C4⋊C4×C73C22⋊C412yes—
110C42×C73—28no—
111C42×C73—24no—
112C42×C73—22no—
113C42×C73—22no—
114C42×C73—21no—
115C42×C73—21no—
116C42×C73—21no—
117C42×C73—42no—
118C42×C73—44no—
119C42×C73—41no—
120C42×C73—41no—
121C42×C73SD1612yescomplemented
122C42×C73C22×C411yescharacteristic
123C42×C73C2×D411yescharacteristic
124C42×C73#4111yescharacteristic
125C42×C73#4111yescharacteristic
126C23×C73#12011yescharacteristic
127C22×D73—41no—
128Q8×C73—24no—
129Q8×C73—22no—
130Q8×C73#10414yescomplemented
131C2×C4×C73—22no—
132C2×C4×C73—24no—
133C2×C4×C73—22no—
134C2×C4×C73—22no—
135C2×C4×C73—22no—
136C2×C4×C73—22no—
137C2×C4×C73—22no—
138C2×C4×C73—24no—
139C2×C4×C73—22no—
140C2×C4×C73—41no—
141C2×C4×C73—41no—
142C2×C4×C73—42no—
143C2×C4×C73—42no—
144C2×C4×C73—41no—
145C2×C4×C73—42no—
146C2×C4×C73C2×C22⋊C411yescharacteristic
147C2×C4×C73C4×D411yescharacteristic
148C2×C4×C73C4×D411yescharacteristic
149C2×C4×C73#12111yescharacteristic
150C2×C4×C73#12111yescharacteristic
151C2×C4×C73#12311yescharacteristic
152C2×C4×C73C2×SD1611yescharacteristic
153C2×C4×C73#13511yescharacteristic
154C2×C4×C73AGL(1,ℤ/8)11yescharacteristic
155C2×C4×C73#13711yescharacteristic
156C22⋊D73—81no—
157C22⋊D73—81no—
158C2×Dic73—41no—
159D292—42no—
160D292—41no—
161D292—41no—
162D292—81no—
163C4×D73—81no—
164C4×D73—81no—
165Dic146—41no—
166Dic146—41no—
167#10596831—42no—
168#10596831—81no—
169#10596831—81no—
170C22×C73—21no—
171C22×C73—21no—
172C22×C73#38611yescharacteristic
173C22×C73#44811yescharacteristic
174C22×C73#44911yescharacteristic
175C2×D73—81no—
176C2×D73—81no—
177C2×D73—81no—
178C4×C73—24no—
179C4×C73—24no—
180C4×C73—21no—
181C4×C73—21no—
182C4×C73—21no—
183C4×C73—42no—
184C4×C73—44no—
185C4×C73—42no—
186C4×C73C2×#10412yes—
187C4×C73C4×SD1611yescharacteristic
188C4×C73#44011yescharacteristic
189C4×C73#46011yescharacteristic
190C4×C73#46411yescharacteristic
191Dic73—81no—
192#9234—731nomaximal, Sylow
193C2×C73—21no—
194C2×C73—21no—
195C2×C73#188011yescharacteristic
196C2×C73#189611yescharacteristic
197C2×C73#190611yescharacteristic
198D73—161no—
199#3103—731no—
200C42×Q8D7311yescharacteristic, complemented
201#1962—731no—
202#1502—731no—
203#1487—731no—
204#1303—731no—
205#1276—731no—
206C73#923411yescharacteristic, complemented, Sylow
207#532—731no—
208C2×C4×Q8—21no—
209C2×C4×Q8C2×D7311yescharacteristic
210#481—1461no—
211#475—732no—
212#460—732no—
213#448—1461no—
214C4×SD16—1462no—
215#431—731no—
216#431—731no—
217#423—731no—
218C2×#104—1461no—
219C2×#104—1461no—
220C4×C4⋊C4—21no—
221C4×C4⋊C4—21no—
222C4×C4⋊C4—22no—
223C4×C4⋊C4C2×D7311yescharacteristic
224C43—21no—
225C43C2×D7311yescharacteristic
226#327—1462no—
227Q8⋊C8—1462no—
228C2×#41—1461no—
229C22×Q8C22⋊D7311yescharacteristic
230C2×SD16—1462no—
231C2×M16—1461no—
232C4⋊Q8—731no—
233#127—731no—
234#124—1461no—
235#121—1461no—
236#121—1461no—
237C4×Q8—28no—
238C4×Q8—21no—
239C4×Q8—21no—
240C4×Q8C4×D7312yes—
241C4×Q8D29212yes—
242C4×D4—1461no—
243C4×D4—1461no—
244C2×C4⋊C4—22no—
245C2×C4⋊C4—21no—
246C2×C4⋊C4—21no—
247C2×C4⋊C4C22⋊D7311yescharacteristic
248C2×C42—22no—
249C2×C42—21no—
250C2×C42—21no—
251C2×C42—22no—
252C2×C42—21no—
253C2×C42—21no—
254C2×C42C22⋊D7311yescharacteristic
255C2×C42C22⋊D7311yescharacteristic
256C2×C42C22×D7311yescharacteristic
257#106—1462no—
258C4⋊C8—732no—
259C4⋊C8—1461no—
260C4⋊C8—1461no—
261C4⋊C8—1461no—
262#104—2922no—
263#104—2922no—
264#103—1462no—
265#102—1462no—
266C22⋊C8—1461no—
267C22⋊C8—1461no—
268C4×C8—1461no—
269C4×C8—1461no—
270#95—22no—
271#95C22⋊D7312yes—
272#41—2921no—
273#41—2921no—
274#41—2921no—
275#41—2921no—
276C2×Q8—24no—
277C2×Q8—21no—
278C2×Q8—1461no—
279C2×Q8#1192400512yes—
280C2×D4—2921no—
281C2×D4—1461no—
282C2×D4—1461no—
283C2×D4—1461no—
284C22×C4—21no—
285C22×C4—21no—
286C22×C4—21no—
287C22×C4—21no—
288C22×C4—22no—
289C22×C4—2921no—
290C22×C4C2×C22⋊D7311yescharacteristic
291C22×C4C2×C22⋊D7311yescharacteristic
292C22×C4C2×C22⋊D7311yescharacteristic
293SD16—2924no—
294M16—2921no—
295M16—2921no—
296C2×C8—2921no—
297C2×C8—2921no—
298C2×C8—1462no—
299C4⋊C4—28no—
300C4⋊C4—24no—
301C4⋊C4—22no—
302C4⋊C4—22no—
303C4⋊C4—21no—
304C4⋊C4—1461no—
305C4⋊C4—1461no—
306C4⋊C4#1192400512yes—
307C22⋊C4—2921no—
308C22⋊C4—2921no—
309C42—28no—
310C42—24no—
311C42—22no—
312C42—22no—
313C42—21no—
314C42—21no—
315C42—21no—
316C42—42no—
317C42—44no—
318C42—41no—
319C42—41no—
320C42#1192401512yescomplemented
321C42C2×C4×D7311yescharacteristic
322C42C2×D29211yescharacteristic
323C42#1192402811yescharacteristic
324C42#1192402811yescharacteristic
325C23—2921no—
326C23#1455656011yescharacteristic
327Q8—24no—
328Q8—22no—
329Q8—2921no—
330Q8—2921no—
331Q8#1455651914yescomplemented
332D4—2921no—
333D4—2921no—
334D4—2922no—
335D4—5841no—
336D4—5841no—
337D4—5841no—
338C2×C4—22no—
339C2×C4—24no—
340C2×C4—22no—
341C2×C4—22no—
342C2×C4—22no—
343C2×C4—22no—
344C2×C4—22no—
345C2×C4—24no—
346C2×C4—22no—
347C2×C4—41no—
348C2×C4—41no—
349C2×C4—42no—
350C2×C4—42no—
351C2×C4—41no—
352C2×C4—42no—
353C2×C4—2921no—
354C2×C4—5841no—
355C2×C4—5841no—
356C2×C4C2×#1192400511yescharacteristic
357C2×C4C4×C22⋊D7311yescharacteristic
358C2×C4C4×C22⋊D7311yescharacteristic
359C2×C4#1455656511yescharacteristic
360C2×C4#1455656511yescharacteristic
361C2×C4#1455657411yescharacteristic
362C2×C4C2×#1192401511yescharacteristic
363C2×C4#1455661111yescharacteristic
364C2×C4#1455661611yescharacteristic
365C2×C4#1455662211yescharacteristic
366C8—2922no—
367C8—5841no—
368C8—5841no—
369C22—21no—
370C22—21no—
371C22—5841no—
372C22—5841no—
373C22—5841no—
374C22#1798846611yescharacteristic
375C22#1798872611yescharacteristic
376C22#1798873211yescharacteristic
377C4—24no—
378C4—21no—
379C4—21no—
380C4—24no—
381C4—21no—
382C4—42no—
383C4—44no—
384C4—42no—
385C4—5841no—
386C4C2×#1455651912yes—
387C4C4×#1192401511yescharacteristic
388C4#1798868511yescharacteristic
389C4#1798878711yescharacteristic
390C4#1798880911yescharacteristic
391C2—21no—
392C2—21no—
393C2—11681no—
394C2#2261028611yescharacteristic
395C2#2261037411yescharacteristic
396C2#2261043211yescharacteristic
subgroupG/H
centerC2×C4#14556565
commutatorC42×C73C22×C4
FrattiniC2×C42C22×D73
FittingC42×Q8×C73C2
socleC22×C73#386
prime radical#17989137C22
cosocleC2×C42×C73C23
hypercenterC42×Q8D73
hypocenterC73#9234
LCSG ▷ C42×C73 ▷ C22×C73 ▷ C73
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ C42×Q8

Complements

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

NH ≅ G/Ntype
4. #22608360C2semidirect
5. #22607834C2semidirect
6. #22607764C2semidirect
7. C42×Q8×C73C2semidirect
10. #17988872C4semidirect
11. #17988787C4semidirect
121. C42×C73SD16semidirect
130. Q8×C73#104semidirect
200. C42×Q8D73semidirect
206. C73#9234semidirect
320. C42#11924015semidirect
331. Q8#14556519semidirect

Maximal subgroups

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

subgroupindex
#226199622
#226107592
#226084252
#226083602
#226078342
#226077642
C42×Q8×C732
#923473