Finite Group Databasebeta

← #32981565

Subgroups — #32981565

Order 18688.

612 conjugacy classes of subgroups.

The subgroup lattice has 412 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#17988644C211yesmaximal, characteristic
2C2×#17988324C212yesmaximal
3#22608158C211yesmaximal, characteristic
4#22607809C212yesmaximal
5C73×#2855C211yesmaximal, characteristic
6#17988644—24no—
7C2×C4⋊#10596831—21no—
8C2×C22⋊#10596831C2212yes—
9C2×#14556502—21no—
10C2×#14556498—21no—
11C2×#14556498—21no—
12C2×#14556498C2212yes—
13#17988324—24no—
14C2×C22⋊Q8×C73C2211yescharacteristic
15C22×C42×C73C2211yescharacteristic
16#391×C73C411yescharacteristic, complemented
17#391×C73C411yescharacteristic, complemented
18#389×C73C411yescharacteristic, complemented
19#389×C73C411yescharacteristic, complemented
20#387×C73—21no—
21#387×C73—21no—
22#385×C73—21no—
23#381×C73—21no—
24#380×C73C2211yescharacteristic
25C23×Dic73—22no—
26C22×#10596831—22no—
27C4⋊#10596831—24no—
28C22⋊#10596831—22no—
29C22⋊#10596831D414yes—
30#14556502—24no—
31#14556498—24no—
32#14556498—22no—
33#14556498—22no—
34#14556498—42no—
35C22×Q8×C73C2×C411yescharacteristic
36C23×C4×C73D411yescharacteristic
37C23×C4×C73D411yescharacteristic
38C23×C4×C73C2311yescharacteristic
39C22⋊Q8×C73—22no—
40C22⋊Q8×C73—22no—
41C2×C4⋊C4×C73—21no—
42C2×C4⋊C4×C73—21no—
43C2×C4⋊C4×C73—21no—
44C2×C4⋊C4×C73C2×C411yescharacteristic
45C2×C4⋊C4×C73C2×C411yescharacteristic
46C2×C4⋊C4×C73C2×C411yescharacteristic
47C2×C22⋊C4×C73—21no—
48C2×C22⋊C4×C73—21no—
49C2×C42×C73—22no—
50C2×C42×C73—22no—
51C2×C42×C73C2×C411yescharacteristic
52C2×C42×C73C2×C411yescharacteristic
53C2×C42×C73D411yescharacteristic
54C2×C42×C73D411yescharacteristic
55#95×C73—21no—
56#95×C73—21no—
57#95×C73—21no—
58#95×C73—21no—
59#95×C73—21no—
60#95×C73—21no—
61#95×C73—21no—
62C22×Dic73—24no—
63C22×Dic73—42no—
64C22×Dic73—42no—
65C22×Dic73—42no—
66C22×Dic73—42no—
67C22×Dic73—42no—
68C22×Dic73—42no—
69C2×#10596831—24no—
70C2×#10596831—42no—
71C2×#10596831—42no—
72C24×C73C2×D411yescharacteristic
73C2×Q8×C73—22no—
74C2×Q8×C73—41no—
75C2×Q8×C73—41no—
76C22×C4×C73—22no—
77C22×C4×C73—22no—
78C22×C4×C73—21no—
79C22×C4×C73—21no—
80C22×C4×C73—22no—
81C22×C4×C73—22no—
82C22×C4×C73—21no—
83C22×C4×C73—21no—
84C22×C4×C73—21no—
85C22×C4×C73—21no—
86C22×C4×C73—41no—
87C22×C4×C73—41no—
88C22×C4×C73—41no—
89C22×C4×C73C22⋊C411yescharacteristic
90C22×C4×C73C22⋊C411yescharacteristic
91C22×C4×C73C22⋊C411yescharacteristic
92C22×C4×C73C22⋊C411yescharacteristic
93C22×C4×C73C22×C411yescharacteristic
94C22×C4×C73C2×D412yes—
95C22×C4×C73C2×D411yescharacteristic
96C22×C4×C73#4112yes—
97C4⋊C4×C73—22no—
98C4⋊C4×C73—41no—
99C4⋊C4×C73—41no—
100C4⋊C4×C73—42no—
101C4⋊C4×C73—42no—
102C4⋊C4×C73—41no—
103C4⋊C4×C73—41no—
104C22⋊C4×C73—42no—
105C22⋊C4×C73—41no—
106C22⋊C4×C73—41no—
107C42×C73—22no—
108C42×C73—22no—
109C42×C73—21no—
110C42×C73—21no—
111C42×C73—21no—
112C42×C73—21no—
113C23×C73—21no—
114C23×C73—21no—
115C23×C73—21no—
116C23×C73—21no—
117C23×C73C2×C22⋊C411yescharacteristic
118C23×C73#12112yes—
119C23×C73AGL(1,ℤ/8)11yescharacteristic
120C23×C73#13711yescharacteristic
121C23×C73#13712yes—
122Q8×C73—82no—
123C2×C4×C73—21no—
124C2×C4×C73—21no—
125C2×C4×C73—21no—
126C2×C4×C73—21no—
127C2×C4×C73—21no—
128C2×C4×C73—21no—
129C2×C4×C73—21no—
130C2×C4×C73—21no—
131C2×C4×C73—21no—
132C2×C4×C73—21no—
133C2×C4×C73—21no—
134C2×C4×C73—21no—
135C2×C4×C73—21no—
136C2×C4×C73—21no—
137C2×C4×C73—21no—
138C2×C4×C73—42no—
139C2×C4×C73—42no—
140C2×C4×C73—44no—
141C2×C4×C73—42no—
142C2×C4×C73—42no—
143C2×C4×C73—41no—
144C2×C4×C73—41no—
145C2×C4×C73—41no—
146C2×C4×C73—41no—
147C2×C4×C73—41no—
148C2×C4×C73—41no—
149C2×C4×C73—41no—
150C2×C4×C73—41no—
151C2×C4×C73—41no—
152C2×C4×C73—41no—
153C2×C4×C73—41no—
154C2×C4×C73—41no—
155C2×C4×C73#10412yes—
156C2×C4×C73C4×D412yes—
157C2×C4×C73#12011yescharacteristic
158C2×C4×C73#12311yescharacteristic
159C2×Dic73—44no—
160C2×Dic73—44no—
161C2×Dic73—44no—
162C2×Dic73—82no—
163C2×Dic73—82no—
164C2×Dic73—82no—
165C2×Dic73—82no—
166#10596831—44no—
167C22×C73—22no—
168C22×C73—22no—
169C22×C73—22no—
170C22×C73—21no—
171C22×C73—21no—
172C22×C73—21no—
173C22×C73—21no—
174C22×C73—22no—
175C22×C73#38611yescharacteristic
176C22×C73#41011yescharacteristic
177C22×C73#41011yescharacteristic
178C22×C73#41711yescharacteristic
179C22×C73#41911yescharacteristic
180C22×C73C2×#10411yescharacteristic
181C22×C73#42111yescharacteristic
182C22×C73#46812yes—
183C22×C73#47012yes—
184C4×C73—22no—
185C4×C73—21no—
186C4×C73—22no—
187C4×C73—21no—
188C4×C73—41no—
189C4×C73—41no—
190C4×C73—41no—
191C4×C73—81no—
192C4×C73—81no—
193C4×C73—81no—
194C4×C73—81no—
195Dic73—84no—
196#9116—731nomaximal, Sylow
197C2×C73—22no—
198C2×C73—22no—
199C2×C73#150611yescharacteristic
200C2×C73#150911yescharacteristic
201C2×C73#151211yescharacteristic
202C2×C73#151511yescharacteristic
203C2×C73#187111yescharacteristic
204C2×C73#189411yescharacteristic
205C2×C73#189911yescharacteristic
206C2×#432—731no—
207#2855D7311yescharacteristic
208C2×#328—732no—
209#1434—731no—
210#1294—732no—
211C73#911611yescharacteristic, complemented, Sylow
212C2×C22⋊Q8C2×D7311yescharacteristic
213C22×C42C2×D7311yescharacteristic
214#432—1464no—
215C2×C4⋊C8—1461no—
216C2×C22⋊C8—732no—
217C2×#97—1461no—
218#391Dic7311yescharacteristic, complemented
219#391Dic7311yescharacteristic, complemented
220#389Dic7311yescharacteristic, complemented
221#389Dic7311yescharacteristic, complemented
222#387—21no—
223#387—21no—
224#385—21no—
225#381—21no—
226#380C2×D7311yescharacteristic
227C2×#95—732no—
228C2×#95—1461no—
229C2×#95—1461no—
230#328—1464no—
231C22×Q8C2×Dic7311yescharacteristic
232C23×C4—1462no—
233C23×C4C22⋊D7311yescharacteristic
234C23×C4C22⋊D7311yescharacteristic
235C23×C4C22×D7311yescharacteristic
236C22×C8—1462no—
237C22⋊Q8—22no—
238C22⋊Q8—22no—
239C2×C4⋊C4—21no—
240C2×C4⋊C4—21no—
241C2×C4⋊C4—21no—
242C2×C4⋊C4C2×Dic7311yescharacteristic
243C2×C4⋊C4C2×Dic7311yescharacteristic
244C2×C4⋊C4C2×Dic7311yescharacteristic
245C2×C22⋊C4—21no—
246C2×C22⋊C4—21no—
247C2×C42—22no—
248C2×C42—22no—
249C2×C42C2×Dic7311yescharacteristic
250C2×C42C2×Dic7311yescharacteristic
251C2×C42C22⋊D7311yescharacteristic
252C2×C42C22⋊D7311yescharacteristic
253C4⋊C8—1464no—
254C22⋊C8—734no—
255C22⋊C8—1462no—
256#97—1464no—
257#95—21no—
258#95—21no—
259#95—21no—
260#95—21no—
261#95—21no—
262#95—21no—
263#95—21no—
264#95—2922no—
265#95—1464no—
266#95—1462no—
267#95—1462no—
268C24C2×C22⋊D7311yescharacteristic
269C2×Q8—22no—
270C2×Q8—41no—
271C2×Q8—41no—
272C22×C4—22no—
273C22×C4—22no—
274C22×C4—21no—
275C22×C4—21no—
276C22×C4—22no—
277C22×C4—22no—
278C22×C4—21no—
279C22×C4—21no—
280C22×C4—21no—
281C22×C4—21no—
282C22×C4—41no—
283C22×C4—41no—
284C22×C4—41no—
285C22×C4—2922no—
286C22×C4—2922no—
287C22×C4—2922no—
288C22×C4—2922no—
289C22×C4—2922no—
290C22×C4—2922no—
291C22×C4—1464no—
292C22×C4C22⋊Dic7311yescharacteristic
293C22×C4C22⋊Dic7311yescharacteristic
294C22×C4C22⋊Dic7311yescharacteristic
295C22×C4C22⋊Dic7311yescharacteristic
296C22×C4C22×Dic7311yescharacteristic
297C22×C4D4×D7312yes—
298C22×C4C2×C22⋊D7311yescharacteristic
299C22×C4#1192402712yes—
300C2×C8—2922no—
301C2×C8—2922no—
302C2×C8—1464no—
303C4⋊C4—22no—
304C4⋊C4—41no—
305C4⋊C4—41no—
306C4⋊C4—42no—
307C4⋊C4—42no—
308C4⋊C4—41no—
309C4⋊C4—41no—
310C22⋊C4—42no—
311C22⋊C4—41no—
312C22⋊C4—41no—
313C42—22no—
314C42—22no—
315C42—21no—
316C42—21no—
317C42—21no—
318C42—21no—
319C23—21no—
320C23—21no—
321C23—21no—
322C23—21no—
323C23C2×C22⋊Dic7311yescharacteristic
324C23#1455656312yes—
325C23#1455661611yescharacteristic
326C23#1455662111yescharacteristic
327C23#1455662212yes—
328Q8—82no—
329C2×C4—21no—
330C2×C4—21no—
331C2×C4—21no—
332C2×C4—21no—
333C2×C4—21no—
334C2×C4—21no—
335C2×C4—21no—
336C2×C4—21no—
337C2×C4—21no—
338C2×C4—21no—
339C2×C4—21no—
340C2×C4—21no—
341C2×C4—21no—
342C2×C4—21no—
343C2×C4—21no—
344C2×C4—42no—
345C2×C4—42no—
346C2×C4—44no—
347C2×C4—42no—
348C2×C4—42no—
349C2×C4—41no—
350C2×C4—41no—
351C2×C4—41no—
352C2×C4—41no—
353C2×C4—41no—
354C2×C4—41no—
355C2×C4—41no—
356C2×C4—41no—
357C2×C4—41no—
358C2×C4—41no—
359C2×C4—41no—
360C2×C4—41no—
361C2×C4—2924no—
362C2×C4—2924no—
363C2×C4—2924no—
364C2×C4—5842no—
365C2×C4—5842no—
366C2×C4—5842no—
367C2×C4—5842no—
368C2×C4#1455651812yes—
369C2×C4D4×Dic7312yes—
370C2×C4#1455655911yescharacteristic
371C2×C4#1455657211yescharacteristic
372C8—2924no—
373C22—22no—
374C22—22no—
375C22—21no—
376C22—21no—
377C22—21no—
378C22—21no—
379C22—22no—
380C22—22no—
381C22#1798846711yescharacteristic
382C22#1798856111yescharacteristic
383C22#1798856111yescharacteristic
384C22#1798858811yescharacteristic
385C22#1798859611yescharacteristic
386C22C2×#1455651811yescharacteristic
387C22#1798860411yescharacteristic
388C22#1798883512yes—
389C22#1798884312yes—
390C4—22no—
391C4—21no—
392C4—22no—
393C4—21no—
394C4—41no—
395C4—41no—
396C4—41no—
397C4—81no—
398C4—81no—
399C4—81no—
400C4—81no—
401C4—5844no—
402C2—22no—
403C2—22no—
404C2#2260844311yescharacteristic
405C2#2260845611yescharacteristic
406C2#2260846911yescharacteristic
407C2#2260848211yescharacteristic
408C2#2261023411yescharacteristic
409C2#2261036511yescharacteristic
410C2#2261039011yescharacteristic
subgroupG/H
centerC23C2×C22⋊Dic73
commutatorC22×C4×C73C22×C4
FrattiniC23×C4C22×D73
FittingC73×#2855C2
socleC23×C73C2×C22⋊C4
prime radicalC24×C73C2×D4
cosocleC23×C4×C73C23
hypercenter#2855D73
hypocenterC73#9116
LCSG ▷ C22×C4×C73 ▷ C22×C73 ▷ C73
UCSC1 ◁ C23 ◁ C22×C42 ◁ #2855

Complements

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

NH ≅ G/Ntype
16. #391×C73C4semidirect
17. #391×C73C4semidirect
18. #389×C73C4semidirect
19. #389×C73C4semidirect
211. C73#9116semidirect
218. #391Dic73semidirect
219. #391Dic73semidirect
220. #389Dic73semidirect
221. #389Dic73semidirect

Maximal subgroups

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

subgroupindex
C2×#179886442
C2×#179883242
#226081582
#226078092
C73×#28552
#911673