Finite Group Databasebeta

← #34008654

Subgroups — #34008654

Order 18688.

854 conjugacy classes of subgroups.

The subgroup lattice has 416 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#14556645C211yesmaximal, characteristic
2C2×#17989461C211yesmaximal, characteristic
3C2×#17989460C212yesmaximal
4#22625108C212yesmaximal
5#340×D73C211yesmaximal, characteristic
6C22×#11924036C2211yescharacteristic
7C22×C4⋊#64408C2212yes—
8C22×C4×#64408C2211yescharacteristic
9C2×#14556645—24no—
10C2×#14556645C414yescomplemented
11C2×(C2×#64408)⋊C4—22no—
12C2×(C2×#64408)⋊C4—22no—
13#17989461C414yescomplemented
14#17989460—24no—
15C22×C8×D73C2211yescharacteristic
16C2×C4⋊C4×D73C2212yes—
17#17988350C411yescharacteristic, complemented
18#17988349C412yescomplemented
19#340×C73C411yescharacteristic, complemented
20C23×#64408—22no—
21C23×#64408D412yes—
22C2×#11924036—22no—
23C2×#11924036—21no—
24C2×#11924036C2×C412yes—
25C2×#11924036D414yes—
26C22×#10596833—21no—
27C2×C4⋊#64408—22no—
28C2×C4⋊#64408—22no—
29C2×C4⋊#64408—22no—
30C2×C4⋊#64408—22no—
31C2×C4⋊#64408—22no—
32C2×C4⋊#64408C2×C414yes—
33C2×C4×#64408—22no—
34C2×C4×#64408—22no—
35C2×C4×#64408C2×C412yes—
36C2×C4×#64408Q812yes—
37#14556645—28no—
38(C2×#64408)⋊C4—44no—
39(C2×#64408)⋊C4—44no—
40C22×C4×D73—22no—
41C22×C4×D73C2311yescharacteristic
42C22×#10596831C2×C411yescharacteristic
43C2×C8×D73—22no—
44C2×C8×D73—22no—
45C2×C8×D73C2×C412yes—
46C2×C8×D73C2×C412yes—
47C2×C4⋊Dic73C2×C412yes—
48C2×#11924007—22no—
49C4⋊C4×D73—22no—
50C4⋊C4×D73—22no—
51C4⋊C4×D73—22no—
52C4⋊C4×D73—22no—
53C22×C8×C73C2×C411yescharacteristic
54C2×C4⋊C4×C73C2×C412yes—
55C22×#64408—22no—
56C22×#64408—22no—
57C22×#64408—22no—
58C22×#64408—22no—
59C22×#64408—22no—
60C22×#64408—22no—
61C22×#64408—24no—
62C22×#64408—42no—
63C22×#64408—42no—
64C22×#64408—42no—
65C22×#64408—42no—
66C22×#64408—42no—
67C22×#64408—42no—
68C22×#64408C4⋊C414yes—
69#11924036—22no—
70#11924036—24no—
71#11924036C22⋊C414yes—
72C2×#10596833—22no—
73C2×#10596833—24no—
74C4⋊#64408—24no—
75C4⋊#64408—24no—
76C4⋊#64408—42no—
77C4⋊#64408—42no—
78C4×#64408—22no—
79C4×#64408—22no—
80C4×#64408—24no—
81C23×D73C2×D411yescharacteristic
82C22×Dic73—22no—
83C22×Dic73#4111yescharacteristic
84C2×C4×D73—42no—
85C2×C4×D73—42no—
86C2×C4×D73—42no—
87C2×C4×D73—42no—
88C2×C4×D73—42no—
89C2×C4×D73—42no—
90C2×C4×D73C22×C411yescharacteristic
91C2×C4×D73C22×C411yescharacteristic
92C2×C4×D73C2×D412yes—
93C2×C4×D73C2×Q811yescharacteristic
94C2×C4×D73#4112yes—
95C2×C4×D73#4112yes—
96C2×C4×D73#4112yes—
97C2×C4×D73#4111yescharacteristic
98C2×#10596831—22no—
99C2×#10596831C4212yes—
100C8×D73—22no—
101C8×D73—22no—
102C8×D73—24no—
103C4⋊Dic73—22no—
104C4⋊Dic73—22no—
105#11924007—42no—
106#11924007—42no—
107C22×C4×C73—22no—
108C22×C4×C73C22×C411yescharacteristic
109C2×C8×C73—22no—
110C2×C8×C73C4212yes—
111C4⋊C4×C73—22no—
112C4⋊C4×C73—22no—
113C23×C73C4×D411yescharacteristic
114C22×D73C2×C4⋊C411yescharacteristic
115C22×D73C4×D411yescharacteristic
116C22×D73#12112yes—
117C22×D73#12412yes—
118C22×D73AGL(1,ℤ/8)12yes—
119C22×D73AGL(1,ℤ/8)12yes—
120C22×D73#13712yes—
121C22×D73#13712yes—
122C2×#64408—24no—
123C2×#64408—24no—
124C2×#64408—24no—
125C2×#64408—42no—
126C2×#64408—42no—
127C2×#64408—42no—
128C2×#64408—42no—
129C2×#64408—44no—
130C2×#64408—44no—
131C2×#64408—44no—
132C2×#64408—82no—
133C2×#64408—82no—
134C2×#64408—82no—
135C2×#64408—82no—
136C2×C4×C73—42no—
137C2×C4×C73—42no—
138C2×C4×C73—42no—
139C2×C4×C73C2×C4211yescharacteristic
140C2×C4×C73#11712yes—
141C2×C4×C73C4×D412yes—
142C2×C4×C73C4×Q811yescharacteristic
143C2×Dic73—42no—
144C2×Dic73—42no—
145C2×Dic73—42no—
146C2×Dic73#11711yescharacteristic
147C2×Dic73C4×Q811yescharacteristic
148C2×Dic73#12312yes—
149C2×Dic73#12612yes—
150C4×D73—82no—
151C4×D73—82no—
152C4×D73—82no—
153C4×D73—82no—
154C4×D73C2×C22⋊C411yescharacteristic
155C4×D73#11711yescharacteristic
156C4×D73#11711yescharacteristic
157C4×D73#11711yescharacteristic
158C4×D73#11712yes—
159C4×D73C4×D412yes—
160C4×D73C22⋊Q812yes—
161C4×D73#12312yes—
162C4×D73#12512yes—
163C4×D73#12612yes—
164#10596833—24no—
165C8×C73—22no—
166#10596831—22no—
167C22×C73C4×C4⋊C411yescharacteristic
168C22×C73#38812yes—
169C22×C73#44012yes—
170C22×C73#44111yescharacteristic
171C22×C73#44211yescharacteristic
172C2×D73#38011yescharacteristic
173C2×D73#38211yescharacteristic
174C2×D73#38812yes—
175C2×D73#42812yes—
176C2×D73#42812yes—
177C2×D73#44012yes—
178C2×D73#44111yescharacteristic
179C2×D73#44211yescharacteristic
180C2×D73#46812yes—
181C2×D73#46812yes—
182C2×D73#46912yes—
183C2×D73#47012yes—
184C2×D73#48912yes—
185C2×D73#48912yes—
186C2×D73#49012yes—
187C2×D73#49112yes—
188#64408—44no—
189#64408—84no—
190C4×C73—82no—
191C4×C73#37611yescharacteristic
192C4×C73C4×C22⋊C411yescharacteristic
193C4×C73#38712yes—
194Dic73—82no—
195Dic73#38111yescharacteristic
196Dic73#38311yescharacteristic
197Dic73#38712yes—
198#10821—731nomaximal, Sylow
199C2×C73#143411yescharacteristic
200C2×C73#177611yescharacteristic
201C2×C73#177611yescharacteristic
202C2×C73#193512yes—
203C2×C73#193612yes—
204D73#180912yes—
205D73#183612yes—
206D73#193512yes—
207D73#193612yes—
208C22×#97—731no—
209C2×#340—731no—
210C2×#340—731no—
211C2×#340—732no—
212C4⋊#95—732no—
213C73#1082111yescharacteristic, complemented, Sylow
214C23×C8—731no—
215C23×C8—731no—
216C22×C4⋊C4—732no—
217C22×C4⋊C4—732no—
218C22×C42—731no—
219C2×#97—734no—
220C2×#97—1464no—
221C2×#95—1462no—
222C2×#95—1462no—
223#340—734no—
224#340—731no—
225#340—732no—
226#340—1464no—
227#340#6440811yescharacteristic, complemented
228C23×C4—731no—
229C23×C4—732no—
230C23×C4—1462no—
231C23×C4—1462no—
232C22×C8—732no—
233C22×C8—734no—
234C22×C8—731no—
235C22×C8—732no—
236C22×C8—732no—
237C22×C8—1462no—
238C22×C8—1462no—
239C22×C8—1461no—
240C22×C8—1461no—
241C22×C8—1462no—
242C22×C8C2×#6440811yescharacteristic
243C2×C4⋊C4—734no—
244C2×C4⋊C4—732no—
245C2×C4⋊C4—1462no—
246C2×C4⋊C4—1462no—
247C2×C4⋊C4—1462no—
248C2×C4⋊C4—1462no—
249C2×C4⋊C4—1462no—
250C2×C4⋊C4—1462no—
251C2×C4⋊C4—1462no—
252C2×C4⋊C4—1462no—
253C2×C4⋊C4—1462no—
254C2×C4⋊C4—1462no—
255C2×C4⋊C4C2×#6440812yes—
256C2×C42—732no—
257C2×C42—732no—
258C2×C42—1462no—
259C2×C42—1462no—
260#97—1468no—
261#95—2924no—
262#95—2924no—
263C24—731no—
264C22×C4—22no—
265C22×C4—2922no—
266C22×C4—2922no—
267C22×C4—2922no—
268C22×C4—2922no—
269C22×C4—2922no—
270C22×C4—2922no—
271C22×C4—2922no—
272C22×C4—2922no—
273C22×C4—2922no—
274C22×C4—2922no—
275C22×C4—2922no—
276C22×C4—2922no—
277C22×C4—731no—
278C22×C4—731no—
279C22×C4—731no—
280C22×C4—731no—
281C22×C4—731no—
282C22×C4—732no—
283C22×C4—732no—
284C22×C4—732no—
285C22×C4—732no—
286C22×C4—734no—
287C22×C4—1464no—
288C22×C4—1462no—
289C22×C4—1462no—
290C22×C4—1462no—
291C22×C4—1462no—
292C22×C4—1462no—
293C22×C4—1462no—
294C22×C4—1462no—
295C22×C4C22×#6440811yescharacteristic
296C2×C8—22no—
297C2×C8—734no—
298C2×C8—732no—
299C2×C8—1462no—
300C2×C8—1464no—
301C2×C8—1462no—
302C2×C8—1462no—
303C2×C8—1464no—
304C2×C8—1462no—
305C2×C8—1464no—
306C2×C8—1462no—
307C2×C8C4×#6440812yes—
308C4⋊C4—22no—
309C4⋊C4—22no—
310C4⋊C4—2922no—
311C4⋊C4—2922no—
312C4⋊C4—2922no—
313C4⋊C4—2922no—
314C4⋊C4—1464no—
315C4⋊C4—1464no—
316C4⋊C4—1462no—
317C4⋊C4—1462no—
318C42—1462no—
319C42—1462no—
320C42—1464no—
321C23—731no—
322C23—731no—
323C23—732no—
324C23—732no—
325C23—732no—
326C23—732no—
327C23—732no—
328C23—732no—
329C23D4×#6440811yescharacteristic
330C2×C4—42no—
331C2×C4—42no—
332C2×C4—42no—
333C2×C4—2922no—
334C2×C4—2922no—
335C2×C4—2922no—
336C2×C4—2922no—
337C2×C4—2924no—
338C2×C4—2924no—
339C2×C4—2924no—
340C2×C4—2922no—
341C2×C4—2922no—
342C2×C4—2922no—
343C2×C4—5842no—
344C2×C4—5842no—
345C2×C4—5842no—
346C2×C4—5842no—
347C2×C4—5842no—
348C2×C4—5842no—
349C2×C4—5842no—
350C2×C4—5842no—
351C2×C4—731no—
352C2×C4—731no—
353C2×C4—732no—
354C2×C4—732no—
355C2×C4—731no—
356C2×C4—732no—
357C2×C4—731no—
358C2×C4—731no—
359C2×C4—732no—
360C2×C4—731no—
361C2×C4—732no—
362C2×C4—732no—
363C2×C4—732no—
364C2×C4—732no—
365C2×C4—1464no—
366C2×C4—1464no—
367C2×C4—1464no—
368C2×C4C2×C4×#6440811yescharacteristic
369C2×C4#1455667012yes—
370C2×C4D4×#6440812yes—
371C2×C4Q8×#6440811yescharacteristic
372C8—22no—
373C8—1464no—
374C8—1462no—
375C22—732no—
376C22—731no—
377C22—731no—
378C22—732no—
379C22—732no—
380C22—731no—
381C22—732no—
382C22—731no—
383C22—732no—
384C22—732no—
385C22—732no—
386C22—732no—
387C22—732no—
388C22—732no—
389C22—732no—
390C22—732no—
391C22C4⋊C4×#6440811yescharacteristic
392C22#1798953712yes—
393C22#1798962612yes—
394C22#1798962711yescharacteristic
395C22#1798962811yescharacteristic
396C4—82no—
397C4—2924no—
398C4—5844no—
399C4—5842no—
400C4—731no—
401C4—731no—
402C4—732no—
403C4#1798952311yescharacteristic
404C4C22⋊C4×#6440811yescharacteristic
405C4#1798953612yes—
406C2—732no—
407C2—732no—
408C2—732no—
409C2—732no—
410C2#2262510611yescharacteristic
411C2#2262574611yescharacteristic
412C2#2262574711yescharacteristic
413C2#2262602212yes—
414C2#2262602312yes—
subgroupG/H
centerC23D4×#64408
commutatorC2×C4×C73C2×C42
FrattiniC22×C4C22×#64408
Fitting#340×C73C4
socleC23×C73C4×D4
prime radicalC23×D73C2×D4
cosocleC22×C4×D73C23
hypercenter#340#64408
hypocenterC73#10821
LCSG ▷ C2×C4×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C23 ◁ C22×C4 ◁ #340

Complements

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

NH ≅ G/Ntype
10. C2×#14556645C4semidirect
13. #17989461C4semidirect
17. #17988350C4semidirect
18. #17988349C4semidirect
19. #340×C73C4semidirect
213. C73#10821semidirect
227. #340#64408semidirect

Maximal subgroups

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

subgroupindex
C22×#145566452
C2×#179894612
C2×#179894602
#226251082
#340×D732
#1082173