Finite Group Databasebeta

← #33115873

Subgroups — #33115873

Order 18688.

988 conjugacy classes of subgroups.

The subgroup lattice has 408 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#17988636C211yesmaximal, characteristic, complemented
2#22618495C212yesmaximal, complemented
3C2×#17988605C212yesmaximal
4#22618375C211yesmaximal, characteristic
5#22611485C218yesmaximal, complemented
6C2×H(3,ℤ/4)×C73C211yesmaximal, characteristic, complemented
7C22×#11924028C2211yescharacteristic
8C22×#11924012C2212yes—
9C22×#11924012C2211yescharacteristic
10#17988636—22no—
11#17988636C2214yescomplemented
12C2×#14556527—22no—
13#17988614—24no—
14#17988614C2218yescomplemented
15#17988611—24no—
16#17988605—24no—
17#17988605C2218yes—
18C2×#14556519—24no—
19#17988555C2214yes—
20#17988553—22no—
21#17988550—22no—
22C22×M16×C73C2211yescharacteristic
23C2×#117×C73C2212yes—
24H(3,ℤ/4)×C73C2214yescomplemented
25C2×#11924028—21no—
26C2×#11924028—22no—
27C2×#11924028—22no—
28C2×#11924028—21no—
29C2×#11924028—22no—
30C2×#11924028—22no—
31C2×#11924028Q812yes—
32C2×#11924028C2312yes—
33C22×Dic146Q811yescharacteristic
34C22×D292Q811yescharacteristic
35C22×C22⋊D73—21no—
36C22×C4×D73—21no—
37C2×#11924012—22no—
38C2×#11924012—22no—
39C2×#11924012—22no—
40C2×#11924012—21no—
41C2×#11924012—21no—
42C2×#11924012—22no—
43C2×#11924012—22no—
44C2×#11924012—22no—
45C2×#11924012—22no—
46C2×#11924012C2314yes—
47C22×#10596831—22no—
48C22×#10596831C2312yes—
49#14556527—28no—
50C4⋊#10596831—216no—
51#14556519—216no—
52#14556503—28no—
53C23×C4×C73C2311yescharacteristic
54C2×M16×C73—22no—
55C2×M16×C73—22no—
56C2×M16×C73D412yes—
57C2×M16×C73C2312yes—
58C22×C8×C73D412yes—
59#117×C73—22no—
60#117×C73—22no—
61#117×C73D414yes—
62#117×C73C2314yes—
63C2×C4⋊C4×C73—22no—
64C2×C22⋊C4×C73—22no—
65C2×C42×C73D414yes—
66C23×D73—21no—
67#11924028—24no—
68#11924028—24no—
69#11924028—24no—
70#11924028—24no—
71#11924028—41no—
72#11924028—41no—
73#11924028—42no—
74#11924028—42no—
75#11924028—41no—
76#11924028—41no—
77C2×Dic146—22no—
78C2×Dic146—22no—
79C2×Dic146—21no—
80C2×Dic146C2×Q812yes—
81C2×D292—22no—
82C2×D292—22no—
83C2×D292—21no—
84C2×D292C2×Q812yes—
85C2×C22⋊D73—22no—
86C2×C22⋊D73—22no—
87C2×C22⋊D73—42no—
88C2×C22⋊D73—42no—
89C22×Dic73—21no—
90C2×C4×D73—22no—
91C2×C4×D73—22no—
92C2×C4×D73—42no—
93C2×C4×D73—42no—
94#11924012—24no—
95#11924012—24no—
96#11924012—41no—
97#11924012—42no—
98#11924012—41no—
99#11924012—42no—
100#11924012—42no—
101C2×#10596831—22no—
102C2×#10596831—24no—
103C2×#10596831—24no—
104C2×#10596831—42no—
105C2×#10596831—42no—
106C24×C73#4111yescharacteristic
107C22×C4×C73—24no—
108C22×C4×C73C2×D412yes—
109C22×C4×C73C2×D412yes—
110C22×C4×C73C2×D411yescharacteristic
111C22×C4×C73C2×D411yescharacteristic
112C22×C4×C73C2×Q811yescharacteristic
113C22×C4×C73C2×Q811yescharacteristic
114C22×C4×C73#4111yescharacteristic
115C22×C4×C73#4112yes—
116C22×C4×C73#4112yes—
117C22×C4×C73C2411yescharacteristic
118M16×C73—24no—
119M16×C73—24no—
120C2×C8×C73—22no—
121C2×C8×C73—22no—
122C2×C8×C73C2×D414yes—
123C4⋊C4×C73—24no—
124C4⋊C4×C73—24no—
125C22⋊C4×C73—24no—
126C22⋊C4×C73—24no—
127C42×C73—24no—
128C42×C73C2×D418yes—
129C23×C73—81no—
130C23×C73C22⋊Q811yescharacteristic
131C23×C73C22⋊Q811yescharacteristic
132C23×C73#12312yes—
133C23×C73#12312yes—
134C23×C73C2×#4111yescharacteristic
135C22×D73—22no—
136C22×D73—42no—
137C22×D73—82no—
138C2×C4×C73—28no—
139C2×C4×C73—84no—
140C2×C4×C73C22⋊Q812yes—
141C2×C4×C73C22⋊Q812yes—
142C2×C4×C73C22⋊Q812yes—
143C2×C4×C73C22⋊Q812yes—
144C2×C4×C73C22⋊Q811yescharacteristic
145C2×C4×C73C22⋊Q811yescharacteristic
146C2×C4×C73#12311yescharacteristic
147C2×C4×C73#12311yescharacteristic
148C2×C4×C73#12312yes—
149C2×C4×C73#12312yes—
150C2×C4×C73#12312yes—
151C2×C4×C73#12711yescharacteristic
152C2×C4×C73#12712yes—
153C2×C4×C73#12711yescharacteristic
154C2×C4×C73C22×D412yes—
155C2×C4×C73C22×D411yescharacteristic
156C2×C4×C73C22×Q811yescharacteristic
157C2×C4×C73C2×#4112yes—
158C22⋊D73—44no—
159C22⋊D73—42no—
160C22⋊D73—42no—
161C2×Dic73—22no—
162C2×Dic73—42no—
163D292—24no—
164D292—41no—
165D292—41no—
166C4×D73—42no—
167C4×D73—42no—
168C4×D73—44no—
169Dic146—24no—
170Dic146—41no—
171Dic146—41no—
172C8×C73—24no—
173#10596831—42no—
174#10596831—44no—
175C22×C73—42no—
176C22×C73—41no—
177C22×C73—41no—
178C22×C73—42no—
179C22×C73—41no—
180C22×C73#39912yes—
181C22×C73#39911yescharacteristic
182C22×C73#39911yescharacteristic
183C22×C73C2×C22⋊Q811yescharacteristic
184C22×C73C2×#12312yes—
185C2×D73—44no—
186C2×D73—42no—
187C2×D73—84no—
188C4×C73—48no—
189C4×C73#39911yescharacteristic
190C4×C73#39911yescharacteristic
191C4×C73#39912yes—
192C4×C73C2×C22⋊Q812yes—
193C4×C73C2×#12311yescharacteristic
194C4×C73C2×#12711yescharacteristic
195Dic73—42no—
196#23740—731nomaximal, Sylow
197C2×C73—22no—
198C2×C73—21no—
199C2×C73—22no—
200C2×C73—21no—
201C2×C73—21no—
202C2×C73C2×#39911yescharacteristic
203D73—82no—
204C2×#430—731no—
205#2905—732no—
206C2×#421—732no—
207#2879—731no—
208#2098—738no—
209C2×H(3,ℤ/4)D7311yescharacteristic, complemented
210C73#2374011yescharacteristic, complemented, Sylow
211C22×#41—731no—
212C22×M16—731no—
213C22×M16—732no—
214C22×M16C2×D7311yescharacteristic
215C2×#117C2×D7312yes—
216#430—734no—
217#430—1462no—
218C2×#108—1462no—
219#424—738no—
220#424—1464no—
221#423—1464no—
222#421—738no—
223#421—1464no—
224C2×#104—1464no—
225#408—734no—
226#408—1462no—
227#407—1462no—
228H(3,ℤ/4)C2×D7314yescomplemented
229C2×#41—732no—
230C2×#41—732no—
231C2×#41—1462no—
232C2×#41—1462no—
233C2×#41—1462no—
234C2×#41—1462no—
235C2×#41—1461no—
236C2×#41—1461no—
237C22×Q8—731no—
238C22×D4—731no—
239C22×D4—1461no—
240C23×C4—1461no—
241C23×C4C22×D7311yescharacteristic
242C2×M16—22no—
243C2×M16—22no—
244C2×M16—734no—
245C2×M16—1462no—
246C2×M16—1462no—
247C2×M16—1462no—
248C2×M16—1462no—
249C2×M16—1461no—
250C2×M16—1461no—
251C2×M16—1462no—
252C2×M16—1462no—
253C2×M16—1462no—
254C2×M16D29212yes—
255C2×M16C22×D7312yes—
256C22×C8—732no—
257C22×C8—1462no—
258C22×C8D29212yes—
259#117—22no—
260#117—22no—
261#117D29214yes—
262#117C22×D7314yes—
263C2×C4⋊C4—22no—
264C2×C22⋊C4—22no—
265C2×C42D29214yes—
266#108—1468no—
267C4⋊C8—14616no—
268#104—14616no—
269C22⋊C8—1468no—
270C24—1461no—
271C24#1192402711yescharacteristic
272#41—2922no—
273#41—2922no—
274#41—2921no—
275#41—2921no—
276#41—2921no—
277#41—2921no—
278#41—1464no—
279#41—1464no—
280#41—1464no—
281#41—1464no—
282C2×Q8—732no—
283C2×Q8—1462no—
284C2×Q8—1462no—
285C2×Q8—1461no—
286C2×D4—2922no—
287C2×D4—2922no—
288C2×D4—732no—
289C2×D4—1462no—
290C2×D4—1462no—
291C2×D4—1461no—
292C2×D4—1462no—
293C2×D4—1462no—
294C22×C4—24no—
295C22×C4—2922no—
296C22×C4—2922no—
297C22×C4—1461no—
298C22×C4—1462no—
299C22×C4—1462no—
300C22×C4C2×D29212yes—
301C22×C4C2×D29212yes—
302C22×C4C2×D29211yescharacteristic
303C22×C4C2×D29211yescharacteristic
304C22×C4Q8×D7311yescharacteristic
305C22×C4Q8×D7311yescharacteristic
306C22×C4#1192402711yescharacteristic
307C22×C4#1192402712yes—
308C22×C4#1192402712yes—
309C22×C4C23×D7311yescharacteristic
310M16—24no—
311M16—24no—
312M16—2922no—
313M16—2922no—
314M16—2921no—
315M16—2922no—
316M16—2921no—
317M16—1464no—
318M16—1464no—
319C2×C8—22no—
320C2×C8—22no—
321C2×C8—2922no—
322C2×C8—2922no—
323C2×C8—1462no—
324C2×C8—1464no—
325C2×C8—1464no—
326C2×C8C2×D29214yes—
327C4⋊C4—24no—
328C4⋊C4—24no—
329C22⋊C4—24no—
330C22⋊C4—24no—
331C42—24no—
332C42C2×D29218yes—
333C23—81no—
334C23—2922no—
335C23—5842no—
336C23—1462no—
337C23#1455657111yescharacteristic
338C23#1455657111yescharacteristic
339C23#1455657512yes—
340C23#1455657512yes—
341C23C2×#1192402711yescharacteristic
342Q8—2921no—
343Q8—2921no—
344Q8—1464no—
345D4—2922no—
346D4—2922no—
347D4—2924no—
348D4—2921no—
349D4—2921no—
350D4—1464no—
351C2×C4—28no—
352C2×C4—84no—
353C2×C4—2922no—
354C2×C4—2922no—
355C2×C4—2924no—
356C2×C4—2922no—
357C2×C4—1462no—
358C2×C4#1455657111yescharacteristic
359C2×C4#1455657111yescharacteristic
360C2×C4#1455657112yes—
361C2×C4#1455657112yes—
362C2×C4#1455657112yes—
363C2×C4#1455657112yes—
364C2×C4#1455657512yes—
365C2×C4#1455657512yes—
366C2×C4#1455657511yescharacteristic
367C2×C4#1455657511yescharacteristic
368C2×C4#1455657512yes—
369C2×C4#1455658811yescharacteristic
370C2×C4#1455658812yes—
371C2×C4#1455658811yescharacteristic
372C2×C4C22×D29212yes—
373C2×C4C22×D29211yescharacteristic
374C2×C4C2×Q8×D7311yescharacteristic
375C2×C4C2×#1192402712yes—
376C8—24no—
377C8—2924no—
378C8—2922no—
379C22—42no—
380C22—41no—
381C22—41no—
382C22—42no—
383C22—41no—
384C22—2924no—
385C22—2922no—
386C22—5844no—
387C22#1798852412yes—
388C22#1798852411yescharacteristic
389C22#1798852411yescharacteristic
390C22C2×#1455657111yescharacteristic
391C22C2×#1455657512yes—
392C4—48no—
393C4—2922no—
394C4#1798852412yes—
395C4#1798852411yescharacteristic
396C4#1798852411yescharacteristic
397C4C2×#1455657112yes—
398C4C2×#1455657511yescharacteristic
399C4C2×#1455658811yescharacteristic
400C2—22no—
401C2—21no—
402C2—22no—
403C2—21no—
404C2—21no—
405C2—5842no—
406C2C2×#1798852411yescharacteristic
subgroupG/H
centerC4C2×#14556588
commutator, cosocleC22×C4×C73C24
FrattiniC22×C4C23×D73
FittingC2×H(3,ℤ/4)×C73C2
socleC2×C73C2×#399
prime radicalC22×#11924028C22
hypercenterC2×H(3,ℤ/4)D73
hypocenterC73#23740
LCSG ▷ C22×C4×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C4 ◁ C23×C4 ◁ C2×H(3,ℤ/4)

Complements

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

NH ≅ G/Ntype
1. C2×#17988636C2semidirect
2. #22618495C2semidirect
5. #22611485C2semidirect
6. C2×H(3,ℤ/4)×C73C2semidirect
11. #17988636C22semidirect
14. #17988614C22semidirect
24. H(3,ℤ/4)×C73C22semidirect
209. C2×H(3,ℤ/4)D73semidirect
210. C73#23740semidirect
228. H(3,ℤ/4)C2×D73semidirect

Maximal subgroups

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

subgroupindex
C2×#179886362
#226184952
C2×#179886052
#226183752
#226114852
C2×H(3,ℤ/4)×C732
#2374073