Finite Group Databasebeta

← #33248557

Subgroups — #33248557

Order 18688.

2124 conjugacy classes of subgroups.

The subgroup lattice has 386 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22613031C216yesmaximal
2#22612991C213yesmaximal
3Q8×C4⋊Dic73C213yesmaximal
4Q8×C4⋊Dic73C212yesmaximal
5Q8×C4⋊Dic73C211yesmaximal, characteristic
6#22612673C216yesmaximal
7C4⋊Q8×Dic73C213yesmaximal
8#22612409C213yesmaximal
9#22612296C212yesmaximal
10C4×Q8×Dic73C211yesmaximal, characteristic
11C2×#557×C73C211yesmaximal, characteristic
12C2×Q8×Dic73C2211yescharacteristic
13C2×Q8×Dic73C2212yes—
14C2×Q8×Dic73C2216yes—
15C2×Q8×Dic73C2211yescharacteristic
16#17988494C2213yes—
17#17988494C2213yes—
18#17988486C2216yes—
19#17988486C2216yes—
20#17988486C2216yes—
21#17988486C22112yes—
22#17988486C2213yes—
23#17988486C2216yes—
24#17988486C2211yescharacteristic
25#17988473C2216yes—
26#17988473C22112yes—
27#17988473C2216yes—
28#17988457C2213yes—
29#17988457C2216yes—
30C4⋊C4×Dic73C2216yes—
31C4⋊C4×Dic73C2216yes—
32C4⋊C4×Dic73C2213yes—
33C4⋊C4×Dic73C2216yes—
34C4⋊C4×Dic73C2213yes—
35C4×C4⋊Dic73C2216yes—
36C4×C4⋊Dic73C2216yes—
37C4×C4⋊Dic73C2213yes—
38C4×C4⋊Dic73C2213yes—
39#17988433C2216yes—
40C42×Dic73C2213yes—
41#557×C73C4116yescomplemented
42C2×C4⋊Q8×C73C2213yes—
43C2×#125×C73C2216yes—
44C2×C4×Q8×C73C2211yescharacteristic
45C2×C4×Q8×C73C2212yes—
46C2×C4×Q8×C73C2213yes—
47Q8×Dic73—24no—
48Q8×Dic73—28no—
49Q8×Dic73—412no—
50Q8×Dic73Q818yescomplemented
51C2×C4⋊Dic73C2312yes—
52C2×C4⋊Dic73C2311yescharacteristic
53C2×C4⋊Dic73C2316yes—
54C2×C4⋊Dic73C2313yes—
55C2×C4⋊Dic73C2313yes—
56C2×C4⋊Dic73C2316yes—
57C2×C4⋊Dic73C2316yes—
58C2×C4⋊Dic73C23112yes—
59C2×C4⋊Dic73C2313yes—
60C2×C4×Dic73C2311yescharacteristic
61C2×C4×Dic73C2313yes—
62C2×C4×Dic73C2313yes—
63C2×C4×Dic73C2313yes—
64C2×C4×Dic73C2316yes—
65C2×C4×Dic73C2313yes—
66C2×C4×Dic73C23112yes—
67C2×C4×Dic73C2313yes—
68#14556498C2312yes—
69#14556498C2316yes—
70#14556498C2316yes—
71#14556498C2316yes—
72#14556498C2316yes—
73#14556498C2316yes—
74#14556498C2316yes—
75#14556498C2316yes—
76C22×Q8×C73C2313yes—
77C22×Q8×C73C2311yescharacteristic
78C4⋊Q8×C73C2×C4124yes—
79#125×C73C2×C4148yes—
80C4×Q8×C73C2×C4116yes—
81C4×Q8×C73C2×C4124yes—
82C4×Q8×C73C2×C418yes—
83C2×C4⋊C4×C73C2313yes—
84C2×C4⋊C4×C73C2316yes—
85C2×C4⋊C4×C73C2316yes—
86C2×C4⋊C4×C73C2316yes—
87C2×C4⋊C4×C73C2311yescharacteristic
88C2×C42×C73C2313yes—
89C2×C42×C73C2316yes—
90C22×Dic73C2412yes—
91C22×Dic73C2413yes—
92C22×Dic73C2416yes—
93C22×Dic73C2411yescharacteristic
94C22×Dic73C2413yes—
95C22×Dic73C2411yescharacteristic
96C4⋊Dic73—212no—
97C4⋊Dic73—26no—
98C4⋊Dic73—26no—
99C4⋊Dic73—22no—
100C4⋊Dic73—24no—
101C4⋊Dic73—46no—
102C4⋊Dic73—46no—
103C4⋊Dic73—412no—
104C4⋊Dic73C2×Q8112yes—
105C4×Dic73—26no—
106C4×Dic73—212no—
107C4×Dic73—26no—
108C4×Dic73—22no—
109C4×Dic73—26no—
110C4×Dic73—43no—
111C4×Dic73—412no—
112C4×Dic73C2×Q8112yes—
113C2×Q8×C73—26no—
114C2×Q8×C73—26no—
115C2×Q8×C73C22×C4112yes—
116C2×Q8×C73C22×C414yes—
117C2×Q8×C73C2×Q814yes—
118C2×Q8×C73#4114yes—
119C22×C4×C73C2412yes—
120C22×C4×C73C2413yes—
121C22×C4×C73C2416yes—
122C22×C4×C73C2413yes—
123C22×C4×C73C2411yescharacteristic
124C4⋊C4×C73C22×C414yes—
125C4⋊C4×C73C22×C4124yes—
126C4⋊C4×C73C22×C4112yes—
127C4⋊C4×C73C22×C4124yes—
128C4⋊C4×C73C22×C4124yes—
129C42×C73C22×C4124yes—
130C42×C73C22×C4112yes—
131C23×C73C2511yescharacteristic
132Q8×C73—224no—
133Q8×C73C4×Q8116yescomplemented
134C2×C4×C73—26no—
135C2×C4×C73—26no—
136C2×C4×C73—23no—
137C2×C4×C73—22no—
138C2×C4×C73—23no—
139C2×C4×C73—22no—
140C2×C4×C73C23×C414yes—
141C2×C4×C73C23×C416yes—
142C2×C4×C73C23×C4112yes—
143C2×C4×C73C23×C416yes—
144C2×C4×C73C23×C412yes—
145C2×C4×C73C22×Q816yes—
146C2×C4×C73C2×#4116yes—
147C2×C4×C73H(5,𝔽2)12yes—
148C2×C4×C732−1+412yes—
149C2×Dic73—21no—
150C2×Dic73—23no—
151C2×Dic73—23no—
152C2×Dic73—21no—
153C2×Dic73—21no—
154C2×Dic73—21no—
155C2×Dic73—22no—
156C2×Dic73—23no—
157C2×Dic73—26no—
158C2×Dic73—23no—
159C2×Dic73—26no—
160C2×Dic73—22no—
161C2×Dic73—22no—
162C2×Dic73—23no—
163C2×Dic73—26no—
164C2×Dic73C22×Q812yes—
165C2×Dic73C22×Q816yes—
166C2×Dic732−1+412yes—
167C22×C73C24×C411yescharacteristic
168C22×C73C23×Q811yescharacteristic
169C22×C73C22×#4111yescharacteristic
170C22×C73C2×H(5,𝔽2)11yescharacteristic
171C22×C73C2×2−1+411yescharacteristic
172C22×C73C2×2−1+411yescharacteristic
173C22×C73#58511yescharacteristic
174C4×C73—212no—
175C4×C73—26no—
176C4×C73—24no—
177C4×C73C2×C4×Q8112yes—
178C4×C73#52014yes—
179Dic73—22no—
180Dic73—26no—
181Dic73—22no—
182Dic73—43no—
183Dic73—46no—
184Dic73—42no—
185#34337—731nomaximal, Sylow
186C2×C73C22×C4×Q811yescharacteristic
187C2×C73C2×#52011yescharacteristic
188C2×C73C4×2−1+411yescharacteristic
189C2×C73#350211yescharacteristic
190C2×C73#350311yescharacteristic
191C2×C73#351711yescharacteristic
192C2×C73#352311yescharacteristic
193C2×#557D7311yescharacteristic
194#2370—736no—
195#2365—733no—
196Q8×C4⋊C4—733no—
197Q8×C4⋊C4—732no—
198Q8×C4⋊C4—731no—
199#2330—736no—
200C4×C4⋊Q8—733no—
201#2295—733no—
202#2276—732no—
203C42×Q8—731no—
204C73#3433711yescharacteristic, complemented, Sylow
205#557Dic73116yescomplemented
206C2×C4⋊Q8C2×D7313yes—
207C2×#125C2×D7316yes—
208C2×C4×Q8—736no—
209C2×C4×Q8—731no—
210C2×C4×Q8—731no—
211C2×C4×Q8—732no—
212C2×C4×Q8C2×D7311yescharacteristic
213C2×C4×Q8C2×D7312yes—
214C2×C4×Q8C2×D7313yes—
215#391—733no—
216#391—733no—
217#389—736no—
218#389—736no—
219#389—736no—
220#389—7312no—
221#389—736no—
222#389—733no—
223#389—731no—
224#387—736no—
225#387—7312no—
226#387—736no—
227#384—733no—
228#384—736no—
229C4×C4⋊C4—736no—
230C4×C4⋊C4—736no—
231C4×C4⋊C4—733no—
232C4×C4⋊C4—736no—
233C4×C4⋊C4—733no—
234C4×C4⋊C4—733no—
235C4×C4⋊C4—733no—
236C4×C4⋊C4—736no—
237C4×C4⋊C4—736no—
238#376—736no—
239C43—733no—
240C22×Q8C22×D7313yes—
241C22×Q8C22×D7311yescharacteristic
242C4⋊Q8C2×Dic73124yes—
243#125C2×Dic73148yes—
244C4×Q8—29212no—
245C4×Q8—738no—
246C4×Q8—1464no—
247C4×Q8—1468no—
248C4×Q8C2×Dic73116yes—
249C4×Q8C2×Dic73124yes—
250C4×Q8C2×Dic7318yes—
251C2×C4⋊C4—7312no—
252C2×C4⋊C4—736no—
253C2×C4⋊C4—736no—
254C2×C4⋊C4—733no—
255C2×C4⋊C4—733no—
256C2×C4⋊C4—733no—
257C2×C4⋊C4—736no—
258C2×C4⋊C4—732no—
259C2×C4⋊C4—731no—
260C2×C4⋊C4C22×D7311yescharacteristic
261C2×C4⋊C4C22×D7313yes—
262C2×C4⋊C4C22×D7316yes—
263C2×C4⋊C4C22×D7316yes—
264C2×C4⋊C4C22×D7316yes—
265C2×C42—7312no—
266C2×C42—733no—
267C2×C42—733no—
268C2×C42—733no—
269C2×C42—733no—
270C2×C42—736no—
271C2×C42—731no—
272C2×C42—733no—
273C2×C42C22×D7316yes—
274C2×C42C22×D7313yes—
275#95—732no—
276#95—736no—
277#95—736no—
278#95—736no—
279#95—736no—
280#95—736no—
281#95—736no—
282#95—736no—
283C2×Q8—26no—
284C2×Q8—26no—
285C2×Q8C22×Dic73112yes—
286C2×Q8C22×Dic7314yes—
287C2×Q8Q8×D7314yes—
288C2×Q8#1192402914yes—
289C22×C4—731no—
290C22×C4—733no—
291C22×C4—731no—
292C22×C4—732no—
293C22×C4—736no—
294C22×C4—733no—
295C22×C4C23×D7313yes—
296C22×C4C23×D7316yes—
297C22×C4C23×D7312yes—
298C22×C4C23×D7313yes—
299C22×C4C23×D7311yescharacteristic
300C4⋊C4—2926no—
301C4⋊C4—2926no—
302C4⋊C4—29212no—
303C4⋊C4—7312no—
304C4⋊C4—1464no—
305C4⋊C4—1466no—
306C4⋊C4—14612no—
307C4⋊C4—1466no—
308C4⋊C4—1462no—
309C4⋊C4C22×Dic7314yes—
310C4⋊C4C22×Dic73124yes—
311C4⋊C4C22×Dic73112yes—
312C4⋊C4C22×Dic73124yes—
313C4⋊C4C22×Dic73124yes—
314C42—2923no—
315C42—29212no—
316C42—7312no—
317C42—1466no—
318C42—14612no—
319C42—1466no—
320C42—1466no—
321C42—1462no—
322C42C22×Dic73124yes—
323C42C22×Dic73112yes—
324C23C24×D7311yescharacteristic
325Q8—224no—
326Q8Q8×Dic73116yescomplemented
327C2×C4—26no—
328C2×C4—23no—
329C2×C4—22no—
330C2×C4—26no—
331C2×C4—23no—
332C2×C4—22no—
333C2×C4—732no—
334C2×C4—736no—
335C2×C4—732no—
336C2×C4—1462no—
337C2×C4—1466no—
338C2×C4—1463no—
339C2×C4—1466no—
340C2×C4—1463no—
341C2×C4—1462no—
342C2×C4—1462no—
343C2×C4—1463no—
344C2×C4—1466no—
345C2×C4—1461no—
346C2×C4—1463no—
347C2×C4—1463no—
348C2×C4—1461no—
349C2×C4—1461no—
350C2×C4—1461no—
351C2×C4C23×Dic7314yes—
352C2×C4C23×Dic7316yes—
353C2×C4C23×Dic73112yes—
354C2×C4C23×Dic7316yes—
355C2×C4C23×Dic7312yes—
356C2×C4C2×Q8×D7316yes—
357C2×C4C2×#1192402916yes—
358C2×C4#1455663712yes—
359C2×C4#1455663812yes—
360C22C24×Dic7311yescharacteristic
361C22C22×Q8×D7311yescharacteristic
362C22C22×#1192402911yescharacteristic
363C22C2×#1455663711yescharacteristic
364C222−1+4×D7311yescharacteristic
365C22C2×#1455663811yescharacteristic
366C22#1798942411yescharacteristic
367C4—212no—
368C4—26no—
369C4—24no—
370C4—2922no—
371C4—2926no—
372C4—2923no—
373C4—1466no—
374C4—1462no—
375C4—1462no—
376C4C2×Q8×Dic73112yes—
377C4#1798907014yes—
378C2C22×Q8×Dic7311yescharacteristic
379C2C2×#1798907011yescharacteristic
380C22−1+4×Dic7311yescharacteristic
381C2#2262373511yescharacteristic
382C2#2262374811yescharacteristic
383C2#2262393811yescharacteristic
384C2#2262402811yescharacteristic
subgroupG/H
center, FrattiniC23C24×D73
commutatorC22×C73C24×C4
FittingC2×#557×C73C2
socle, prime radical, cosocleC23×C73C25
hypercenterC2×#557D73
hypocenterC73#34337
LCSG ▷ C22×C73 ▷ C73
UCSC1 ◁ C23 ◁ C2×#557

Complements

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

NH ≅ G/Ntype
41. #557×C73C4semidirect
50. Q8×Dic73Q8semidirect
133. Q8×C73C4×Q8semidirect
204. C73#34337semidirect
205. #557Dic73semidirect
326. Q8Q8×Dic73semidirect

Maximal subgroups

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

subgroupindex
#226130312
#226129912
Q8×C4⋊Dic732
#226126732
C4⋊Q8×Dic732
#226124092
#226122962
C4×Q8×Dic732
C2×#557×C732
#3433773