Finite Group Databasebeta

← #34016341

Subgroups — #34016341

Order 18688.

884 conjugacy classes of subgroups.

The subgroup lattice has 428 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22626831C212yesmaximal
2#22625204C212yesmaximal
3#22625203C214yesmaximal
4#22625194C212yesmaximal
5#22625193C211yesmaximal, characteristic
6#22625174C212yesmaximal
7#22625173C211yesmaximal, characteristic
8C2×#17989312C211yesmaximal, characteristic
9#17989639—24no—
10#17989637—22no—
11#17989636—22no—
12#17989608C2212yes—
13#17989607C2214yes—
14#17989604C2212yes—
15#17989603C2214yes—
16#17989433C2212yes—
17#17989433C2212yes—
18#17989432C2211yescharacteristic
19#17989432C2211yescharacteristic
20#17989432C2212yes—
21#17989432C2214yes—
22#17989432C2214yes—
23#17989312—22no—
24#17989312—22no—
25#17989312—24no—
26C2×#14556589C412yes—
27C2×#14556582C411yescharacteristic
28C2×#14556582C2211yescharacteristic
29C2×#14556581C412yes—
30C2×#14556581C2212yes—
31C2×Q8×Dic73C2212yes—
32C2×C4×Dic146C411yescharacteristic
33C2×C4×Dic146C2211yescharacteristic
34C2×#14556556C411yescharacteristic
35C2×#14556556C2211yescharacteristic
36C2×C4⋊Q8×C73C411yescharacteristic
37#14556658—24no—
38#14556658—22no—
39C4⋊#10596833—22no—
40C4⋊#10596833—24no—
41C22⋊#10596833C2312yes—
42C22⋊#10596833C2314yes—
43C22⋊#10596833C2312yes—
44#14556646—24no—
45#14556646—22no—
46C4×#10596833—22no—
47C4×#10596833—24no—
48C22×Dic146C2×C411yescharacteristic
49C22×Dic146C2×C411yescharacteristic
50#14556589—24no—
51#14556589—24no—
52#14556582—22no—
53#14556582—22no—
54#14556582—22no—
55#14556582—22no—
56#14556581—24no—
57#14556581—24no—
58#14556581—24no—
59#14556581—24no—
60Q8×Dic73—24no—
61Q8×Dic73—24no—
62C4×Dic146—22no—
63C4×Dic146—21no—
64C4×Dic146—21no—
65C4×Dic146—22no—
66C4×Dic146—22no—
67#14556556—22no—
68#14556556—21no—
69#14556556—21no—
70#14556556D412yes—
71#14556556D412yes—
72#14556556D414yes—
73C2×C4⋊Dic73C2×C411yescharacteristic
74C2×C4⋊Dic73C2×C412yes—
75C2×C4⋊Dic73C2311yescharacteristic
76C2×C4⋊Dic73C2312yes—
77C2×#11924007C2×C412yes—
78C2×#11924007C2×C412yes—
79C2×#11924007C2×C412yes—
80C2×#11924007C2×C412yes—
81C2×#11924007C2×C411yescharacteristic
82C2×#11924007C2×C412yes—
83C2×#11924007C2311yescharacteristic
84C2×C4×Dic73C2×C412yes—
85C2×C4×Dic73C2×C411yescharacteristic
86C2×C4×Dic73C2×C412yes—
87C2×C4×Dic73C2311yescharacteristic
88C2×C4×Dic73C2312yes—
89C22×Q8×C73C2×C412yes—
90C4⋊Q8×C73—22no—
91C4⋊Q8×C73—22no—
92C2×C4⋊C4×C73C2×C411yescharacteristic
93C2×C4⋊C4×C73C2×C412yes—
94C2×C4⋊C4×C73C2×C411yescharacteristic
95C2×C42×C73C2×C411yescharacteristic
96C2×#10596833—22no—
97C2×#10596833—24no—
98C2×#10596833—22no—
99C2×Dic146—24no—
100C2×Dic146—22no—
101C2×Dic146—22no—
102C2×Dic146—41no—
103C2×Dic146C22⋊C411yescharacteristic
104C2×Dic146C22⋊C412yes—
105C2×Dic146C22⋊C411yescharacteristic
106C22×Dic73C22×C411yescharacteristic
107C22×Dic73C22×C412yes—
108C22×Dic73C22×C411yescharacteristic
109C22×Dic73C22×C411yescharacteristic
110C22×Dic73C22×C412yes—
111C22×Dic73C2411yescharacteristic
112C4⋊Dic73—21no—
113C4⋊Dic73—21no—
114C4⋊Dic73—24no—
115C4⋊Dic73—22no—
116C4⋊Dic73—24no—
117#11924007—24no—
118#11924007—22no—
119#11924007—22no—
120#11924007—22no—
121#11924007—22no—
122#11924007—22no—
123#11924007—24no—
124#11924007—22no—
125#11924007C22⋊C414yes—
126#11924007C22⋊C414yes—
127C4×Dic73—21no—
128C4×Dic73—21no—
129C4×Dic73—22no—
130C4×Dic73—22no—
131C4×Dic73—22no—
132C4×Dic73—22no—
133C4×Dic73C2×D412yes—
134C4×Dic73C2×D412yes—
135C4×Dic73C2×D414yes—
136C4×Dic73C2×D414yes—
137C2×Q8×C73—24no—
138C2×Q8×C73—24no—
139C2×Q8×C73—42no—
140C22×C4×C73C22×C412yes—
141C22×C4×C73C22×C411yescharacteristic
142C22×C4×C73C22×C412yes—
143C22×C4×C73C22×C411yescharacteristic
144C22×C4×C73C22×C411yescharacteristic
145C4⋊C4×C73—21no—
146C4⋊C4×C73—21no—
147C4⋊C4×C73—24no—
148C4⋊C4×C73C22⋊C412yes—
149C4⋊C4×C73C22⋊C412yes—
150C42×C73—21no—
151C42×C73—21no—
152C23×C73C23×C411yescharacteristic
153Q8×C73—48no—
154C2×C4×C73—21no—
155C2×C4×C73—22no—
156C2×C4×C73—22no—
157C2×C4×C73—21no—
158C2×C4×C73—22no—
159C2×C4×C73—21no—
160C2×C4×C73—22no—
161C2×C4×C73—22no—
162C2×C4×C73—22no—
163C2×C4×C73—41no—
164C2×C4×C73C2×C22⋊C411yescharacteristic
165C2×C4×C73C2×C22⋊C414yes—
166C2×C4×C73C2×C22⋊C411yescharacteristic
167C2×C4×C73#13112yes—
168C2×Dic73—21no—
169C2×Dic73—21no—
170C2×Dic73—24no—
171C2×Dic73—21no—
172C2×Dic73—21no—
173C2×Dic73—22no—
174C2×Dic73—21no—
175C2×Dic73—21no—
176C2×Dic73—41no—
177C2×Dic73—41no—
178C2×Dic73—42no—
179C2×Dic73C2×C22⋊C412yes—
180C2×Dic73C2×C22⋊C412yes—
181C2×Dic73C2×C22⋊C412yes—
182C2×Dic73C22×D412yes—
183Dic146—22no—
184Dic146—24no—
185Dic146—44no—
186Dic146#10114yes—
187#10596833—42no—
188#10596833—44no—
189#10596833—42no—
190C22×C73—21no—
191C22×C73—21no—
192C22×C73C22×C22⋊C411yescharacteristic
193C22×C73C2×#13111yescharacteristic
194C22×C73#56811yescharacteristic
195C4×C73—22no—
196C4×C73—24no—
197C4×C73—21no—
198C4×C73—42no—
199C4×C73—41no—
200C4×C73C2×#10112yes—
201Dic73—22no—
202Dic73—22no—
203Dic73—22no—
204Dic73—41no—
205Dic73—41no—
206Dic73—42no—
207Dic73C2×#10114yes—
208#17430—731nomaximal, Sylow
209C2×C73—21no—
210C2×C73—21no—
211C2×C73#288011yescharacteristic
212C2×C73#288411yescharacteristic
213C2×C73C22×#10111yescharacteristic
214C2×#557—731no—
215#2984—732no—
216#1470—734no—
217#1470—732no—
218#1467—731no—
219#1467—732no—
220#1461—731no—
221#1461—732no—
222C73#1743011yescharacteristic, complemented, Sylow
223#557—1462no—
224#557—1462no—
225#557—1464no—
226C2×C4⋊Q8—732no—
227C2×C4⋊Q8#6440811yescharacteristic
228C2×#125—732no—
229C2×#125—731no—
230C2×#125—732no—
231C2×#125—731no—
232C2×C4×Q8—732no—
233C2×C4×Q8—731no—
234C2×C4×Q8—731no—
235C2×C4×Q8—731no—
236C2×C4×Q8—731no—
237#446—1462no—
238#446—1464no—
239C8×Q8—1462no—
240#432—734no—
241#432—732no—
242#431—734no—
243#431—732no—
244#324—731no—
245#324—731no—
246#324—732no—
247#324—734no—
248#324—734no—
249#324—732no—
250#324—732no—
251C22×Q8—731no—
252C22×Q8—731no—
253C22×Q8C2×#6440812yes—
254C4⋊Q8—22no—
255C4⋊Q8—22no—
256C4⋊Q8—1464no—
257C4⋊Q8—1464no—
258#125—1464no—
259#125—1464no—
260#125—1462no—
261#125—1462no—
262#125—1464no—
263#125—1464no—
264#125—1462no—
265#125—1462no—
266C4×Q8—732no—
267C4×Q8—732no—
268C4×Q8—734no—
269C4×Q8—1462no—
270C4×Q8—1462no—
271C4×Q8—1461no—
272C4×Q8—1462no—
273C4×Q8—1461no—
274C4×Q8—1464no—
275C4×Q8—1464no—
276C4×Q8—1461no—
277C4×Q8—1462no—
278C4×Q8—1461no—
279C2×C4⋊C4—732no—
280C2×C4⋊C4—731no—
281C2×C4⋊C4—731no—
282C2×C4⋊C4—732no—
283C2×C4⋊C4—731no—
284C2×C4⋊C4—732no—
285C2×C4⋊C4—732no—
286C2×C4⋊C4—731no—
287C2×C4⋊C4—732no—
288C2×C4⋊C4—732no—
289C2×C4⋊C4—732no—
290C2×C4⋊C4C2×#6440811yescharacteristic
291C2×C4⋊C4C2×#6440812yes—
292C2×C4⋊C4C2×#6440811yescharacteristic
293C2×C42—732no—
294C2×C42—731no—
295C2×C42—732no—
296C2×C42—731no—
297C2×C42—732no—
298C2×C42C2×#6440811yescharacteristic
299C4⋊C8—1464no—
300C4⋊C8—1462no—
301C4⋊C8—1462no—
302C4⋊C8—1464no—
303C22⋊C8—732no—
304C22⋊C8—734no—
305C22⋊C8—732no—
306#97—1464no—
307#97—1462no—
308C4×C8—1464no—
309C4×C8—1462no—
310C2×Q8—24no—
311C2×Q8—24no—
312C2×Q8—42no—
313C2×Q8—2921no—
314C2×Q8—731no—
315C2×Q8—732no—
316C2×Q8—731no—
317C2×Q8—1462no—
318C2×Q8—1464no—
319C2×Q8—1462no—
320C22×C4—731no—
321C22×C4—731no—
322C22×C4—732no—
323C22×C4—732no—
324C22×C4—731no—
325C22×C4—731no—
326C22×C4C22×#6440812yes—
327C22×C4C22×#6440811yescharacteristic
328C22×C4C22×#6440811yescharacteristic
329C22×C4C22×#6440812yes—
330C22×C4C22×#6440811yescharacteristic
331C2×C8—1462no—
332C2×C8—1464no—
333C2×C8—1462no—
334C4⋊C4—21no—
335C4⋊C4—21no—
336C4⋊C4—24no—
337C4⋊C4—734no—
338C4⋊C4—734no—
339C4⋊C4—1462no—
340C4⋊C4—1462no—
341C4⋊C4—1462no—
342C4⋊C4—1462no—
343C4⋊C4—1462no—
344C4⋊C4—1462no—
345C4⋊C4—1464no—
346C4⋊C4—1462no—
347C4⋊C4—1464no—
348C4⋊C4—1464no—
349C4⋊C4—1464no—
350C4⋊C4—1461no—
351C4⋊C4—1461no—
352C4⋊C4C22⋊#6440812yes—
353C4⋊C4C22⋊#6440812yes—
354C42—21no—
355C42—21no—
356C42—734no—
357C42—732no—
358C42—734no—
359C42—732no—
360C42—1462no—
361C42—1462no—
362C42—1462no—
363C42—1462no—
364C42—1461no—
365C42—1461no—
366C23C23×#6440811yescharacteristic
367Q8—48no—
368Q8—2924no—
369Q8—734no—
370Q8—1464no—
371Q8—1462no—
372C2×C4—21no—
373C2×C4—22no—
374C2×C4—21no—
375C2×C4—22no—
376C2×C4—22no—
377C2×C4—21no—
378C2×C4—22no—
379C2×C4—22no—
380C2×C4—22no—
381C2×C4—41no—
382C2×C4—2921no—
383C2×C4—2922no—
384C2×C4—2921no—
385C2×C4—732no—
386C2×C4—732no—
387C2×C4—732no—
388C2×C4—732no—
389C2×C4—1461no—
390C2×C4—1462no—
391C2×C4—1461no—
392C2×C4—1461no—
393C2×C4—1461no—
394C2×C4—1464no—
395C2×C4—1461no—
396C2×C4—1461no—
397C2×C4C2×C22⋊#6440811yescharacteristic
398C2×C4C2×C22⋊#6440814yes—
399C2×C4C2×C22⋊#6440811yescharacteristic
400C2×C4#1455667812yes—
401C8—2922no—
402C8—2924no—
403C8—2922no—
404C22—21no—
405C22—21no—
406C22C22×C22⋊#6440811yescharacteristic
407C22C2×#1455667811yescharacteristic
408C22#1798966511yescharacteristic
409C4—22no—
410C4—24no—
411C4—21no—
412C4—42no—
413C4—41no—
414C4—2921no—
415C4—2921no—
416C4—2922no—
417C4—734no—
418C4—1462no—
419C4—1462no—
420C4—1462no—
421C4D73⋊#10112yes—
422C2—21no—
423C2—21no—
424C2#2262651911yescharacteristic
425C2#2262653211yescharacteristic
426C2C2×D73⋊#10111yescharacteristic
subgroupG/H
centerC22C22×C22⋊#64408
commutator, prime radicalC23×C73C23×C4
FrattiniC23C23×#64408
FittingC2×C4⋊Q8×C73C4
socleC22×C73C22×C22⋊C4
cosocleC22×Dic73C24
hypercenterC2×C4⋊Q8#64408
hypocenterC73#17430
LCSG ▷ C23×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C22 ◁ C2×C4⋊C4 ◁ C2×C4⋊Q8

Complements

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

NH ≅ G/Ntype
222. C73#17430semidirect

Maximal subgroups

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

subgroupindex
#226268312
#226252042
#226252032
#226251942
#226251932
#226251742
#226251732
C2×#179893122
#1743073