Finite Group Databasebeta

← #32982190

Subgroups — #32982190

Order 18688.

578 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
1C2×#17988789C211yesmaximal, characteristic
2C2×#17988664C211yesmaximal, characteristic
3#22610893C211yesmaximal, characteristic, complemented
4#22609932C211yesmaximal, characteristic
5#22607815C211yesmaximal, characteristic, complemented
6#22607765C211yesmaximal, characteristic, complemented
7C2×#332×C73C211yesmaximal, characteristic, complemented
8C22×#11924015—21no—
9C22×#11924011—21no—
10C2×C4×D292C2211yescharacteristic
11#17988789—22no—
12#17988789C414yescomplemented
13#17988664—22no—
14#17988664—22no—
15C2×C4⋊#10596831C2211yescharacteristic
16C2×#14556515—21no—
17C2×#14556513—21no—
18C2×#14556503—21no—
19C2×C4×#10596831C2211yescharacteristic
20#17988457C2211yescharacteristic
21#17988350—21no—
22#17988349—21no—
23#17988349—21no—
24C2×C4⋊Q8×C73C2211yescharacteristic
25C2×C4⋊C8×C73C2211yescharacteristic
26C2×C4⋊C8×C73C2211yescharacteristic
27#332×C73—22no—
28C22×D292D411yescharacteristic
29C22×C4×D73—21no—
30C2×#11924015—24no—
31C2×#11924015—42no—
32C2×#11924015—42no—
33C2×#11924011—24no—
34C2×#11924011—42no—
35C2×#11924011—42no—
36C22×#10596831—21no—
37C22×#10596831D411yescharacteristic
38C22×#10596831D411yescharacteristic
39C4×D292—21no—
40C4×D292—21no—
41C4×D292C2×C412yes—
42C4×D292D412yes—
43C2×C4⋊Dic73—21no—
44C2×C4⋊Dic73—21no—
45C2×C4⋊Dic73—21no—
46C2×C4⋊Dic73D411yescharacteristic
47C2×#11924005—21no—
48C4⋊#10596831—21no—
49C4⋊#10596831C2×C412yes—
50#14556515—24no—
51#14556513—42no—
52#14556503—24no—
53C4×#10596831—21no—
54C4×#10596831—21no—
55C22×Q8×C73—21no—
56C22×C8×C73—21no—
57C22×C8×C73—21no—
58C4⋊Q8×C73—21no—
59C4⋊Q8×C73—21no—
60C4⋊Q8×C73—21no—
61C4⋊Q8×C73C2×C412yes—
62C2×C4⋊C4×C73—21no—
63C2×C4⋊C4×C73—21no—
64C2×C42×C73C2311yescharacteristic
65C4⋊C8×C73—21no—
66C4⋊C8×C73—21no—
67C4⋊C8×C73—21no—
68C4⋊C8×C73D412yes—
69C23×D73—21no—
70C2×D292—21no—
71C2×D292—21no—
72C2×D292—21no—
73C2×D292—21no—
74C2×D292—21no—
75C2×D292C22⋊C412yes—
76C22×Dic73—21no—
77C22×Dic73—41no—
78C2×C4×D73—22no—
79C2×C4×D73—22no—
80C2×C4×D73—41no—
81C2×C4×D73—41no—
82C2×C4×D73—41no—
83C2×C4×D73—41no—
84#11924015—48no—
85#11924011—48no—
86C2×#10596831—21no—
87C2×#10596831—21no—
88C2×#10596831—22no—
89C2×#10596831—21no—
90C2×#10596831—21no—
91C2×#10596831—21no—
92C2×#10596831—41no—
93C2×#10596831—41no—
94C2×#10596831SD1612yescomplemented
95C4⋊Dic73—21no—
96C4⋊Dic73—41no—
97C4⋊Dic73—41no—
98C4⋊Dic73—41no—
99C4⋊Dic73—41no—
100C4⋊Dic73—81no—
101C4⋊Dic73C22⋊C412yes—
102#11924005—22no—
103#11924005—22no—
104C2×Q8×C73—22no—
105C2×Q8×C73—41no—
106C2×Q8×C73—41no—
107C2×Q8×C73—41no—
108C2×Q8×C73—41no—
109C2×Q8×C73—41no—
110C22×C4×C73—41no—
111C22×C4×C73C2×D411yescharacteristic
112C22×C4×C73C2×D411yescharacteristic
113C22×C4×C73#4111yescharacteristic
114C2×C8×C73—22no—
115C2×C8×C73—41no—
116C2×C8×C73—41no—
117C2×C8×C73—41no—
118C2×C8×C73—41no—
119C2×C8×C73—41no—
120C4⋊C4×C73—22no—
121C4⋊C4×C73—22no—
122C4⋊C4×C73—41no—
123C4⋊C4×C73—41no—
124C42×C73C22×C411yescharacteristic
125C42×C73C2×D411yescharacteristic
126C42×C73C2×D411yescharacteristic
127C42×C73#4111yescharacteristic
128C23×C73#12111yescharacteristic
129C22×D73—22no—
130C22×D73—41no—
131C22×D73—41no—
132C22×D73—41no—
133C22×D73—41no—
134C22×D73—41no—
135C22×D73—41no—
136Q8×C73—42no—
137Q8×C73—42no—
138Q8×C73—81no—
139Q8×C73—81no—
140C2×C4×C73—21no—
141C2×C4×C73—21no—
142C2×C4×C73—42no—
143C2×C4×C73—81no—
144C2×C4×C73—81no—
145C2×C4×C73C2×C22⋊C411yescharacteristic
146C2×C4×C73C4×D411yescharacteristic
147C2×C4×C73C4×D411yescharacteristic
148C2×C4×C73#12011yescharacteristic
149C2×C4×C73#12111yescharacteristic
150C2×C4×C73#12311yescharacteristic
151C2×C4×C73C2×SD1611yescharacteristic
152C2×C4×C73#13511yescharacteristic
153C2×C4×C73AGL(1,ℤ/8)11yescharacteristic
154C2×C4×C73AGL(1,ℤ/8)11yescharacteristic
155C2×C4×C73AGL(1,ℤ/8)11yescharacteristic
156C2×C4×C73AGL(1,ℤ/8)11yescharacteristic
157C2×C4×C73#13711yescharacteristic
158C2×C4×C73#13711yescharacteristic
159C2×Dic73—22no—
160C2×Dic73—41no—
161C2×Dic73—41no—
162C2×Dic73—81no—
163C2×Dic73—81no—
164C2×Dic73—81no—
165D292—22no—
166D292—22no—
167D292—41no—
168D292—41no—
169C4×D73—42no—
170C4×D73—42no—
171C4×D73—42no—
172C4×D73—42no—
173C8×C73—42no—
174C8×C73—81no—
175#10596831—22no—
176#10596831—42no—
177#10596831—41no—
178C22×C73#38611yescharacteristic
179C22×C73#46111yescharacteristic
180C22×C73#46311yescharacteristic
181C22×C73#46811yescharacteristic
182C22×C73#46911yescharacteristic
183C22×C73#47111yescharacteristic
184C22×C73#47311yescharacteristic
185C2×D73—42no—
186C2×D73—42no—
187C2×D73—42no—
188C2×D73—81no—
189C2×D73—81no—
190C2×D73—81no—
191C2×D73—81no—
192C4×C73—21no—
193C4×C73—21no—
194C4×C73—82no—
195C4×C73#41711yescharacteristic
196C4×C73#41811yescharacteristic
197C4×C73C4×SD1611yescharacteristic
198C4×C73#44011yescharacteristic
199C4×C73#45011yescharacteristic
200C4×C73#45211yescharacteristic
201C4×C73#46811yescharacteristic
202C4×C73#46911yescharacteristic
203Dic73—42no—
204Dic73—161no—
205#9225—731nomaximal, Sylow
206C2×C73#169111yescharacteristic
207C2×C73#169311yescharacteristic
208C2×C73#169911yescharacteristic
209C2×C73#170111yescharacteristic
210C2×C73#187711yescharacteristic
211C2×C73#189511yescharacteristic
212C2×C73#190811yescharacteristic
213D73—82no—
214C2×#461—731no—
215C2×#436—731no—
216#1990—731no—
217#1802—731no—
218C2×#332D7311yescharacteristic, complemented
219#1296—731no—
220#1277—731no—
221C73#922511yescharacteristic, complemented, Sylow
222C22×SD16—1461no—
223C22×M16—1461no—
224C2×C4⋊Q8C2×D7311yescharacteristic
225C2×C4×D4—731no—
226#461—734no—
227#461—1462no—
228#436—1462no—
229#436—1462no—
230C2×C4⋊C8—731no—
231C2×C4⋊C8C2×D7311yescharacteristic
232C2×C4⋊C8C2×D7311yescharacteristic
233C2×#103—1461no—
234C2×#102—1461no—
235C2×C22⋊C8—1461no—
236C2×C4×C8—731no—
237#384—731no—
238#340—1461no—
239#340—1461no—
240#340—1461no—
241#332—22no—
242C22×Q8—21no—
243C22×D4—731no—
244C23×C4—1461no—
245C2×SD16—2922no—
246C2×SD16—2922no—
247C2×SD16—1464no—
248C2×M16—2922no—
249C2×M16—2922no—
250C2×M16—1464no—
251C22×C8—21no—
252C22×C8—21no—
253C22×C8—731no—
254C22×C8—731no—
255C22×C8—1461no—
256C4⋊Q8—21no—
257C4⋊Q8—21no—
258C4⋊Q8—21no—
259C4⋊Q8C4×D7312yes—
260C4×D4—732no—
261C4×D4—732no—
262C4×D4—1461no—
263C4×D4—1461no—
264C2×C4⋊C4—21no—
265C2×C4⋊C4—21no—
266C2×C4⋊C4—731no—
267C2×C4⋊C4—1461no—
268C2×C4⋊C4—1461no—
269C2×C4⋊C4—1461no—
270C2×C22⋊C4—1461no—
271C2×C42C22×D7311yescharacteristic
272C4⋊C8—21no—
273C4⋊C8—21no—
274C4⋊C8—21no—
275C4⋊C8—732no—
276C4⋊C8—1461no—
277C4⋊C8C22⋊D7312yes—
278#103—1464no—
279#102—2922no—
280C22⋊C8—1464no—
281C4×C8—1461no—
282C4×C8—1461no—
283C24—1461no—
284C2×Q8—22no—
285C2×Q8—41no—
286C2×Q8—41no—
287C2×Q8—41no—
288C2×Q8—41no—
289C2×Q8—41no—
290C2×D4—732no—
291C2×D4—1461no—
292C2×D4—1461no—
293C2×D4—1461no—
294C2×D4—1461no—
295C2×D4—1461no—
296C22×C4—41no—
297C22×C4—2921no—
298C22×C4—2921no—
299C22×C4—2921no—
300C22×C4—2921no—
301C22×C4—2921no—
302C22×C4—1461no—
303C22×C4—1462no—
304C22×C4—1462no—
305C22×C4D4×D7311yescharacteristic
306C22×C4D4×D7311yescharacteristic
307C22×C4#1192402811yescharacteristic
308SD16—2928no—
309M16—2928no—
310C2×C8—22no—
311C2×C8—41no—
312C2×C8—41no—
313C2×C8—41no—
314C2×C8—41no—
315C2×C8—41no—
316C2×C8—2921no—
317C2×C8—2921no—
318C2×C8—732no—
319C2×C8—1462no—
320C2×C8—1461no—
321C2×C8—1461no—
322C2×C8—1461no—
323C2×C8—1461no—
324C2×C8—1461no—
325C4⋊C4—22no—
326C4⋊C4—22no—
327C4⋊C4—41no—
328C4⋊C4—41no—
329C4⋊C4—2921no—
330C4⋊C4—2921no—
331C4⋊C4—2921no—
332C4⋊C4—2921no—
333C4⋊C4—5841no—
334C4⋊C4—732no—
335C4⋊C4—1461no—
336C22⋊C4—1462no—
337C22⋊C4—1462no—
338C42C2×C4×D7311yescharacteristic
339C42D4×D7311yescharacteristic
340C42C2×C22⋊D7311yescharacteristic
341C42#1192402711yescharacteristic
342C23—2921no—
343C23—2921no—
344C23—2921no—
345C23—2921no—
346C23—2921no—
347C23—2921no—
348C23—1462no—
349C23#1455656411yescharacteristic
350Q8—42no—
351Q8—42no—
352Q8—81no—
353Q8—81no—
354D4—2921no—
355D4—2921no—
356D4—1462no—
357D4—1462no—
358C2×C4—21no—
359C2×C4—21no—
360C2×C4—42no—
361C2×C4—81no—
362C2×C4—81no—
363C2×C4—2921no—
364C2×C4—2921no—
365C2×C4—2922no—
366C2×C4—2922no—
367C2×C4—2922no—
368C2×C4—2922no—
369C2×C4—5841no—
370C2×C4—5841no—
371C2×C4—5841no—
372C2×C4—1462no—
373C2×C4C22⋊C4×D7311yescharacteristic
374C2×C4#1455655111yescharacteristic
375C2×C4C4×C22⋊D7311yescharacteristic
376C2×C4#1455655911yescharacteristic
377C2×C4#1455656211yescharacteristic
378C2×C4#1455657311yescharacteristic
379C2×C4SD16×D7311yescharacteristic
380C2×C4#1455661411yescharacteristic
381C2×C4#1455661511yescharacteristic
382C2×C4#1455661511yescharacteristic
383C2×C4#1455661811yescharacteristic
384C2×C4#1455661811yescharacteristic
385C2×C4#1455662011yescharacteristic
386C2×C4#1455662311yescharacteristic
387C8—42no—
388C8—81no—
389C8—2921no—
390C8—2922no—
391C8—1462no—
392C22—2922no—
393C22—2922no—
394C22—2922no—
395C22—5841no—
396C22—5841no—
397C22—5841no—
398C22—5841no—
399C22#1798846311yescharacteristic
400C22#1798879111yescharacteristic
401C22#1798880511yescharacteristic
402C22#1798883211yescharacteristic
403C22#1798883911yescharacteristic
404C22#1798885311yescharacteristic
405C22#1798886111yescharacteristic
406C4—21no—
407C4—21no—
408C4—82no—
409C4—2922no—
410C4—11681no—
411C4#1798858511yescharacteristic
412C4#1798859011yescharacteristic
413C4#1798867211yescharacteristic
414C4#1798869011yescharacteristic
415C4#1798873611yescharacteristic
416C4#1798874811yescharacteristic
417C4#1798883011yescharacteristic
418C4#1798883711yescharacteristic
419C2—5842no—
420C2#2260942911yescharacteristic
421C2#2260944311yescharacteristic
422C2#2260948111yescharacteristic
423C2#2260949511yescharacteristic
424C2#2261026211yescharacteristic
425C2#2261036611yescharacteristic
426C2#2261044311yescharacteristic
subgroupG/H
centerC23#14556564
commutatorC42×C73C22×C4
FrattiniC2×C42C22×D73
FittingC2×#332×C73C2
socleC23×C73#121
prime radicalC22×D292D4
cosocleC2×C42×C73C23
hypercenterC2×#332D73
hypocenterC73#9225
LCSG ▷ C42×C73 ▷ C22×C73 ▷ C73
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#332

Complements

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

NH ≅ G/Ntype
3. #22610893C2semidirect
5. #22607815C2semidirect
6. #22607765C2semidirect
7. C2×#332×C73C2semidirect
12. #17988789C4semidirect
94. C2×#10596831SD16semidirect
218. C2×#332D73semidirect
221. C73#9225semidirect

Maximal subgroups

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

subgroupindex
C2×#179887892
C2×#179886642
#226108932
#226099322
#226078152
#226077652
C2×#332×C732
#922573