Finite Group Databasebeta

← #32980608

Subgroups — #32980608

Order 18688.

566 conjugacy classes of subgroups.

The subgroup lattice has 456 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C4×#14556616C211yesmaximal, characteristic
2#22610724C211yesmaximal, characteristic, complemented
3#22608321C211yesmaximal, characteristic, complemented
4#22608313C211yesmaximal, characteristic, complemented
5#22607781C211yesmaximal, characteristic, complemented
6#22607767C211yesmaximal, characteristic, complemented
7C73×#2988C211yesmaximal, characteristic, complemented
8C2×#14556616C411yescharacteristic
9C2×C4×D292C2211yescharacteristic
10#17988697—21no—
11#17988685—21no—
12C4×#11924015—21no—
13C4×#11924013—21no—
14#17988640C2211yescharacteristic
15#17988630C411yescharacteristic
16#17988591C411yescharacteristic
17#17988588C411yescharacteristic
18C2×#14556503—22no—
19C4×#11924012C2211yescharacteristic
20C2×C4×#10596831C2211yescharacteristic
21Q8⋊#10596831—21no—
22D4⋊#10596831—21no—
23#17988314—21no—
24#17988314—21no—
25C4×#41×C73C2211yescharacteristic
26C8×Q8×C73—21no—
27C8×D4×C73—22no—
28C8×D4×C73—21no—
29#431×C73—21no—
30#431×C73C2211yescharacteristic
31C2×C4⋊C8×C73—21no—
32C2×C4⋊C8×C73C2211yescharacteristic
33C22×D292C2×C411yescharacteristic
34C22×C4×D73—21no—
35#14556616—22no—
36#14556616C814yescomplemented
37C2×#11924015—21no—
38C2×#11924013—21no—
39C2×#11924012C2×C411yescharacteristic
40C2×#11924012C2×C411yescharacteristic
41C22×#10596831—21no—
42C4×D292—21no—
43C4×D292—21no—
44C4×D292—21no—
45C4×D292—21no—
46C2×C4⋊Dic73C2×C411yescharacteristic
47C2×#11924005—21no—
48#14556524—21no—
49#14556522—21no—
50C4⋊#10596831—21no—
51#14556515—21no—
52#14556512—21no—
53#14556511—21no—
54#14556511—21no—
55C22⋊#10596831—21no—
56#14556503—44no—
57#14556502—21no—
58C4×#10596831—21no—
59C4×#10596831—21no—
60C4×#10596831—21no—
61C4×#10596831—21no—
62C2×#41×C73C2×C411yescharacteristic
63C22×C8×C73—22no—
64C22×C8×C73D412yes—
65C4×Q8×C73D411yescharacteristic
66C4×Q8×C73D411yescharacteristic
67C4×D4×C73—21no—
68C4×D4×C73—21no—
69C4×D4×C73D411yescharacteristic
70C4×D4×C73D411yescharacteristic
71#117×C73—21no—
72#117×C73C2×C411yescharacteristic
73C2×C42×C73—21no—
74C2×C42×C73C2311yescharacteristic
75C4⋊C8×C73—22no—
76C4⋊C8×C73—21no—
77C4⋊C8×C73—21no—
78C4⋊C8×C73D411yescharacteristic
79C4⋊C8×C73D411yescharacteristic
80C22⋊C8×C73—22no—
81C22⋊C8×C73—21no—
82C4×C8×C73—22no—
83C4×C8×C73—21no—
84C23×D73—21no—
85C2×D292—21no—
86C2×D292—21no—
87C2×D292—21no—
88C2×D292—21no—
89C2×D292—21no—
90C2×D292C2×C812yescomplemented
91C22×Dic73—21no—
92C2×C4×D73—42no—
93C2×C4×D73—42no—
94C2×C4×D73—41no—
95C2×C4×D73—41no—
96#11924015—24no—
97#11924013—24no—
98#11924012—21no—
99#11924012—21no—
100#11924012—21no—
101#11924012C2×C812yescomplemented
102C2×#10596831—21no—
103C2×#10596831—22no—
104C2×#10596831—21no—
105C2×#10596831—41no—
106C2×#10596831—41no—
107C2×#10596831—41no—
108C4⋊Dic73—21no—
109C4⋊Dic73—21no—
110#11924005—41no—
111#11924005—41no—
112#41×C73—21no—
113#41×C73—21no—
114#41×C73C2×C812yes—
115#41×C73M1612yes—
116C2×Q8×C73C22⋊C411yescharacteristic
117C2×D4×C73—21no—
118C2×D4×C73C22⋊C411yescharacteristic
119C22×C4×C73—21no—
120C22×C4×C73—21no—
121C22×C4×C73—21no—
122C22×C4×C73C22×C411yescharacteristic
123C22×C4×C73C2×D411yescharacteristic
124C22×C4×C73#4111yescharacteristic
125C2×C8×C73—21no—
126C2×C8×C73—22no—
127C2×C8×C73—21no—
128C2×C8×C73—42no—
129C2×C8×C73—82no—
130C4⋊C4×C73—21no—
131C4⋊C4×C73—21no—
132C4⋊C4×C73C22⋊C411yescharacteristic
133C4⋊C4×C73C22⋊C411yescharacteristic
134C22⋊C4×C73—21no—
135C22⋊C4×C73—21no—
136C22⋊C4×C73—21no—
137C42×C73—21no—
138C42×C73—21no—
139C42×C73—21no—
140C42×C73C2×D411yescharacteristic
141C42×C73C2×D411yescharacteristic
142C42×C73C2×D411yescharacteristic
143C42×C73#4111yescharacteristic
144C23×C73—21no—
145C23×C73C4×D411yescharacteristic
146C22×D73—22no—
147C22×D73—42no—
148C22×D73—42no—
149C22×D73—41no—
150C22×D73—41no—
151Q8×C73—21no—
152Q8×C73C22⋊C812yescomplemented
153D4×C73—22no—
154D4×C73—22no—
155D4×C73—21no—
156D4×C73C22⋊C812yescomplemented
157C2×C4×C73—21no—
158C2×C4×C73—22no—
159C2×C4×C73—22no—
160C2×C4×C73—21no—
161C2×C4×C73—21no—
162C2×C4×C73—21no—
163C2×C4×C73—21no—
164C2×C4×C73—41no—
165C2×C4×C73—41no—
166C2×C4×C73—41no—
167C2×C4×C73—81no—
168C2×C4×C73C2×C22⋊C411yescharacteristic
169C2×C4×C73C4×D411yescharacteristic
170C2×C4×C73#12011yescharacteristic
171C2×C4×C73#12111yescharacteristic
172C2×C4×C73#12111yescharacteristic
173C2×C4×C73#12311yescharacteristic
174C2×C4×C73C22×C811yescharacteristic
175C2×C4×C73C2×M1611yescharacteristic
176C2×C4×C73#13111yescharacteristic
177C2×C4×C73#13111yescharacteristic
178C2×Dic73—42no—
179C2×Dic73—41no—
180D292—22no—
181D292—22no—
182D292—41no—
183D292—41no—
184C4×D73—82no—
185C4×D73—81no—
186C4×D73—81no—
187C8×C73—42no—
188C8×C73—41no—
189C8×C73—41no—
190#10596831—22no—
191#10596831—42no—
192#10596831—41no—
193#10596831—81no—
194C22×C73—21no—
195C22×C73—21no—
196C22×C73—22no—
197C22×C73—41no—
198C22×C73—41no—
199C22×C73#38611yescharacteristic
200C22×C73C8×D411yescharacteristic
201C22×C73#43511yescharacteristic
202C2×D73—44no—
203C2×D73—42no—
204C2×D73—82no—
205C2×D73—81no—
206C2×D73—81no—
207C4×C73—22no—
208C4×C73—21no—
209C4×C73—21no—
210C4×C73—21no—
211C4×C73—41no—
212C4×C73—41no—
213C4×C73C2×C22⋊C811yescharacteristic
214C4×C73#40811yescharacteristic
215C4×C73C8×D411yescharacteristic
216C4×C73#43511yescharacteristic
217C4×C73#45311yescharacteristic
218C4×C73#45411yescharacteristic
219Dic73—81no—
220#8954—731nomaximal, Sylow
221C2×C73—21no—
222C2×C73—21no—
223C2×C73—42no—
224C2×C73#186911yescharacteristic
225C2×C73#188111yescharacteristic
226C2×C73#188311yescharacteristic
227D73—82no—
228#2988D7311yescharacteristic, complemented
229C4×AGL(1,ℤ/8)—731no—
230#1954—731no—
231#1478—731no—
232#1476—731no—
233#1283—731no—
234#1278—731no—
235C73#895411yescharacteristic, complemented, Sylow
236C2×AGL(1,ℤ/8)—731no—
237C4×#41C2×D7311yescharacteristic
238C2×C4×D4—731no—
239C8×Q8—21no—
240#442—1461no—
241#440—1461no—
242C4×SD16—1461no—
243C4×D8—1461no—
244C8×D4—22no—
245C8×D4—21no—
246#431—21no—
247#431—731no—
248#431C2×D7311yescharacteristic
249#428—731no—
250C2×C4⋊C8—21no—
251C2×C4⋊C8C2×D7311yescharacteristic
252#418—731no—
253#417—731no—
254C2×C22⋊C8—1462no—
255C4×M16—731no—
256C2×C4×C8—731no—
257Q8⋊C8—1461no—
258D4⋊C8—1461no—
259D4⋊C8—1461no—
260D4⋊C8—1461no—
261C2×#41C4×D7311yescharacteristic
262C22×D4—731no—
263C23×C4—1461no—
264AGL(1,ℤ/8)—734no—
265AGL(1,ℤ/8)—1462no—
266C2×SD16—1461no—
267C2×D8—1461no—
268C2×M16—731no—
269C2×M16—731no—
270C22×C8—22no—
271C22×C8—1461no—
272C22×C8C22⋊D7312yes—
273C4×Q8D29211yescharacteristic
274C4×Q8C22⋊D7311yescharacteristic
275C4×D4—21no—
276C4×D4—21no—
277C4×D4—1461no—
278C4×D4—1461no—
279C4×D4—1461no—
280C4×D4—1461no—
281C4×D4D29211yescharacteristic
282C4×D4C22⋊D7311yescharacteristic
283#117—21no—
284#117C4×D7311yescharacteristic
285C2×C4⋊C4—731no—
286C2×C22⋊C4—1461no—
287C2×C42—21no—
288C2×C42C22×D7311yescharacteristic
289#107—1461no—
290#106—1461no—
291C4⋊C8—22no—
292C4⋊C8—21no—
293C4⋊C8—21no—
294C4⋊C8—1461no—
295C4⋊C8C22⋊D7311yescharacteristic
296C4⋊C8C22⋊D7311yescharacteristic
297#103—1461no—
298#102—1461no—
299#102—1461no—
300#102—1461no—
301C22⋊C8—22no—
302C22⋊C8—21no—
303C22⋊C8—2924no—
304C22⋊C8—1461no—
305#97—1461no—
306C4×C8—22no—
307C4×C8—21no—
308C4×C8—1461no—
309C4×C8—1461no—
310C4×C8—1461no—
311C4×C8—1461no—
312C24—1461no—
313#41—21no—
314#41—21no—
315#41C8×D7312yes—
316#41#1192401112yes—
317C2×Q8#1192400511yescharacteristic
318C2×D4—21no—
319C2×D4—732no—
320C2×D4—1461no—
321C2×D4—1461no—
322C2×D4—1461no—
323C2×D4—1461no—
324C2×D4—1461no—
325C2×D4#1192400511yescharacteristic
326C22×C4—21no—
327C22×C4—21no—
328C22×C4—21no—
329C22×C4—2922no—
330C22×C4—2922no—
331C22×C4—2921no—
332C22×C4—2921no—
333C22×C4—1461no—
334C22×C4C2×C4×D7311yescharacteristic
335C22×C4C2×C22⋊D7311yescharacteristic
336C22×C4#1192402811yescharacteristic
337SD16—1464no—
338D8—1464no—
339M16—732no—
340M16—1461no—
341M16—1461no—
342M16—1461no—
343C2×C8—21no—
344C2×C8—22no—
345C2×C8—21no—
346C2×C8—42no—
347C2×C8—82no—
348C2×C8—2921no—
349C2×C8—2921no—
350C2×C8—2921no—
351C2×C8—1462no—
352C2×C8—1461no—
353C2×C8—1461no—
354C4⋊C4—21no—
355C4⋊C4—21no—
356C4⋊C4—1461no—
357C4⋊C4—1461no—
358C4⋊C4#1192400511yescharacteristic
359C4⋊C4#1192400511yescharacteristic
360C22⋊C4—21no—
361C22⋊C4—21no—
362C22⋊C4—21no—
363C22⋊C4—2921no—
364C22⋊C4—2921no—
365C42—21no—
366C42—21no—
367C42—21no—
368C42C2×C22⋊D7311yescharacteristic
369C42C2×C22⋊D7311yescharacteristic
370C42C2×D29211yescharacteristic
371C42#1192402811yescharacteristic
372C23—21no—
373C23—2922no—
374C23—2922no—
375C23—2921no—
376C23—2921no—
377C23—1462no—
378C23C4×C22⋊D7311yescharacteristic
379Q8—21no—
380Q8#1455650312yescomplemented
381D4—22no—
382D4—22no—
383D4—21no—
384D4—2921no—
385D4—2921no—
386D4—1462no—
387D4—1462no—
388D4#1455650312yescomplemented
389C2×C4—21no—
390C2×C4—22no—
391C2×C4—22no—
392C2×C4—21no—
393C2×C4—21no—
394C2×C4—21no—
395C2×C4—21no—
396C2×C4—41no—
397C2×C4—41no—
398C2×C4—41no—
399C2×C4—81no—
400C2×C4—2922no—
401C2×C4—2921no—
402C2×C4—5842no—
403C2×C4—5841no—
404C2×C4—5841no—
405C2×C4C2×#1192400511yescharacteristic
406C2×C4C4×C22⋊D7311yescharacteristic
407C2×C4#1455656011yescharacteristic
408C2×C4#1455656511yescharacteristic
409C2×C4#1455656511yescharacteristic
410C2×C4#1455657411yescharacteristic
411C2×C4C2×C8×D7311yescharacteristic
412C2×C4C2×#1192401111yescharacteristic
413C2×C4#1455659711yescharacteristic
414C2×C4#1455659711yescharacteristic
415C8—42no—
416C8—41no—
417C8—41no—
418C8—2922no—
419C8—2921no—
420C8—5841no—
421C8—1462no—
422C22—21no—
423C22—21no—
424C22—22no—
425C22—41no—
426C22—41no—
427C22—2924no—
428C22—2922no—
429C22—5842no—
430C22—5841no—
431C22—5841no—
432C22#1798846611yescharacteristic
433C22C8×C22⋊D7311yescharacteristic
434C22#1798865811yescharacteristic
435C4—22no—
436C4—21no—
437C4—21no—
438C4—21no—
439C4—41no—
440C4—41no—
441C4—5841no—
442C4C2×#1455650311yescharacteristic
443C4#1798855311yescharacteristic
444C4C8×C22⋊D7311yescharacteristic
445C4#1798865811yescharacteristic
446C4#1798875511yescharacteristic
447C4#1798876011yescharacteristic
448C2—21no—
449C2—21no—
450C2—42no—
451C2—5842no—
452C2#2261022311yescharacteristic
453C2#2261029311yescharacteristic
454C2#2261030711yescharacteristic
subgroupG/H
centerC2×C4#14556560
commutatorC2×C4×C73C22×C8
FrattiniC2×C42C22×D73
FittingC73×#2988C2
socleC22×C73#386
prime radicalC2×#14556616C4
cosocleC2×C42×C73C23
hypercenter#2988D73
hypocenterC73#8954
LCSG ▷ C2×C4×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #2988

Complements

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

NH ≅ G/Ntype
2. #22610724C2semidirect
3. #22608321C2semidirect
4. #22608313C2semidirect
5. #22607781C2semidirect
6. #22607767C2semidirect
7. C73×#2988C2semidirect
36. #14556616C8semidirect
90. C2×D292C2×C8semidirect
101. #11924012C2×C8semidirect
152. Q8×C73C22⋊C8semidirect
156. D4×C73C22⋊C8semidirect
228. #2988D73semidirect
235. C73#8954semidirect
380. Q8#14556503semidirect
388. D4#14556503semidirect

Maximal subgroups

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

subgroupindex
C4×#145566162
#226107242
#226083212
#226083132
#226077812
#226077672
C73×#29882
#895473