Finite Group Databasebeta

← #32984942

Subgroups — #32984942

Order 18688.

600 conjugacy classes of subgroups.

The subgroup lattice has 496 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C4×#14556580C211yesmaximal, characteristic
2C4⋊C8×Dic73C211yesmaximal, characteristic, complemented
3C4×#14556520C211yesmaximal, characteristic, complemented
4C2×Q8⋊#10596831C211yesmaximal, characteristic, complemented
5C2×#17988316C211yesmaximal, characteristic
6C2×#17988314C211yesmaximal, characteristic
7C2×Q8⋊C8×C73C211yesmaximal, characteristic, complemented
8C2×#14556580C411yescharacteristic
9C2×C4×D292C2211yescharacteristic
10C2×C4⋊#10596831C2211yescharacteristic
11C2×#14556520—21no—
12C2×C4×#10596831C2211yescharacteristic
13C2×C8×Dic73—21no—
14#17988494C411yescharacteristic
15#17988488C411yescharacteristic
16#17988477—21no—
17#17988477—21no—
18#17988452C411yescharacteristic
19C4×#11924005—21no—
20C4×#11924005—21no—
21C42×Dic73C2211yescharacteristic
22#17988341—21no—
23#17988341—21no—
24Q8⋊#10596831—22no—
25#17988316—22no—
26#17988314—22no—
27C2×C4×Q8×C73C2211yescharacteristic
28C2×C4⋊C8×C73C2211yescharacteristic
29C2×C4×C8×C73C2211yescharacteristic
30Q8⋊C8×C73—22no—
31C22×D292C2×C411yescharacteristic
32C22×C4×D73—21no—
33C22×#10596831—21no—
34C22×#10596831—21no—
35#14556580—22no—
36#14556580C814yescomplemented
37C4×D292—21no—
38C4×D292—21no—
39C4×D292—21no—
40C4×D292—21no—
41C2×C4⋊Dic73C2×C411yescharacteristic
42C2×#11924005—21no—
43C2×#11924005—21no—
44C2×#11924005—21no—
45C2×#11924005—21no—
46C2×#11924005—21no—
47C2×C4×Dic73C2×C411yescharacteristic
48C2×C4×Dic73C2×C411yescharacteristic
49C2×C4×Dic73D411yescharacteristic
50C2×C4×Dic73D411yescharacteristic
51C2×C4×Dic73D411yescharacteristic
52C2×C4×Dic73D411yescharacteristic
53C4⋊#10596831—21no—
54C4⋊#10596831D412yes—
55#14556520—24no—
56C4×#10596831—21no—
57C4×#10596831—21no—
58C8×Dic73—42no—
59#14556498—21no—
60#14556498—21no—
61#14556498—21no—
62#14556498—21no—
63C22×Q8×C73C2×C411yescharacteristic
64C22×C8×C73—21no—
65C22×C8×C73—21no—
66C4×Q8×C73—21no—
67C4×Q8×C73—21no—
68C4×Q8×C73—21no—
69C4×Q8×C73—21no—
70C2×C4⋊C4×C73—21no—
71C2×C4⋊C4×C73C2×C411yescharacteristic
72C2×C42×C73—21no—
73C2×C42×C73C2311yescharacteristic
74C4⋊C8×C73—21no—
75C4⋊C8×C73D412yes—
76C4×C8×C73—21no—
77C4×C8×C73—21no—
78C23×D73—21no—
79C2×D292—21no—
80C2×D292—21no—
81C2×D292—21no—
82C2×D292—21no—
83C2×D292—21no—
84C2×D292C2×C812yes—
85C22×Dic73—21no—
86C22×Dic73C22⋊C411yescharacteristic
87C22×Dic73C22⋊C411yescharacteristic
88C22×Dic73C22⋊C411yescharacteristic
89C22×Dic73C22⋊C411yescharacteristic
90C2×C4×D73—41no—
91C2×C4×D73—41no—
92C2×C4×D73—41no—
93C2×C4×D73—41no—
94C2×C4×D73—41no—
95C2×C4×D73—41no—
96C2×#10596831—22no—
97C2×#10596831—22no—
98C2×#10596831—41no—
99C2×#10596831—41no—
100C2×#10596831—41no—
101C2×#10596831—41no—
102C4⋊Dic73—21no—
103C4⋊Dic73—21no—
104#11924005—24no—
105#11924005—24no—
106#11924005—42no—
107#11924005—41no—
108#11924005—41no—
109#11924005—42no—
110C4×Dic73—21no—
111C4×Dic73—21no—
112C4×Dic73—21no—
113C4×Dic73—21no—
114C4×Dic73—21no—
115C4×Dic73—21no—
116C4×Dic73—22no—
117C4×Dic73—41no—
118C4×Dic73C2×C812yes—
119C4×Dic73M1612yes—
120C2×Q8×C73—21no—
121C2×Q8×C73—21no—
122C2×Q8×C73—21no—
123C2×Q8×C73—21no—
124C2×Q8×C73—21no—
125C2×Q8×C73C2×C812yes—
126C22×C4×C73—21no—
127C22×C4×C73—21no—
128C22×C4×C73C22×C411yescharacteristic
129C22×C4×C73C2×D411yescharacteristic
130C22×C4×C73C2×D411yescharacteristic
131C2×C8×C73—22no—
132C2×C8×C73—22no—
133C2×C8×C73—41no—
134C2×C8×C73—41no—
135C2×C8×C73—41no—
136C2×C8×C73—41no—
137C4⋊C4×C73—21no—
138C4⋊C4×C73—21no—
139C4⋊C4×C73—41no—
140C4⋊C4×C73—41no—
141C42×C73—41no—
142C42×C73—41no—
143C42×C73C2×D411yescharacteristic
144C42×C73C2×D411yescharacteristic
145C42×C73#4111yescharacteristic
146C42×C73#4111yescharacteristic
147C23×C73C2×C22⋊C411yescharacteristic
148C22×D73—22no—
149C22×D73—41no—
150C22×D73—41no—
151C22×D73—41no—
152C22×D73—41no—
153C22×D73—41no—
154C22×D73—41no—
155Q8×C73—22no—
156Q8×C73—22no—
157Q8×C73—41no—
158Q8×C73—41no—
159C2×C4×C73—21no—
160C2×C4×C73—21no—
161C2×C4×C73—22no—
162C2×C4×C73—41no—
163C2×C4×C73—41no—
164C2×C4×C73—41no—
165C2×C4×C73—41no—
166C2×C4×C73—41no—
167C2×C4×C73C4×D411yescharacteristic
168C2×C4×C73C4×D411yescharacteristic
169C2×C4×C73#12111yescharacteristic
170C2×C4×C73#12111yescharacteristic
171C2×C4×C73#12411yescharacteristic
172C2×C4×C73#12711yescharacteristic
173C2×C4×C73C22×C811yescharacteristic
174C2×C4×C73C2×M1611yescharacteristic
175C2×C4×C73#13111yescharacteristic
176C2×C4×C73#13111yescharacteristic
177C2×C4×C73#13511yescharacteristic
178C2×C4×C73#13511yescharacteristic
179C2×C4×C73AGL(1,ℤ/8)11yescharacteristic
180C2×C4×C73#13711yescharacteristic
181C2×Dic73—21no—
182C2×Dic73—21no—
183C2×Dic73—21no—
184C2×Dic73—21no—
185C2×Dic73—21no—
186C2×Dic73—21no—
187C2×Dic73—21no—
188C2×Dic73—21no—
189C2×Dic73—21no—
190C2×Dic73—41no—
191C2×Dic73—41no—
192C2×Dic73—41no—
193C2×Dic73C22⋊C812yes—
194C2×Dic73C22⋊C812yes—
195C2×Dic73#10412yes—
196D292—22no—
197D292—22no—
198D292—41no—
199D292—41no—
200C4×D73—81no—
201C4×D73—81no—
202C4×D73—81no—
203C4×D73—81no—
204C8×C73—42no—
205C8×C73—42no—
206#10596831—42no—
207#10596831—42no—
208C22×C73#39011yescharacteristic
209C22×C73C2×C22⋊C811yescharacteristic
210C22×C73#40811yescharacteristic
211C22×C73#41611yescharacteristic
212C22×C73#41711yescharacteristic
213C22×C73C2×#10411yescharacteristic
214C22×C73#42111yescharacteristic
215C2×D73—42no—
216C2×D73—42no—
217C2×D73—42no—
218C2×D73—81no—
219C2×D73—81no—
220C2×D73—81no—
221C2×D73—81no—
222C4×C73—21no—
223C4×C73—21no—
224C4×C73—42no—
225C4×C73—81no—
226C4×C73C8×D411yescharacteristic
227C4×C73C8×D411yescharacteristic
228C4×C73#43611yescharacteristic
229C4×C73#43611yescharacteristic
230C4×C73#46311yescharacteristic
231C4×C73#46311yescharacteristic
232C4×C73#46411yescharacteristic
233C4×C73#46411yescharacteristic
234Dic73—22no—
235Dic73—22no—
236Dic73—22no—
237Dic73—41no—
238Dic73—81no—
239#9675—731nomaximal, Sylow
240C2×C73#147611yescharacteristic
241C2×C73#147711yescharacteristic
242C2×C73#148211yescharacteristic
243C2×C73#148311yescharacteristic
244C2×C73#195411yescharacteristic
245C2×C73#198011yescharacteristic
246C2×C73#198311yescharacteristic
247D73—82no—
248C4×#124—731no—
249C4×C4⋊C8—731no—
250C4×C4⋊C8—731no—
251C2×Q8⋊C8—731no—
252C2×Q8⋊C8D7311yescharacteristic, complemented
253C2×D4⋊C8—731no—
254C2×D4⋊C8—731no—
255C73#967511yescharacteristic, complemented, Sylow
256C2×#124—731no—
257C2×C4×Q8C2×D7311yescharacteristic
258C2×C4×D4—731no—
259C2×C4⋊C8—731no—
260C2×C4⋊C8—1461no—
261C2×C4⋊C8C2×D7311yescharacteristic
262C2×C4×C8—731no—
263C2×C4×C8—1461no—
264C2×C4×C8C2×D7311yescharacteristic
265#391—731no—
266#390—731no—
267#388—1461no—
268#388—1461no—
269#382—731no—
270C4×C22⋊C4—1461no—
271C4×C22⋊C4—1461no—
272C43—731no—
273#336—1461no—
274#336—1461no—
275Q8⋊C8—22no—
276Q8⋊C8—1462no—
277D4⋊C8—1462no—
278D4⋊C8—1462no—
279C22×Q8C4×D7311yescharacteristic
280C22×D4—731no—
281C23×C4—1461no—
282C22×C8—21no—
283C22×C8—21no—
284C22×C8—1461no—
285C22×C8—1461no—
286#124—734no—
287#124—1462no—
288C4×Q8—21no—
289C4×Q8—21no—
290C4×Q8—21no—
291C4×Q8—21no—
292C4×D4—1461no—
293C4×D4—1461no—
294C4×D4—1461no—
295C4×D4—1461no—
296C2×C4⋊C4—21no—
297C2×C4⋊C4—731no—
298C2×C4⋊C4C4×D7311yescharacteristic
299C2×C22⋊C4—1461no—
300C2×C22⋊C4—1461no—
301C2×C22⋊C4—1461no—
302C2×C22⋊C4—1461no—
303C2×C22⋊C4—1461no—
304C2×C42—21no—
305C2×C42—731no—
306C2×C42—731no—
307C2×C42—731no—
308C2×C42—731no—
309C2×C42—731no—
310C2×C42—731no—
311C2×C42C22×D7311yescharacteristic
312C4⋊C8—21no—
313C4⋊C8—732no—
314C4⋊C8—1464no—
315C4⋊C8—1461no—
316C4⋊C8C22⋊D7312yes—
317C4×C8—21no—
318C4×C8—21no—
319C4×C8—2922no—
320C4×C8—1461no—
321C4×C8—1461no—
322#95—1461no—
323#95—1461no—
324#95—1461no—
325#95—1461no—
326C24—1461no—
327C2×Q8—21no—
328C2×Q8—21no—
329C2×Q8—21no—
330C2×Q8—21no—
331C2×Q8—21no—
332C2×Q8C8×D7312yes—
333C2×D4—732no—
334C2×D4—1461no—
335C2×D4—1461no—
336C2×D4—1461no—
337C2×D4—1461no—
338C2×D4—1461no—
339C22×C4—21no—
340C22×C4—21no—
341C22×C4—2921no—
342C22×C4—2921no—
343C22×C4—2921no—
344C22×C4—2921no—
345C22×C4—2921no—
346C22×C4—2921no—
347C22×C4—731no—
348C22×C4—731no—
349C22×C4—731no—
350C22×C4—731no—
351C22×C4—1461no—
352C22×C4C2×C4×D7311yescharacteristic
353C22×C4D4×D7311yescharacteristic
354C22×C4D4×D7311yescharacteristic
355C2×C8—22no—
356C2×C8—22no—
357C2×C8—41no—
358C2×C8—41no—
359C2×C8—41no—
360C2×C8—41no—
361C2×C8—2921no—
362C2×C8—2921no—
363C2×C8—2921no—
364C2×C8—2921no—
365C2×C8—1462no—
366C2×C8—1462no—
367C4⋊C4—21no—
368C4⋊C4—21no—
369C4⋊C4—41no—
370C4⋊C4—41no—
371C4⋊C4—1461no—
372C4⋊C4—1461no—
373C22⋊C4—2922no—
374C22⋊C4—2922no—
375C22⋊C4—2921no—
376C22⋊C4—2921no—
377C22⋊C4—1464no—
378C22⋊C4—1464no—
379C42—41no—
380C42—41no—
381C42—2921no—
382C42—732no—
383C42—732no—
384C42—1462no—
385C42—1461no—
386C42—1461no—
387C42—1461no—
388C42—1461no—
389C42—1461no—
390C42—1461no—
391C42D4×D7311yescharacteristic
392C42C2×C22⋊D7311yescharacteristic
393C42#1192402711yescharacteristic
394C42#1192402811yescharacteristic
395C23—2921no—
396C23—2921no—
397C23—2921no—
398C23—2921no—
399C23—2921no—
400C23—2921no—
401C23—1462no—
402C23C22⋊C4×D7311yescharacteristic
403Q8—22no—
404Q8—22no—
405Q8—41no—
406Q8—41no—
407D4—2921no—
408D4—2921no—
409D4—1462no—
410D4—1462no—
411C2×C4—21no—
412C2×C4—21no—
413C2×C4—22no—
414C2×C4—41no—
415C2×C4—41no—
416C2×C4—41no—
417C2×C4—41no—
418C2×C4—41no—
419C2×C4—2921no—
420C2×C4—2921no—
421C2×C4—2921no—
422C2×C4—5841no—
423C2×C4—5841no—
424C2×C4—5841no—
425C2×C4—5841no—
426C2×C4—732no—
427C2×C4—732no—
428C2×C4—732no—
429C2×C4—1461no—
430C2×C4—1461no—
431C2×C4—1461no—
432C2×C4—1461no—
433C2×C4—1461no—
434C2×C4—1461no—
435C2×C4—1461no—
436C2×C4—1461no—
437C2×C4—1461no—
438C2×C4#1455655111yescharacteristic
439C2×C4C4×C22⋊D7311yescharacteristic
440C2×C4#1455656311yescharacteristic
441C2×C4#1455656411yescharacteristic
442C2×C4#1455657711yescharacteristic
443C2×C4#1455658711yescharacteristic
444C2×C4C2×C8×D7311yescharacteristic
445C2×C4M16×D7311yescharacteristic
446C2×C4#1455659711yescharacteristic
447C2×C4#1455659911yescharacteristic
448C2×C4#1455661211yescharacteristic
449C2×C4#1455661411yescharacteristic
450C2×C4#1455661811yescharacteristic
451C2×C4#1455662311yescharacteristic
452C8—42no—
453C8—42no—
454C8—2922no—
455C8—2922no—
456C22—2922no—
457C22—2922no—
458C22—2922no—
459C22—5841no—
460C22—5841no—
461C22—5841no—
462C22—5841no—
463C22#1798848711yescharacteristic
464C22C22⋊C8×D7311yescharacteristic
465C22#1798855211yescharacteristic
466C22#1798858211yescharacteristic
467C22#1798858711yescharacteristic
468C22#104×D7311yescharacteristic
469C22#1798860211yescharacteristic
470C4—21no—
471C4—21no—
472C4—42no—
473C4—81no—
474C4—2921no—
475C4—5841no—
476C4—1462no—
477C4—1462no—
478C4—1462no—
479C4#1798865011yescharacteristic
480C4C8×C22⋊D7311yescharacteristic
481C4#1798866211yescharacteristic
482C4#1798866311yescharacteristic
483C4#1798880411yescharacteristic
484C4#1798880511yescharacteristic
485C4#1798881111yescharacteristic
486C4#1798881211yescharacteristic
487C2—5842no—
488C2#2260831211yescharacteristic
489C2#2260831711yescharacteristic
490C2#2260833811yescharacteristic
491C2#2260834311yescharacteristic
492C2#2261072311yescharacteristic
493C2#2261084811yescharacteristic
494C2#2261086011yescharacteristic
subgroupG/H
centerC22×C4D4×D73
commutatorC2×C4×C73C22×C8
FrattiniC2×C42C22×D73
FittingC2×Q8⋊C8×C73C2
socleC23×C73C2×C22⋊C4
prime radicalC22×D292C2×C4
cosocleC2×C42×C73C23
hypercenterC2×Q8⋊C8D73
hypocenterC73#9675
LCSG ▷ C2×C4×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C22×C4 ◁ C2×C42 ◁ C2×Q8⋊C8

Complements

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

NH ≅ G/Ntype
2. C4⋊C8×Dic73C2semidirect
3. C4×#14556520C2semidirect
4. C2×Q8⋊#10596831C2semidirect
7. C2×Q8⋊C8×C73C2semidirect
36. #14556580C8semidirect
252. C2×Q8⋊C8D73semidirect
255. C73#9675semidirect

Maximal subgroups

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

subgroupindex
C4×#145565802
C4⋊C8×Dic732
C4×#145565202
C2×Q8⋊#105968312
C2×#179883162
C2×#179883142
C2×Q8⋊C8×C732
#967573