Finite Group Databasebeta

← #30748504

Subgroups — #30748504

Order 17152.

678 conjugacy classes of subgroups.

The subgroup lattice has 478 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22560189C211yesmaximal, characteristic, complemented
2#22560189C211yesmaximal, characteristic, complemented
3C2×#16863120C212yesmaximal
4#22558052C212yesmaximal, complemented
5C2×#388×C67C211yesmaximal, characteristic
6C2×C4⋊(C67⋊C8)—21no—
7C2×C4⋊(C67⋊C8)—21no—
8C2×C22⋊(C67⋊C8)C2212yes—
9C2×C22⋊(C67⋊C8)C2212yes—
10#16863151—21no—
11#16863151—21no—
12#16863120—24no—
13C22×C4⋊C4×C67C411yescharacteristic
14C22×C22⋊C4×C67C411yescharacteristic
15C22×C22⋊C4×C67C411yescharacteristic
16C22×C22⋊C4×C67C2211yescharacteristic
17C22×C42×C67C2211yescharacteristic
18#388×C67—21no—
19#388×C67—21no—
20#388×C67—21no—
21#388×C67—21no—
22C2×#95×C67C411yescharacteristic
23C2×#95×C67C2211yescharacteristic
24C22×C67⋊C8—22no—
25C22×C67⋊C8—22no—
26C4⋊(C67⋊C8)—42no—
27C4⋊(C67⋊C8)—42no—
28C22⋊(C67⋊C8)—22no—
29C22⋊(C67⋊C8)—22no—
30C22⋊(C67⋊C8)—22no—
31C22⋊(C67⋊C8)—22no—
32C25×C67C2×C411yescharacteristic
33C23×C4×C67C2×C411yescharacteristic
34C23×C4×C67C2×C411yescharacteristic
35C23×C4×C67C2×C411yescharacteristic
36C23×C4×C67C2×C411yescharacteristic
37C23×C4×C67C2×C411yescharacteristic
38C23×C4×C67C2311yescharacteristic
39C2×C4⋊C4×C67—21no—
40C2×C4⋊C4×C67—21no—
41C2×C4⋊C4×C67—42no—
42C2×C22⋊C4×C67—21no—
43C2×C22⋊C4×C67—21no—
44C2×C22⋊C4×C67—21no—
45C2×C22⋊C4×C67—21no—
46C2×C22⋊C4×C67—21no—
47C2×C22⋊C4×C67—21no—
48C2×C22⋊C4×C67—21no—
49C2×C22⋊C4×C67—21no—
50C2×C22⋊C4×C67—42no—
51C2×C22⋊C4×C67—42no—
52C2×C22⋊C4×C67D412yes—
53C2×C22⋊C4×C67Q812yes—
54C2×C42×C67—42no—
55C2×C42×C67D411yescharacteristic
56C2×C42×C67D411yescharacteristic
57C2×C42×C67D411yescharacteristic
58C2×C42×C67D411yescharacteristic
59#95×C67—21no—
60#95×C67—21no—
61#95×C67—21no—
62#95×C67—21no—
63C2×C67⋊C8—42no—
64C2×C67⋊C8—42no—
65C2×C67⋊C8—42no—
66C2×C67⋊C8—42no—
67C2×C67⋊C8—42no—
68C2×C67⋊C8—42no—
69C24×C67—21no—
70C24×C67—21no—
71C24×C67—22no—
72C24×C67—22no—
73C24×C67—41no—
74C24×C67—41no—
75C24×C67—41no—
76C24×C67—41no—
77C24×C67C4⋊C412yes—
78C24×C67C22×C411yescharacteristic
79C22×C4×C67—21no—
80C22×C4×C67—21no—
81C22×C4×C67—22no—
82C22×C4×C67—22no—
83C22×C4×C67—22no—
84C22×C4×C67—22no—
85C22×C4×C67—21no—
86C22×C4×C67—21no—
87C22×C4×C67—22no—
88C22×C4×C67—22no—
89C22×C4×C67—41no—
90C22×C4×C67—41no—
91C22×C4×C67—41no—
92C22×C4×C67—41no—
93C22×C4×C67—41no—
94C22×C4×C67—41no—
95C22×C4×C67—41no—
96C22×C4×C67—41no—
97C22×C4×C67—41no—
98C22×C4×C67—41no—
99C22×C4×C67C22⋊C411yescharacteristic
100C22×C4×C67C22⋊C411yescharacteristic
101C22×C4×C67C22⋊C411yescharacteristic
102C22×C4×C67C22⋊C411yescharacteristic
103C22×C4×C67C4⋊C412yes—
104C22×C4×C67C2×D411yescharacteristic
105C22×C4×C67C2×D411yescharacteristic
106C22×C4×C67C2×D411yescharacteristic
107C22×C4×C67C2×Q811yescharacteristic
108C22×C4×C67#4111yescharacteristic
109C22×C4×C67#4111yescharacteristic
110C4⋊C4×C67—42no—
111C4⋊C4×C67—42no—
112C22⋊C4×C67—22no—
113C22⋊C4×C67—22no—
114C22⋊C4×C67—22no—
115C22⋊C4×C67—22no—
116C22⋊C4×C67—42no—
117C22⋊C4×C67—42no—
118C22⋊C4×C67—42no—
119C22⋊C4×C67—42no—
120C42×C67—22no—
121C42×C67—22no—
122C42×C67—41no—
123C42×C67—41no—
124C23×C67—21no—
125C23×C67—21no—
126C23×C67—21no—
127C23×C67—21no—
128C23×C67—22no—
129C23×C67—24no—
130C23×C67—42no—
131C23×C67—42no—
132C23×C67—42no—
133C23×C67—42no—
134C23×C67—41no—
135C23×C67—41no—
136C23×C67—41no—
137C23×C67—41no—
138C23×C67—41no—
139C23×C67—41no—
140C23×C67—42no—
141C23×C67—82no—
142C23×C67—82no—
143C23×C67—82no—
144C23×C67—82no—
145C23×C67C2×C22⋊C411yescharacteristic
146C23×C67C2×C4⋊C411yescharacteristic
147C23×C67#11711yescharacteristic
148C23×C67#13112yes—
149C23×C67#13112yes—
150C2×C4×C67—21no—
151C2×C4×C67—21no—
152C2×C4×C67—21no—
153C2×C4×C67—21no—
154C2×C4×C67—21no—
155C2×C4×C67—21no—
156C2×C4×C67—21no—
157C2×C4×C67—21no—
158C2×C4×C67—22no—
159C2×C4×C67—21no—
160C2×C4×C67—21no—
161C2×C4×C67—21no—
162C2×C4×C67—21no—
163C2×C4×C67—21no—
164C2×C4×C67—21no—
165C2×C4×C67—41no—
166C2×C4×C67—41no—
167C2×C4×C67—41no—
168C2×C4×C67—41no—
169C2×C4×C67—44no—
170C2×C4×C67—44no—
171C2×C4×C67—42no—
172C2×C4×C67—41no—
173C2×C4×C67—41no—
174C2×C4×C67—42no—
175C2×C4×C67—41no—
176C2×C4×C67—41no—
177C2×C4×C67—41no—
178C2×C4×C67—41no—
179C2×C4×C67—41no—
180C2×C4×C67—41no—
181C2×C4×C67—82no—
182C2×C4×C67—82no—
183C2×C4×C67#9912yes—
184C2×C4×C67#10012yes—
185C2×C4×C67#12111yescharacteristic
186C2×C4×C67#12111yescharacteristic
187C2×C4×C67C22⋊Q811yescharacteristic
188C2×C4×C67C22⋊Q811yescharacteristic
189C67⋊C8—82no—
190C67⋊C8—82no—
191C22×C67—21no—
192C22×C67—21no—
193C22×C67—21no—
194C22×C67—21no—
195C22×C67—22no—
196C22×C67—22no—
197C22×C67—22no—
198C22×C67—22no—
199C22×C67—42no—
200C22×C67—42no—
201C22×C67—42no—
202C22×C67—84no—
203C22×C67—84no—
204C22×C67—81no—
205C22×C67—81no—
206C22×C67—81no—
207C22×C67—81no—
208C22×C67#38011yescharacteristic
209C22×C67#40811yescharacteristic
210C22×C67#40811yescharacteristic
211C22×C67C2×#9911yescharacteristic
212C22×C67#41011yescharacteristic
213C22×C67C2×#10011yescharacteristic
214C22×C67#41311yescharacteristic
215C22×C67#42412yes—
216C22×C67#43312yes—
217C4×C67—21no—
218C4×C67—21no—
219C4×C67—21no—
220C4×C67—21no—
221C4×C67—22no—
222C4×C67—22no—
223C4×C67—42no—
224C4×C67—41no—
225C4×C67—41no—
226C4×C67—81no—
227C4×C67—81no—
228#8561—671nomaximal, Sylow
229C2×C67—22no—
230C2×C67—22no—
231C2×C67—82no—
232C2×C67#146811yescharacteristic
233C2×C67#146911yescharacteristic
234C2×C67#147111yescharacteristic
235C2×C67#147311yescharacteristic
236C2×C67#179811yescharacteristic
237C2×C67#179911yescharacteristic
238C2×C67#180011yescharacteristic
239C2×#388D6711yescharacteristic
240#1797—671no—
241#1797—671no—
242C2×#323—672no—
243#1280—672no—
244C67#856111yescharacteristic, complemented, Sylow
245C22×C4⋊C4Dic6711yescharacteristic
246C22×C22⋊C4Dic6711yescharacteristic
247C22×C22⋊C4Dic6711yescharacteristic
248C22×C22⋊C4C2×D6711yescharacteristic
249C22×C42C2×D6711yescharacteristic
250C2×C4⋊C8—1341no—
251C2×C4⋊C8—1341no—
252C2×C22⋊C8—672no—
253C2×C22⋊C8—672no—
254#388—21no—
255#388—21no—
256#388—21no—
257#388—21no—
258C2×#95Dic6711yescharacteristic
259C2×#95C2×D6711yescharacteristic
260#336—1341no—
261#336—1341no—
262#323—1344no—
263C25C2×Dic6711yescharacteristic
264C23×C4C2×Dic6711yescharacteristic
265C23×C4C2×Dic6711yescharacteristic
266C23×C4C2×Dic6711yescharacteristic
267C23×C4C2×Dic6711yescharacteristic
268C23×C4C2×Dic6711yescharacteristic
269C23×C4C22×D6711yescharacteristic
270C22×C8—1342no—
271C22×C8—1342no—
272C2×C4⋊C4—21no—
273C2×C4⋊C4—21no—
274C2×C4⋊C4—42no—
275C2×C22⋊C4—21no—
276C2×C22⋊C4—21no—
277C2×C22⋊C4—21no—
278C2×C22⋊C4—21no—
279C2×C22⋊C4—21no—
280C2×C22⋊C4—21no—
281C2×C22⋊C4—21no—
282C2×C22⋊C4—21no—
283C2×C22⋊C4—42no—
284C2×C22⋊C4—42no—
285C2×C22⋊C4Dic13412yes—
286C2×C22⋊C4D26812yes—
287C2×C42—42no—
288C2×C42C22⋊D6711yescharacteristic
289C2×C42C22⋊D6711yescharacteristic
290C2×C42C22⋊D6711yescharacteristic
291C2×C42C22⋊D6711yescharacteristic
292C4⋊C8—2682no—
293C4⋊C8—2682no—
294C22⋊C8—1342no—
295C22⋊C8—1342no—
296C22⋊C8—1342no—
297C22⋊C8—1342no—
298#95—21no—
299#95—21no—
300#95—21no—
301#95—21no—
302C24—21no—
303C24—21no—
304C24—22no—
305C24—22no—
306C24—41no—
307C24—41no—
308C24—41no—
309C24—41no—
310C24C4⋊Dic6712yes—
311C24C22×Dic6711yescharacteristic
312C22×C4—21no—
313C22×C4—21no—
314C22×C4—22no—
315C22×C4—22no—
316C22×C4—22no—
317C22×C4—22no—
318C22×C4—21no—
319C22×C4—21no—
320C22×C4—22no—
321C22×C4—22no—
322C22×C4—41no—
323C22×C4—41no—
324C22×C4—41no—
325C22×C4—41no—
326C22×C4—41no—
327C22×C4—41no—
328C22×C4—41no—
329C22×C4—41no—
330C22×C4—41no—
331C22×C4—41no—
332C22×C4C22⋊Dic6711yescharacteristic
333C22×C4C22⋊Dic6711yescharacteristic
334C22×C4C22⋊Dic6711yescharacteristic
335C22×C4C22⋊Dic6711yescharacteristic
336C22×C4C4⋊Dic6712yes—
337C22×C4C2×C22⋊D6711yescharacteristic
338C22×C4C2×C22⋊D6711yescharacteristic
339C22×C4C2×D26811yescharacteristic
340C22×C4C2×Dic13411yescharacteristic
341C22×C4#1176162811yescharacteristic
342C22×C4#1176162811yescharacteristic
343C2×C8—2682no—
344C2×C8—2682no—
345C2×C8—2682no—
346C2×C8—2682no—
347C2×C8—2682no—
348C2×C8—2682no—
349C4⋊C4—42no—
350C4⋊C4—42no—
351C22⋊C4—22no—
352C22⋊C4—22no—
353C22⋊C4—22no—
354C22⋊C4—22no—
355C22⋊C4—42no—
356C22⋊C4—42no—
357C22⋊C4—42no—
358C22⋊C4—42no—
359C42—22no—
360C42—22no—
361C42—41no—
362C42—41no—
363C23—21no—
364C23—21no—
365C23—21no—
366C23—21no—
367C23—22no—
368C23—24no—
369C23—42no—
370C23—42no—
371C23—42no—
372C23—42no—
373C23—41no—
374C23—41no—
375C23—41no—
376C23—41no—
377C23—41no—
378C23—41no—
379C23—42no—
380C23—82no—
381C23—82no—
382C23—82no—
383C23—82no—
384C23C2×C22⋊Dic6711yescharacteristic
385C23C2×C4⋊Dic6711yescharacteristic
386C23#1452400211yescharacteristic
387C23#1452405012yes—
388C23#1452405012yes—
389C2×C4—21no—
390C2×C4—21no—
391C2×C4—21no—
392C2×C4—21no—
393C2×C4—21no—
394C2×C4—21no—
395C2×C4—21no—
396C2×C4—21no—
397C2×C4—22no—
398C2×C4—21no—
399C2×C4—21no—
400C2×C4—21no—
401C2×C4—21no—
402C2×C4—21no—
403C2×C4—21no—
404C2×C4—41no—
405C2×C4—41no—
406C2×C4—41no—
407C2×C4—41no—
408C2×C4—44no—
409C2×C4—44no—
410C2×C4—42no—
411C2×C4—41no—
412C2×C4—41no—
413C2×C4—42no—
414C2×C4—41no—
415C2×C4—41no—
416C2×C4—41no—
417C2×C4—41no—
418C2×C4—41no—
419C2×C4—41no—
420C2×C4—82no—
421C2×C4—82no—
422C2×C4#1452395812yes—
423C2×C4#1452396012yes—
424C2×C4#1452401711yescharacteristic
425C2×C4#1452401711yescharacteristic
426C2×C4#1452402211yescharacteristic
427C2×C4#1452402211yescharacteristic
428C8—5362no—
429C8—5362no—
430C22—22no—
431C22—22no—
432C22—22no—
433C22—22no—
434C22—21no—
435C22—21no—
436C22—21no—
437C22—21no—
438C22—42no—
439C22—42no—
440C22—42no—
441C22—84no—
442C22—84no—
443C22—81no—
444C22—81no—
445C22—81no—
446C22—81no—
447C22#1686325611yescharacteristic
448C22#1686336311yescharacteristic
449C22#1686336311yescharacteristic
450C22C2×#1452395811yescharacteristic
451C22#1686337011yescharacteristic
452C22C2×#1452396011yescharacteristic
453C22#1686338011yescharacteristic
454C22#1686342312yes—
455C22#1686345712yes—
456C4—22no—
457C4—22no—
458C4—21no—
459C4—21no—
460C4—21no—
461C4—21no—
462C4—41no—
463C4—41no—
464C4—42no—
465C4—81no—
466C4—81no—
467C2—22no—
468C2—22no—
469C2—82no—
470C2#2255855711yescharacteristic
471C2#2255856111yescharacteristic
472C2#2255856911yescharacteristic
473C2#2255857811yescharacteristic
474C2#2256019311yescharacteristic
475C2#2256019711yescharacteristic
476C2#2256020111yescharacteristic
subgroupG/H
centerC23C2×C22⋊Dic67
commutatorC24×C67C22×C4
FrattiniC23×C4C22×D67
FittingC2×#388×C67C2
socleC23×C67C2×C22⋊C4
prime radicalC25×C67C2×C4
cosocleC23×C4×C67C23
hypercenterC2×#388D67
hypocenterC67#8561
LCSG ▷ C24×C67 ▷ C2×C67 ▷ C67
UCSC1 ◁ C23 ◁ C22×C42 ◁ C2×#388

Complements

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

NH ≅ G/Ntype
1. #22560189C2semidirect
2. #22560189C2semidirect
4. #22558052C2semidirect
244. C67#8561semidirect

Maximal subgroups

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

subgroupindex
#225601892
C2×#168631202
#225580522
C2×#388×C672
#856167