Finite Group Databasebeta

← #10823739

Subgroups — #10823739

Order 768.

994 conjugacy classes of subgroups.

The subgroup lattice has 502 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#83768C211yesmaximal, characteristic
2C2×#5132C211yesmaximal, characteristic
3C22×#822C211yesmaximal, characteristic, complemented
4C2×#5119C211yesmaximal, characteristic, complemented
5C2×#5002C211yesmaximal, characteristic
6#82207C211yesmaximal, characteristic, complemented
7C3×#1742C211yesmaximal, characteristic, complemented
8#68999C214yesmaximal, complemented
9#68996C214yesmaximal, complemented
10#23601—31nomaximal, Sylow
11C22×#235C2211yescharacteristic
12C22×C3×M16C2211yescharacteristic
13C22×C3×M16C2211yescharacteristic
14C2×C3×#117C2211yescharacteristic
15C2×#926C2211yescharacteristic
16C22×#208C2211yescharacteristic
17C23×C3⋊C8C2211yescharacteristic
18C2×#824C2212yes—
19#5133—22no—
20#5132—24no—
21#5131C2212yes—
22C2×#823C2212yes—
23C2×#822—24no—
24C2×#822C2214yes—
25#5120C2212yes—
26#5119—22no—
27#5119C2214yes—
28#5113—22no—
29C2×Dic3⋊C8—22no—
30#5002—22no—
31#5002—22no—
32#4998—22no—
33#4995—22no—
34C2×#810C2212yes—
35C2×#809C2212yes—
36#4964C2214yes—
37C3×#343C2214yes—
38C22×#108—31no—
39#2911—31no—
40C2×#424—31no—
41#2896—31no—
42C2×#416—31no—
43C2×#413—31no—
44#2064—34no—
45#2063—34no—
46#1742D311yescharacteristic, complemented
47C23×C3×C4C2311yescharacteristic
48C22×#67—21no—
49C2×#235—22no—
50C2×#235—21no—
51C2×#235—21no—
52C2×#235—21no—
53C2×#235—21no—
54C2×#235—22no—
55C2×#235—22no—
56C2×#235Q814yes—
57C22×D12C2311yescharacteristic
58C22×C4×D3—21no—
59C22×Dic6C2311yescharacteristic
60C2×C3×M16—21no—
61C2×C3×M16—21no—
62C2×C3×M16—22no—
63C2×C3×M16—22no—
64C2×C3×M16—22no—
65C2×C3×M16—22no—
66C2×C3×M16C2312yes—
67C2×C3×M16C2312yes—
68C22×C3×C8—21no—
69C22×C3×C8—21no—
70C3×#117—21no—
71C3×#117—21no—
72C3×#117—21no—
73C3×#117—21no—
74C2×C3×C4⋊C4C2312yes—
75C2×C3×C22⋊C4—21no—
76C2×C3×C42—21no—
77C2×#217—21no—
78#926—21no—
79#926—21no—
80#926D412yes—
81#926C2312yes—
82C2×C4⋊Dic3D412yes—
83C2×C4×Dic3—21no—
84C2×#208—21no—
85C2×#208—22no—
86C2×#208D414yes—
87C2×#208C2312yes—
88C22×C3⋊C8—22no—
89C22×C3⋊C8—21no—
90C22×C3⋊C8—22no—
91C22×C3⋊C8D412yes—
92C22×C3⋊C8D412yes—
93C22×C3⋊C8C2312yes—
94#824—24no—
95#823—24no—
96#822—28no—
97#821—24no—
98#816—24no—
99Dic3⋊C8—28no—
100#810—24no—
101#809—24no—
102#808—24no—
103#807—24no—
104C22×#41—31no—
105C22×M16—31no—
106C22×M16C2×D311yescharacteristic
107C22×M16C2×D311yescharacteristic
108C23×C8—31no—
109C2×#117—31no—
110C2×#117C2×D311yescharacteristic
111C2×#108—34no—
112C2×#108—64no—
113#428—34no—
114#428—62no—
115#427—62no—
116#424—34no—
117#424—62no—
118C2×C4⋊C8—62no—
119#419—32no—
120#418—32no—
121#417—62no—
122#416—62no—
123#416—62no—
124#416—62no—
125C2×#103—32no—
126C2×#102—32no—
127#413—64no—
128C2×#101—32no—
129C2×#100—32no—
130#343C2×D314yes—
131C24×C3#4111yescharacteristic
132C23×D3—21no—
133C22×C3×C4—22no—
134C22×C3×C4C2×D412yes—
135C22×C3×C4C2×D411yescharacteristic
136C22×C3×C4C2×D411yescharacteristic
137C22×C3×C4C2×D411yescharacteristic
138C22×C3×C4C2×D411yescharacteristic
139C22×C3×C4C2×Q812yes—
140C22×C3×C4#4111yescharacteristic
141C22×C3×C4#4112yes—
142C22×C3×C4#4112yes—
143C22×C3×C4C2411yescharacteristic
144C2×#67—24no—
145C2×#67—41no—
146C2×#67—41no—
147C2×#67—42no—
148C22×Dic3—22no—
149C22×Dic3—21no—
150#235—24no—
151#235—24no—
152#235—24no—
153#235—24no—
154#235—42no—
155#235—42no—
156#235—42no—
157#235—42no—
158C2×D12—21no—
159C2×D12—21no—
160C2×D12—21no—
161C2×D12—21no—
162C2×D12—21no—
163C2×D12C2×Q812yes—
164C2×C4×D3—24no—
165C2×C4×D3—41no—
166C2×C4×D3—41no—
167C2×C4×D3—42no—
168C2×Dic6—21no—
169C2×Dic6—21no—
170C2×Dic6—21no—
171C2×Dic6—21no—
172C2×Dic6—21no—
173C2×Dic6C2×Q812yes—
174C3×M16—22no—
175C3×M16—22no—
176C3×M16—22no—
177C3×M16—22no—
178C3×M16—42no—
179C3×M16—42no—
180C2×C3×C8—22no—
181C2×C3×C8—22no—
182C2×C3×C8—42no—
183C2×C3×C8—42no—
184C3×C4⋊C4—22no—
185C3×C4⋊C4—22no—
186C3×C22⋊C4—41no—
187C3×C22⋊C4—41no—
188C3×C42—41no—
189C3×C42—41no—
190#217—22no—
191#217—22no—
192C4⋊Dic3—22no—
193C4⋊Dic3C2×D414yes—
194C4×Dic3—22no—
195C4×Dic3—22no—
196#208—24no—
197#208—22no—
198#208C2×D414yes—
199C2×C3⋊C8—24no—
200C2×C3⋊C8—24no—
201C2×C3⋊C8—24no—
202C2×C3⋊C8—22no—
203C2×C3⋊C8—22no—
204C2×C3⋊C8—22no—
205C2×C3⋊C8C2×D414yes—
206C2×#41—34no—
207C2×#41—62no—
208C2×#41—62no—
209C2×#41—62no—
210C2×#41—61no—
211C2×#41—61no—
212C2×#41—61no—
213C2×#41—61no—
214C22×Q8—31no—
215C22×D4—31no—
216C22×D4—61no—
217C23×C4—61no—
218C23×C4C22×D311yescharacteristic
219C2×M16—21no—
220C2×M16—21no—
221C2×M16—22no—
222C2×M16—22no—
223C2×M16—22no—
224C2×M16—22no—
225C2×M16—34no—
226C2×M16—32no—
227C2×M16—62no—
228C2×M16—61no—
229C2×M16C22×D312yes—
230C2×M16C22×D312yes—
231C22×C8—21no—
232C22×C8—21no—
233C22×C8—32no—
234C22×C8—32no—
235C22×C8—32no—
236C22×C8—62no—
237C22×C8—62no—
238C22×C8—61no—
239#117—21no—
240#117—21no—
241#117—21no—
242#117—21no—
243#117—32no—
244#117—32no—
245#117—61no—
246#117—61no—
247C2×C4⋊C4—32no—
248C2×C4⋊C4C22×D312yes—
249C2×C22⋊C4—21no—
250C2×C22⋊C4—61no—
251C2×C42—21no—
252C2×C42—61no—
253#108—68no—
254#107—64no—
255#106—64no—
256C4⋊C8—68no—
257#103—64no—
258#103—64no—
259#102—64no—
260#102—64no—
261#101—64no—
262#100—64no—
263C23×C3—42no—
264C23×C3#12111yescharacteristic
265C23×C3#12111yescharacteristic
266C23×C3C22⋊Q812yes—
267C23×C3#12612yes—
268C23×C3C2×#4111yescharacteristic
269C22×D3—22no—
270C22×D3—41no—
271C22×D3—41no—
272C22×D3—82no—
273C2×C3×C4—42no—
274C2×C3×C4—82no—
275C2×C3×C4#12112yes—
276C2×C3×C4#12112yes—
277C2×C3×C4#12112yes—
278C2×C3×C4#12112yes—
279C2×C3×C4#12111yescharacteristic
280C2×C3×C4#12111yescharacteristic
281C2×C3×C4C22⋊Q812yes—
282C2×C3×C4C22⋊Q812yes—
283C2×C3×C4C22⋊Q812yes—
284C2×C3×C4C22⋊Q812yes—
285C2×C3×C4C22⋊Q812yes—
286C2×C3×C4#12612yes—
287C2×C3×C4C22×D411yescharacteristic
288C2×C3×C4C22×D411yescharacteristic
289C2×C3×C4C22×D411yescharacteristic
290C2×C3×C4C22×Q811yescharacteristic
291C2×C3×C4C2×#4111yescharacteristic
292C2×C3×C4C2×#4111yescharacteristic
293#67—44no—
294#67—44no—
295C2×Dic3—24no—
296C2×Dic3—22no—
297C2×Dic3—44no—
298C2×Dic3—41no—
299C2×Dic3—41no—
300D12—22no—
301D12—22no—
302D12—41no—
303D12—41no—
304C4×D3—44no—
305C4×D3—44no—
306Dic6—22no—
307Dic6—22no—
308Dic6—41no—
309Dic6—41no—
310C3×C8—42no—
311C3×C8—42no—
312C3⋊C8—24no—
313C3⋊C8—24no—
314C24—61no—
315C24#23511yescharacteristic
316#41—64no—
317#41—64no—
318#41—64no—
319#41—64no—
320#41—122no—
321#41—122no—
322#41—122no—
323#41—122no—
324C2×Q8—32no—
325C2×Q8—61no—
326C2×Q8—61no—
327C2×Q8—61no—
328C2×Q8—61no—
329C2×Q8—61no—
330C2×D4—32no—
331C2×D4—64no—
332C2×D4—61no—
333C2×D4—61no—
334C2×D4—61no—
335C2×D4—61no—
336C2×D4—61no—
337C2×D4—122no—
338C2×D4—121no—
339C2×D4—121no—
340C22×C4—22no—
341C22×C4—62no—
342C22×C4—61no—
343C22×C4—64no—
344C22×C4—122no—
345C22×C4—121no—
346C22×C4—121no—
347C22×C4#23511yescharacteristic
348C22×C4#23512yes—
349C22×C4#23512yes—
350C22×C4D3×D412yes—
351C22×C4D3×D411yescharacteristic
352C22×C4D3×D411yescharacteristic
353C22×C4D3×D411yescharacteristic
354C22×C4D3×D411yescharacteristic
355C22×C4D3×Q812yes—
356C22×C4C23×D311yescharacteristic
357M16—22no—
358M16—22no—
359M16—22no—
360M16—22no—
361M16—34no—
362M16—42no—
363M16—42no—
364M16—64no—
365M16—62no—
366C2×C8—22no—
367C2×C8—22no—
368C2×C8—34no—
369C2×C8—42no—
370C2×C8—42no—
371C2×C8—64no—
372C2×C8—64no—
373C2×C8—64no—
374C2×C8—62no—
375C2×C8—62no—
376C2×C8—62no—
377C4⋊C4—22no—
378C4⋊C4—22no—
379C4⋊C4—34no—
380C4⋊C4—62no—
381C22⋊C4—41no—
382C22⋊C4—41no—
383C22⋊C4—62no—
384C22⋊C4—62no—
385C42—41no—
386C42—41no—
387C42—62no—
388C42—62no—
389C22×C3—22no—
390C22×C3—22no—
391C22×C3—22no—
392C22×C3—42no—
393C22×C3—42no—
394C22×C3#39612yes—
395C22×C3#39612yes—
396C22×C3C2×#12111yescharacteristic
397C22×C3C2×C22⋊Q811yescharacteristic
398C22×C3C2×#12611yescharacteristic
399C2×D3—42no—
400C2×D3—82no—
401C2×D3—82no—
402C2×D3—82no—
403C3×C4—82no—
404C3×C4#39612yes—
405C3×C4#39612yes—
406C3×C4C2×#12111yescharacteristic
407C3×C4C2×#12111yescharacteristic
408C3×C4C2×C22⋊Q811yescharacteristic
409C3×C4C2×C22⋊Q811yescharacteristic
410Dic3—42no—
411Dic3—44no—
412C23—42no—
413C23—62no—
414C23—121no—
415C23—121no—
416C23—242no—
417C23#87612yes—
418C23#88411yescharacteristic
419C23#88411yescharacteristic
420C23#89612yes—
421C23C2×#23511yescharacteristic
422Q8—62no—
423Q8—62no—
424Q8—121no—
425Q8—121no—
426D4—62no—
427D4—62no—
428D4—124no—
429D4—124no—
430D4—121no—
431D4—121no—
432C2×C4—42no—
433C2×C4—64no—
434C2×C4—62no—
435C2×C4—82no—
436C2×C4—124no—
437C2×C4—121no—
438C2×C4—121no—
439C2×C4—124no—
440C2×C4—124no—
441C2×C4#87612yes—
442C2×C4#88411yescharacteristic
443C2×C4#88412yes—
444C2×C4#88412yes—
445C2×C4#88412yes—
446C2×C4#88412yes—
447C2×C4#88411yescharacteristic
448C2×C4#89612yes—
449C2×C4#89612yes—
450C2×C4#89612yes—
451C2×C4#89612yes—
452C2×C4#89612yes—
453C2×C4C2×#23511yescharacteristic
454C2×C4C2×#23511yescharacteristic
455C2×C4C2×D3×D411yescharacteristic
456C2×C4C2×D3×D411yescharacteristic
457C2×C4C2×D3×D411yescharacteristic
458C2×C4C2×D3×Q811yescharacteristic
459C8—42no—
460C8—42no—
461C8—64no—
462C8—64no—
463C2×C3—21no—
464C2×C3—21no—
465C2×C3—22no—
466C2×C3—22no—
467C2×C3C2×#39611yescharacteristic
468C2×C3#306712yes—
469D3—82no—
470C22—22no—
471C22—22no—
472C22—22no—
473C22—42no—
474C22—42no—
475C22—122no—
476C22—242no—
477C22—242no—
478C22—242no—
479C22#467612yes—
480C22#467612yes—
481C22C2×#87611yescharacteristic
482C22C2×#88411yescharacteristic
483C22C2×#89611yescharacteristic
484C4—82no—
485C4—124no—
486C4—122no—
487C4#467612yes—
488C4#467612yes—
489C4C2×#88411yescharacteristic
490C4C2×#88411yescharacteristic
491C4C2×#89611yescharacteristic
492C4C2×#89611yescharacteristic
493C3#2360111yescharacteristic, complemented, Sylow
494C2—21no—
495C2—21no—
496C2—22no—
497C2—22no—
498C2—242no—
499C2C2×#467611yescharacteristic
500C2#7871912yes—
subgroupG/H
centerC2×C4#884
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#1742C2
socleC22×C3C2×#121
prime radicalC22×#235C22
hypercenter#1742D3
hypocenterC3#23601
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C2×C4 ◁ C23×C4 ◁ #1742

Complements

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

NH ≅ G/Ntype
3. C22×#822C2semidirect
4. C2×#5119C2semidirect
6. #82207C2semidirect
7. C3×#1742C2semidirect
8. #68999C2semidirect
9. #68996C2semidirect
46. #1742D3semidirect
493. C3#23601semidirect

Maximal subgroups

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

subgroupindex
#837682
C2×#51322
C22×#8222
C2×#51192
C2×#50022
#822072
C3×#17422
#689992
#689962
#236013