Finite Group Databasebeta

← #31915784

Subgroups — #31915784

Order 18176.

490 conjugacy classes of subgroups; only the 107 normal subgroups are shown.

normal subgroups only

This group has 490 conjugacy classes of subgroups — too many to draw. Only its 107 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#17985798C212yesmaximal, complemented
2C2×#17985796C212yesmaximal
3#22588336C211yesmaximal, characteristic, complemented
4C4×#14556047C211yesmaximal, characteristic, complemented
5C71×#2545C211yesmaximal, characteristic
6#17986023—21no—
7C2×#14556047C2212yes—
8C2×#14556047C2212yes—
9C4×#11765849—21no—
10#17985798—24no—
11#17985796—24no—
12#17985768—21no—
13#17985768—21no—
14C2×C4⋊Q8×C71C2211yescharacteristic
15C2×#124×C71C411yescharacteristic
16C2×#124×C71C2211yescharacteristic
17C2×C4×Q8×C71C2211yescharacteristic
18#396×C71—21no—
19#396×C71—21no—
20#391×C71C411yescharacteristic
21#390×C71C411yescharacteristic
22#388×C71—21no—
23#388×C71—21no—
24C4×C4⋊C4×C71C411yescharacteristic
25C2×#11765849—22no—
26C2×#11765849—22no—
27C4⋊(C71⋊C8)—41no—
28C4⋊(C71⋊C8)—41no—
29#14556047—22no—
30#14556047—22no—
31#14556047—22no—
32#14556047D414yes—
33#14556039—41no—
34C4×C71⋊C8—41no—
35C22×Q8×C71C2×C411yescharacteristic
36C22×Q8×C71C2311yescharacteristic
37C22×D4×C71C2×C411yescharacteristic
38C4⋊Q8×C71—22no—
39C4⋊Q8×C71—22no—
40#124×C71—22no—
41#124×C71—22no—
42#124×C71—41no—
43#124×C71—41no—
44C4×Q8×C71—41no—
45C4×Q8×C71—41no—
46C2×C4⋊C4×C71—21no—
47C2×C4⋊C4×C71—21no—
48C2×C4⋊C4×C71—21no—
49C2×C4⋊C4×C71C2×C411yescharacteristic
50C2×C4⋊C4×C71D411yescharacteristic
51C2×C4⋊C4×C71D411yescharacteristic
52C2×C22⋊C4×C71—21no—
53C2×C22⋊C4×C71—21no—
54C2×C22⋊C4×C71—21no—
55C2×C22⋊C4×C71—21no—
56C2×C22⋊C4×C71—21no—
57C2×C22⋊C4×C71—21no—
58C2×C42×C71C2×C411yescharacteristic
59C2×C42×C71C2×C411yescharacteristic
60C2×C42×C71C2×C411yescharacteristic
61C2×C42×C71D411yescharacteristic
62C2×C42×C71D411yescharacteristic
63#95×C71—21no—
64#95×C71—21no—
65#95×C71—21no—
66#11765849—24no—
67#11765849—42no—
68#11765849—42no—
69#11765849—42no—
70C2×C71⋊C8—42no—
71C2×C71⋊C8—42no—
72C24×C71—21no—
73C2×Q8×C71—22no—
74C2×Q8×C71—41no—
75C2×Q8×C71—41no—
76C2×Q8×C71—41no—
77C2×Q8×C71—41no—
78C2×Q8×C71C2×D412yes—
79C2×Q8×C71#4112yes—
80C2×D4×C71—22no—
81C2×D4×C71—41no—
82C2×D4×C71—41no—
83C22×C4×C71—21no—
84C22×C4×C71—21no—
85C22×C4×C71—21no—
86C22×C4×C71C22⋊C411yescharacteristic
87C22×C4×C71C22⋊C411yescharacteristic
88C22×C4×C71C22⋊C411yescharacteristic
89C22×C4×C71C22⋊C411yescharacteristic
90C22×C4×C71C22×C411yescharacteristic
91C22×C4×C71C2×D411yescharacteristic
92C22×C4×C71C2×D411yescharacteristic
93C4⋊C4×C71—41no—
94C4⋊C4×C71—41no—
95C4⋊C4×C71—41no—
96C4⋊C4×C71—42no—
97C4⋊C4×C71—42no—
98C4⋊C4×C71—41no—
99C4⋊C4×C71—41no—
100C22⋊C4×C71—42no—
101C22⋊C4×C71—42no—
102C22⋊C4×C71—42no—
103C22⋊C4×C71—41no—
104C22⋊C4×C71—41no—
105C22⋊C4×C71—42no—
106C22⋊C4×C71—41no—
107C22⋊C4×C71—41no—
108C42×C71—21no—
109C42×C71—21no—
110C42×C71—21no—
111C42×C71—21no—
112C42×C71—21no—
113C42×C71—21no—
114C42×C71—22no—
115C42×C71—22no—
116C23×C71—41no—
117C23×C71—41no—
118C23×C71—41no—
119C23×C71—41no—
120C23×C71—41no—
121C23×C71—41no—
122C23×C71—41no—
123C23×C71C2×C22⋊C411yescharacteristic
124Q8×C71—42no—
125Q8×C71—42no—
126Q8×C71—82no—
127D4×C71—82no—
128C2×C4×C71—21no—
129C2×C4×C71—21no—
130C2×C4×C71—21no—
131C2×C4×C71—21no—
132C2×C4×C71—21no—
133C2×C4×C71—21no—
134C2×C4×C71—22no—
135C2×C4×C71—21no—
136C2×C4×C71—21no—
137C2×C4×C71—21no—
138C2×C4×C71—41no—
139C2×C4×C71—41no—
140C2×C4×C71—41no—
141C2×C4×C71—41no—
142C2×C4×C71—41no—
143C2×C4×C71—41no—
144C2×C4×C71—41no—
145C2×C4×C71—41no—
146C2×C4×C71—41no—
147C2×C4×C71—41no—
148C2×C4×C71—41no—
149C2×C4×C71—41no—
150C2×C4×C71#9911yescharacteristic
151C2×C4×C71#9911yescharacteristic
152C2×C4×C71C4×D412yes—
153C2×C4×C71#12112yes—
154C2×C4×C71#12411yescharacteristic
155C2×C4×C71#12711yescharacteristic
156C71⋊C8—44no—
157C71⋊C8—82no—
158C22×C71—21no—
159C22×C71—21no—
160C22×C71—81no—
161C22×C71—81no—
162C22×C71—81no—
163C22×C71—81no—
164C22×C71—81no—
165C22×C71—81no—
166C22×C71—81no—
167C22×C71#39011yescharacteristic
168C22×C71C2×#9911yescharacteristic
169C22×C71#41011yescharacteristic
170C4×C71—21no—
171C4×C71—21no—
172C4×C71—22no—
173C4×C71—21no—
174C4×C71—22no—
175C4×C71—21no—
176C4×C71—42no—
177C4×C71—41no—
178C4×C71—41no—
179C4×C71—41no—
180C4×C71—81no—
181C4×C71—81no—
182#13228—711nomaximal, Sylow
183C2×C71—21no—
184C2×C71—21no—
185C2×C71—161no—
186C2×C71#196411yescharacteristic
187C2×C71#213211yescharacteristic
188C2×C71#213311yescharacteristic
189#2545D7111yescharacteristic
190C2×#356—712no—
191C2×#355—712no—
192#1801—711no—
193C4×#101—711no—
194C71#1322811yescharacteristic, complemented, Sylow
195C2×C4⋊Q8C2×D7111yescharacteristic
196C2×#124Dic7111yescharacteristic
197C2×#124C2×D7111yescharacteristic
198C2×C4×Q8C2×D7111yescharacteristic
199#423—1421no—
200C2×#101—712no—
201C2×#101—712no—
202C4×M16—1421no—
203#396—21no—
204#396—21no—
205#391Dic7111yescharacteristic
206#390Dic7111yescharacteristic
207#388—21no—
208#388—21no—
209C4×C4⋊C4Dic7111yescharacteristic
210#356—1424no—
211#355—1424no—
212#343—1421no—
213#343—1421no—
214C22×Q8C2×Dic7111yescharacteristic
215C22×Q8C22×D7111yescharacteristic
216C22×D4C2×Dic7111yescharacteristic
217C2×M16—1422no—
218C2×M16—1422no—
219C4⋊Q8—22no—
220C4⋊Q8—22no—
221#124—22no—
222#124—22no—
223#124—41no—
224#124—41no—
225C4×Q8—41no—
226C4×Q8—41no—
227C2×C4⋊C4—21no—
228C2×C4⋊C4—21no—
229C2×C4⋊C4—21no—
230C2×C4⋊C4C2×Dic7111yescharacteristic
231C2×C4⋊C4C22⋊D7111yescharacteristic
232C2×C4⋊C4C22⋊D7111yescharacteristic
233C2×C22⋊C4—21no—
234C2×C22⋊C4—21no—
235C2×C22⋊C4—21no—
236C2×C22⋊C4—21no—
237C2×C22⋊C4—21no—
238C2×C22⋊C4—21no—
239C2×C42C2×Dic7111yescharacteristic
240C2×C42C2×Dic7111yescharacteristic
241C2×C42C2×Dic7111yescharacteristic
242C2×C42C22⋊D7111yescharacteristic
243C2×C42C22⋊D7111yescharacteristic
244C4⋊C8—2841no—
245C4⋊C8—2841no—
246#101—714no—
247#101—1422no—
248#101—1422no—
249#101—1422no—
250#97—2841no—
251C4×C8—2841no—
252#95—21no—
253#95—21no—
254#95—21no—
255C24—21no—
256C2×Q8—22no—
257C2×Q8—41no—
258C2×Q8—41no—
259C2×Q8—41no—
260C2×Q8—41no—
261C2×Q8D4×D7112yes—
262C2×Q8#1176586412yes—
263C2×D4—22no—
264C2×D4—41no—
265C2×D4—41no—
266C22×C4—21no—
267C22×C4—21no—
268C22×C4—21no—
269C22×C4C22⋊Dic7111yescharacteristic
270C22×C4C22⋊Dic7111yescharacteristic
271C22×C4C22⋊Dic7111yescharacteristic
272C22×C4C22⋊Dic7111yescharacteristic
273C22×C4C22×Dic7111yescharacteristic
274C22×C4C2×C22⋊D7111yescharacteristic
275C22×C4C2×C22⋊D7111yescharacteristic
276M16—2842no—
277M16—2842no—
278M16—2842no—
279M16—1424no—
280C2×C8—2842no—
281C2×C8—2842no—
282C4⋊C4—41no—
283C4⋊C4—41no—
284C4⋊C4—41no—
285C4⋊C4—42no—
286C4⋊C4—42no—
287C4⋊C4—41no—
288C4⋊C4—41no—
289C22⋊C4—42no—
290C22⋊C4—42no—
291C22⋊C4—42no—
292C22⋊C4—41no—
293C22⋊C4—41no—
294C22⋊C4—42no—
295C22⋊C4—41no—
296C22⋊C4—41no—
297C42—21no—
298C42—21no—
299C42—21no—
300C42—21no—
301C42—21no—
302C42—21no—
303C42—22no—
304C42—22no—
305C23—41no—
306C23—41no—
307C23—41no—
308C23—41no—
309C23—41no—
310C23—41no—
311C23—41no—
312C23C2×C22⋊Dic7111yescharacteristic
313Q8—42no—
314Q8—42no—
315Q8—82no—
316D4—82no—
317C2×C4—21no—
318C2×C4—21no—
319C2×C4—21no—
320C2×C4—21no—
321C2×C4—21no—
322C2×C4—21no—
323C2×C4—22no—
324C2×C4—21no—
325C2×C4—21no—
326C2×C4—21no—
327C2×C4—41no—
328C2×C4—41no—
329C2×C4—41no—
330C2×C4—41no—
331C2×C4—41no—
332C2×C4—41no—
333C2×C4—41no—
334C2×C4—41no—
335C2×C4—41no—
336C2×C4—41no—
337C2×C4—41no—
338C2×C4—41no—
339C2×C4#1455604311yescharacteristic
340C2×C4#1455604311yescharacteristic
341C2×C4D4×Dic7112yes—
342C2×C4#1455609912yes—
343C2×C4#1455611511yescharacteristic
344C2×C4#1455612411yescharacteristic
345C8—2844no—
346C8—5682no—
347C22—21no—
348C22—21no—
349C22—81no—
350C22—81no—
351C22—81no—
352C22—81no—
353C22—81no—
354C22—81no—
355C22—81no—
356C22#1798590211yescharacteristic
357C22C2×#1455604311yescharacteristic
358C22#1798597311yescharacteristic
359C4—21no—
360C4—21no—
361C4—22no—
362C4—21no—
363C4—22no—
364C4—21no—
365C4—42no—
366C4—41no—
367C4—41no—
368C4—41no—
369C4—81no—
370C4—81no—
371C2—21no—
372C2—21no—
373C2—161no—
374C2#2258917711yescharacteristic
375C2#2259002511yescharacteristic
376C2#2259002911yescharacteristic
subgroupG/H
centerC22C2×#14556043
commutatorC22×C4×C71C22×C4
FrattiniC22×Q8C22×D71
FittingC71×#2545C2
socleC22×C71C2×#99
prime radicalC22×D4×C71C2×C4
cosocleC22×Q8×C71C23
hypercenter#2545D71
hypocenterC71#13228
LCSG ▷ C22×C4×C71 ▷ C22×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4×Q8 ◁ #2545

Complements

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

NH ≅ G/Ntype
1. C2×#17985798C2semidirect
3. #22588336C2semidirect
4. C4×#14556047C2semidirect
194. C71#13228semidirect

Maximal subgroups

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

subgroupindex
C2×#179857982
C2×#179857962
#225883362
C4×#145560472
C71×#25452
#1322871